This document discusses multiset path orderings and their application in proving termination of term rewriting systems. It provides background on term rewriting systems and defines key concepts such as rewrite rules, termination, and reduction orders. The main point is that multiset path orderings allow the use of relatively simple termination functions to prove termination of a class of term rewriting systems.
This document provides a simplified proof of Kruskal's tree theorem. It begins with preliminaries that define
well-founded orders, homeomorphic embedding, and well-partial orders. It then states Kruskal's theorem for finite
signatures and proves the general version of the theorem. The proof uses two lemmas and shows that any infinite
sequence of terms contains indices such that the terms are related by homeomorphic embedding. This implies
homeomorphic embedding is a well-partial order. The document concludes that simplification orders are
well-founded over finite signatures due to Kruskal's theorem.
FURTHER CHARACTERIZATIONS AND SOME APPLICATIONS OF UPPER AND LOWER WEAKLY QUA...IJESM JOURNAL
In his paper we characterize upper and lower weakly quasi continuous fuzzy multifunction’s [3] be a new type of convergence of a net in a topological space and also characterize lower weakly quasi continuous fuzzy multifunction by a newly defined convergence of a fuzzy net. Again a new concept of regularity in a topological space has been introduced and characterized and using this regularity several applications of upper weakly quasi continuous fuzzy multifunctions have been shown.
This document presents a new systematic construction of zero correlation zone (ZCZ) sequences based on interleaved perfect sequences. It begins with background on perfect sequences, ZCZ sequences, and the relationship between interleaved sequences and their associated shift sequences. It then describes a four-step procedure to construct ZCZ sequence sets from a perfect sequence and orthogonal matrix by generating an appropriate shift sequence. Two theorems establish the auto- and cross-correlation properties of the resulting sequences. Three classes of almost optimal shift sequences are deduced from prior works. A new general construction of shift sequences is presented and proven to generate almost optimal ZCZ sequence sets for any sequence lengths.
Master Thesis on the Mathematial Analysis of Neural NetworksAlina Leidinger
Master Thesis submitted on June 15, 2019 at TUM's chair of Applied Numerical Analysis (M15) at the Mathematics Department.The project was supervised by Prof. Dr. Massimo Fornasier. The thesis took a detailed look at the existing mathematical analysis of neural networks focusing on 3 key aspects: Modern and classical results in approximation theory, robustness and Scattering Networks introduced by Mallat, as well as unique identification of neural network weights. See also the one page summary available on Slideshare.
On The Transition Probabilities for the Fuzzy States of a Fuzzy Markov ChainIJERA Editor
In this paper the theory of fuzzy logic is mixed with the theory of Markov systems and the abstraction of a
Markov system with fuzzy states introduced. The notions such as fuzzy transient, fuzzy recurrent etc., were
introduced. The results based on these notions are introduced.
The Minimum Hamming Distances of the Irreducible Cyclic Codes of Length inventionjournals
This document discusses irreducible cyclic codes of length n over finite fields. It begins by reviewing some preliminary concepts regarding characters of finite abelian groups. It then provides theorems that give the irreducible factorization of polynomials of the form x^n-1 over finite fields under certain conditions on n and the field size. Finally, it determines the primitive idempotents in the ring Fq[x]/(x^n-1) and obtains the check polynomials, dimensions, and minimum Hamming distances of the irreducible cyclic codes generated by these idempotents. The key results are the theorems providing the irreducible factorizations of x^n-1 and determining the corresponding primitive idempotents.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
This document provides a simplified proof of Kruskal's tree theorem. It begins with preliminaries that define
well-founded orders, homeomorphic embedding, and well-partial orders. It then states Kruskal's theorem for finite
signatures and proves the general version of the theorem. The proof uses two lemmas and shows that any infinite
sequence of terms contains indices such that the terms are related by homeomorphic embedding. This implies
homeomorphic embedding is a well-partial order. The document concludes that simplification orders are
well-founded over finite signatures due to Kruskal's theorem.
FURTHER CHARACTERIZATIONS AND SOME APPLICATIONS OF UPPER AND LOWER WEAKLY QUA...IJESM JOURNAL
In his paper we characterize upper and lower weakly quasi continuous fuzzy multifunction’s [3] be a new type of convergence of a net in a topological space and also characterize lower weakly quasi continuous fuzzy multifunction by a newly defined convergence of a fuzzy net. Again a new concept of regularity in a topological space has been introduced and characterized and using this regularity several applications of upper weakly quasi continuous fuzzy multifunctions have been shown.
This document presents a new systematic construction of zero correlation zone (ZCZ) sequences based on interleaved perfect sequences. It begins with background on perfect sequences, ZCZ sequences, and the relationship between interleaved sequences and their associated shift sequences. It then describes a four-step procedure to construct ZCZ sequence sets from a perfect sequence and orthogonal matrix by generating an appropriate shift sequence. Two theorems establish the auto- and cross-correlation properties of the resulting sequences. Three classes of almost optimal shift sequences are deduced from prior works. A new general construction of shift sequences is presented and proven to generate almost optimal ZCZ sequence sets for any sequence lengths.
Master Thesis on the Mathematial Analysis of Neural NetworksAlina Leidinger
Master Thesis submitted on June 15, 2019 at TUM's chair of Applied Numerical Analysis (M15) at the Mathematics Department.The project was supervised by Prof. Dr. Massimo Fornasier. The thesis took a detailed look at the existing mathematical analysis of neural networks focusing on 3 key aspects: Modern and classical results in approximation theory, robustness and Scattering Networks introduced by Mallat, as well as unique identification of neural network weights. See also the one page summary available on Slideshare.
On The Transition Probabilities for the Fuzzy States of a Fuzzy Markov ChainIJERA Editor
In this paper the theory of fuzzy logic is mixed with the theory of Markov systems and the abstraction of a
Markov system with fuzzy states introduced. The notions such as fuzzy transient, fuzzy recurrent etc., were
introduced. The results based on these notions are introduced.
The Minimum Hamming Distances of the Irreducible Cyclic Codes of Length inventionjournals
This document discusses irreducible cyclic codes of length n over finite fields. It begins by reviewing some preliminary concepts regarding characters of finite abelian groups. It then provides theorems that give the irreducible factorization of polynomials of the form x^n-1 over finite fields under certain conditions on n and the field size. Finally, it determines the primitive idempotents in the ring Fq[x]/(x^n-1) and obtains the check polynomials, dimensions, and minimum Hamming distances of the irreducible cyclic codes generated by these idempotents. The key results are the theorems providing the irreducible factorizations of x^n-1 and determining the corresponding primitive idempotents.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
A dominating set is a split dominating
set in . If the induced subgraph is
disconnected in The split domination number of
is denoted by , is the minimum cardinality of
a split dominating set in . In this paper, some results on
were obtained in terms of vertices, blocks, and other
different parameters of but not members of
Further, we develop its relationship with other different
domination parameters of
This seminar will cover the basics of complex manifolds and geometry through readings and presentations on topics from reference book [2]. The seminar will be organized as a reading group, with participants reading assigned material each week and one person giving a talk on a particular topic. Some of the topics to be covered include holomorphic functions, complex manifolds, vector bundles, divisors, projective space, differential forms, Kähler manifolds, and applications to curves.
This document discusses extremally β-disconnectedness in smooth fuzzy β-centered systems. It introduces the concept of maximal smooth fuzzy β-centered systems and defines the space (R) as the collection of all such systems belonging to a space R. Properties of maximal smooth fuzzy β-centered systems are proved. The space (R) is given a smooth fuzzy maximal structure and shown to be a smooth fuzzy β-compact space with a base of r-fuzzy β-open and β-closed neighborhoods. Finally, smooth fuzzy extremal β-disconnectedness is defined for a smooth fuzzy β-Hausdorff space R as the β-closure of any r-fuzzy β-open set
1. The document presents two theorems regarding conditions under which a topological space is metrizable.
2. Theorem 1 states that if a topological space satisfies four conditions, including being a T1 space and having a neighborhood basis with certain properties, then it is metrizable.
3. Theorem 2 also provides conditions for a space to be metrizable, including being a T1 space and having a neighborhood basis at each point with one additional property. The proof shows that Theorem 1 implies the conditions of Theorem 2.
This document discusses conflicts between two approaches to modeling belief revision: relevance-sensitive belief revision and iterated belief revision. It summarizes the key postulates of each approach and then presents the main result, which is that the postulate for relevance-sensitive belief revision proposed by Parikh is inconsistent with each of the postulates for iterated belief revision proposed by Darwiche and Pearl. This incompatibility is significant because both relevance and iteration are important concepts in modeling belief revision, yet the commonly accepted formalizations of each approach are inconsistent. The document analyzes the implications and significance of this result.
This document discusses propositional logic inference rules and their properties. It introduces several common rules of inference like modus ponens, and introduction, and elimination. It also discusses the relationship between inference and entailment, and defines important properties of soundness, completeness, and decidability for logical systems. Examples are provided to demonstrate proving goals using rules of inference and a truth table is used to check entailment.
The document describes how to compute the suffix array of a string in linear time. It involves sorting suffixes into groups based on their value modulo some number v. The suffixes in each group are sorted recursively to determine their relative order, which provides the sorted order of all suffixes when combined. By using a difference cover set of size O(sqrt(v)), this process can be done in O(n) time overall.
Common Fixed Point Theorem for Weakly Compatible Maps in Intuitionistic Fuzzy...inventionjournals
In this paper, we prove some common fixed point theorem for weakly compatible maps in intuitionistic fuzzy metric space for two, four and six self mapping.
This document introduces the concept of fuzzy sets, which are classes of objects that have a continuum of grades of membership rather than crisp criteria for inclusion. Fuzzy sets are characterized by membership functions that assign each object a value between 0 and 1 indicating its grade of membership. Operations like union, intersection, and complementation are extended from ordinary sets to fuzzy sets. Basic properties of these operations are established, including identities analogous to those for ordinary sets. Fuzzy sets provide a framework for dealing with imprecisely defined classes that play an important role in human thinking and domains like pattern recognition.
The document discusses integration rules and techniques for rational functions using the log rule. It provides examples of using the log rule with substitutions, recognizing quotients in disguised forms, using long division to reveal quotients, and changing variables. It emphasizes mastering the "form-fitting" nature of integration to apply the appropriate rules compared to differentiation which is more straightforward. Students are encouraged to memorize the integration and derivative rules.
This document discusses properties of the complement of fuzzy graphs. It begins by introducing fuzzy graphs and defining the complement of a fuzzy graph. It then presents three theorems about the structure of complements:
1) If a fuzzy graph G is a cycle with 5 or more vertices, then the underlying graph of G's complement (Gc)* will be a block.
2) If a fuzzy graph G satisfies a certain connectivity property, then (Gc)* will be a block.
3) If a fuzzy graph G satisfies a certain degree and connectivity property, then Gc will be a fuzzy block.
The document concludes by stating that the results are sufficient but not necessary conditions for the structural properties of
This document provides an introduction to First Order Predicate Logic (FOPL). It discusses the differences between propositional logic and FOPL, the parts and syntax of FOPL including terms, atomic sentences, quantifiers and rules of inference. The semantics of FOPL are also explained. Pros and cons are provided, such as FOPL's ability to represent individual entities and generalizations compared to propositional logic. Applications include using FOPL as a framework for formulating theories.
This document summarizes recent results from Mircea-Dan Hernest's PhD thesis on optimizing proof-theoretic techniques used by Ulrich Kohlenbach for extracting bounds from proofs in analysis. Specifically, it explores adapting Kohlenbach's "light monotone Dialectica" interpretation to proofs involving "non-computational" quantifiers. It describes how ε-arithmetization, elimination of extensionality, and model interpretation can be applied in this "non-computational" setting while maintaining certain restrictions. The goal is to more efficiently extract moduli from a larger class of non-trivial analytical proofs.
This document discusses inference rules in first-order logic, specifically resolution. It defines resolution in propositional logic and first-order logic, and proves it is a sound and complete proof procedure. It also discusses how to use resolution to answer true/false and fill-in-the-blank questions by adding clauses representing the question and searching for resolutions that prove or answer the question.
This document discusses techniques for evaluating indefinite integrals using substitution, including pattern recognition, change of variables, and the general power rule. It explains that substitution helps evaluate more complex integrals that involve a composite function. Several examples demonstrate identifying the pattern f(g(x))g'(x) and performing u-substitutions to evaluate the integral. It also discusses the need to sometimes multiply and divide by a constant or perform a formal change of variables if the pattern does not exactly match.
This document discusses techniques for integration using substitution. It explains that substitution allows for the integration of more complex composite functions by breaking them into an inside and outside part. The main techniques are pattern recognition, where the integrand fits the standard pattern of f(g(x))g'(x), and change of variables, where a substitution u is made to simplify the integrand. It provides examples of applying both pattern recognition and change of variables substitutions to evaluate definite and indefinite integrals.
Proof-Theoretic Semantics: Point-free meaninig of first-order systemsMarco Benini
This document summarizes a talk on providing a semantics for first-order logical theories using logical categories. The semantics interprets formulae as objects in a category and proofs as morphisms, without assuming elements exist. Quantifiers are interpreted using stars and costars. A logical category is a prelogical category where stars and costars exist to interpret all formulae. This semantics is sound and complete - a formula is true if a proof morphism exists. The semantics can interpret many other approaches and inconsistent theories have "trivial" models.
First-order logic (FOL) is a formal system used in mathematics, philosophy, linguistics, and computer science to represent knowledge about domains involving objects and relations. FOL extends propositional logic with quantifiers and predicates to describe properties of and relations between objects. Well-formed formulas in FOL involve constants, variables, functions, predicates, quantifiers, and logical connectives. The meaning and truth of FOL statements is determined with respect to a structure called a model that specifies a domain of objects and interpretations of symbols. FOL can be used to represent knowledge about many different domains and perform logical inference.
This document discusses techniques for implicitly differentiating second derivatives and provides examples of implicit differentiation. It notes that the second derivative can often be simplified by making substitutions and avoiding full simplification. It also discusses identifying when to use the product rule and chain rule during implicit differentiation and distinguishing constants as coefficients versus terms. Examples provided include finding the normal line equation to a circle at a given point.
Sentient Arithmetic and Godel's Incompleteness TheoremsKannan Nambiar
For me, there is only one logic that we rational human beings are able to accept and appreciate, and that is the mathematical logic of ZF theory. But in the last century we found that ZF theory is not in a position to provide all that we want, and went in search of a new mode of thinking and got one which we called meta mathematics. My question is: if we can put the unambiguous logic of ZF theory on paper, why can't we do the same with meta mathematics. This paper is my feeble attempt in that direction.
This document discusses diabetes mellitus in the Arab world. It defines diabetes as a metabolic disorder resulting from defects in insulin production or insulin resistance. The document notes the increasing prevalence of type 2 diabetes in the Arab world due to factors like obesity, physical inactivity, and nutrition changes associated with westernization. Complications of diabetes discussed include cardiovascular disease. Management involves glycemic control, lifestyle changes, and treatment of other risk factors. The document calls for national diabetes prevention programs in the Arab world to help curb the growing epidemic.
El documento describe las principales empresas petroleras estatales de varios países de América Latina. Menciona empresas como YPF de Argentina, YPFB de Bolivia, Petrobras de Brasil, ENAP de Chile, Ecopetrol de Colombia, Petroecuador de Ecuador, Pemex de México, Petroperú del Perú, ANCAP de Uruguay y PDVSA de Venezuela, detallando brevemente su historia y operaciones.
A dominating set is a split dominating
set in . If the induced subgraph is
disconnected in The split domination number of
is denoted by , is the minimum cardinality of
a split dominating set in . In this paper, some results on
were obtained in terms of vertices, blocks, and other
different parameters of but not members of
Further, we develop its relationship with other different
domination parameters of
This seminar will cover the basics of complex manifolds and geometry through readings and presentations on topics from reference book [2]. The seminar will be organized as a reading group, with participants reading assigned material each week and one person giving a talk on a particular topic. Some of the topics to be covered include holomorphic functions, complex manifolds, vector bundles, divisors, projective space, differential forms, Kähler manifolds, and applications to curves.
This document discusses extremally β-disconnectedness in smooth fuzzy β-centered systems. It introduces the concept of maximal smooth fuzzy β-centered systems and defines the space (R) as the collection of all such systems belonging to a space R. Properties of maximal smooth fuzzy β-centered systems are proved. The space (R) is given a smooth fuzzy maximal structure and shown to be a smooth fuzzy β-compact space with a base of r-fuzzy β-open and β-closed neighborhoods. Finally, smooth fuzzy extremal β-disconnectedness is defined for a smooth fuzzy β-Hausdorff space R as the β-closure of any r-fuzzy β-open set
1. The document presents two theorems regarding conditions under which a topological space is metrizable.
2. Theorem 1 states that if a topological space satisfies four conditions, including being a T1 space and having a neighborhood basis with certain properties, then it is metrizable.
3. Theorem 2 also provides conditions for a space to be metrizable, including being a T1 space and having a neighborhood basis at each point with one additional property. The proof shows that Theorem 1 implies the conditions of Theorem 2.
This document discusses conflicts between two approaches to modeling belief revision: relevance-sensitive belief revision and iterated belief revision. It summarizes the key postulates of each approach and then presents the main result, which is that the postulate for relevance-sensitive belief revision proposed by Parikh is inconsistent with each of the postulates for iterated belief revision proposed by Darwiche and Pearl. This incompatibility is significant because both relevance and iteration are important concepts in modeling belief revision, yet the commonly accepted formalizations of each approach are inconsistent. The document analyzes the implications and significance of this result.
This document discusses propositional logic inference rules and their properties. It introduces several common rules of inference like modus ponens, and introduction, and elimination. It also discusses the relationship between inference and entailment, and defines important properties of soundness, completeness, and decidability for logical systems. Examples are provided to demonstrate proving goals using rules of inference and a truth table is used to check entailment.
The document describes how to compute the suffix array of a string in linear time. It involves sorting suffixes into groups based on their value modulo some number v. The suffixes in each group are sorted recursively to determine their relative order, which provides the sorted order of all suffixes when combined. By using a difference cover set of size O(sqrt(v)), this process can be done in O(n) time overall.
Common Fixed Point Theorem for Weakly Compatible Maps in Intuitionistic Fuzzy...inventionjournals
In this paper, we prove some common fixed point theorem for weakly compatible maps in intuitionistic fuzzy metric space for two, four and six self mapping.
This document introduces the concept of fuzzy sets, which are classes of objects that have a continuum of grades of membership rather than crisp criteria for inclusion. Fuzzy sets are characterized by membership functions that assign each object a value between 0 and 1 indicating its grade of membership. Operations like union, intersection, and complementation are extended from ordinary sets to fuzzy sets. Basic properties of these operations are established, including identities analogous to those for ordinary sets. Fuzzy sets provide a framework for dealing with imprecisely defined classes that play an important role in human thinking and domains like pattern recognition.
The document discusses integration rules and techniques for rational functions using the log rule. It provides examples of using the log rule with substitutions, recognizing quotients in disguised forms, using long division to reveal quotients, and changing variables. It emphasizes mastering the "form-fitting" nature of integration to apply the appropriate rules compared to differentiation which is more straightforward. Students are encouraged to memorize the integration and derivative rules.
This document discusses properties of the complement of fuzzy graphs. It begins by introducing fuzzy graphs and defining the complement of a fuzzy graph. It then presents three theorems about the structure of complements:
1) If a fuzzy graph G is a cycle with 5 or more vertices, then the underlying graph of G's complement (Gc)* will be a block.
2) If a fuzzy graph G satisfies a certain connectivity property, then (Gc)* will be a block.
3) If a fuzzy graph G satisfies a certain degree and connectivity property, then Gc will be a fuzzy block.
The document concludes by stating that the results are sufficient but not necessary conditions for the structural properties of
This document provides an introduction to First Order Predicate Logic (FOPL). It discusses the differences between propositional logic and FOPL, the parts and syntax of FOPL including terms, atomic sentences, quantifiers and rules of inference. The semantics of FOPL are also explained. Pros and cons are provided, such as FOPL's ability to represent individual entities and generalizations compared to propositional logic. Applications include using FOPL as a framework for formulating theories.
This document summarizes recent results from Mircea-Dan Hernest's PhD thesis on optimizing proof-theoretic techniques used by Ulrich Kohlenbach for extracting bounds from proofs in analysis. Specifically, it explores adapting Kohlenbach's "light monotone Dialectica" interpretation to proofs involving "non-computational" quantifiers. It describes how ε-arithmetization, elimination of extensionality, and model interpretation can be applied in this "non-computational" setting while maintaining certain restrictions. The goal is to more efficiently extract moduli from a larger class of non-trivial analytical proofs.
This document discusses inference rules in first-order logic, specifically resolution. It defines resolution in propositional logic and first-order logic, and proves it is a sound and complete proof procedure. It also discusses how to use resolution to answer true/false and fill-in-the-blank questions by adding clauses representing the question and searching for resolutions that prove or answer the question.
This document discusses techniques for evaluating indefinite integrals using substitution, including pattern recognition, change of variables, and the general power rule. It explains that substitution helps evaluate more complex integrals that involve a composite function. Several examples demonstrate identifying the pattern f(g(x))g'(x) and performing u-substitutions to evaluate the integral. It also discusses the need to sometimes multiply and divide by a constant or perform a formal change of variables if the pattern does not exactly match.
This document discusses techniques for integration using substitution. It explains that substitution allows for the integration of more complex composite functions by breaking them into an inside and outside part. The main techniques are pattern recognition, where the integrand fits the standard pattern of f(g(x))g'(x), and change of variables, where a substitution u is made to simplify the integrand. It provides examples of applying both pattern recognition and change of variables substitutions to evaluate definite and indefinite integrals.
Proof-Theoretic Semantics: Point-free meaninig of first-order systemsMarco Benini
This document summarizes a talk on providing a semantics for first-order logical theories using logical categories. The semantics interprets formulae as objects in a category and proofs as morphisms, without assuming elements exist. Quantifiers are interpreted using stars and costars. A logical category is a prelogical category where stars and costars exist to interpret all formulae. This semantics is sound and complete - a formula is true if a proof morphism exists. The semantics can interpret many other approaches and inconsistent theories have "trivial" models.
First-order logic (FOL) is a formal system used in mathematics, philosophy, linguistics, and computer science to represent knowledge about domains involving objects and relations. FOL extends propositional logic with quantifiers and predicates to describe properties of and relations between objects. Well-formed formulas in FOL involve constants, variables, functions, predicates, quantifiers, and logical connectives. The meaning and truth of FOL statements is determined with respect to a structure called a model that specifies a domain of objects and interpretations of symbols. FOL can be used to represent knowledge about many different domains and perform logical inference.
This document discusses techniques for implicitly differentiating second derivatives and provides examples of implicit differentiation. It notes that the second derivative can often be simplified by making substitutions and avoiding full simplification. It also discusses identifying when to use the product rule and chain rule during implicit differentiation and distinguishing constants as coefficients versus terms. Examples provided include finding the normal line equation to a circle at a given point.
Sentient Arithmetic and Godel's Incompleteness TheoremsKannan Nambiar
For me, there is only one logic that we rational human beings are able to accept and appreciate, and that is the mathematical logic of ZF theory. But in the last century we found that ZF theory is not in a position to provide all that we want, and went in search of a new mode of thinking and got one which we called meta mathematics. My question is: if we can put the unambiguous logic of ZF theory on paper, why can't we do the same with meta mathematics. This paper is my feeble attempt in that direction.
This document discusses diabetes mellitus in the Arab world. It defines diabetes as a metabolic disorder resulting from defects in insulin production or insulin resistance. The document notes the increasing prevalence of type 2 diabetes in the Arab world due to factors like obesity, physical inactivity, and nutrition changes associated with westernization. Complications of diabetes discussed include cardiovascular disease. Management involves glycemic control, lifestyle changes, and treatment of other risk factors. The document calls for national diabetes prevention programs in the Arab world to help curb the growing epidemic.
El documento describe las principales empresas petroleras estatales de varios países de América Latina. Menciona empresas como YPF de Argentina, YPFB de Bolivia, Petrobras de Brasil, ENAP de Chile, Ecopetrol de Colombia, Petroecuador de Ecuador, Pemex de México, Petroperú del Perú, ANCAP de Uruguay y PDVSA de Venezuela, detallando brevemente su historia y operaciones.
Undang-undang ini membentuk Kabupaten Minahasa Selatan dan Kota Tomohon di Provinsi Sulawesi Utara dengan memisahkan wilayah dari Kabupaten Minahasa. Kabupaten Minahasa Selatan terdiri dari 13 kecamatan sedangkan Kota Tomohon terdiri dari 3 kecamatan. Undang-undang ini juga mengatur tentang batas wilayah, ibu kota, kewenangan, pemerintahan daerah, dan ketentuan peralihan dari Kabupaten Minahasa ke Kabupaten Minahasa
This document discusses convolutional codes. It begins by introducing convolutional codes and noting they are different from block codes in that they encode an entire data stream continuously rather than segmenting into blocks. It then provides details on convolutional code encoding including using a rate 1/2 encoder as an example and discussing encoder representations through state diagrams and trellis diagrams. Finally, it discusses maximum likelihood decoding and how the Viterbi algorithm can be used for convolutional code decoding in an efficient manner by finding the highest probability path through the trellis.
The document discusses the growing awakening of masses of people around the world to the true nature of world events and power structures. It notes that mass movements are beginning to turn the tide from destructive power to more honest and humanitarian relationships. If the masses respond to power structures with clear communication and without hostility, they can help transform ("transmute") those power structures. The document encourages continuing to spread truthful information from reliable sources and shortwave radio to help sort out understanding of current events. It also notes that the letter writing campaign against nuclear weapons and war is having an effect and "freaking out" those in power, revealing the scope of public awakening.
Este documento describe el proyecto ConCiencia del CSIC para integrar la gestión y comunicación de la producción científica a través de una plataforma que recopila y normaliza datos de publicaciones de investigadores del CSIC de bases de datos externas como Scopus y Web of Science. El proyecto busca mejorar los procesos de investigación y soporte a través de una política y actuaciones coordinadas, y el uso de tecnología como APIs de Scopus. Digital.CSIC juega un papel clave en la colaboración con proveedores externos y el diseño
The document appears to show the amino acid sequence of a protein or peptide. It consists of 21 amino acids in the following order: MET, ASP, ACID, LEU, ISO, THR, PRO, GLY, STOP. The sequence starts with methionine (MET) and ends with a stop codon (STOP).
The document provides a weekly market outlook from UBS. It notes that additional turbulence is expected in the coming week due to heavy economic data releases and policy uncertainty. However, equity markets appear oversold and may be poised for a modest relief rally. Any sustained recovery will require confirmation that the economic recovery is on track, earnings impacts are modest and transitory, and policy remains supportive of growth. The document also discusses the challenges facing monetary policymakers and expectations that additional stimulus is unlikely, meaning markets will need to find comfort with the existing policy mix.
Quickpoint allows users to open, view, edit, and present PowerPoint presentations on an iPad. It supports both 2003 and 2007 PowerPoint file formats. Presentations can be viewed in portrait or landscape mode and slides can be zoomed, panned, and navigated. Editing options include adding, deleting, and reordering slides. Images, shapes, and text boxes on slides can be manipulated and formatting options are available. Graphics and media can be inserted. Presentations can be shown in slideshow mode on the iPad screen or an external display.
This document discusses bibliometrics and their use at Cardiff University. It begins with an introduction to bibliometric measures like citations, impact factors, and altmetrics. It then discusses how bibliometric data is presented in Cardiff's institutional repository and how it was used to provide context for research evaluations in the UK's REF2014 assessment exercise. The document concludes by outlining Cardiff's trial of the SciVal analytics tool and plans for a new research information system to better integrate bibliometric and altmetric data.
Teori pemisahan kekuasaan menurut John Locke membagi kekuasaan pemerintahan menjadi tiga cabang: legislatif untuk membuat undang-undang, eksekutif untuk menjalankan undang-undang, dan yudikatif untuk mengadili pelanggaran undang-undang.
Sm rhayent pj (PERANAN BIOREMOVAL PADA RESTORASI LAHAN TERKONTAMINASI LOGAM B...Rian vebrianto
Teknik bioremoval memainkan peran penting dalam restorasi lahan terkontaminasi logam berat dengan menggunakan mikroorganisme. Terdapat dua mekanisme utama bioremoval yaitu passive uptake dan active uptake, di mana passive uptake dapat mengakumulasi logam lebih besar dibandingkan active uptake. Faktor-faktor seperti jenis mikroorganisme, waktu kontak, dan pertimbangan biaya perlu diperhatikan dalam mendesain proses bioremoval unt
The document provides information for participants attending the training course "Art as a tool against discrimination" taking place in Lviv, Ukraine from October 11-21, 2013. The training will bring together 30 youth workers and artists from 6 European and Caucasus countries to discuss using art to promote intercultural dialogue and inclusion through non-formal educational activities like workshops and presentations. Participants will be equipped with new skills and tools for implementing youth projects on these themes.
Destination Wirksworth is a grassroots campaign to promote the town of Wirksworth, Duffield, and the Ecclesbourne Valley as a tourist destination. The campaign aims to capitalize on the maturing Wirksworth-Duffield railway and differentiate the area from typical "Visit Arcadia" promotions. The initiative faces challenges from local competition and a historically unfocused approach in Wirksworth. However, the railway provides a traffic-free gateway to the Peak District and the valley offers leisure opportunities. If stakeholders unite behind Destination Wirksworth, they have an opportunity to coordinate promotion and welcome visitors arriving by train.
Wordfast es un motor de memoria de traducción diseñado para Microsoft Word que permite la traducción de documentos de Word, Excel y PowerPoint utilizando memorias de traducción previas y gestión de terminología. La traducción asistida por computadora (CAT) utiliza programas de software para facilitar el proceso de traducción mediante el uso de memorias de traducción y glosarios para obtener documentos multilingües homogéneos.
Overview of the Resource Management Act 1991Raewyn Peart
This is a presentation by Environmental Defence Society Lawyer Nicola de Wit given at an EDS community workshop held in Gisborne on 13 April 2013. It provides a description of the purpose and principles of the RMA, responsibilities of different management agencies, policies and plans and resource consenting.
This document provides a simplified proof of Kruskal's tree theorem. It begins with preliminaries that define concepts like well-founded orders, homeomorphic embedding, and well-partial orders. It then states Kruskal's theorem for finite terms and the general version. The main part of the document proves the general version of Kruskal's theorem by constructing a minimal bad sequence and reaching a contradiction. It concludes that every simplification order over a finite signature is well-founded, which has applications in proving termination of term rewriting systems.
AN ALGORITHM FOR OPTIMIZED SEARCHING USING NON-OVERLAPPING ITERATIVE NEIGHBOR...IJCSEA Journal
The document presents an algorithm for optimized searching using non-overlapping iterative neighbor intervals. The algorithm aims to reduce the number of checked conditions by saving the frequency of replicated words and using non-overlapping intervals based on the plane sweep algorithm. It does this by focusing the search on a smaller subspace of relevant intervals near the minimum frequency keyword. The algorithm iterates through ranges, eliminating unsatisfied keywords to detect relevant ranges. This improves efficiency and reduces the number of comparisons compared to previous methods.
This document summarizes various stemming algorithms used for information retrieval. It discusses rule-based stemming algorithms like the Porter stemmer and Lovins stemmer which use language rules to remove suffixes and extract word stems. It also describes statistical stemming methods which use machine learning on text corpora to identify morphological variants. Finally, it analyzes different techniques for conflating words during stemming, such as affix removal, successor variety, table lookup, and n-gram methods.
A survey of Stemming Algorithms for Information Retrievaliosrjce
IOSR Journal of Computer Engineering (IOSR-JCE) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of computer engineering and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in computer technology. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Dictionaries and Tolerant Retrieval.pptManimaran A
The document discusses dictionaries and tolerant retrieval in information retrieval systems. It describes different data structures that can be used to store term dictionaries for inverted indexes, including arrays, hash tables, binary trees, and B-trees. It also discusses how to handle wildcard queries using techniques like permuterm indexes and k-gram indexes. The document explains methods for spell checking documents and queries, such as edit distance, weighted edit distance, n-gram overlap, and Soundex.
The document discusses using clustering techniques like K-means, LDA, and PAM to analyze topics in a large dataset of Wikipedia documents. It explores preprocessing steps, compares different clustering algorithms, and analyzes the results. K-means identified around 250 topics using the elbow method. LDA was able to identify coherent topics based on word co-occurrence. PAM using bigrams found some meaningful word pairs but the clusters did not separate well. The techniques revealed topics related to music, politics, war and more.
The document discusses computing the longest common prefix array (LCP array) directly from the Burrows-Wheeler transform (BWT) of a string, rather than first constructing the suffix array. It presents the first algorithm that computes the LCP array in linear time O(n) directly from the wavelet tree of the BWT-transformed string, requiring approximately 2.2n bytes of space. This is more efficient than previous approaches that first constructed the suffix array and then derived the LCP array, requiring at least 5n bytes of space.
Extractive Document Summarization - An Unsupervised ApproachFindwise
1. This paper presents and evaluates an unsupervised extractive document summarization system that uses TextRank, K-means clustering, and one-class SVM algorithms for sentence ranking.
2. The system achieves state-of-the-art performance on the DUC 2002 English dataset with a ROUGE score of 0.4797 and can also summarize Swedish documents.
3. Domain knowledge is added through sentence boosting to improve summarization of news articles, and similarities between sentences are calculated to avoid redundancy for multi-document summarization.
This study tested the proposed translation universal of convergence, which claims that translated texts tend to be more similar than non-translated texts, using Spanish medical and technical texts. Stylistic features showed some evidence of convergence in translated texts, but syntax comparisons found no evidence of convergence. Overall, the results did not provide strong support that convergence is a valid universal of translation. The study highlights limitations in making definitive claims about translation universals based on empirical evidence.
Strict intersection types for the lambda calculusunyil96
This article discusses strict intersection types for the lambda calculus. It focuses on an essential intersection type assignment system (E) that is almost syntax directed. The system E is shown to satisfy all major properties of the Barendregt-Coppo-Dezani type system (BCD), including the approximation theorem, characterization of normalization, completeness of type assignment using filter semantics, strong normalization for cut-elimination, and the principal pair property. Some proofs of these properties for E are new. E is a true restriction of BCD and provides a less complicated approach than BCD while achieving the same results.
This document compares the Vector Space Model (VSM) and Latent Semantic Indexing (LSI) techniques for information retrieval. VSM represents documents as vectors in a multi-dimensional space, where cosine similarity between vectors indicates document similarity. LSI builds on VSM by extracting concepts from terms and representing documents based on these concepts, allowing matching of documents using different vocabularies. While VSM is simpler, LSI can handle synonymy and polysemy better. Both are commonly used in search engines.
This document describes a theorem proving procedure that combines resolution and term rewriting approaches. It restricts paramodulation to only allow replacement of subterms on the left-hand side of an equation. This strongly oriented character leads the procedure to be viewed as a form of superposition. The procedure represents clauses as equational clauses, where each literal is an equation, and shows that superposition on these clauses embodies paramodulation and binary resolution. It incorporates term simplification and subsumption optimizations. Experimental results and applications are also discussed.
SIMILAR THESAURUS BASED ON ARABIC DOCUMENT: AN OVERVIEW AND COMPARISONIJCSEA Journal
The massive grow of the modern information retrieval system (IRS), especially in natural languages
becomes more difficult. The search in Arabic languages, as natural language, is not good enough yet. This
paper will try to build similar thesaurus based on Arabic language in two mechanisms, the first one is full
word mechanisms and the other is stemmed mechanisms, and then to compare between them.
The comparison made by this study proves that the similar thesaurus using stemmed mechanisms get more
better results than using traditional in the same mechanisms and similar thesaurus improved more the
recall and precision than traditional information retrieval system at recall and precision levels.
Formal treatments of inheritance are rather scarce and those that do exist are often more suited for
analysis of existing systems than as guides to language designers. One problem that adds complexity to
previous efforts is the need to pass a reference to the original invoking object throughout the method call
tree. In this paper, a novel specification of inheritance semantics is given. The approach dispenses with
self-reference, instead using static and dynamic scope to accomplish similar behaviour. The result is a
methodology that is simpler than previous specification attempts, easy to understand, and sufficiently
expressive. Moreover, an inheritance system based on this approach can be implemented with relatively
few lines of code in environment-passing interpreters.
A Survey of String Matching AlgorithmsIJERA Editor
The concept of string matching algorithms are playing an important role of string algorithms in finding a place where one or several strings (patterns) are found in a large body of text (e.g., data streaming, a sentence, a paragraph, a book, etc.). Its application covers a wide range, including intrusion detection Systems (IDS) in computer networks, applications in bioinformatics, detecting plagiarism, information security, pattern recognition, document matching and text mining. In this paper we present a short survey for well-known and recent updated and hybrid string matching algorithms. These algorithms can be divided into two major categories, known as exact string matching and approximate string matching. The string matching classification criteria was selected to highlight important features of matching strategies, in order to identify challenges and vulnerabilities.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Commentz-Walter: Any Better than Aho-Corasick for Peptide Identification? IJORCS
An algorithm for locating all occurrences of a finite number of keywords in an arbitrary string, also known as multiple strings matching, is commonly required in information retrieval (such as sequence analysis, evolutionary biological studies, gene/protein identification and network intrusion detection) and text editing applications. Although Aho-Corasick was one of the commonly used exact multiple strings matching algorithm, Commentz-Walter has been introduced as a better alternative in the recent past. Comments-Walter algorithm combines ideas from both Aho-Corasick and Boyer Moore. Large scale rapid and accurate peptide identification is critical in computational proteomics. In this paper, we have critically analyzed the time complexity of Aho-Corasick and Commentz-Walter for their suitability in large scale peptide identification. According to the results we obtained for our dataset, we conclude that Aho-Corasick is performing better than Commentz-Walter as opposed to the common beliefs.
This document summarizes a research paper that proposes a new algorithm called Constraint Based Frequent Motif Mining (CBFMM) to efficiently find frequent patterns in biological sequence data in the presence of variations. CBFMM uses a frequent pattern tree structure to flexibly handle different similarity definitions and can be used for protein and nucleotide pattern mining to identify potential malfunctions and diseases. The paper describes related work on sequence mining algorithms and motif finding approaches. It then presents the CBFMM algorithm and experimental results demonstrating its effectiveness at mining biological domains for cryptic sequence repeats compared to other methods.
TIME-ABSTRACTING BISIMULATION FOR MARKOVIAN TIMED AUTOMATAijseajournal
Markovian timed automata (MTA) has been proposed as an expressive formalism for specification of realtime
properties in Markovian stochastic processes. In this paper, we define bisimulation relation for
deterministic MTA. This definition provides a basis for developing effective algorithms for deciding
bisimulation relation for such automata.
The document discusses data structures and algorithms. It defines arrays as a series of objects of the same size and type, where each object is an element that can be accessed via an index. Algorithms are described as finite sequences of instructions to solve problems, with analysis of algorithms determining the resources like time and storage required.
Similar to Multiset path orderings and their application to termination of term rewriting systems (20)
Abnormalities of hormones and inflammatory cytokines in women affected with p...Alexander Decker
Women with polycystic ovary syndrome (PCOS) have elevated levels of hormones like luteinizing hormone and testosterone, as well as higher levels of insulin and insulin resistance compared to healthy women. They also have increased levels of inflammatory markers like C-reactive protein, interleukin-6, and leptin. This study found these abnormalities in the hormones and inflammatory cytokines of women with PCOS ages 23-40, indicating that hormone imbalances associated with insulin resistance and elevated inflammatory markers may worsen infertility in women with PCOS.
A usability evaluation framework for b2 c e commerce websitesAlexander Decker
This document presents a framework for evaluating the usability of B2C e-commerce websites. It involves user testing methods like usability testing and interviews to identify usability problems in areas like navigation, design, purchasing processes, and customer service. The framework specifies goals for the evaluation, determines which website aspects to evaluate, and identifies target users. It then describes collecting data through user testing and analyzing the results to identify usability problems and suggest improvements.
A universal model for managing the marketing executives in nigerian banksAlexander Decker
This document discusses a study that aimed to synthesize motivation theories into a universal model for managing marketing executives in Nigerian banks. The study was guided by Maslow and McGregor's theories. A sample of 303 marketing executives was used. The results showed that managers will be most effective at motivating marketing executives if they consider individual needs and create challenging but attainable goals. The emerged model suggests managers should provide job satisfaction by tailoring assignments to abilities and monitoring performance with feedback. This addresses confusion faced by Nigerian bank managers in determining effective motivation strategies.
A unique common fixed point theorems in generalized dAlexander Decker
This document presents definitions and properties related to generalized D*-metric spaces and establishes some common fixed point theorems for contractive type mappings in these spaces. It begins by introducing D*-metric spaces and generalized D*-metric spaces, defines concepts like convergence and Cauchy sequences. It presents lemmas showing the uniqueness of limits in these spaces and the equivalence of different definitions of convergence. The goal of the paper is then stated as obtaining a unique common fixed point theorem for generalized D*-metric spaces.
A trends of salmonella and antibiotic resistanceAlexander Decker
This document provides a review of trends in Salmonella and antibiotic resistance. It begins with an introduction to Salmonella as a facultative anaerobe that causes nontyphoidal salmonellosis. The emergence of antimicrobial-resistant Salmonella is then discussed. The document proceeds to cover the historical perspective and classification of Salmonella, definitions of antimicrobials and antibiotic resistance, and mechanisms of antibiotic resistance in Salmonella including modification or destruction of antimicrobial agents, efflux pumps, modification of antibiotic targets, and decreased membrane permeability. Specific resistance mechanisms are discussed for several classes of antimicrobials.
A transformational generative approach towards understanding al-istifhamAlexander Decker
This document discusses a transformational-generative approach to understanding Al-Istifham, which refers to interrogative sentences in Arabic. It begins with an introduction to the origin and development of Arabic grammar. The paper then explains the theoretical framework of transformational-generative grammar that is used. Basic linguistic concepts and terms related to Arabic grammar are defined. The document analyzes how interrogative sentences in Arabic can be derived and transformed via tools from transformational-generative grammar, categorizing Al-Istifham into linguistic and literary questions.
A time series analysis of the determinants of savings in namibiaAlexander Decker
This document summarizes a study on the determinants of savings in Namibia from 1991 to 2012. It reviews previous literature on savings determinants in developing countries. The study uses time series analysis including unit root tests, cointegration, and error correction models to analyze the relationship between savings and variables like income, inflation, population growth, deposit rates, and financial deepening in Namibia. The results found inflation and income have a positive impact on savings, while population growth negatively impacts savings. Deposit rates and financial deepening were found to have no significant impact. The study reinforces previous work and emphasizes the importance of improving income levels to achieve higher savings rates in Namibia.
A therapy for physical and mental fitness of school childrenAlexander Decker
This document summarizes a study on the importance of exercise in maintaining physical and mental fitness for school children. It discusses how physical and mental fitness are developed through participation in regular physical exercises and cannot be achieved solely through classroom learning. The document outlines different types and components of fitness and argues that developing fitness should be a key objective of education systems. It recommends that schools ensure pupils engage in graded physical activities and exercises to support their overall development.
A theory of efficiency for managing the marketing executives in nigerian banksAlexander Decker
This document summarizes a study examining efficiency in managing marketing executives in Nigerian banks. The study was examined through the lenses of Kaizen theory (continuous improvement) and efficiency theory. A survey of 303 marketing executives from Nigerian banks found that management plays a key role in identifying and implementing efficiency improvements. The document recommends adopting a "3H grand strategy" to improve the heads, hearts, and hands of management and marketing executives by enhancing their knowledge, attitudes, and tools.
This document discusses evaluating the link budget for effective 900MHz GSM communication. It describes the basic parameters needed for a high-level link budget calculation, including transmitter power, antenna gains, path loss, and propagation models. Common propagation models for 900MHz that are described include Okumura model for urban areas and Hata model for urban, suburban, and open areas. Rain attenuation is also incorporated using the updated ITU model to improve communication during rainfall.
A synthetic review of contraceptive supplies in punjabAlexander Decker
This document discusses contraceptive use in Punjab, Pakistan. It begins by providing background on the benefits of family planning and contraceptive use for maternal and child health. It then analyzes contraceptive commodity data from Punjab, finding that use is still low despite efforts to improve access. The document concludes by emphasizing the need for strategies to bridge gaps and meet the unmet need for effective and affordable contraceptive methods and supplies in Punjab in order to improve health outcomes.
A synthesis of taylor’s and fayol’s management approaches for managing market...Alexander Decker
1) The document discusses synthesizing Taylor's scientific management approach and Fayol's process management approach to identify an effective way to manage marketing executives in Nigerian banks.
2) It reviews Taylor's emphasis on efficiency and breaking tasks into small parts, and Fayol's focus on developing general management principles.
3) The study administered a survey to 303 marketing executives in Nigerian banks to test if combining elements of Taylor and Fayol's approaches would help manage their performance through clear roles, accountability, and motivation. Statistical analysis supported combining the two approaches.
A survey paper on sequence pattern mining with incrementalAlexander Decker
This document summarizes four algorithms for sequential pattern mining: GSP, ISM, FreeSpan, and PrefixSpan. GSP is an Apriori-based algorithm that incorporates time constraints. ISM extends SPADE to incrementally update patterns after database changes. FreeSpan uses frequent items to recursively project databases and grow subsequences. PrefixSpan also uses projection but claims to not require candidate generation. It recursively projects databases based on short prefix patterns. The document concludes by stating the goal was to find an efficient scheme for extracting sequential patterns from transactional datasets.
A survey on live virtual machine migrations and its techniquesAlexander Decker
This document summarizes several techniques for live virtual machine migration in cloud computing. It discusses works that have proposed affinity-aware migration models to improve resource utilization, energy efficient migration approaches using storage migration and live VM migration, and a dynamic consolidation technique using migration control to avoid unnecessary migrations. The document also summarizes works that have designed methods to minimize migration downtime and network traffic, proposed a resource reservation framework for efficient migration of multiple VMs, and addressed real-time issues in live migration. Finally, it provides a table summarizing the techniques, tools used, and potential future work or gaps identified for each discussed work.
A survey on data mining and analysis in hadoop and mongo dbAlexander Decker
This document discusses data mining of big data using Hadoop and MongoDB. It provides an overview of Hadoop and MongoDB and their uses in big data analysis. Specifically, it proposes using Hadoop for distributed processing and MongoDB for data storage and input. The document reviews several related works that discuss big data analysis using these tools, as well as their capabilities for scalable data storage and mining. It aims to improve computational time and fault tolerance for big data analysis by mining data stored in Hadoop using MongoDB and MapReduce.
1. The document discusses several challenges for integrating media with cloud computing including media content convergence, scalability and expandability, finding appropriate applications, and reliability.
2. Media content convergence challenges include dealing with the heterogeneity of media types, services, networks, devices, and quality of service requirements as well as integrating technologies used by media providers and consumers.
3. Scalability and expandability challenges involve adapting to the increasing volume of media content and being able to support new media formats and outlets over time.
This document surveys trust architectures that leverage provenance in wireless sensor networks. It begins with background on provenance, which refers to the documented history or derivation of data. Provenance can be used to assess trust by providing metadata about how data was processed. The document then discusses challenges for using provenance to establish trust in wireless sensor networks, which have constraints on energy and computation. Finally, it provides background on trust, which is the subjective probability that a node will behave dependably. Trust architectures need to be lightweight to account for the constraints of wireless sensor networks.
This document discusses private equity investments in Kenya. It provides background on private equity and discusses trends in various regions. The objectives of the study discussed are to establish the extent of private equity adoption in Kenya, identify common forms of private equity utilized, and determine typical exit strategies. Private equity can involve venture capital, leveraged buyouts, or mezzanine financing. Exits allow recycling of capital into new opportunities. The document provides context on private equity globally and in developing markets like Africa to frame the goals of the study.
This document discusses a study that analyzes the financial health of the Indian logistics industry from 2005-2012 using Altman's Z-score model. The study finds that the average Z-score for selected logistics firms was in the healthy to very healthy range during the study period. The average Z-score increased from 2006 to 2010 when the Indian economy was hit by the global recession, indicating the overall performance of the Indian logistics industry was good. The document reviews previous literature on measuring financial performance and distress using ratios and Z-scores, and outlines the objectives and methodology used in the current study.
UiPath Test Automation using UiPath Test Suite series, part 6DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 6. In this session, we will cover Test Automation with generative AI and Open AI.
UiPath Test Automation with generative AI and Open AI webinar offers an in-depth exploration of leveraging cutting-edge technologies for test automation within the UiPath platform. Attendees will delve into the integration of generative AI, a test automation solution, with Open AI advanced natural language processing capabilities.
Throughout the session, participants will discover how this synergy empowers testers to automate repetitive tasks, enhance testing accuracy, and expedite the software testing life cycle. Topics covered include the seamless integration process, practical use cases, and the benefits of harnessing AI-driven automation for UiPath testing initiatives. By attending this webinar, testers, and automation professionals can gain valuable insights into harnessing the power of AI to optimize their test automation workflows within the UiPath ecosystem, ultimately driving efficiency and quality in software development processes.
What will you get from this session?
1. Insights into integrating generative AI.
2. Understanding how this integration enhances test automation within the UiPath platform
3. Practical demonstrations
4. Exploration of real-world use cases illustrating the benefits of AI-driven test automation for UiPath
Topics covered:
What is generative AI
Test Automation with generative AI and Open AI.
UiPath integration with generative AI
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Driving Business Innovation: Latest Generative AI Advancements & Success StorySafe Software
Are you ready to revolutionize how you handle data? Join us for a webinar where we’ll bring you up to speed with the latest advancements in Generative AI technology and discover how leveraging FME with tools from giants like Google Gemini, Amazon, and Microsoft OpenAI can supercharge your workflow efficiency.
During the hour, we’ll take you through:
Guest Speaker Segment with Hannah Barrington: Dive into the world of dynamic real estate marketing with Hannah, the Marketing Manager at Workspace Group. Hear firsthand how their team generates engaging descriptions for thousands of office units by integrating diverse data sources—from PDF floorplans to web pages—using FME transformers, like OpenAIVisionConnector and AnthropicVisionConnector. This use case will show you how GenAI can streamline content creation for marketing across the board.
Ollama Use Case: Learn how Scenario Specialist Dmitri Bagh has utilized Ollama within FME to input data, create custom models, and enhance security protocols. This segment will include demos to illustrate the full capabilities of FME in AI-driven processes.
Custom AI Models: Discover how to leverage FME to build personalized AI models using your data. Whether it’s populating a model with local data for added security or integrating public AI tools, find out how FME facilitates a versatile and secure approach to AI.
We’ll wrap up with a live Q&A session where you can engage with our experts on your specific use cases, and learn more about optimizing your data workflows with AI.
This webinar is ideal for professionals seeking to harness the power of AI within their data management systems while ensuring high levels of customization and security. Whether you're a novice or an expert, gain actionable insights and strategies to elevate your data processes. Join us to see how FME and AI can revolutionize how you work with data!
Building Production Ready Search Pipelines with Spark and MilvusZilliz
Spark is the widely used ETL tool for processing, indexing and ingesting data to serving stack for search. Milvus is the production-ready open-source vector database. In this talk we will show how to use Spark to process unstructured data to extract vector representations, and push the vectors to Milvus vector database for search serving.
Goodbye Windows 11: Make Way for Nitrux Linux 3.5.0!SOFTTECHHUB
As the digital landscape continually evolves, operating systems play a critical role in shaping user experiences and productivity. The launch of Nitrux Linux 3.5.0 marks a significant milestone, offering a robust alternative to traditional systems such as Windows 11. This article delves into the essence of Nitrux Linux 3.5.0, exploring its unique features, advantages, and how it stands as a compelling choice for both casual users and tech enthusiasts.
For the full video of this presentation, please visit: https://www.edge-ai-vision.com/2024/06/building-and-scaling-ai-applications-with-the-nx-ai-manager-a-presentation-from-network-optix/
Robin van Emden, Senior Director of Data Science at Network Optix, presents the “Building and Scaling AI Applications with the Nx AI Manager,” tutorial at the May 2024 Embedded Vision Summit.
In this presentation, van Emden covers the basics of scaling edge AI solutions using the Nx tool kit. He emphasizes the process of developing AI models and deploying them globally. He also showcases the conversion of AI models and the creation of effective edge AI pipelines, with a focus on pre-processing, model conversion, selecting the appropriate inference engine for the target hardware and post-processing.
van Emden shows how Nx can simplify the developer’s life and facilitate a rapid transition from concept to production-ready applications.He provides valuable insights into developing scalable and efficient edge AI solutions, with a strong focus on practical implementation.
HCL Notes and Domino License Cost Reduction in the World of DLAUpanagenda
Webinar Recording: https://www.panagenda.com/webinars/hcl-notes-and-domino-license-cost-reduction-in-the-world-of-dlau/
The introduction of DLAU and the CCB & CCX licensing model caused quite a stir in the HCL community. As a Notes and Domino customer, you may have faced challenges with unexpected user counts and license costs. You probably have questions on how this new licensing approach works and how to benefit from it. Most importantly, you likely have budget constraints and want to save money where possible. Don’t worry, we can help with all of this!
We’ll show you how to fix common misconfigurations that cause higher-than-expected user counts, and how to identify accounts which you can deactivate to save money. There are also frequent patterns that can cause unnecessary cost, like using a person document instead of a mail-in for shared mailboxes. We’ll provide examples and solutions for those as well. And naturally we’ll explain the new licensing model.
Join HCL Ambassador Marc Thomas in this webinar with a special guest appearance from Franz Walder. It will give you the tools and know-how to stay on top of what is going on with Domino licensing. You will be able lower your cost through an optimized configuration and keep it low going forward.
These topics will be covered
- Reducing license cost by finding and fixing misconfigurations and superfluous accounts
- How do CCB and CCX licenses really work?
- Understanding the DLAU tool and how to best utilize it
- Tips for common problem areas, like team mailboxes, functional/test users, etc
- Practical examples and best practices to implement right away
How to Get CNIC Information System with Paksim Ga.pptxdanishmna97
Pakdata Cf is a groundbreaking system designed to streamline and facilitate access to CNIC information. This innovative platform leverages advanced technology to provide users with efficient and secure access to their CNIC details.
Observability Concepts EVERY Developer Should Know -- DeveloperWeek Europe.pdfPaige Cruz
Monitoring and observability aren’t traditionally found in software curriculums and many of us cobble this knowledge together from whatever vendor or ecosystem we were first introduced to and whatever is a part of your current company’s observability stack.
While the dev and ops silo continues to crumble….many organizations still relegate monitoring & observability as the purview of ops, infra and SRE teams. This is a mistake - achieving a highly observable system requires collaboration up and down the stack.
I, a former op, would like to extend an invitation to all application developers to join the observability party will share these foundational concepts to build on:
Threats to mobile devices are more prevalent and increasing in scope and complexity. Users of mobile devices desire to take full advantage of the features
available on those devices, but many of the features provide convenience and capability but sacrifice security. This best practices guide outlines steps the users can take to better protect personal devices and information.
In his public lecture, Christian Timmerer provides insights into the fascinating history of video streaming, starting from its humble beginnings before YouTube to the groundbreaking technologies that now dominate platforms like Netflix and ORF ON. Timmerer also presents provocative contributions of his own that have significantly influenced the industry. He concludes by looking at future challenges and invites the audience to join in a discussion.
In the rapidly evolving landscape of technologies, XML continues to play a vital role in structuring, storing, and transporting data across diverse systems. The recent advancements in artificial intelligence (AI) present new methodologies for enhancing XML development workflows, introducing efficiency, automation, and intelligent capabilities. This presentation will outline the scope and perspective of utilizing AI in XML development. The potential benefits and the possible pitfalls will be highlighted, providing a balanced view of the subject.
We will explore the capabilities of AI in understanding XML markup languages and autonomously creating structured XML content. Additionally, we will examine the capacity of AI to enrich plain text with appropriate XML markup. Practical examples and methodological guidelines will be provided to elucidate how AI can be effectively prompted to interpret and generate accurate XML markup.
Further emphasis will be placed on the role of AI in developing XSLT, or schemas such as XSD and Schematron. We will address the techniques and strategies adopted to create prompts for generating code, explaining code, or refactoring the code, and the results achieved.
The discussion will extend to how AI can be used to transform XML content. In particular, the focus will be on the use of AI XPath extension functions in XSLT, Schematron, Schematron Quick Fixes, or for XML content refactoring.
The presentation aims to deliver a comprehensive overview of AI usage in XML development, providing attendees with the necessary knowledge to make informed decisions. Whether you’re at the early stages of adopting AI or considering integrating it in advanced XML development, this presentation will cover all levels of expertise.
By highlighting the potential advantages and challenges of integrating AI with XML development tools and languages, the presentation seeks to inspire thoughtful conversation around the future of XML development. We’ll not only delve into the technical aspects of AI-powered XML development but also discuss practical implications and possible future directions.
Communications Mining Series - Zero to Hero - Session 1DianaGray10
This session provides introduction to UiPath Communication Mining, importance and platform overview. You will acquire a good understand of the phases in Communication Mining as we go over the platform with you. Topics covered:
• Communication Mining Overview
• Why is it important?
• How can it help today’s business and the benefits
• Phases in Communication Mining
• Demo on Platform overview
• Q/A
“An Outlook of the Ongoing and Future Relationship between Blockchain Technologies and Process-aware Information Systems.” Invited talk at the joint workshop on Blockchain for Information Systems (BC4IS) and Blockchain for Trusted Data Sharing (B4TDS), co-located with with the 36th International Conference on Advanced Information Systems Engineering (CAiSE), 3 June 2024, Limassol, Cyprus.
Unlocking Productivity: Leveraging the Potential of Copilot in Microsoft 365, a presentation by Christoforos Vlachos, Senior Solutions Manager – Modern Workplace, Uni Systems
Maruthi Prithivirajan, Head of ASEAN & IN Solution Architecture, Neo4j
Get an inside look at the latest Neo4j innovations that enable relationship-driven intelligence at scale. Learn more about the newest cloud integrations and product enhancements that make Neo4j an essential choice for developers building apps with interconnected data and generative AI.
HCL Notes und Domino Lizenzkostenreduzierung in der Welt von DLAUpanagenda
Webinar Recording: https://www.panagenda.com/webinars/hcl-notes-und-domino-lizenzkostenreduzierung-in-der-welt-von-dlau/
DLAU und die Lizenzen nach dem CCB- und CCX-Modell sind für viele in der HCL-Community seit letztem Jahr ein heißes Thema. Als Notes- oder Domino-Kunde haben Sie vielleicht mit unerwartet hohen Benutzerzahlen und Lizenzgebühren zu kämpfen. Sie fragen sich vielleicht, wie diese neue Art der Lizenzierung funktioniert und welchen Nutzen sie Ihnen bringt. Vor allem wollen Sie sicherlich Ihr Budget einhalten und Kosten sparen, wo immer möglich. Das verstehen wir und wir möchten Ihnen dabei helfen!
Wir erklären Ihnen, wie Sie häufige Konfigurationsprobleme lösen können, die dazu führen können, dass mehr Benutzer gezählt werden als nötig, und wie Sie überflüssige oder ungenutzte Konten identifizieren und entfernen können, um Geld zu sparen. Es gibt auch einige Ansätze, die zu unnötigen Ausgaben führen können, z. B. wenn ein Personendokument anstelle eines Mail-Ins für geteilte Mailboxen verwendet wird. Wir zeigen Ihnen solche Fälle und deren Lösungen. Und natürlich erklären wir Ihnen das neue Lizenzmodell.
Nehmen Sie an diesem Webinar teil, bei dem HCL-Ambassador Marc Thomas und Gastredner Franz Walder Ihnen diese neue Welt näherbringen. Es vermittelt Ihnen die Tools und das Know-how, um den Überblick zu bewahren. Sie werden in der Lage sein, Ihre Kosten durch eine optimierte Domino-Konfiguration zu reduzieren und auch in Zukunft gering zu halten.
Diese Themen werden behandelt
- Reduzierung der Lizenzkosten durch Auffinden und Beheben von Fehlkonfigurationen und überflüssigen Konten
- Wie funktionieren CCB- und CCX-Lizenzen wirklich?
- Verstehen des DLAU-Tools und wie man es am besten nutzt
- Tipps für häufige Problembereiche, wie z. B. Team-Postfächer, Funktions-/Testbenutzer usw.
- Praxisbeispiele und Best Practices zum sofortigen Umsetzen
HCL Notes und Domino Lizenzkostenreduzierung in der Welt von DLAU
Multiset path orderings and their application to termination of term rewriting systems
1. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.5, 2013
96
Multiset Path Orderings and Their Application to Termination of
Term Rewriting Systems
Dasharath Singh1
and Ali Maianguwa Shuaibu2*
1
Department of Mathematics, Ahmadu Bello University, Zaria-Nigeria. Email: mathdss@yahoo.com.
2
Department of Mathematics, Statistics and Computer science, Kaduna Polytechnic, P.M.B. 2021, Kaduna,
Nigeria.
*
Corresponding Author Email: shuaibuali16@gmail.com.
Abstract
In this expository paper, a comprehensive study of multiset orderings, nested multiset orderings and multiset
path orderings is presented. In particular, it is illustrated how multiset path orderings admit the use of relatively
simple and intuitive termination functions that lead to termination of a class of term rewriting systems.
1. Introduction
Termination is one of the most important properties of term rewriting systems (trss, for short) which, in general,
is known to be undecidable (Huet and Lankford, 1978; Dershowitz, 1987). In the recent years, some powerful
methods have been developed to prove termination of a large class of trss. Broadly, these methods can be
divided into direct and transformational methods (Zankl, 2006). The direct methods are further classified into
syntactical and semantical fragments. The recursive path orders (rpos), such as lexicographic path orders (lpos)
and multiset path orders (mpos), are purely syntactical; while Polynomial orders and Knuth-Bendix order (kbo)
are semantical. In transformational method, termination proof of a given trs is accomplished by reducing it to an
appropriate equivalent form for which proving termination is relatively easier. Some examples of this class are
the dependency pair method, semantic labeling and freezing (Terese, 2003; Singh et al. 2012a).
More often than not, designing an appropriate reduction order and termination function for trss is found quite an
involved problem. The main objective of this paper is to demonstrate how the application of mpos permits the
use of relatively simple and intuitive termination functions that help achieving termination of a class of trss.
2. Preliminaries
In what follows, abstracting from various expositions on trss, specially from (Ohlebusch, 2002; Dershowitz,
1982), we describe some basic concepts in order to make the paper self-contained.
Let or denote a signature (a finite set of function symbols with natural numbers as their arities) and a
countably infinite set of variables with . A constant is a function symbol having no arguments. A
term is formed from function symbols, constants and variables. A term without variables is called a ground or
closed term and, the set of ground terms is denoted by . Also, let denote the set of all terms built
over and . It is assumed that contains at least one constant. The set of variables occurring in a term is
denoted by . Terms are usually denoted by , possibly extended by subscripts. Occasionally,
we write to denote a sequence of terms and, to denote a sequence of function symbols
. The size of a term , denoted by | |, represents the number of symbols in . A trs is called
length preserving if | | | | for all rules and all ground substitutions . It is not difficult to
prove that any length preserving trs is simply terminating.
A rewrite rule is an ordered pair of terms and such that and the variables which occur in the right-
hand side , also occur in the left-hand side . A rewrite rule is usually written as . A term rewriting
system is a pair consisting of a signature and a set of rewrite rules between terms in . A trs
is often presented as a set of rewrite rules, without explicitly mentioning its signature, assuming that the
signature consists of the function symbols occurring in the rewrite rules. We assume that is finite, unless stated
otherwise. Also, , where is a substitution. A subterm is successively replaced by an
equal term until no further rewriting is possible.
Essentially, the main objective of the method of term rewriting is to apply a set of rewrite rules to terms to
reduce them to their simplest forms. Formally, a trs is called terminating if there is no infinite rewriting
2. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.5, 2013
97
sequence . In other words, a trs is terminating if and only if all terms have only rewrite
sequences of finite length. A rewrite relation that is also a partial order is called a rewrite order. A well-founded
rewrite order is called a reduction order. More explicitly, a reduction order is an order which is well founded,
monotonic, and stable (closed under contexts and substitution). A trs and a partial order on
are said to be compatible if is contained in ; that is, for every rewrite rule of . The rewrite
relation induced by a rewrite system is denoted by and, its transitive closure by . It is easy to see
that a trs is terminating if and only if it is compatible with a reduction order. The classical approach to prove
termination of a particular trs consists in constructing a reduction order such that for each rewrite
step .
Definition 1.
A partial ordering is a simplification ordering for a set of terms if it possesses the following three
properties:
(i) implies . . . (replacement)
(ii) . . . (subterm)
(iii) . . . (deletion)
where belong to .
Definition 2.
A partial ordering is called monotonic if it possesses the subterm property , and the
deletion property , for all terms in .
A well-founded monotonic order satisfying the subterm property is called a simplification ordering. By iterating
the subterm property, it is easy to see that every term is also greater than any (not necessarily immediate) of its
subterms. The deletion condition implies that deleting subterms of an operator of variable arity reduces the size
of the term in the ordering; if an operator is of fixed arity, the deletion condition is superfluous. Further,
simultaneously holding of these conditions imply that syntactically simpler terms are smaller in the ordering
(Dershowitz, 1982; Singh et al. 2012b).
A trs over a finite signature is called simply terminating if is compatible with a simplification order.
Definition 3. Let be a set
of variables. The homeomorphic embedding , a binary relation on , is defined as follows:
if and only if one of :
1. for a variable .
2. and for a function symbol , and
.
3. for a function symbol , and for some
For example,
( )
Definition 4.
An infinite sequence of terms is self-embedding if there exist such that
. Homeomorphic embedding
could also be defined as the reduction relation induced by the rewrite system
{ | .
Since is obviously terminating, this shows that is a well-founded partial order. In fact, in
view of Kruskal’s Tree Theorem (Kruskal, 1960), satisfies a stronger property called well partial order
3. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.5, 2013
98
(Baader and Nipkow, 1998 ) for finite and . Simplification orderings cannot be
used to prove termination of self- embedding systems (Dershowitz, 1987).
Lemma 1
Let be a simplification ordering on a set of terms , and . Then implies .
Proof Assume that
. We consider the three cases in the definition of , and prove by induction on | |.
(i) If then , because is reflexive. (ii) Assume
that and for a function symbol and ,
. By induction, we obtain . Since is a rewrite order, we have
. (iii) Assume that
for a function symbol and for some . By induction, we
obtain . In addition, the subterm property of yields , and thus .
Lemma 2
Let be a trs over a finite signature . Then every simplification order is a reduction order.
Proof.
By definition of
simplification orders, it remains to be shown that every simplification order is well-founded. Assume that is a
simplification order on , and is an infinite chain in .
First, we show by contradiction that holds. Assume that there exists a
variable . Define a substitution { such that on one hand, (since
does not occur in ) and (since is a rewrite order). On the other hand, since is a
subterm of , it follows from the subterm property that . If we combine the two inequalities,
we obtain , which is a contradiction. The first part of the proof shows that, for the finite set
, all terms in the sequence belong to . Since and are finite, Kruskal’s Theorem
implies that this sequence is good. i.e., there exist such that . Now, Lemma 1 yields ,
which is a contradiction since we know that .
A direct consequence of Kruskal’s theorem (Kruskal, 1960) is that any simplification order over a finite
signature is well-founded as shown above.
Theorem 1.
Simplification orders are well-founded on terms over finite signature .
Proof follows by Kruskal’s Tree Theorem and Lemma 2 above.
3. The Recursive Path Ordering
An important syntactical technique to prove termination of term rewriting is by using the recursive path ordering
(rpo) defined by Dershowitz (Dershowitz, 1987). For defining rpo, there underlies a well-founded ordering on
the set of function symbols. At the first step, two terms are compared by comparing their root symbols, and then
recursively, the collections of their immediate subterms are compared. These collections can be seen as
unordered multisets (giving rise to the notion of multiset path order), introduced by Dershowitz (1982), or as
ordered tuples (giving rise to the notion of lexicographic path order), introduced by Kamin and Levy (1980), or
one can employ a combination of the two (giving rise to a recursive path order with status). In this work, we
confine ourselves to the case where the arguments are compared as multisets.
In order to determine if a term is greater than a term using rpo, the outmost operators of the two terms are
compared first. If the outermost operators happen to be equal, then those (immediate) subterms of that are not
also subterms of must each be smaller(recursively in the term ordering) than some subterm of . If the outmost
operator of is greater than that of , then must be greater than each subterm of ; while if the outmost
4. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.5, 2013
99
operator of is neither equal to nor greater than that of ,then some subterm of must be greater than or equal
to .
Formally, rpo can be defined as follows:
Definition 5.
Let be a signature and be a strict ordering (precedence) on . Then the rpo, on the set of
terms over is defined recursively as follows:
,
if and only if
(i) and { { , or
(ii) and { { , or
(iii) and { { .
In other words, if either,
(i) equals but the tuple accompanying in is bigger than the one accompanying in (i.e.,
, or
(ii) is bigger than all the immediate subterms of when , or
(iii) when the outmost operator of is neither equal to nor greater than that of , then some subterm of must
be bigger than or equal to .
Example .
For the set of operators { with and the precedence given
by , we have:
(i) because (this is because is a subterm of ), and
.
(ii) .
On one hand, , because is a subterm of and
(because and ).
On the other hand, we have, .
The mpo method for proving termination is based on the following:
3.1 Multiset Orderings
Multiset ordering was invented in the ‘70s to prove termination of programs (Dershowitz and Manna, 1979). It
has been used in devising termination techniques like mpo, rpo and recently, in combination with the size-
change principle of Lee et al. (2001), in the form of SCNP (SCT in NP) reduction pairs.
A multiset (mset, for short) over a set can be defined as a function from into , the set of natural numbers
including zero. Let denote the number of occurrences of an object of in called the multiplicity of
in . If for finitely many in , then is called a finite mset.
Let denote the set of all finite msets built on . The additive union (or sum) denoted of two msets
and is defined as follows:
, for all .
5. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.5, 2013
100
The difference of and is defined as follows:
{ , for all .
Following (Dershowitz and Manna, 1979), a partial ordering on a set may be extended to a partial ordering
on finite multisets of elements of . In this extended ordering, , for two finite multisets
and over , if can be obtained from by replacing one or more elements in by any (finite) number
of elements taken from , each of which is smaller than one of the replaced elements. They also proved that
every well-founded relation on a set induces a well-founded relation on . Note that well-foundedness is
an essential property of orderings for proving termination of TRSs. Formally, given a partially ordered set
, the multiset ordering is defined on as follows:
,
if , where ,
and for all , such that .
Thus, if is an ordering, we have if we can obtain from by either (i) removing some
(possibly duplicated) elements , or (ii) replacing some elements by new but smaller elements
.
For example,
{ { , { { ,{ { ,
{ { , { { , etc.
Theorem 2.
If is irreflexive and transitive, then is also irreflexive and transitive.
Proof.
To show irreflexivity, we must show that there can be no multiset such that .
Suppose that , then there would be some nonempty finite multiset , such that and
In other words, for every element of there would be a distinct elements of
greater than it, which is impossible for a finite
To show transitivity of , consider the following irreflexive relation on multisets in
{ if In other words, a finite multiset is reduced in the relation by
replacing a single element with zero or more smaller elements. Note that the multiset ordering is the
transitive closure of the relation i.e., if and only if can be obtained from by replacing
elements in one by one. It follows that is transitive.
The following theorem implies that multiset orderings yield much simpler proof of termination of trss.
Theorem 3.
The multiset ordering over (S is well-founded if and only if ( is well-founded.
Proof.
(a) “only if” part. Suppose ( is not well-founded, then there exists an infinite decreasing sequence
of elements in . The corresponding sequence of singletons
{ { { forms an infinite decreasing sequence of elements in and thus
is not well-founded.
6. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.5, 2013
101
(b) “if” part. Assume that is not well-founded. We first extend by adding to it an element , and extend
the ordering on to make the least element i.e., for every element in , Thus, clearly remains
well-founded.
Now, suppose that is not well-founded. Thus, there exists an infinite decreasing sequence
of multisets from We derive a contradiction by constructing the
following tree. Each node in the tree is labelled with some element of and at each stage of the construction;
the set of all terminal nodes in the tree forms a multiset in
Let us begin with a root node with children corresponding to each element of .
Then, since , there must exist multisets , such that are not empty,
and . Then for each , add a “son” labelled to the corresponding node. In
addition, grow a child from each of the elements of . Since is nonempty, growing ensures that even if
is empty, at least one node is added to the tree. Since is finite, the nodes corresponding to each have a finite
number of sons. Repeat the process for , and so on.
Since at least one node is added to the tree for each multiset , in the sequence, were the sequence infinite, the
tree corresponding to the sequence would also be infinite. But, by Konig’s Infinity Lemma, an infinite tree with a
finite number of children for each node must have an infinite path. On the other hand, by our construction, all
paths in the tree are descending in the well-founded ordering on , and must be finite. Thus, we have derived
a contradiction, implying there cannot be an infinite sequence of multisets satisfying
Definition 6.
Given a quasi-ordered set , the multiset quasi-ordering on is defined as follows:
if and only if, for some multisets and , , and for all there is an
such that , where the two multisets are considered equivalent if the equivalence classes of their
elements (under ) are the same.
Remark 1.
An ordering over a set can be extended to an ordering on tuples in (for some ) as follows:
if { { .
If ( is totally ordered, then for any two multisets , one may decide whether
by first sorting the elements of both and in descending order (with respect to the relation ) and then
comparing the two sorted sequences lexicographically.
For example, in order to compare the multisets { and { , one may compare the sorted
sequences and . Since is lexicographically greater than , it
follows that { { .
Multisets ordering enjoys the following minimality property:
Theorem 4 (Lescanne and Jouannaud, 1982):
For a given partial ordering on a set , any partial ordering on that satisfies the property
implies { {
is contained in the multiset ordering .
Remark 2.
If ( is of order type , then the multiset ordering ( over ( is of order type .This
follows from the fact that there exists a mapping from onto that is one-to-one and order-
preserving, i.e., if for ) ,then the ordinal is greater than . Such a
7. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.5, 2013
102
mapping can be defined ∑ where denotes the natural (commutative) sum of ordinals,
and is the one-to-one order-preserving map from onto
Remark 3.
Let us consider the special case where there is a bound k on the number of replacement elements i.e., | | .
Any termination proof using this bounded multiset ordering over may be translated into a proof using .
This may be done using the order-preserving function
∑
which maps multisets over the natural numbers into the natural numbers by summing the number for every
natural number in a multiset . Two special cases of interest are the following:
(i) If | | | | i.e., the size of the multiset is not increased, then the simpler function
∑ | |
is order-preserving.
(ii) If | | | | i.e., the size of the multiset is constant, then
∑
is order-preserving.
3.2 Nested Multiset Ordering
By a nested multiset, we mean that the elements of the multiset may be elements of the ground set , or may be
multisets of elements of , or may be multisets containing both elements of and multisets of elements of ,
and so on.
For example,
{{ {{ } is a nested multiset.
Further, given a partially ordered set ), a nested multiset over is either an element of or else it is a finite
multiset of nested multisets over . Let denote the set of nested multisets over .
Nested multiset ordering on which is a recursive version of the standard multiset ordering, is
defined as follows:
Definition 7 (Dershowitz and Manna, 1979):
For any two elements , if
and i.e., two elements of the base set are compared using , or
and i.e., a multiset is greater than an element of the base set, or
, and for some where , and
and .
For example,
8. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.5, 2013
103
{{ {{ } is greater than {{ {{ } } since { is greater than both { and ,
and also {{ {{ } is greater than { {{ { since {{ } is greater than each of the
three elements {{}, 1, 2} , {5, 5, 2} and 5 .
Remark 4.
Let denote the set of all nested multisets of depth . In other words,
and contains the multisets whose elements are taken from
, with at least one element taken from . Thus, the set is the infinite union of the disjoint sets
The following property holds:
Theorem 5
For nested multisets and , if the depth of M is greater than the depth of , then . That is,
the elements of are all greater than the elements of for any .
Proof.
The proof follows by induction on depth. It trivially holds for of depth . For the inductive step, let us assume
that the nested multisets of depth are greater than the nested multisets of depth less than . That is, we need to
show that a nested multiset of depth is greater than any nested multiset of lesser depth. If the depth of
is 0, then while , and therefore , as desired. If the depth of is less than but
greater than 0 , then each of the elements in is of depth less than . The nested multiset , on the other
hand, is of depth and must therefore contain some element of depth , which by the inductive hypothesis,
must be greater than each of the elements in . It follows that .
It is easy to see that the partial ordering is irreflexive and transitive. The proof that it is well-founded is
the following theorem:
Theorem 6
The nested multiset ordering over is well-founded if and only if is well-
founded.
Proof.
(a) “only if” part. If is not well-founded, then there exists an infinite decreasing sequence
of elements in . This sequence is also an infinite decreasing sequence of elements in u n d e r
, a n d is therefore not well-founded.
(b) “if” part. In order to show that is well-founded, it suffices to show that each is itself
well-founded under . If were not well-founded, then there would exist an infinite decreasing
sequence of nested multisets . By theorem 5 above, it follows that the depth of any
nested multiset in the sequence cannot be greater than the depth of its predecessor . Since the sequence
is infinite, it must have an infinite subsequence of nested multisets all of the same depth , which contradicts the
well-foundedness of .
We prove that each is well-founded by induction on : The ordering on is
simply the ordering on and hence it follows that is well-founded. For the inductive step,
assume that each is well-founded (note that each of the elements of is a member
of the union of . By the induction hypothesis, each of these is well-
founded under .Therefore their union under also is well-founded. Furthermore, the ordering
on a pair of nested multisets from is exactly the standard multiset ordering over their union and
since the union is well-founded, is well-founded.
9. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.5, 2013
104
Remark 5.
We have seen earlier that for of order type , the multiset ordering is of order type
In a similar manner, it can be shown that the order type of is
}
,
the limit of which is the ordinal , provided is less than . Consequently, if is of order type less than
, then is of order type (Dershowitz, 1982).
Definition 8.
Let and be terms , then and are said to be equivalent up to a
permutation of arguments if both and are variables, or if
(i) the top-level function symbols of and are identitical, and
(ii) there is a one- to – one correspondence between the multisets { and { such that if
and correspond to each other, then and are equivalent up to a permutation of arguments.
We write if and are identical up to a permutation of arguments.
For example, and are equivalent up to a permutation of arguments.
Remark 6.
It is observed that the multiset ordering, nested multiset ordering (Dershowitz and Manna, 1979), and the
simple path ordering (Plaisted ,1978a) can be considered as a special case of the recursive path ordering, in
which the multiset constructor { is greater than all other operators involved. The nested multiset ordering is
a recursive path ordering on all terms constructed from one varyadic operator and with just that one operator of
the order type . It was pointed out in (Dershowitz, 1987 and Paulson, 1984) that the nested multiset ordering
has all the properties of simplification orderings.
3.3 Multiset Path Ordering
Plaisted (1978b) introduced a syntactic order on terms which consists in comparing terms by first comparing
their root symbols according to a given precedence, and, in case of equality, recursively comparing the multisets
of their immediate subterms. As these multisets ignore the positions of the subterms, this order is not sensitive to
permutations of subterms. Multiset path ordering (mpo) is a binary relation on the first order terms. It was first
introduced by Dershowitz (Dershowitz, 1982) for proving termination of rewriting systems. The mpo is a
simplification ordering which is transitive and irreflexive, closed both under context application and substitution,
and as well possesses the subterm property. Moreover, the subterm property contains homeomorphic embedding.
Definition 9.
If a term is of the form then is called the root symbol of . Also, { are called
the top-level subterms of .
Definition 10.
A path is a sequence of operators, beginning with outmost one of the whole terms (say, the root, on viewing
terms as trees) and taking subterms until a constant (leaf) is reached. In other words, if is a ground term of the
form , then a path in is a sequence beginning with and followed by a path from some top-
level subterm of . A path of subterms of is the sequence consisting of itself followed by a path of subterms
for for some , . If is a variable, then itself is the only path of subterms for Thus, a path of
subterms for is the sequence , and . Also, a path of subterms of the term
10. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.5, 2013
105
is the sequence , and . A path order is a well-founded relation on a set of
terms; for example, mpo and lpo.
Definition 11.
A TRS is called strict mpo-terminating if and only if there exists a strict precedence such that
for all .
Lemma 3.
Whenever is a strict precedence on a signature then is a simplification order.
Definition 12.
Let be a strict precedence and . If then and
.The mpo, is defined if one of the following holds:
(i) and , for a { , or
(ii) and , for all { , or
(iii) and { {
where stands for multiset extension of .
3.4 Application of mpo in Proving Termination of TRSs
In what follows, a number of illustrations are provided to emphasize that mpo are amongst the simplest recursive
path orderings developed so far for proving termination of trss.
Example 1. Consider the trs
.
It is immediate to see that lpo with precedence cannot orient the last rule and mpo with precedence
orients all rules from left to right. We have
{ { .
Example 2. The trs , consisting of the rule
,
is mpo-terminating, but not lpo-terminating.
Since neither nor , both (i) and (ii) of definition 14 do not hold, we conclude
that is not lpo-terminating. Since the root symbols are equal case (iii) might apply and
the arguments of the two terms could be compared lexicographically, but as and are incomparable, it does
not lead to termination. For mpo, (iii) applies and {{ } {{ is satisfied because { a
{ such that . That is, , (rule (ii)).
Example 3. Consider the set of arithmetic expressions constructed from some set of symbols and the single
operator . The trs
over , contains just one rewrite rule which reparenthesizes a sum by associating to the right. For example, the
expression ( ) becomes either ( ) or , both
of which become ( ) . Since the size of the expression remains constant when the rule is
applied, some other measure is required to prove termination. This is as follows (Dershowitz and Mann, 1979):
11. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.5, 2013
106
Let us consider the multiset ordering over the natural numbers, . Let the termination function
return the multiset of the sizes | | of all the subexpressions of the form in the expression
( . i.e.,
{| | .
For example,
( ( )) { , since the left operands of the operator are and .
Also, ( ( )) { .
The value of the termination function decreases with each application of the rewrite rule,
i.e., ( ) .
Thus, the system terminates.
Example 4. Determine if , where terms are denoted as trees (Dershowitz, 1982 ),
we have
…
This is an rpo over set of terms of natural numbers, , with the operators ordered by . Since the two terms
have the same outermost operator, using the definition of , we only need to compare the multisets of their
subterms, viz:
.
Since 2 > 1, in order that the former be greater than the latter, we must have:
…...
.
t
2
0
S
3333
333
1
3
0 3
0
1
3
2 2
3
0
3
0
3 0
0
2
0
3 1
0 3
0
and
1
3
2 2
0
3
0
3 0
0
2
0
3 1
0 3
0
3
2 2
0
3
0
3 0,
0
12. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.5, 2013
107
Since 2 = 2, we must now compare
in the multiset ordering .
Finally, since
is greater than both
we indeed have .
For ordering operators, we take to be greater than all other operators, as shown below.
Example 5 (Dershowitz and Manna, 1979).
The following rewrite rules symbolically differentiates an expression with respect to .
⁄ ⁄⁄
⁄
( ( ( ))) (( ) )
To prove termination, we use the multiset over sequences of natural numbers. The termination function is
{ is an occurrence of a symbol in ,where is the number of operators
between and the th position enclosing .
For example, consider the expression
( ) or ( )
with the numbered (for expository purposes) in tree form:
0 and ,
3
0
1
3
0
0
1
0
0 3
,
0
3
0,
0
3
0
3
,
0
3
with and
13. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.5, 2013
108
There are three atoms and , which constitutes three paths. The leftmost atom contributes the element
to the multiset. This is because there are no operators between and , there are two operators
between and , and there are three operators between and . Similarly, the other two remaining atoms
and contribute elements and respectively to the multiset. Thus,
{ , or,
{ .
Applying the rewrite rule
to , yields
( )
with the labeling of the retained, and thereby,
{
or, { .
D1
z1
D2
x
D3
y
+
y D4
D5
x
14. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.5, 2013
109
Representing in the form of a tree, we have
can be seen as follows:
Remark 7.
The mpo defined here is only for strict precedences. This is because in multiset difference for quasi-precedences,
equivalent terms have to be removed. This is known to be problematic because when the constraints for mpo are
encoded in propositional logic, the equivalence relation on terms is still unknown. A foreseable way out would
be to encode all possible multiset differences (exponential number) if that is computationally feasible which is an
open question at the moment (Zankl, 2006).
4. Concluding remarks and some further research directions
The paper illustrates that, for a class of trss, mpos are easy to handle and amenable to mechanization (Leclerc,
1995, for details). However, an mpo, being a simplification ordering, can handle only simply terminating systems
(Borralleras and Rubio, 2001). Several techniques have been developed for overcoming this weakness of rpos. A
very recent such technique is the monotonic semantic path ordering (mspo), a simple and easily automatable
ordering which generalizes other simplification methods (Borralleras and Rubio, 2001). Application of the
techniques of size-change to mpos is another promising area of research. The Size-change principle when
compared with classical simplification orders can simulate a certain form of lexicographic and multiset
comparison. Hence, the size-change principle in connection with mpo can often prove termination of trss where
one would otherwise need more complex orders (Lee, Jone and Ben-Amram, 2001).
It is known (Hofbauer, 1992 ) that termination proofs using multiset path orderings yield a primitive recursive
upper bound on the length of derivations measured in the size of the starting term. This is as well true for many
other path orderings as long as status is restricted to multiset status only. In general, the use of mpo or
simplification orders is critically limited as shown by the analysis of induced derivational complexity (Hofbauer,
1992). In recent years, a less restricted variant of mpo has been developed (Avanzini and Moser, 2008). It has
been shown in (Avanzini and Moser, 2008) that Polynomial path order ( ) on terms induces polynomial
derivation height for innermost rewriting. is closely related to the light multiset path order (lmpo, for
short) introduced by Marion (Marion, 2003). Besides, greater challenges lie in the area of automation of
termination techniques. The current direction of research is largely concerned with resolving problems related to
automation of termination analysis for trss and developing thereby competing termination tools. The increasing
interest in automated termination analysis of trss has led to an annual International Competition of Termination
Tools initiated in 2004 (March ́ and Zantema, 2007, for details). It aims at identifying most talented competitors
y
D3
D2
x
D5
D4y
y
D3
D2
D1
y
y
D5
D4
15. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.5, 2013
110
who could obtain an assigned task by applying appropriate choices of termination proving techniques within a
time limit of seconds. Using automata techniques is a relatively new and elegant approach of automatically
proving the termination of rewrite systems. Initially proposed for string rewriting by Geser, Hofbauer, and
Waldmann (2004), the method has recently been extended to left-linear trss. Variations and improvements are in
progress (Endrullis, 2006; Geser, Hofbauer, and Waldmann, 2004, for details). The fact that the method has been
implemented in several different termination provers (Waldmann, 2004; Zantema, 2005) is a clear indication of
the success of the approach.
References
Avanzini, M. & Moser, G. (2008), “Complexity Analysis by Rewriting”, Technical Report, University of
Innsbruck.
Baader, F. & Nipkow, T. (1998). Term Rewriting and All That, Cambridge University press.
Borralleras, C. & Rubio, A. (2001), “A Monotonic Higher-order Semantic Path Ordering”, Proceedings of
LPAR, 531-547.
Dershowitz, N. (1982), “Ordering for Term Rewriting Systems”, Journal of Theoretical Computer Science,
17(3): 279-301.
Dershowitz, N. (1987), “Termination of Term Rewriting”, Journal of Symbolic Computations, 3(1&2): 69-115.
Dershowitz, N. & Manna, Z. (1979), “Proving termination with multiset orderings”, Comm. ACM, 22(8): 465-
476.
Endrullis, J. (2006), Jambox: Automated Termination Proofs for String/ Term Rewriting, Available: http:
//joerg.endrullis.de /, 2006.
Geser, A., Hofbauer, D., & Waldmann, J. (2004), Match-bounded string rewriting systems, AAECC, 15(3-
4):149-171.
[9] Hofbauer, D. (1992), “Termination of Proofs by Multiset Path Orderings imply primitive Recursive
Derivation Lengths”, Theoretical Computer Science, 105, 129 -140.
Huet, G. & Lankford, D.S. (1978), “On the uniform halting problem for Term Rewriting Systems”, Technical
Report 283, IRIA.
Kamin, S., & Levy, J.J. (1980), “Two Generalizations of the Recursive Path Ordering”, Dept. of Computer
Science, University of Illinois, USA.
Kruskal’s, J.B. (1960), “Well-quasi ordering, the tree theorem, and Vazsonyi’s Conjecture”, Transactions, AMS,
95, 210-225.
Leclerc, F. (1995), “Termination Proof of Term Rewriting System with Multiset Path Ordering, A Complete
Development in the System Coq”, Proceedings of Second International Conference on Typed Lambda Calculi
and Applications,TLCA ’95, LNCS 902, Springer, 312-327.
Lee, C.S, Jone, N.D & Ben-Amram, A.M (2001), “The size-Change Principle for Program termination”, Proc.
POPL, 81-92.
Lescanne, P. & Jouannaud, J.P. (1982), “On Multiset Orderings”, Information Processing Letters, 16(2): 57-62.
March ́, C. & Zantema, H. (2007), “The Termination Competition”, Franz Baader (ed.), Proceedings of the 18th
International Conference on Rewriting Techniques and Applications, LNCS 4533, Springer Verlag, 303-313.
Marion, J. (2003), “Analysing the implicit complexity of programs”, Inform. and Comput. 183, 2-18.
Ohlebusch, E. (2002), “Advanced Topics in Term Rewriting”, Springer- Verlag.
Paulson, L.C. (1984), “Constructing Recursion Operations in Intuitionistic Type Theory”,Technical Report 57,
Computer Laboratory, University of Cambridge, UK.
16. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.5, 2013
111
Plaisted, D.A. (1978a), “Well- founded Ordering for Proving Termination of Systems of Rewrite Rules”, Report
R-78-932, Department of Computer Science, University of Illinois, USA.
Plaisted, D.A. (1978b), “A Recursively Defined Ordering for Proving Termination of Term Rewriting Systems”,
Report R-78-943, Department of Computer Science, University of Illinois, USA.
Singh, D., Shuaibu, M.A. & Ibrahim, M.A. (2012a), “An Overview of Term Rewriting Systems”, African
Journal of Mathematics and Computer Science Research, 5(9):153-157.
Singh, D., Shuaibu, M.A. & Ibrahim, M.A. (2012b), “Transformation Method: Making Termination Easier”,
IOSR Journal of Mathematics, 1(5): 25-30.
Terese (2003), Term Rewriting Systems, In: M. Bezem, J. W. Klop & R. Vrijer (eds.), Vol.55 of Cambridge
Tracts in Theoretical Computer Science, Cambridge University Press,.
Waldmann, J. (2004), “Matchbox: A Tool for Match-bounded String Rewriting”, Proceedings, 15th
RTA, LNCS,
Volume 3091, 85-94.
Zankl, H. (2006), “BDD and SAT Techniques for Precedences Based Orders”, Master Thesis, University of
Innsbruck.
Zantema, H. (2005), “Termination of Rewriting proved Automatically”, JAR, 34, 105-139.