Introduction to complexity theory that solves your assignment problem it contains about complexity class,deterministic class,big- O notation ,proof by mathematical induction, L-Space ,N-Space and characteristics functions of set and so on
In the field of artificial intelligence (AI), planning refers to the process of developing a sequence of actions or steps that an intelligent agent should take to achieve a specific goal or solve a particular problem. AI planning is a fundamental component of many AI systems and has applications in various domains, including robotics, autonomous systems, scheduling, logistics, and more. Here are some key aspects of planning in AI:
Definition of Planning: Planning involves defining a problem, specifying the initial state, setting a goal state, and finding a sequence of actions or a plan that transforms the initial state into the desired goal state while adhering to certain constraints.
State-Space Representation: In AI planning, the problem is often represented as a state-space, where each state represents a snapshot of the system, and actions transform one state into another. The goal is to find a path through this state-space from the initial state to the goal state.
Search Algorithms: AI planning typically relies on search algorithms to explore the state-space efficiently. Uninformed search algorithms, such as depth-first search and breadth-first search, can be used, as well as informed search algorithms, like A* search, which incorporates heuristics to guide the search.
Heuristics: Heuristics are used in planning to estimate the cost or distance from a state to the goal. Heuristic functions help inform the search algorithms by providing an estimate of how close a state is to the solution. Good heuristics can significantly improve the efficiency of the search.
Plan Execution: Once a plan is generated, the next step is plan execution, where the agent carries out the actions in the plan to achieve the desired goal. This often requires monitoring the environment to ensure that the actions are executed as planned.
Temporal and Hierarchical Planning: In more complex scenarios, temporal planning deals with actions that have temporal constraints, and hierarchical planning involves creating plans at multiple levels of abstraction, making planning more manageable in complex domains.
Partial and Incremental Planning: Sometimes, it may not be necessary to create a complete plan from scratch. Partial and incremental planning allows agents to adapt and modify existing plans to respond to changing circumstances.
Applications: Planning is used in a wide range of applications, from manufacturing and logistics (e.g., scheduling production and delivery) to robotics (e.g., path planning for robots) and game playing (e.g., chess and video games).
Challenges: Challenges in AI planning include dealing with large search spaces, handling uncertainty, addressing resource constraints, and optimizing plans for efficiency and performance.
AI planning is a critical component in creating intelligent systems that can autonomously make decisions and solve complex problems.
Knowledge representation and Predicate logicAmey Kerkar
This presentation is specifically designed for the in depth coverage of predicate logic and the inference mechanism :resolution algorithm.
feel free to write to me at : amecop47@gmail.com
In the field of artificial intelligence (AI), planning refers to the process of developing a sequence of actions or steps that an intelligent agent should take to achieve a specific goal or solve a particular problem. AI planning is a fundamental component of many AI systems and has applications in various domains, including robotics, autonomous systems, scheduling, logistics, and more. Here are some key aspects of planning in AI:
Definition of Planning: Planning involves defining a problem, specifying the initial state, setting a goal state, and finding a sequence of actions or a plan that transforms the initial state into the desired goal state while adhering to certain constraints.
State-Space Representation: In AI planning, the problem is often represented as a state-space, where each state represents a snapshot of the system, and actions transform one state into another. The goal is to find a path through this state-space from the initial state to the goal state.
Search Algorithms: AI planning typically relies on search algorithms to explore the state-space efficiently. Uninformed search algorithms, such as depth-first search and breadth-first search, can be used, as well as informed search algorithms, like A* search, which incorporates heuristics to guide the search.
Heuristics: Heuristics are used in planning to estimate the cost or distance from a state to the goal. Heuristic functions help inform the search algorithms by providing an estimate of how close a state is to the solution. Good heuristics can significantly improve the efficiency of the search.
Plan Execution: Once a plan is generated, the next step is plan execution, where the agent carries out the actions in the plan to achieve the desired goal. This often requires monitoring the environment to ensure that the actions are executed as planned.
Temporal and Hierarchical Planning: In more complex scenarios, temporal planning deals with actions that have temporal constraints, and hierarchical planning involves creating plans at multiple levels of abstraction, making planning more manageable in complex domains.
Partial and Incremental Planning: Sometimes, it may not be necessary to create a complete plan from scratch. Partial and incremental planning allows agents to adapt and modify existing plans to respond to changing circumstances.
Applications: Planning is used in a wide range of applications, from manufacturing and logistics (e.g., scheduling production and delivery) to robotics (e.g., path planning for robots) and game playing (e.g., chess and video games).
Challenges: Challenges in AI planning include dealing with large search spaces, handling uncertainty, addressing resource constraints, and optimizing plans for efficiency and performance.
AI planning is a critical component in creating intelligent systems that can autonomously make decisions and solve complex problems.
Knowledge representation and Predicate logicAmey Kerkar
This presentation is specifically designed for the in depth coverage of predicate logic and the inference mechanism :resolution algorithm.
feel free to write to me at : amecop47@gmail.com
Knowledge representation In Artificial IntelligenceRamla Sheikh
facts, information, and skills acquired through experience or education; the theoretical or practical understanding of a subject.
Knowledge = information + rules
EXAMPLE
Doctors, managers.
This file contains the concepts of Class P, Class NP, NP- completeness, Travelling Salesman Person problem, Clique Problem, Vertex cover problem, Hamiltonian problem, FFT and DFT.
Artificial Intelligence: Introduction, Typical Applications. State Space Search: Depth Bounded
DFS, Depth First Iterative Deepening. Heuristic Search: Heuristic Functions, Best First Search,
Hill Climbing, Variable Neighborhood Descent, Beam Search, Tabu Search. Optimal Search: A
*
algorithm, Iterative Deepening A*
, Recursive Best First Search, Pruning the CLOSED and OPEN
Lists
Presents features of ARM Processors, ARM architecture variants and Processor families. Further presents, ARM v4T architecture, ARM7-TDMI processor: Register organization, pipelining, modes, exception handling, bus architecture, debug architecture and interface signals.
This presentation gives a brief over view of Embedded Systems. It describes the common characteristics of Embedded systems, the design metrics, processor technologies and also summarizes differences between Microcontrollers and Microprocessors.
this is a briefer overview about the Big O Notation. Big O Notaion are useful to check the Effeciency of an algorithm and to check its limitation at higher value. with big o notation some examples are also shown about its cases and some functions in c++ are also described.
As part of the highly successful lunchtime talk series, the contemporary Tavistock Institute of Human Relations (TIHR) food-for-thought programme, Eliat Aram, the Institute’s CEO introduced staff and guests to some key concepts and philosophical underpinning of Complexity theory and its implications to understanding organisational praxis.
Knowledge representation In Artificial IntelligenceRamla Sheikh
facts, information, and skills acquired through experience or education; the theoretical or practical understanding of a subject.
Knowledge = information + rules
EXAMPLE
Doctors, managers.
This file contains the concepts of Class P, Class NP, NP- completeness, Travelling Salesman Person problem, Clique Problem, Vertex cover problem, Hamiltonian problem, FFT and DFT.
Artificial Intelligence: Introduction, Typical Applications. State Space Search: Depth Bounded
DFS, Depth First Iterative Deepening. Heuristic Search: Heuristic Functions, Best First Search,
Hill Climbing, Variable Neighborhood Descent, Beam Search, Tabu Search. Optimal Search: A
*
algorithm, Iterative Deepening A*
, Recursive Best First Search, Pruning the CLOSED and OPEN
Lists
Presents features of ARM Processors, ARM architecture variants and Processor families. Further presents, ARM v4T architecture, ARM7-TDMI processor: Register organization, pipelining, modes, exception handling, bus architecture, debug architecture and interface signals.
This presentation gives a brief over view of Embedded Systems. It describes the common characteristics of Embedded systems, the design metrics, processor technologies and also summarizes differences between Microcontrollers and Microprocessors.
this is a briefer overview about the Big O Notation. Big O Notaion are useful to check the Effeciency of an algorithm and to check its limitation at higher value. with big o notation some examples are also shown about its cases and some functions in c++ are also described.
As part of the highly successful lunchtime talk series, the contemporary Tavistock Institute of Human Relations (TIHR) food-for-thought programme, Eliat Aram, the Institute’s CEO introduced staff and guests to some key concepts and philosophical underpinning of Complexity theory and its implications to understanding organisational praxis.
A set of slides initially designed to help students revise and consolidate their understanding of complexity theory and its application to work and our management of work.
Of the complex, the simple and the non-complexRicardo Alvira
It reviews the meaning/definition of terms 'complex' and 'simple' from an etimological perspective, in order to help set the basic definitions to formulate a Unified Theory of Complexity.
Management and Complexity Theory Lecture (Anglia Ruskin, Oct 2010)Dr. Michael B. Duignan
Presentation for Anglia Ruskin University 3rd year undergraduates, on the slightly left field concept of complexity theory, and concepts like the 'edge of chaos' and 'emergence' to management thought.
October, 2010 - Cambridge, UK
Blog: www.olympicresearcher.wordpress.com
Academic profile: http://www.anglia.ac.uk/ruskin/en/home/faculties/aibs/staff_profiles/all_staff/michael_duignan.html
Applying design thinking and complexity theory in agile organizations AgileSparks
"Applying design thinking and complexity theory in Agile organizations"
By Jean Tabaka @ Agile Israel 2012
http://agilesparks.com/DesignThinking-JeanTabaka
Although, chaos/complexity theory and SLA have commonplaces, they seem to be different in that chaos/complexity theory offers the wider perspective that has served SLA in the past. As opposed to SLA, chaos/complexity theory encourages linguists to think in relational terms. It refers to the fact that by accepting participation metaphor/language use/emergent grammar position, chaos/complexity theory does not reject psychological perspective. As it is stated , chaos complexity theory like socialists focus on the following issues: 1. all languages are static 2. there are mechanisms for language change 3.language and learning are seen as an open systems.
Nevertheless, C/CT never rejects the following characteristics which psychological perspective focuses on: 1. languages are sensitive to initial conditions 2. there are systemic patterns with dynamic paths.
An overview of Systems Thinking, and how to apply the ideas of Complexity Theory to management of systems, with the results being called "Complexity Thinking".
This presentation is part of the Management 3.0 course created by Jurgen Appelo.
http://www.management30.com/course-introduction/
NP completeness. Classes P and NP are two frequently studied classes of problems in computer science. Class P is the set of all problems that can be solved by a deterministic Turing machine in polynomial time.
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.
Formal and Computational Representations
The Semantics of First-Order Logic
Event Representations
Description Logics & the Web Ontology Language
Compositionality
Lamba calculus
Corpus-based approaches:
Latent Semantic Analysis
Topic models
Distributional Semantics
This paper reviews algorithmic information theory, which is an attempt to apply information-theoretic and probabilistic
ideas to recursive function theory. Typical concerns in this approach are, for example, the number of bits of information required to
specify an algorithm, or the probability that a program whose bits are chosen by coin flipping produces a given output. During the past
few years the definitions of algorithmic information theory have been reformulated. The basic features of the new formalism are presented
here and certain results of R. M. Solovay are reported.
Fractal Dimension of Space-time Diagrams and the Runtime Complexity of Small ...Hector Zenil
Complexity measures are designed to capture complex behaviour and to quantify how complex that particular behaviour is. If a certain phenomenon is genuinely complex this means that it does not all of a sudden becomes simple by just translating the phenomenon to a different setting or framework with a different complexity value. It is in this sense that we expect different complexity measures from possibly entirely different fields to be related to each other. This work presents our work on a beautiful connection between the fractal dimension of space-time diagrams of Turing machines and their time complexity. Presented at Machines, Computations and Universality (MCU) 2013, Zurich, Switzerland.
Fractal dimension versus Computational ComplexityHector Zenil
We investigate connections and tradeoffs between two important complexity measures: fractal dimension and computational (time) complexity. We report exciting results applied to space-time diagrams of small Turing machines with precise mathematical relations and formal conjectures connecting these measures. The preprint of the paper is available at: http://arxiv.org/abs/1309.1779
Neural Model-Applying Network (Neuman): A New Basis for Computational Cognitionaciijournal
NeuMAN represents a new model for computational cognition synthesizing important results across AI, psychology, and neuroscience. NeuMAN is based on three important ideas: (1) neural mechanisms perform all requirements for intelligence without symbolic reasoning on finite sets, thus avoiding exponential matching algorithms; (2) the network reinforces hierarchical abstraction and composition for sensing and acting; and (3) the network uses learned sequences within contextual frames to make predictions, minimize reactions to expected events, and increase responsiveness to high-value information. These systems exhibit both automatic and deliberate processes. NeuMAN accords with a wide variety of findings in neural and cognitive science and will supersede symbolic reasoning as a foundation for AI and as a model of human intelligence. It will likely become the principal mechanism for engineering intelligent systems.
Similar to Introduction to complexity theory assignment (20)
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
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How to Make a Field invisible in Odoo 17Celine George
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Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
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http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
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Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
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Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
1. Bahirdar University
Bahirdar institute of technology
Faculty of computing
Department of computer science
Complexity Theory Assignment
TESFAHUNEGN MINWUYELET
Date of Submission: 13/05/2016 G.C
2. i
Table of Contents
1.What is the characteristicfunctionof set?Explainitwithashortnote;elaborate itwithnotlessthan3
example.............................................................................................................................................1
2.What is meant by complexity class? Define the basic deterministic complexity classes. .....................10
Time and Space Complexity Classes ...............................................................................................11
The following are the canonical complexity classes:........................................................................11
L (complexity)...............................................................................................................................11
P (complexity)...............................................................................................................................12
3.Define Big-O notation and illustrate it with notless than 3 examples................................................12
Formal definition:- .......................................................................................................................14
Theorems you can use without proof.............................................................................................14
4.Write a shortnote aboutthe similarityanddifference betweenTuringmachineandrandomaccess
machine?.........................................................................................................................................16
5.Prove by mathematical induction that n!>2n
, for n≥4. ......................................................................19
3. 1
1.What is the characteristic function of set? Explain it with a short note; elaborate it with
not less than 3 example.
12. 10
2.What is meant by complexity class? Define the basic deterministic complexity classes.
Typically, a complexity class is defined by (1) a model of computation, (2) a resource (or
collection
of resources), and (3) a function known as the complexity bound for each resource. The models
used to define complexity classes fall into two main categories: (a) machine based models, and
(b) circuit-based models. Turing machines (TMs) and random-access machines (RAMs) are the
two principal families of machine models. When we wish to model real computations,
deterministic machines and circuits are our closest links to reality. Then why consider the other
kinds of machines? There are two main reasons. The most potent reason comes from the
computational problems whose complexity we are trying to understand. The most notorious
examples are the hundreds of natural NP-complete problems. To the extent that we understand
anything about the complexity of these problems, it is because of the model of nondeterministic
Turing machines. Nondeterministic machines do not model physical computation devices, but
they do model real computational problems. There are many other examples where a particular
model of computation has been introduced in order to capture some well-known computational
problem in a complexity class. The second reason is related to the rest. Our desire to understand
real computational problems has forced upon us a repertoire of models of computation and
resource bounds. In order to understand the relationships between these models and bounds, we
combine and mix them and attempt to discover their relative power. Consider, for example,
nondeterminism. By considering the complements of languages accepted by nondeterministic
machines, researchers were naturally led to the notion of alternating machines. When alternating
machines and deterministic machines were compared, a surprising virtual identity of
deterministic space and alternating time emerged. Subsequently, alternation was found to be a
useful way to model efficient parallel computation. This phenomenon, whereby models of
computation are generalized and modified in order to clarify their relative complexity, has
occurred often through the brief history of complexity theory, and has generated some of the
most important new insights. Other underlying principles in complexity theory emerge from the
major theorems showing relationships between complexity classes. These theorems fall into two
broad categories. Simulation theorems show that computations in one class can be simulated by
13. 11
computations that meet the defining resource bounds of another class. The containment of
nondeterministic logarithmic space (NL) in polynomial time (P), and the equality of the class P
with alternating logarithmic space, are simulation theorems. Separation theorems show that
certain complexity classes are distinct. Complexity theory currently has precious few of these.
The main tool used in those separation theorems we have is called diagonalization.
Time and Space Complexity Classes
DTIME[t(n)] is the class of languages decided by deterministic Turing machines of time com-
plexity t(n).
DSPACE[s(n)] is the class of languages decided by deterministic Turing machines of space
complexity s(n).
The following are the canonical complexity classes:
L (complexity)
In computational complexity theory, L (also known as LSPACE) is the complexity class
containing decision problems that can be solved by a deterministic Turing machine using a
logarithmic amount of memory space. Logarithmic space is sufficient to hold a constant number
of pointers into the input and a logarithmic number of Boolean flags and many basic log space
algorithms use the memory in this way.
L is a subclass of NL, which is the class of languages decidable in logarithmic space on a
nondeterministic Turing machine. A problem in NL may be transformed into a problem of
reachability in a directed graph representing states and state transitions of the nondeterministic
machine, and the logarithmic space bound implies that this graph has a polynomial number of
vertices and edges, from which it follows that NL is contained in the complexity class P of
problems solvable in deterministic polynomial time. Thus L ⊆ NL ⊆ P. The inclusion of L into P
can also be proved more directly: a decider using O(log n) space cannot use more than
2O(log n) = nO(1) time, because this is the total number of possible configurations.
L further relates to the class NC in the following way: NC1 ⊆ L ⊆ NL ⊆ NC2. In words, given a
parallel computer C with a polynomial number O(nk) of processors for some constant k, any
problem that can be solved on C in O(log n) time is in L, and any problem in L can be solved in
O(log2 n) time on C.
14. 12
Important open problems include whether L = P, and whether L = NL.
L is low for itself, because it can simulate log-space oracle queries (roughly speaking, "function
calls which use log space") in log space, reusing the same space for each query.
The relatedclassof functionproblems isFL.FL isoftenusedto define logspace reductions.
P (complexity)
In computational complexity theory, P, also known as PTIME or DTIME (nO(1)), is one of the
most fundamental complexity classes. It contains all decision problems that can be solved by a
deterministic Turing machine using a polynomial amount of computation time, or polynomial
time.
A language L is in P if and only if there exists a deterministic Turing machine M, such that
M runs for polynomial time on all inputs
For all x in L, M outputs 1
For all x not in L, M outputs 0
P can also be viewed as a uniform family of Boolean circuits. A language L is in P if and only if
there exists a polynomial-time uniform family of Boolean circuits such that The circuit definition
can be weakened to use only a log space uniform family without changing the complexity class.
deterministicTuringmachine T∈TT∈T canberepresentedasa tuple ⟨Q,Σ,δ,s⟩⟨Q,Σ,δ,s⟩ where QQ is
a finite set of internal states, ΣΣ is a finite tape alphabet, s∈Qs∈Q is TT’s start state, and δδ is a
transition function mapping state-symbol pairs ⟨q,σ⟩⟨q,σ⟩ into state-action pairs ⟨q,a⟩⟨q,a⟩.
Here aa is chosen from the set of actions {σ,⇐,⇒}{σ,⇐,⇒} – i.e. write the symbol σ∈Σσ∈Σ on
the current square, move the head left, or move the head right. Such a function is hence of
type δ:Q×Σ→Q×αδ:Q×Σ→Q×α. On the other hand, a non-deterministic Turing
machine N∈NN∈N is of the form ⟨Q, Σ,Δ,s⟩⟨Q,Σ,Δ,s⟩ where Q,ΣQ,Σ, and ss are as before
but ΔΔ is now only required to be a relation – i.e. Δ⊆(Q×Σ)×(Q×α)Δ⊆(Q×Σ)×(Q×α). As a
consequence, a machine configuration in which NN is in state qq and reading symbol σσ can
lead to finitely many distinct successor configurations –
e.g. it is possible that ΔΔ relates ⟨q,σ⟩⟨q,σ⟩ to both ⟨q′,a′⟩⟨q′,a′⟩ and ⟨q′′,a′′⟩⟨q″,a″⟩ for distinct
states q′q′ and q′′q″ and actions a′a′ and a′′a″.
3.Define Big-O notation and illustrate it with not less than 3 examples.
15. 13
Big O notation (with a capital letter O, not a zero), also called Landau’s symbol, is a symbolism
used in complexity theory, computer science, and mathematics to describe the asymptotic
behavior of functions. Basically, it tells you how fast a function grows or declines. Landau’s
symbol comes from the name of the German number theoretician Edmund Landau who invented
the notation. The letter O is used because the rate of growth of a function is also called its order.
For example, when analyzing some algorithm, one might find that the time (or the number of
steps) it takes to complete a problem of size n is given by T(n) = 4 n2- 2 n + 2. If we ignore
constants (which makes sense because those depend on the particular hardware the program is
run on) and slower growing terms, we could say “T(n) grows at the order of n2" and write T(n)
=O(n2). In mathematics, it is often important to get a handle on the error term of an
approximation. For instance, people will write ex= 1 + x + x2/ 2 + O(x3) for x -> 0 to express
the fact that the error is smaller in absolute value than some constant times x3 if x is close enough
to 0. For the formal definition, suppose f(x) and g(x) are two functions defined on some subset of
the real numbers. We write f(x) = O(g(x)) (or f(x) = O(g(x)) for x -> ∞ to be more precise) if and
only if there exist constants N and C such that |f(x)| ≤ C |g(x)| for all x>Intuitively, this means
that f does not grow faster than g. If a is some real number, we write f(x) = O(g(x)) for x -> a if
and only if there exist constants d > 0 and C such that |f(x)| C |g(x)| for all x with |x-a| < d.
The first definition is the only one used in computer science (where typically only positive
functions with a natural number n as argument are considered; the absolute
values can then be ignored), while both usages appear in mathematics. Here is a list of classes of
functions that are commonly encountered when analyzing algorithms. The slower growing
functions are listed first. c is some arbitrary constant.
Notation name
O(1) constant
O(log(n)) logarithmic
O((log(n)) c) polylogarithmic
O(n) linear
O(n2) quadratic
O(nc) polynomial
O(cn) exponential
16. 14
Note that O(nc) and O(cn) are very different. The latter grows much, much faster, no matter how
big the constant c is. A function that grows faster than any power of n is called super
polynomial. One that grows slower than an exponential function of the form cn is called sub
exponential. An algorithm can require time that is both super polynomial and sub exponential;
examples of this include the fastest algorithms known for integer factorization. Note, too, that O
(log n) is exactly the same as O (log (nc)). The logarithms differ only by a constant factor, and
the big O notation ignores that. Similarly, logs with different constant bases are equivalent. The
above list is useful because of the following fact: if a function f(n) is a sum of functions, one of
which grows faster than the others, then the faster growing one determines the order of f(n).
Example: If f(n) = 10 log(n) + 5 (log(n))3+ 7 n + 3 n2+ 6 n3, then f(n) = O(n3).
One caveat here: the number of summands has to be constant and may not depend on n. This
notation can also be used with multiple variables and with other expressions on the right side of
the equal sign. The notation: f(n,m) = n2+ m3+ O(n+m) represents the statement:∃C ∃ N ∀
n,m>N : f(n,m)n2+m3+C(n+m)
Formal definition: -
Given f, g : N → R+, we say that f ∈ O(g) if there exists some constants c >0, n0 ≥ 0 such that
for every n ≥ n 0, f (n) ≤ cg(n). That is, for sufficiently large n, the rate of growth of f is bounded
by g, up to a constant c. f, g might represent arbitrary functions, or the running time or space
complexity of a program or algorithm.
Theorems you can use without proof
17. 15
Example 1: Prove that running time T(n) = n 3 + 20n + 1 is O(n 3 ) Proof: by the Big-Oh
definition, T(n) is O(n 3 ) if T(n) ≤ c·n 3 for some n ≥ n0 . Let us check this condition: if n 3 +
20n + 1 ≤ c·n 3 then c n n + + ≤ 2 3 20 1 1 . Therefore, the Big-Oh condition holds for n ≥ n0 = 1
and c ≥ 22 (= 1 + 20 + 1). Larger values of n0 result in smaller factors c (e.g., for n0 = 10 c ≥
1.201 and so on) but in any case the above statement is valid.
Example 2: Prove that running time T(n) = n 3 + 20n + 1 is not O(n 2 ) Proof: by the Big-Oh
definition, T(n) is O(n 2 ) if T(n) ≤ c·n 2 for some n ≥ n0 . Let us check this condition: if n 3 +
20n + 1 ≤ c·n 2 then c n n n + + ≤ 2 20 1 . Therefore, the Big-Oh condition cannot hold (the left
side of the latter inequality is growing infinitely, so that there is no such constant factor c).
18. 16
Example 3: Prove that running time T(n) = n 3 + 20n + 1 is O (n 4 ) Proof: by the Big-Oh
definition, T(n) is O(n 4 ) if T(n) ≤ c·n 4 for some n ≥ n0 . Let us check this condition: if n 3 +
20n + 1 ≤ c·n 4 then c n n n + + ≤ 3 4 1 20 1 .Therefore, the Big-Oh condition holds for n ≥ n0 =
1 and c ≥ 22 (= 1 + 20 + 1). Larger values of n0 result in smaller factors c (e.g., for n0 = 10 c ≥
0.10201 and so on) but in any case the above statement is valid.
Example 4: Prove that running time T(n) = n 3 + 20n is Ω(n 2 ) Proof: by the Big-Omega
definition, T(n) is Ω(n 2 ) if T(n) ≥ c·n 2 for some n ≥ n0 . Let us check this condition: if n 3 +
20n ≥ c·n 2 then c n n + ≥ 20 . The left side of this inequality has the minimum value of 8.94 for
n = 20 ≅ 4.47 Therefore, the Big-Omega condition holds for n ≥ n0 = 5 and c ≤ 9. Larger values
of n0 result in larger factors c (e.g., for n0 = 10 c ≤ 12.01) but in any case the above statement is
valid.
4.Write a short note about the similarity and difference between Turing machine and
random access machine?
define the basic deterministic complexity classes, L, P and PSPACE.
Definition 1: Given a set A, we say that A ∈ L iff there is a Turing machine which
computes the characteristic function of A in space O(log n).
big O notation usually only provides an upper bound on the growth rate of the function
Definition 2: Given a set A, we say that A ∈ P iff there is a Turing machine which for
some constant k computes the characteristic function of A in time O(nk)
Definition 3: Given a set A, we say that A ∈ PSPACE iff there is a Turing machine which
for some constant k computes the characteristic function of A in space O(nk)
Theorem 1: L ⊂ PSPACE
Theorem 2: P ⊆ PSPACE
since a Turing machine cannot use more space than time.
Theorem 3: L ⊆ P
a machine which runs in logarithmic space also runs in polynomial time.
Turing machine seems incredibly inefficient and thus we will compare it to a model of
computation which is more or less a normal computer (programmed in assembly
language).
This type of computer is called a Random Access Machine (RAM)
19. 17
A RAM has a finite control and infinite number of registers and two accumulators.
Both the registers and the accumulators can hold arbitrarily large integers.
We will let r(i) be the content of register i and ac1 and ac2 the contents of the
accumulators.
The finite control can read a program and has a read-only input-tape and a write-only
output tape.
In one step a RAM can carry out the following instructions.
Add, subtract, divide or multiply the two numbers in ac1 and ac2,the result ends up in ac1.
Make conditional and unconditional jumps. (Condition ac1 > 0 or ac1 = 0).
Load something into an accumulator, e.g. ac1 = r(k) for constant k or ac1= r(ac1),similarly
for ac2.
Store the content of an accumulator, e.g. r(k) = ac1 for constant k or r(ac2) = ac1, similarly
for ac2.
Read input ac1 = input(ac2).
Write an output.
Use constants in the program.
Halt
Definition 4: The time to do a particular instruction on a RAM is1+ log(k + 1) where k is
the least upper bound on the integers involved in the instruction. The time for a
computation on a RAM is the sum of the times for the individual instructions.
Definition 5:
20. 18
Intuitively the RAM is more powerful than a Turing machine.
The size of a computer word is bounded by a constant and operations on larger numbers
require us to access a number of memory cells which is proportional to logarithm of the
number used.
Theorem 4: If a Turing machine can compute a function in time T(n) and space S(n), for
T(n) ≥ n and S(n) ≥ log n then the same function can be computed in time O(T2(n)) and
space O(S(n)) on a RAM.
fact a Turing machine is not that much less powerful than a RAM.
Theorem 5:- If a function f can be computed by a RAM in time T(n) and space S(n) then
f can be computed in time O(T2(n)) and space S(n) on a Turing machine.
Example 1:- Given two n-digit numbers x and y written in binary, write the instruction
that computes their sum.
it can be done in logarithmic space.
We have x = ∑i=0
n-1xi2i and y = ∑i=0
n-1yi2i
x + y is computed by the following instruction
carry= 0
For i = 0 to n − 1
bit = xi + yi + carry
carry = 0
If bit ≥ 2 then carry = 1, bit = bit − 2
write bit
next i
write carry
This can clearly be done in O(log n) space and thus addition belongs to L.
Example 2:- Given two n-digit numbers x and y written in binary, write a machine
instruction that compute their product.
◦ This can be done in P time o(n2)
carry= 0
21. 19
For i = 0 to 2n − 2
low = max(0, i − (n − 1))
high = min(n − 1, i)
For j = low to high, carry = carry + xj ∗ yi−j
write lsb(carry)
carry = carry/2
next i
write carry with least significant bit first
5.Prove by mathematical induction that n!>2n, for n≥4.
Basis: 4! = 24 > 16 = 24.
Induction:
IH: n! > 2 n
NTS: (n + 1)! > 2 n+1
(n + 1)! = n! · (n + 1) (definition of !)
> 2 n · (n + 1) (IH)
> 2 n · 2 (n ≥ 4)
= 2n+1