- Process algebra originated in the early 1970s as researchers studied parallel and distributed systems.
- Hans Bekiç made early contributions including modeling programs as automata and introducing quasi-parallel composition. This abandoned the input/output function model and was a precursor to the expansion law, an important concept in process algebra.
- CCS, CSP, and ACP were later developed as the first process algebras, establishing process algebra as a separate area of research by the early 1980s.
The Potency of Formalism Logical Operations of Truth Tables StudyIOSR Journals
In this article, we used the automata theory to highlight the syntactic property representing the formal knowledge, and also we can use other contexts to represent the semantic property.
This document provides an overview of the theory of computation. It discusses the following key points:
1. Computation involves executing programs on computers to perform input-output transformations. Programs are algorithms expressed in programming languages.
2. The theory of computation classifies problems by their computability and complexity. It studies models of computation like Turing machines.
3. The theory has three main components: computability theory, complexity theory, and automata theory/formal languages.
4. Proofs in the theory use techniques like mathematical induction to show properties are true for all cases.
The document introduces the Word-Sensibility Model as a way to represent commonsense knowledge for AI. It consists of several components, including quadranyms, micro-topics, and an ecological perspective. Quadranyms are four-part constructs that represent virtual units of orientation and constraint. Micro-topics help organize lexical information and abstract human contextual expectations. The model takes an ecological view of representing dynamic relationships between an agent's internal responses and external occurrences across different contextual levels.
Logic is the study of correct reasoning through analyzing propositions, which are simple statements that can be either true or false. An expression tree visually represents how a formula is constructed from propositions and logical operators, while logic gates are physical devices that perform operations to describe computer circuits.
The document provides an introduction to calculus concepts including:
- Sets can represent collections of objects and are denoted using curly brackets. There are five regular solids that follow the Euler formula relating the number of faces, vertices and edges of each solid.
- The preface outlines the main purposes and organization of the two-volume calculus text, which includes fundamental concepts, applications, proofs, and multiple approaches.
- Volume I contains 5 chapters covering sets, functions, graphs, limits, differential calculus, integral calculus, sequences, summations, and applications of calculus.
This document provides an introduction to algorithms and data structures. It defines algorithms and describes their key properties including being finite, definite, and executable sequences of steps. It also discusses structured programming using sequences, selections, and iterations. The document presents two common algorithm description methods - flow diagrams and pseudocode - and provides examples of each. It then classifies algorithms into four types based on their inputs and outputs and provides example algorithms for each type. The document also discusses special algorithms like recursive and backtracking algorithms, and provides examples of each. Finally, it introduces analysis of algorithms and some common data structures like arrays, linked lists, stacks, queues, binary search trees, and graphs.
Swarm Intelligence Based Algorithms: A Critical AnalysisXin-She Yang
This document summarizes and analyzes swarm intelligence based algorithms. It discusses how these algorithms can be viewed as iterative processes, self-organizing systems, or Markov chains. The key components of exploration and exploitation are also analyzed. Evolutionary algorithms like genetic algorithms are discussed in terms of their crossover, mutation, and selection operators. Overall, the document provides a critical analysis of swarm intelligence algorithms from different perspectives to understand how they work and can be improved.
This document discusses the application of finite element exterior calculus to evolution problems. It introduces the concept of Hilbert complexes, which allow the abstraction of essential properties of function spaces and their approximations. Differential forms and operators like the exterior derivative are used to elegantly formulate elliptic problems like the heat, wave, Poisson, and Maxwell equations. Approximation theory for subcomplexes and non-subcomplexes is developed. Finally, methods are discussed for the semidiscretization and discretization of parabolic problems like the heat equation using these concepts.
The Potency of Formalism Logical Operations of Truth Tables StudyIOSR Journals
In this article, we used the automata theory to highlight the syntactic property representing the formal knowledge, and also we can use other contexts to represent the semantic property.
This document provides an overview of the theory of computation. It discusses the following key points:
1. Computation involves executing programs on computers to perform input-output transformations. Programs are algorithms expressed in programming languages.
2. The theory of computation classifies problems by their computability and complexity. It studies models of computation like Turing machines.
3. The theory has three main components: computability theory, complexity theory, and automata theory/formal languages.
4. Proofs in the theory use techniques like mathematical induction to show properties are true for all cases.
The document introduces the Word-Sensibility Model as a way to represent commonsense knowledge for AI. It consists of several components, including quadranyms, micro-topics, and an ecological perspective. Quadranyms are four-part constructs that represent virtual units of orientation and constraint. Micro-topics help organize lexical information and abstract human contextual expectations. The model takes an ecological view of representing dynamic relationships between an agent's internal responses and external occurrences across different contextual levels.
Logic is the study of correct reasoning through analyzing propositions, which are simple statements that can be either true or false. An expression tree visually represents how a formula is constructed from propositions and logical operators, while logic gates are physical devices that perform operations to describe computer circuits.
The document provides an introduction to calculus concepts including:
- Sets can represent collections of objects and are denoted using curly brackets. There are five regular solids that follow the Euler formula relating the number of faces, vertices and edges of each solid.
- The preface outlines the main purposes and organization of the two-volume calculus text, which includes fundamental concepts, applications, proofs, and multiple approaches.
- Volume I contains 5 chapters covering sets, functions, graphs, limits, differential calculus, integral calculus, sequences, summations, and applications of calculus.
This document provides an introduction to algorithms and data structures. It defines algorithms and describes their key properties including being finite, definite, and executable sequences of steps. It also discusses structured programming using sequences, selections, and iterations. The document presents two common algorithm description methods - flow diagrams and pseudocode - and provides examples of each. It then classifies algorithms into four types based on their inputs and outputs and provides example algorithms for each type. The document also discusses special algorithms like recursive and backtracking algorithms, and provides examples of each. Finally, it introduces analysis of algorithms and some common data structures like arrays, linked lists, stacks, queues, binary search trees, and graphs.
Swarm Intelligence Based Algorithms: A Critical AnalysisXin-She Yang
This document summarizes and analyzes swarm intelligence based algorithms. It discusses how these algorithms can be viewed as iterative processes, self-organizing systems, or Markov chains. The key components of exploration and exploitation are also analyzed. Evolutionary algorithms like genetic algorithms are discussed in terms of their crossover, mutation, and selection operators. Overall, the document provides a critical analysis of swarm intelligence algorithms from different perspectives to understand how they work and can be improved.
This document discusses the application of finite element exterior calculus to evolution problems. It introduces the concept of Hilbert complexes, which allow the abstraction of essential properties of function spaces and their approximations. Differential forms and operators like the exterior derivative are used to elegantly formulate elliptic problems like the heat, wave, Poisson, and Maxwell equations. Approximation theory for subcomplexes and non-subcomplexes is developed. Finally, methods are discussed for the semidiscretization and discretization of parabolic problems like the heat equation using these concepts.
This document discusses constructive modal logics and open questions in the field. It describes two main families of constructive modal logics, CK and IK, which differ in their proof-theoretical properties. Developing satisfactory proof theories for these logics has been challenging, requiring augmentations to sequent systems. The document also notes that while IK logics have better model-theoretic properties, CK logics are better suited for lambda calculus interpretations. Overall, the document advocates for further work to develop a unified framework that can capture both families of logics along with categorical semantics.
The document discusses modalities in linear logic and dialectica categories. It motivates studying (co)monads and (co)algebras as constructive modalities in linear logic. It describes how the linear logic bang modality ! can be modeled as a comonad in dialectica categories. Specifically, in the dialectica category Dial2(C), ! is modeled as a cofree comonad. This provides a model of intuitionistic linear logic with both linear and non-linear connectives. In the simpler category DDial2(C), modeling the bang requires composing two comonads.
The document discusses categorical semantics for explicit substitutions. It begins by motivating the need for categorical semantics of syntactic calculi to provide mathematical models and ensure correctness. It then discusses different categorical structures that can provide semantics for calculi with explicit substitutions, including indexed categories, context-handling categories, and E-categories/L-categories. These categorical models impose equations on explicit substitutions that correspond to the intended behavior. The document also discusses how additional type structures like functions, tensors, and the exponential/bang type can be modeled using these categorical structures. Overall, the document advocates for the use of category theory to guide the design of calculi with explicit substitutions and ensure their semantics are well-behaved.
This document outlines the key points that will be covered in a presentation on dialectica categories as models of linear logic. It discusses four main theorems: 1) The original dialectica category DC is a model of intuitionistic linear logic without the exponential bang (!). 2) Adding a co-free monoidal comonad makes DC a model of full intuitionistic linear logic. 3) The simplified dialectica category GC is a model of classical linear logic without bang and question mark (?). 4) Adding a complicated monoidal comonad makes GC a model of full classical linear logic. The document provides brief motivations and outlines of the constructions and proofs involved in these main results. It also discusses some applications and areas for further development
This document provides an introduction to categorical proof theory and linear type theory. It discusses how categorical proof theory models derivations and proofs using categories. Several linear type theories with categorical models are mentioned, including the linear lambda calculus, DILL system, and LNL system. It also discusses the challenges of modeling linear logic categorically and different varieties of linear logic and type theory. Finally, it introduces intuitionistic and linear type theory (ILT) and its categorical models using fibrations.
This document provides an overview of modeling the Lambek Calculus using Dialectica Categories. It begins with introductions to the Lambek Calculus and Dialectica Categories. It then discusses how Dialectica Categories can be used to model proof theory categorically. Specifically, it presents the non-commutative Dialectica category DialM(Sets) which can model the non-commutative multiplicative fragment of the Lambek Calculus. It concludes by noting the advantages of this approach over previous work and areas for future work.
In this article we present a brief history and some applications of semirings, the structure of compact monothetic c semirings. The classification of these semirings be based on known description of discrete cyclic semirings and compact monothetic semirings. Boris Tanana "Compact Monothetic C-semirings" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-2 , February 2021, URL: https://www.ijtsrd.com/papers/ijtsrd38612.pdf Paper Url: https://www.ijtsrd.com/mathemetics/algebra/38612/compact-monothetic-csemirings/boris-tanana
The document discusses the history and connections between logic, proofs, programs, and category theory. It notes that Hilbert's program to formalize mathematics led to the development of proof theory and Gentzen's natural deduction and sequent calculus systems. The Curry-Howard correspondence showed that proofs and programs are closely related through the use of lambda calculus. Category theory provides a unified framework where types represent logical formulas, terms represent proofs, and reductions represent proof normalization. Categorical proof theory models proofs as first-class citizens.
A Stochastic Iteration Method for A Class of Monotone Variational Inequalitie...CSCJournals
We examined a general method for obtaining a solution to a class of monotone variational inequalities in Hilbert space. Let H be a real Hilbert space, and Let T : H -> H be a continuous linear monotone operator and K be a non empty closed convex subset of H. From an initial arbitrary point x0 ∈ K. We proposed and obtained iterative method that converges in norm to a solution of the class of monotone variational inequalities. A stochastic scheme {xn} is defined as follows: x(n+1) = xn - anF* (xn), n≥0, F*(xn), n ≥ 0 is a strong stochastic approximation of Txn - b, for all b (possible zero) ∈ H and an ∈ (0,1).
Automated Education Propositional Logic Tool (AEPLT): Used For Computation in...CSCJournals
This document describes an Automated Education Propositional Logic Tool (AEPLT) that was designed to simplify and automate the calculation of propositional logic for compound propositions involving conjunction, disjunction, conditional, and bi-conditional statements. The AEPLT has an architecture that allows a user to enter propositional variables, which are then mapped to logical connectives by a parser to form formulas. These formulas are evaluated against assumption statements to determine truth values, which are recorded in a truth table. The tool is intended to provide students with accurate results in a user-friendly interface, avoiding human errors that can occur in manual calculations.
This document provides an outline for a lecture on discrete mathematics. It introduces topics like logic, set theory, mathematical reasoning/proof techniques, propositional/predicate calculus, Boolean algebra, induction, algorithms, recursion, counting/probability, and graph theory. The lecture begins with an introduction to logic, including statements, propositions, truth values, and logical operators like negation, conjunction, disjunction, implication, and biconditional. It provides examples of combining propositions using logical operators and truth tables. It also discusses propositional functions, quantification, and counterexamples.
This document provides an introduction to Latent Semantic Analysis (LSA) and Probabilistic Latent Semantic Analysis (PLSA). It discusses some challenges with traditional information retrieval based on lexical matching. It then introduces LSA and PLSA as statistical approaches to address these challenges by discovering the latent semantic structure in word usage. The key aspects of LSA include using singular value decomposition to reduce the dimensionality of a term-document matrix. PLSA builds upon LSA by introducing a probabilistic model with a latent class variable to represent topics. The document contrasts the objective functions and interpretations of the reduced dimensional spaces between LSA and PLSA.
This document proposes a standardized logic notation for use in mathematics classrooms. It describes using the symbols "→" and "↔" as connectives that can return true or false values, and "⟹" and "⟺" only when the corresponding connectives always return true. This addresses inconsistencies in current logic notation usage. Examples are given showing how the standardized notation clarifies logical relationships in calculus, algebra, and other mathematics topics. Adopting this standardized notation is argued to benefit students by making logic portable between courses and instructors.
The document summarizes a talk given by Valeria de Paiva on intuitionistic modal logic 15 years after its initial development. It discusses the early work developing systems of intuitionistic modal logic like Constructive S4 and their proof theories. It also describes the formation of the Intuitionistic Modal Logic and Applications association aimed at bringing together researchers from different fields to share tools and information. However, the goal of this association being fully realized, with communities still largely talking past each other, is assessed as not having been attained so far.
Unit I of the syllabus covers propositional logic and counting theory. It introduces concepts such as propositions, logical connectives like conjunction, disjunction, negation, implication and biconditional. It discusses how to represent compound statements using these connectives and their truth tables. The unit also covers topics like predicate logic, methods of proof, mathematical induction and fundamental counting principles like permutations and combinations. It aims to provide the logical foundations for discrete mathematics concepts that will be useful in computer science and information technology.
Bat Algorithm: Literature Review and ApplicationsXin-She Yang
This document provides a review of the bat algorithm, which is a bio-inspired optimization algorithm developed in 2010 based on the echolocation behavior of microbats. The paper summarizes the basic behavior and formulation of the bat algorithm, reviews variants that have been developed, and highlights diverse applications that have been studied. It also discusses the essence of algorithms and links between algorithms and self-organization, noting that optimization algorithms can be viewed as complex dynamical systems that self-organize to select optimal solutions.
In this paper, the underlying principles about the theory of relativity are briefly introduced and reviewed. The mathematical prerequisite needed for the understanding of general relativity and of Einstein field equations are discussed. Concepts such as the principle of least action will be included and its explanation using the Lagrange equations will be given. Where possible, the mathematical details and rigorous analysis of the subject has been given in order to ensure a more precise and thorough understanding of the theory of relativity. A brief mathematical analysis of how to derive the Einstein’s field’s equations from the Einstein-Hilbert action and the Schwarzschild solution was also given.
This document discusses constructive modal logics and open questions in the field. It describes two main families of constructive modal logics, CK and IK, which differ in their proof-theoretical properties. Developing satisfactory proof theories for these logics has been challenging, requiring augmentations to sequent systems. The document also notes that while IK logics have better model-theoretic properties, CK logics are better suited for lambda calculus interpretations. Overall, the document advocates for further work to develop a unified framework that can capture both families of logics along with categorical semantics.
The document discusses modalities in linear logic and dialectica categories. It motivates studying (co)monads and (co)algebras as constructive modalities in linear logic. It describes how the linear logic bang modality ! can be modeled as a comonad in dialectica categories. Specifically, in the dialectica category Dial2(C), ! is modeled as a cofree comonad. This provides a model of intuitionistic linear logic with both linear and non-linear connectives. In the simpler category DDial2(C), modeling the bang requires composing two comonads.
The document discusses categorical semantics for explicit substitutions. It begins by motivating the need for categorical semantics of syntactic calculi to provide mathematical models and ensure correctness. It then discusses different categorical structures that can provide semantics for calculi with explicit substitutions, including indexed categories, context-handling categories, and E-categories/L-categories. These categorical models impose equations on explicit substitutions that correspond to the intended behavior. The document also discusses how additional type structures like functions, tensors, and the exponential/bang type can be modeled using these categorical structures. Overall, the document advocates for the use of category theory to guide the design of calculi with explicit substitutions and ensure their semantics are well-behaved.
This document outlines the key points that will be covered in a presentation on dialectica categories as models of linear logic. It discusses four main theorems: 1) The original dialectica category DC is a model of intuitionistic linear logic without the exponential bang (!). 2) Adding a co-free monoidal comonad makes DC a model of full intuitionistic linear logic. 3) The simplified dialectica category GC is a model of classical linear logic without bang and question mark (?). 4) Adding a complicated monoidal comonad makes GC a model of full classical linear logic. The document provides brief motivations and outlines of the constructions and proofs involved in these main results. It also discusses some applications and areas for further development
This document provides an introduction to categorical proof theory and linear type theory. It discusses how categorical proof theory models derivations and proofs using categories. Several linear type theories with categorical models are mentioned, including the linear lambda calculus, DILL system, and LNL system. It also discusses the challenges of modeling linear logic categorically and different varieties of linear logic and type theory. Finally, it introduces intuitionistic and linear type theory (ILT) and its categorical models using fibrations.
This document provides an overview of modeling the Lambek Calculus using Dialectica Categories. It begins with introductions to the Lambek Calculus and Dialectica Categories. It then discusses how Dialectica Categories can be used to model proof theory categorically. Specifically, it presents the non-commutative Dialectica category DialM(Sets) which can model the non-commutative multiplicative fragment of the Lambek Calculus. It concludes by noting the advantages of this approach over previous work and areas for future work.
In this article we present a brief history and some applications of semirings, the structure of compact monothetic c semirings. The classification of these semirings be based on known description of discrete cyclic semirings and compact monothetic semirings. Boris Tanana "Compact Monothetic C-semirings" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-2 , February 2021, URL: https://www.ijtsrd.com/papers/ijtsrd38612.pdf Paper Url: https://www.ijtsrd.com/mathemetics/algebra/38612/compact-monothetic-csemirings/boris-tanana
The document discusses the history and connections between logic, proofs, programs, and category theory. It notes that Hilbert's program to formalize mathematics led to the development of proof theory and Gentzen's natural deduction and sequent calculus systems. The Curry-Howard correspondence showed that proofs and programs are closely related through the use of lambda calculus. Category theory provides a unified framework where types represent logical formulas, terms represent proofs, and reductions represent proof normalization. Categorical proof theory models proofs as first-class citizens.
A Stochastic Iteration Method for A Class of Monotone Variational Inequalitie...CSCJournals
We examined a general method for obtaining a solution to a class of monotone variational inequalities in Hilbert space. Let H be a real Hilbert space, and Let T : H -> H be a continuous linear monotone operator and K be a non empty closed convex subset of H. From an initial arbitrary point x0 ∈ K. We proposed and obtained iterative method that converges in norm to a solution of the class of monotone variational inequalities. A stochastic scheme {xn} is defined as follows: x(n+1) = xn - anF* (xn), n≥0, F*(xn), n ≥ 0 is a strong stochastic approximation of Txn - b, for all b (possible zero) ∈ H and an ∈ (0,1).
Automated Education Propositional Logic Tool (AEPLT): Used For Computation in...CSCJournals
This document describes an Automated Education Propositional Logic Tool (AEPLT) that was designed to simplify and automate the calculation of propositional logic for compound propositions involving conjunction, disjunction, conditional, and bi-conditional statements. The AEPLT has an architecture that allows a user to enter propositional variables, which are then mapped to logical connectives by a parser to form formulas. These formulas are evaluated against assumption statements to determine truth values, which are recorded in a truth table. The tool is intended to provide students with accurate results in a user-friendly interface, avoiding human errors that can occur in manual calculations.
This document provides an outline for a lecture on discrete mathematics. It introduces topics like logic, set theory, mathematical reasoning/proof techniques, propositional/predicate calculus, Boolean algebra, induction, algorithms, recursion, counting/probability, and graph theory. The lecture begins with an introduction to logic, including statements, propositions, truth values, and logical operators like negation, conjunction, disjunction, implication, and biconditional. It provides examples of combining propositions using logical operators and truth tables. It also discusses propositional functions, quantification, and counterexamples.
This document provides an introduction to Latent Semantic Analysis (LSA) and Probabilistic Latent Semantic Analysis (PLSA). It discusses some challenges with traditional information retrieval based on lexical matching. It then introduces LSA and PLSA as statistical approaches to address these challenges by discovering the latent semantic structure in word usage. The key aspects of LSA include using singular value decomposition to reduce the dimensionality of a term-document matrix. PLSA builds upon LSA by introducing a probabilistic model with a latent class variable to represent topics. The document contrasts the objective functions and interpretations of the reduced dimensional spaces between LSA and PLSA.
This document proposes a standardized logic notation for use in mathematics classrooms. It describes using the symbols "→" and "↔" as connectives that can return true or false values, and "⟹" and "⟺" only when the corresponding connectives always return true. This addresses inconsistencies in current logic notation usage. Examples are given showing how the standardized notation clarifies logical relationships in calculus, algebra, and other mathematics topics. Adopting this standardized notation is argued to benefit students by making logic portable between courses and instructors.
The document summarizes a talk given by Valeria de Paiva on intuitionistic modal logic 15 years after its initial development. It discusses the early work developing systems of intuitionistic modal logic like Constructive S4 and their proof theories. It also describes the formation of the Intuitionistic Modal Logic and Applications association aimed at bringing together researchers from different fields to share tools and information. However, the goal of this association being fully realized, with communities still largely talking past each other, is assessed as not having been attained so far.
Unit I of the syllabus covers propositional logic and counting theory. It introduces concepts such as propositions, logical connectives like conjunction, disjunction, negation, implication and biconditional. It discusses how to represent compound statements using these connectives and their truth tables. The unit also covers topics like predicate logic, methods of proof, mathematical induction and fundamental counting principles like permutations and combinations. It aims to provide the logical foundations for discrete mathematics concepts that will be useful in computer science and information technology.
Bat Algorithm: Literature Review and ApplicationsXin-She Yang
This document provides a review of the bat algorithm, which is a bio-inspired optimization algorithm developed in 2010 based on the echolocation behavior of microbats. The paper summarizes the basic behavior and formulation of the bat algorithm, reviews variants that have been developed, and highlights diverse applications that have been studied. It also discusses the essence of algorithms and links between algorithms and self-organization, noting that optimization algorithms can be viewed as complex dynamical systems that self-organize to select optimal solutions.
In this paper, the underlying principles about the theory of relativity are briefly introduced and reviewed. The mathematical prerequisite needed for the understanding of general relativity and of Einstein field equations are discussed. Concepts such as the principle of least action will be included and its explanation using the Lagrange equations will be given. Where possible, the mathematical details and rigorous analysis of the subject has been given in order to ensure a more precise and thorough understanding of the theory of relativity. A brief mathematical analysis of how to derive the Einstein’s field’s equations from the Einstein-Hilbert action and the Schwarzschild solution was also given.
May 21-25, 2000: "Why We Don't Understand Complex Systems". Poster session, presented at the International Conference on Complex Systems, sponsored by the New England Complex Systems Institute.
István Dienes Lecture For Unified Theories 2006Istvan Dienes
The document proposes a model called the Consciousness-Holomatrix to describe the physics of consciousness and logical mind. It suggests that consciousness is a topological energy field where thoughts are topological structures. Physical models and matrix logic are used to develop this idea. Consciousness and physical reality may be two holographically mapped fields within a quantum holographic Holomatrix that represents all information and is projected by logical membranes (L-branes) created in the projection process.
How deeply can you understand what is happening inside your application? In modern, microservices-based applications, it’s critical to have end-to-end observability of each component and the communications between them in order to quickly identify and debug issues. In this session, we show how to have the necessary instrumentation and how to use the data you collect to have a better grasp of your production environment. On AWS, CloudWatch collects monitoring and operational data in the form of logs, metrics, and events, providing you with a unified view of AWS resources, applications, and services. With AWS X-Ray, you can understand how your application and its underlying services are performing to identify and troubleshoot the root cause of performance issues and errors. X-Ray provides an end-to-end view of requests as they travel through your application, and shows a map of your application’s underlying components. AWS App Mesh standardizes how your microservices communicate, giving you end-to-end visibility and helping to ensure high-availability for your applications.
Cuckoo Search: Recent Advances and ApplicationsXin-She Yang
This document summarizes recent advances and applications of the cuckoo search algorithm, a nature-inspired metaheuristic optimization algorithm developed in 2009. Cuckoo search mimics the brood parasitism breeding behavior of some cuckoo species. It uses a combination of local and global search achieved through random walks and Levy flights to efficiently explore the search space. Studies show cuckoo search often finds optimal solutions faster than genetic algorithms and particle swarm optimization. The algorithm has been applied to diverse optimization problems and continues to be improved and extended to multi-objective optimization.
Metaheuristic Optimization: Algorithm Analysis and Open ProblemsXin-She Yang
This document analyzes metaheuristic optimization algorithms and discusses open problems in their analysis. It reviews convergence analyses that have been done for simulated annealing and particle swarm optimization. It also provides a novel convergence analysis for the firefly algorithm, showing that it can converge for certain parameter values but also exhibit chaos which can be advantageous for exploration. The document outlines the need for further mathematical analysis of convergence and efficiency in metaheuristics.
This document provides a summary of parallel evolutionary algorithms (PEAs). It discusses how PEAs can be implemented by either structuring populations into subpopulations (distributed EAs/island models) or neighborhoods (cellular EAs), or by parallelizing panmictic EAs without population structure. The document reviews the history of PEAs and discusses theoretical issues. It also outlines different types of structured PEAs and how they can be parallelized on different hardware. The summary provides an overview of PEAs and discusses open problems in the field.
This document contains summaries of key concepts in knowledge-based agents, propositional logic, and theorem proving/inference:
1. Knowledge-based agents maintain an internal knowledge base and use an inference system to reason over that knowledge, update their knowledge based on observations, and determine appropriate actions. They represent the world using a formal knowledge representation language.
2. Propositional logic uses propositional variables and logical connectives to construct well-formed formulas that can be assigned true or false values. Truth tables are used to systematically evaluate the truth values of complex formulas.
3. Theorem proving demonstrates the validity of a mathematical statement by constructing a logical proof based on axioms, definitions, and previously proven theorems.
Neuro Quantology is an international, interdisciplinary, open-access, peer-reviewed journal that publishes original research and review articles on the interface between quantum physics and neuroscience. The journal focuses on the exploration of the neural mechanisms underlying consciousness, cognition, perception, and behavior from a quantum perspective. Neuro Quantology is published monthly.
This document provides an overview of engineering analysis and design (modelling and simulation) with a focus on systems and simulation. It defines a system as a collection of components organized for a common purpose. Simulation is defined as imitating the operation of a real-world process over time by generating artificial history. The key benefits of simulation are that it allows experimenting with models to study the effects of changes before implementing them in the real world and to help with system analysis, design, and prediction.
This document provides an introduction to algorithms and algorithm analysis. It defines an algorithm as a set of unambiguous instructions to solve a problem in a finite amount of time. The most famous early algorithm is Euclid's algorithm for calculating greatest common divisors. Algorithm analysis involves proving an algorithm's correctness and analyzing its running time and space complexity. Common notations for analyzing complexity include Big-O, which provides upper bounds, Big-Omega, which provides lower bounds, and Big-Theta, which provides tight bounds. The goal of analysis is to determine the most efficient algorithm by evaluating performance as problem size increases.
Improving Tools in Artificial IntelligenceBogdan Patrut
This document provides an overview of tools used in artificial intelligence for knowledge representation. It discusses logic-based approaches like rules, frames and scripts. It also discusses search methods like breadth-first, depth-first and heuristic search. Additionally, it introduces fuzzy logic as an approach to handle non-monotonic or uncertain reasoning needed for problems in the real world. Fuzzy sets allow degrees of membership between 0 and 1 rather than classical binary set membership. Fuzzy relations can then be defined through membership functions and combined through operations like minimum.
Time complexity analysis is used to determine the most efficient algorithm by comparing how the running time of each algorithm grows with the size of the input. The time complexity of an algorithm can be expressed using asymptotic notations like Big O, which defines the worst-case running time. For example, the time complexity of binary search is O(log n), meaning it grows logarithmically with the input size. Comparing the time complexities of sorting and searching algorithms can help identify the most efficient approach for different situations.
Mathematical analysis is a branch of mathematics that studies concepts like differentiation, integration, limits, and infinite series. It evolved from calculus and can be applied to spaces with defined distances or nearness. Key areas include real analysis of real functions and sequences, complex analysis of functions of complex variables, and functional analysis which studies topological vector spaces. Measure theory assigns a size or number to subsets of a set, generalizing concepts like length, area and volume. Numerical analysis uses numerical approximations rather than symbolic manipulations to solve problems in mathematical analysis.
How deeply can you understand what is happening inside your application? In modern, microservices-based applications, it’s critical to have end-to-end observability of each component and the communications between them in order to quickly identify and debug issues. In this session, we show how to have the necessary instrumentation and how to use the data you collect to have a better grasp of your production environment. On AWS, CloudWatch collects monitoring and operational data in the form of logs, metrics, and events, providing you with a unified view of AWS resources, applications, and services. With AWS X-Ray, you can understand how your application and its underlying services are performing to identify and troubleshoot the root cause of performance issues and errors. X-Ray provides an end-to-end view of requests as they travel through your application, and shows a map of your application’s underlying components. AWS App Mesh standardizes how your microservices communicate, giving you end-to-end visibility and helping to ensure high-availability for your applications.
Mc0079 computer based optimization methods--phpapp02Rabby Bhatt
This document discusses mathematical models and provides examples of different types of mathematical models. It begins by defining a mathematical model as a description of a system using mathematical concepts and language. It then classifies mathematical models in several ways, such as linear vs nonlinear, deterministic vs probabilistic, static vs dynamic, discrete vs continuous, and deductive vs inductive vs floating. The document provides examples and explanations of each type of model. It also discusses using finite queuing tables to analyze queuing systems with a finite population size. In summary, the document outlines different ways to classify mathematical models and provides examples of applying various types of models.
Introduction to Ethology Part 2 Quantifying BehaviorBio 112 Lab.docxmariuse18nolet
Introduction to Ethology Part 2: Quantifying Behavior
Bio 112 Lab, Spring 2016
Describing Action Patterns
In order to study and measure behavior you must break the behavior up into units, called action patterns. These become evident to you when you watch an animal. As you become familiar with its behavior, you immediately see that its behavior is not continuous. Instead, the animal performs discrete, repeatable acts that you can identify. These discrete, repeated acts or bodily postures are what we call action patterns.
Action Patterns, then are:
• discrete, complex muscular movements involving many muscular contractions that are closely associated temporally
• repeated in essentially the same manner time after time (i.e. they are stereotyped) and
• they are similar among individuals of the same species (i.e. they are species-typical).
Action patterns are often defined in terms of the following characteristics:
• the form of the action.
• the velocity of the action -- how rapid are the movements of the action pattern?
• the duration or how long the movements are held in different positions.
• the amplitude or the height of the action.
• the orientation of the behavior towards objects in the environment, including conspecifics.
Look at the action patterns for mallard ducks depicted below. One is a posture (head-round) while the others all involve motions from simple head shaking to more complex motions that involve much of the body:
Major Approaches to Quantifying Behavior: Survey vs. Focal
Once you have completed your catalog of behaviors, you are ready to quantify how your animal spends its time. There are two main methods to do this:
Survey approach: Watch many individuals at the same time. At systematic (or randomized) times, you count the number of individuals performing each possible behavior that you previously cataloged. For example, if you are watching a flock of ten geese in a lake, you might set your timer to alert you at one minute intervals. When the timer goes off, you note what each of the geese in the flock are doing at that instant, as well as the context and consequences of their actions. You will repeat this every minute for your entire observation period. Taking survey data in the field also is facilitated by creating a data form that can be filled in quickly. We are less likely to use the survey method in this course, but it can be used if necessary.
Focal approach: Locate a single individual and follow its behavior for a standard time (or as long as possible up to that time). If a focal individual moves out of your view, then you start a new sequence of observations on a new focal individual. Selecting the focal animal can be systematic (e.g. only watch males) or randomized (select a random number from a table, then follow the nth individual encountered). During a focal study, you should record the following data as they occur:
· the context (date, time, location, weather, habitat, social context)
· the focal an.
Similar to A brief history of process algebra (20)
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A Brief History of Process Algebra
J.C.M. Baeten
Department of Computer Science, Technische Universiteit Eindhoven,
P.O. Box 513, 5600 MB Eindhoven, The Netherlands
josb@win.tue.nl
Abstract. This note addresses the history of process algebra as an area
of research in concurrency theory, the theory of parallel and distributed
systems in computer science. Origins are traced back to the early sev-
enties of the twentieth century, and developments since that time are
sketched. The author gives his personal views on these matters. He also
considers the present situation, and states some challenges for the future.
Note: Based on [9]. Also report CS-R 04-02, Department of Mathematics
and Computer Science, Technische Universiteit Eindhoven,
http://www.win.tue.nl/fm/pubbaeten.html.
1 Introduction
Computer science is a very young science, that has its origins in the middle of the
twentieth century. For this reason, describing its history, or the history of an area
of research in computer science, is an activity that receives scant attention. The
organizers of a workshop on process algebra in Bertinoro in July 2003 (see [2])
aimed at describing the current state of affairs and future directions of research
in process algebra, an active area of research in computer science. In planning
the workshop, the question came up how the present situation came about. I
was asked to investigate the history of process algebra.
This note addresses the history of process algebra as an area of research in
concurrency theory, the theory of parallel and distributed systems in computer
science. Origins are traced back to the early seventies of the twentieth century,
and developments since that time are sketched.
First of all, I define the term ‘process algebra’ and consider possible dif-
ferences of scope that people may have. In the rest of the note, I use a strict
definition but also make some remarks about related matters. Then, I sketch
the state of research in the early seventies, and state which breakthroughs were
needed in order for the theories to appear. I consider the development of CCS,
CSP and ACP. In Section 4, I sketch the main developments since then.
Let me stress that I give my personal views on what are the central issues
and the important breakthroughs. Others may well have very different views. I
welcome discussion, in order to get a better and more complete account. I briefly
consider the present situation, and state some challenges for the future.
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Acknowledgement
The author gratefully acknowledges the discussions at the Process Algebra: Open
Problems and Future Directions Workshop in Bertinoro in July 2003 (see [2]).
In particular, I would like to mention input from Jan Friso Groote [44] and Luca
Aceto.
2 What is a process algebra?
The term process algebra is used in different meanings. First of all, let me con-
sider the word ‘process’. It refers to behaviour of a system. A system is anything
showing behaviour, in particular the execution of a software system, the actions
of a machine or even the actions of a human being. Behaviour is the total of
events or actions that a system can perform, the order in which they can be exe-
cuted and maybe other aspects of this execution such as timing or probabilities.
Always, we describe certain aspects of behaviour, disregarding other aspects, so
we are considering an abstraction or idealization of the ‘real’ behaviour. Rather,
we can say that we have an observation of behaviour, and an action is the chosen
unit of observation. Usually, the actions are thought to be discrete: occurrence
is at some moment in time, and different actions are separated in time. This is
why a process is sometimes also called a discrete event system.
The word ‘algebra’ denotes that we take an algebraic/axiomatic approach in
talking about behaviour. That is, we use the methods and techniques of universal
algebra (see e.g. [61]). In order to make a comparison, consider the definition of
a group in mathematical algebra.
Definition 1. A group has a signature (G, ∗, u,−1 ) with the laws or axioms
– a ∗ (b ∗ c) = (a ∗ b) ∗ c
– u∗a=a=a∗u
– a ∗ a−1 = a−1 ∗ a = u
So, a group is any mathematical structure with operators satisfying the group
axioms. Stated differently: a group is any model of the equational theory of
groups. Likewise, we can say that a process algebra is any mathematical structure
satisfying the axioms given for the basic operators. A process is an element of a
process algebra. By using the axioms, we can perform calculations with processes.
The simplest model of behaviour is to see behaviour as an input/output
function. A value or input is given at the beginning of the process, and at some
moment there is a(nother) value as outcome or output. This model was used
to advantage as the simplest model of the behaviour of a computer program in
computer science, from the start of the subject in the middle of the twentieth
century. It was instrumental in the development of (finite state) automata theory.
In automata theory, a process is modeled as an automaton. An automaton has a
number of states and a number of transitions, going from one state to a(nother)
state. A transition denotes the execution of an (elementary) action, the basic
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Brief History of Process Algebra 3
unit of behaviour. Besides, there is an initial state (sometimes, more than one)
and a number of final states. A behaviour is a run, i.e. a path from initial state
to final state. Important from the beginning is when to consider two automata
equal, expressed by a notion of equivalence. On automata, the basic notion of
equivalence is language equivalence: a behaviour is characterized by the set of
executions from the initial state to a final state. An algebra that allows equational
reasoning about automata is the algebra of regular expressions (see e.g. [58]).
Later on, this model was found to be lacking in several situations. Basically,
what is missing is the notion of interaction: during the execution from initial
state to final state, a system may interact with another system. This is needed
in order to describe parallel or distributed systems, or so-called reactive sys-
tems. When dealing with interacting systems, we say we are doing concurrency
theory, so concurrency theory is the theory of interacting, parallel and/or dis-
tributed systems. When talking about process algebra, we usually consider it as
an approach to concurrency theory, so a process algebra will usually (but not
necessarily) have parallel composition as a basic operator.
Thus, we can say that process algebra is the study of the behaviour of parallel
or distributed systems by algebraic means. It offers means to describe or specify
such systems, and thus it has means to talk about parallel composition. Besides
this, it can usually also talk about alternative composition (choice) and sequen-
tial composition (sequencing). Moreover, we can reason about such systems using
algebra, i.e. equational reasoning. By means of this equational reasoning, we can
do verification, i.e. we can establish that a system satisfies a certain property.
What are these basic laws of process algebra? We can list some, that are
usually called structural or static laws. We start out from a given set of atomic
actions, and use the basic operators to compose these into more complicated pro-
cesses. As basic operators, we use + denoting alternative composition, ; denoting
sequential composition and denoting parallel composition. Usually, there are
also neutral elements for some or all of these operators, but we do not consider
these here. Some basic laws are the following (+ binding weakest, ; binding
strongest).
– x + y = y + x (commutativity of alternative composition)
– x + (y + z) = (x + y) + z (associativity of alternative composition)
– x + x = x (idempotency of alternative composition)
– (x + y); z = x; z + y; z (right distributivity of + over ;)
– (x; y); z = x; (y; z) (associativity of sequential composition)
– x y = y x (commutativity of parallel composition)
– (x y) z = x (y z) (associativity of parallel composition)
These laws are called static laws because they do not mention action execu-
tion explicitly, but only list general properties of the operators involved.
We can see that there is a law connecting alternative and sequential compo-
sition. In some cases, other connections are considered. On the other hand, we
list no law connecting parallel composition to the other operators. It turns out
such a connection is at the heart of process algebra, and it is the tool that makes
calculation possible. In most process algebras, this law allows to express parallel
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composition in terms of the other operators, and is called an expansion theorem.
As this law involves action execution explicitly, it is an example of a so-called
dynamic law. Process algebras with an expansion theorem are called interleaving
process algebras, those without are called partial order or true concurrency. For
a discussion concerning this dichotomy, see [8].
So, we can say that any mathematical structure with three binary operations
satisfying these 7 laws is a process algebra. Most often, these structures are
formulated in terms of automata, in this case mostly called transition systems.
This means a transition system has a number of states and transitions between
them, an initial state and a number of final states. The notion of equivalence
studied is usually not language equivalence. Prominent among the equivalences
studied is the notion of bisimulation. Often, the study of transition systems,
ways to define them and equivalences on them are also considered part of process
algebra, even in the case no equational theory is present.
In this note, I will be more precise: process algebra is the study of pertinent
equational theories with their models, while the wider field that also includes
the study of transition systems and related structures, ways to define them and
equivalences on them will be called process theory.
3 History
The history of process algebra will be traced back to the early seventies of the
twentieth century. At that point, the only part of concurrency theory that existed
is the theory of Petri nets, conceived by Petri starting from his thesis in 1962 [84].
In 1970, we can distinguish three main styles of formal reasoning about computer
programs, focusing on giving semantics (meaning) to programming languages.
1. Operational semantics. A computer program is modeled as an execution of
an abstract machine. A state of such a machine is a valuation of variables, a
transition between states is an elementary program instruction. Pioneer of
this field is McCarthy [64].
2. Denotational semantics. More abstract than operational semantics, com-
puter programs are usually modeled by a function transforming input into
output. Most well-known are Scott and Strachey [92].
3. Axiomatic semantics. Here, emphasis is put on proof methods proving pro-
grams correct. Central notions are program assertions, proof triples consist-
ing of precondition, program statement and postcondition, and invariants.
Pioneers are Floyd [38] and Hoare [53].
Then, the question was raised how to give semantics to programs contain-
ing a parallel operator. It was found that this is difficult using the methods of
denotational, operational or axiomatic semantics, although several attempts are
made (for instance, see [50]).
There are two paradigm shifts that need to made, before a theory of parallel
programs in terms of a process algebra can be developed. First of all, the idea of
a behaviour as an input/output function needed to be abandoned. A program
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Brief History of Process Algebra 5
could still be modeled as an automaton, but the notion of language equivalence
is no longer appropriate. This is because the interaction a process has between
input and output influences the outcome, disrupting functional behaviour. Sec-
ondly, the notion of global variables needs to be overcome. Using global variables,
a state of a modeling automaton is given as a valuation of the program variables,
that is, a state is determined by the values of the variables. The independent
execution of parallel processes makes it difficult or impossible to determine the
values of global variables at a given moment. It turns out to be simpler to let
each process have its own local variables, and to denote exchange of information
explicitly.
In the rest of this section, we describe the history of process algebra from the
early seventies to the early eighties, by focusing on the central people involved.
By the early eighties, we can say process algebra is established as a separate area
of research. Section 4 will consider the developments since the early eighties until
the present time by subarea.
3.1 Bekiˇ
c
One of the people studying the semantics of parallel programs in the early seven-
ties was Hans Bekiˇ. He was born in 1936, and died due to a mountain accident
c
in 1982. In the period we are speaking of, he worked at the IBM lab in Vienna,
Austria. The lab was well-known in the sixties and seventies for the work on the
definition and semantics of programming languages, and Bekiˇ played a part in
c
this, working on the denotational semantics of ALGOL and PL/I. Growing out
of his work on PL/I, the problem arose how to give a denotational semantics
for parallel composition. Bekiˇ tackled this problem in [18]. This internal report,
c
and indeed all the work of Bekiˇ is made accessible to us through the work of
c
Cliff Jones [19]. On this book, I base the following remarks.
In [18], Bekiˇ addresses the semantics of what he calls “quasi-parallel execu-
c
tion of processes”. From the introduction, I quote:
Our plan to develop an algebra of processes may be viewed as a high-level
approach: we are interested in how to compose complex processes from
simpler (still arbitrarily complex) ones.
Bekiˇ uses global variables, so a state ξ is a valuation of variables, and a
c
program determines an action A, which gives, in a state (non-deterministically)
either null iff it is an end-state, or an elementary step f , giving a new state f ξ and
rest-action A . Further, there are and cases denoting alternative composition,
; denoting sequential composition, and // denoting (quasi-)parallel composition.
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On page 183 in [19], we see the following law for quasi-parallel composition:
(A//B)ξ =
(cases Aξ : null → Bξ
(f, A ) → f, (A //B))
(cases Bξ : null → Aξ
(g, B ) → g, (A//B ))
and this is called the “unspecified merging” of the elementary steps of A and B.
This is definitely a pre-cursor of what later would be called the expansion law of
process algebra. It also makes explicit that Bekiˇ has made the first paradigm
c
shift: the next step in a merge is not determined, so we have abandoned the idea
of a program as a function.
The book [19] goes on with clarifications of [18] from a lecture in Amsterdam
in 1972. Here, Bekiˇ states that an action is tree-like, behaves like a scheduler, so
c
that for instance f ; (g h) is not the same as (f ; g) (f ; h) for elementary steps
f, g, h, another example of non-functional behaviour. In a letter to Peter Lucas
from 1975, Bekiˇ is still struggling with his notion of an action, and writes:
c
These actions still contain enough information so that the normal oper-
ations can be defined between them, but on the other hand little enough
information to fulfil certain desirable equivalences, such as:
a; 0 = a a; (b; c) = (a; b); c a//b = b//a
etc.
In a lecture on the material in 1974 in Newcastle, Bekiˇ has changed the
c
notation of // to and calls the operator parallel composition. In giving the
equations, we even encounter a “left-parallel” operator, with laws, with the same
meaning that Bergstra and Klop will later give to their left-merge operator [20].
Concluding, we can say that Bekiˇ contributed a number of basic ingredients
c
to the emergence of process algebra, but we see no coherent comprehensive theory
yet.
3.2 CCS
The central person in the history of process algebra without a doubt is Robin
Milner. A.J.R.G. Milner, born in 1934, developed his process theory CCS (Cal-
culus of Communicating Systems) over the years 1973 to 1980, culminating in
the publication of the book [72] in 1980.
The oldest publications concerning the semantics of parallel composition are
[67, 68], formulated within the framework of denotational semantics, using so-
called transducers. He considers the problems caused by non-terminating pro-
grams, with side effects, and non-determinism. He uses the operations ∗ for
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Brief History of Process Algebra 7
sequential composition, ? for alternative composition and for parallel compo-
sition. He refers to [18] as related work.
Next, chronologically, are the articles [71, 66]. Here, Milner introduces flow
graphs, with ports where a named port synchronizes with the port with its co-
name. Operators are | for parallel composition, restriction and relabeling. The
symbol is now reserved for restricted parallel composition. Static laws are
stated for these operators.
The following two papers are [69, 70], putting in place most of CCS as we
know it. The dynamic operators prefixing and alternative composition are added
and provided with laws. Synchronization trees are used as a model. The prefix τ
occurs as a communication trace (what remains of a synchronization of a name
and a co-name). The paradigm of message passing is taken over from [54]. Inter-
leaving is introduced as the observation of a single observer of a communicating
system, and the expansion law is stated. Sequential composition is not a basic
operator, but a derived one, using communication, abstraction and restriction.
The paper [49], with Matthew Hennessy, formulates basic CCS, with obser-
vational equivalence and strong equivalence defined inductively. Also, so-called
Hennessy-Milner logic is introduced, which provides a logical characterization of
process equivalence. Next, the book [72] is the standard process algebra refer-
ence. Here we have for the first time in history a complete process algebra, with a
set of equations and a semantical model. In fact, Milner talks about process cal-
culus everywhere in his work, emphasizing the calculational aspect. He presents
the equational laws as truths about his chosen semantical domain, rather than
considering the laws as primary, and investigating the range of models that they
have.
An important contribution that was realized just after the appearance of
[72] is the formulation of bisimulation by David Park [83]. This became a central
notion in process theory subsequently. The book [72] was later updated in [75].
A related development is the birth of structural operational semantics in [86].
More can be read about this in the recent history paper [87].
3.3 CSP
A very important contributor to the development of process algebra is Tony
Hoare. C.A.R. Hoare, born in 1934, published the influential paper [54] as a
technical report in 1976. The important step is that he does away completely with
global variables, and adopts the message passing paradigm of communication,
thus realizing the second paragdigm shift. The language CSP (Communicating
Sequential Processes) described in [54] has synchronous communication and is a
guarded command language (based on [35]). No model or semantics is provided.
This paper inspired Milner to treat message passing in CCS in the same way.
A model for CSP was elaborated in [55]. This is a model based on trace
theory, i.e. on the sequences of actions a process can perform. Later on, it was
found that this model was lacking, for instance because deadlock behaviour is
not preserved. For this reason, a new model based on failure pairs was presented
in [29] for the language that was then called TCSP (Theoretical CSP ). Later,
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8 J.C.M. Baeten
TCSP was called CSP again. Some time later it was established that the failure
model is the least discriminating model that preserves deadlock behaviour. In
the language, due to the presence of two alternative composition operators, it is
possible to do without a silent step like τ altogether. The book [56] gives a good
overview of CSP.
3.4 Some Other Process Theories
Around 1980, concurrency theory and in particular process theory is a vibrant
field with a lot of activity world wide. We already mentioned research on Petri
nets, that is an active area [85]. Another partial order process theory is given
in [63]. Research on temporal logic has started, see e.g. [88].
Some other process theories can be mentioned. We already remarked that
Hoare investigated trace theory. More work was done in this direction, e.g. by
Rem [90]. There is also the invariants calculus [6].
Another process theory that should be mentioned is the metric approach by
De Bakker and Zucker [16, 17]. There is a notion of distance between processes:
processes that do not differ in behaviour before the nth step have a distance of
at most 2−n . This turns the domain of processes into a metric space, that can
be completed, and solutions to guarded recursive equations exist by application
of Banach’s fixed point theorem.
3.5 ACP
Jan Bergstra and Jan Willem Klop in 1982 started work on a question of De
Bakker as to what can be said about solutions of unguarded recursive equations.
As a result, they wrote the paper [20]. In this paper, the phrase “process algebra”
is used for the first time. We quote:
A process algebra over a set of atomic actions A is a structure A =
A, +, ·, , ai (i ∈ I) where A is a set containing A, the ai are constant
symbols corresponding to the ai ∈ A, and + (union), · (concatenation
or composition, left out in the axioms), (left merge) satisfy for all
x, y, z ∈ A and a ∈ A the following axioms:
PA1 x+y =y+x
PA2 x + (y + z) = (x + y) + z
PA3 x+x=x
PA4 (xy)z = x(yz)
PA5 (x + y)z = xz + yz
PA6 (x + y) z = x z + y z
PA7 ax y = a(x y + y x)
PA8 a y = ay
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Brief History of Process Algebra 9
This clearly establishes process algebra in the strict sense as I set it out in
Section 2. In the paper, process algebra was defined with alternative, sequential
and parallel composition, but without communication. A model was established
based on projective sequences (a process is given by a sequence of approximations
by finite terms), and in this model, it is established that all recursive equations
have a solution. In adapted form, this paper was later published as [23]. In [22],
this process algebra PA was extended with communication to yield the theory
ACP (Algebra of Communicating Processes). The book [15] gives an overview
of ACP.
Comparing the three most well-known process algebras CCS, CSP and ACP,
we can say there is a considerable amount of work and applications realized in
all three of them. Historically, CCS was the first with a complete theory. Differ-
ent from the other two, CSP has a least distinguishing equational theory. More
than the other two, ACP emphasizes the algebraic aspect: there is an equational
theory with a range of semantical models. Also, ACP has a more general com-
munication scheme: in CCS, communication is combined with abstraction, in
CSP, communication is combined with restriction.
3.6 Further Remarks
In later years, other process algebras were developed. We can mention SCCS [73],
CIRCAL [65], MEIJE [7], the process algebra of Hennessy [48].
We see that over the years many process algebras have been developed, each
making its own set of choices in the different possibilities. The reader may won-
der whether this is something to be lamented. In the paper [11], it is argued that
this is actually a good thing, as long as there is a good exchange of information
between the different groups, as each different process algebra will have its own
set of advantages and disadvantages. I do think it is lamentable that the com-
munity still has not been able to come up with a uniform notation, denoting the
same thing in the same way.
4 Developments
In this section, I will address a number of important developments that have
occurred since the formulation of the basic process algebras CCS, CSP and
ACP.
4.1 Theory
A nice overview of the most important theoretical results since the start of
process algebra is the recent paper [1]. I already mentioned the formulation of the
notion of bisimulation by Park [83]. Strong bisimulation and weak bisimulation
became the central notion of equivalence in process theory. As an alternative
to weak bisimulation, branching bisimulation was proposed in [42] with certain
advantages [40]. In between bisimulation and trace equivalence there is a whole
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lattice of other equivalences, see the overview in [41]. See also [39]. For process
algebra based on partial order semantics, see [27].
There is a wealth of results concerning process algebra extended with some
form of recursion, see e.g. the complete axiomatization of regular processes by
Milner [74] or the overview on decidability in [30].
Structural operational semantics (SOS) has become the dominant way of
providing a model for a process algebra. It is standard practice to provide a
new operation with SOS rules, even before an equational characterization is
attempted. There are many results based on the formats of these rules. An
overview can be found in [3].
Finally, there is a whole range of expressiveness results, some examples can
be found in [21].
As is also stated in [1], there are many nice results, but also many remaining
open problems.
4.2 Tooling
Over the years, several software systems have been developed in order to fa-
cilitate the application of process algebra in the analysis of systems. In this
subsection, I only mention general process algebra tools. Tools that deal with
specific extensions are mentioned below.
The most well-known general tool is the Concurrency Workbench, see [80],
dealing with CCS type process algebra. There is also the variant CWB-NC,
see [98] for the current state of affairs. There is the French set of tools CADP, see
e.g. [37]. In the CSP tradition, there is the FDR tool (see http://www.fsel.com/).
The challenge in tool development is to combine an attractive user interface
with a powerful and fast back engine.
4.3 Verification
A measure of success of process algebra is the systems that have been success-
fully verified by means of techniques that come from process algebra. A good
overview of the current state of affairs is [46]. Process algebra focuses on equa-
tional reasoning. Other successful techniques are model checking and theorem
proving. Combination of these different approaches proves to be very promising.
4.4 Data
Process algebra is very successful in describing the dynamic behaviour of sys-
tems. In describing the static aspects, treatment of data is very important. Ac-
tions and processes are parametrized with data elements. The combination of
processes and data has received much attention over the years. A standardized
formal description technique is LOTOS, see [28]. Another combination of pro-
cesses and data is PSF, see [62] with associated tooling. The process algebra
with data µCRL has tooling focusing on equational verification, see e.g. [45].
In model checking, so-called symbolic model checking is employed in order to
deal with data parameters.
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Brief History of Process Algebra 11
4.5 Time
Research on process algebra extended with a quantitative notion of time started
with the work of Reed and Roscoe in the CSP context, see [89]. A textbook in
this tradition is [91].
There are many variants of CCS with timing, see e.g. [96], [81].
In the ACP tradition, work starts with [10]. An integrated theory, involving
both discrete and dense time, both relative and absolute time, is presented in
the book [13].
Also the theory ATP can be mentioned, see [82].
There are many formulations, in these works and elsewhere, of bisimulation
that takes into account timing information. Mostly, the passage of time is treated
as the occurrence of an action: bisimilar processes must have exactly matching
timing behaviour. Recent protocol verification has shown that this may be too
strict: perhaps a less discriminating equivalence should be used with respect to
timing information (see [14]). More research is needed in this direction. Also a
notion of approximation would be very useful. More experience with verification
of functional behaviour and timing behaviour is certainly needed.
Tooling has been developed for processes with timing mostly in terms of
timed automata, see e.g. UPPAAL [57] or KRONOS [97]. Equational reasoning
is investigated for µCRL with timing [93].
4.6 Mobility
Research on networks of processes where processes are mobile and configuration
of communication links is dynamic has been dominated by the π-calculus. An
early reference is [36], the standard reference is [79] and the textbook is [77].
The associated tool is the Mobility Workbench, see [94]. Also in this domain, it
is important to gain more experience with protocol verification. On the theory
side, there are a number of different equivalences that have been defined, and it
is not clear which is the ‘right’ one to use.
More recently, other calculi concerning mobility have been developed, notably
the ambient calculus, see [31]. As to unifying frameworks for different mobile
calculi, Milner investigates action calculus [76] and bigraphs [78].
4.7 Probabilities and Stochastics
Process algebras extended with probabilistic or stochastic information have gen-
erated a lot of research in recent years. An early reference is [47]. In the CSP
tradition, there is [59], in the CCS tradition [52], in the ACP tradition [12]. There
is the process algebra TIPP with associated tool, see e.g. [43], and EMPA, see
e.g. [26].
Recently, the insight that both (unquantified) alternative composition and
probabilistic choice are needed for a useful theory has gained attention, see e.g.
the work in [33] or [5].
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12 J.C.M. Baeten
Notions of abstraction are still a matter of continued research. The goal
is to combine functional verification with performance analysis. A notion of
approximation is very important here, for an attempt see [34]. Much further
work needs to be done on probabilistic and stochastic process algebra.
4.8 Hybrid Systems
Systems that in their behaviour depend on continuously changing variables other
than time are the latest challenge to be addressed by process algebra. System
descriptions involve differential algebraic equations, connections with dynamic
control theory are very important. Here, process algebra research is just at the
beginning. The attempts [24, 32] are worth mentioning.
In process theory, work centres around hybrid automata [4] and hybrid I/O
automata [60]. A tool is HyTech, see [51]. A connection with process algebra can
be found in [95].
5 Conclusion
In this note, a brief history is sketched of process algebra. The early work cen-
tred around giving semantics to programming languages involving a parallel
construct. Here, two breakthroughs were needed: first of all, abandoning the
idea that a program is a transformation from input to output, replacing this by
an approach where all intermediate states are important, and, secondly, replac-
ing the notion of global variables by the paradigm of message passing and local
variables.
In the seventies of the twentieth century, both these steps were taken, and
the process algebras CCS and CSP evolved. In doing so, process algebra became
an underlying theory of all parallel and distributed systems, extending formal
language and automata theory with the central ingredient of interaction.
In the following years, much work has been done, and many process alge-
bras have been formulated, extended with data, time, mobility, probabilities
and stochastics. The work is not finished, however. I formulated some challenges
for the future. More can be found in [2].
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