0.0 Introduction to theory of computation
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The document provides an introduction to the theory of computation. It discusses how the theory of computation deals with how efficiently problems can be solved using algorithms on computational models. The field is divided into three main branches: automata theory and language, computability theory, and computational complexity theory. The objectives of the course are to understand computational models like finite state machines, pushdown automata, and Turing machines, and to learn about the decidability and undecidability of problems. The course will cover topics like finite automata, grammars, pushdown automata, Turing machines, unsolvable problems, and computational complexity.
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Course Structure*
Block- 1 Introduction to Algorithms
Unit 1: Basics of an Algorithm and its
properties
- Introduction
- Objective
- Example of an Algorithm
- Basics building blocks of Algorithms
- A survey of common running time
- Analysis & Complexity of Algorithm
- Types of problems
- Problem Solving Techniques
- Deterministic and Stochastic
Algorithms
- Summary
- Solutions/Answers
- Further Readings
Unit 2: Some pre-requisites and
Asymptotic Bounds
Introduction
Objectives
Some Useful Mathematical
Functions &Notations
Functions & Notations
Modular Arithmetic/Mod
Function
Mathematical Expectation
Principle of Mathematical
Induction
Concept of Efficiency of an Algorithm
Well Known Asymptotic Functions &
Notations
Summary
Solutions/Answers
Unit 3: Analysis of Simple Algorithm
Introduction
Objectives
complexity Analysis of Algorithms
Euclid Algorithm for GCD
Polynomial Evaluation Algorithm
Exponent Evaluation
Sorting Algorithm
3.3 Analysis of Non-Recursive Control
Structures
Sequencing
for Construct
While and Repeat Constructs
Recursive Constructs
Summary
Solutions/Answers
Further Readings
22
Unit 4: Solving Recurrences
- Introduction
- Objective
- Substitution Methods
- Iteration Methods
- Recursive Tree Methods
- Master Methods
- Summary
- Solution/Answers
- Further Readings
Block- 2 Design Techniques-I
Unit 1: Greedy Technique
Some Examples to understand Greedy
Techniques
Formalization of Greedy Techniques
An overview of local and global
optima
Fractional Knapsack problem
Huffman Codes
A task scheduling algorithm
Unit 2: Divide & Conquer Technique
General Issues in Divide and Conquer
Technique
Binary Search Algorithm
Sorting Algorithm
o Merge Sort
o Quick Sort
Matrix Multiplication Algorithm
Unit 3: Graph Algorithm -I Basic Definition and terminologies Graph Representation
o Adjacency Matrix
o Adjacency List Graph Traversal Algorithms
o Depth First Search
o Breadth First Search Topological Sort Strongly Connected Components
Bl
Course file for theory of computation dt 08 08-2016.
Course file for theory of computation dt 08 08-2016.sumit jain The document provides details of the course plan for Theory of Computation at Acropolis Technical Campus in Indore, India. It lists the course code, semester, tutors, and course overview. The course aims to cover finite automata, pushdown automata, context-free grammars, and Turing machines. It outlines 6 course learning objectives and 6 course outcomes. It also maps the course outcomes to program outcomes and program specific outcomes. The document provides information on topic delivery, time schedules, books, syllabus, and lab work objectives.
Syllabus for Bachelors in Engineering Information Science
Syllabus for Bachelors in Engineering Information Sciencesyed qutubuddin Information Science Syllabus
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5.2 primitive recursive functions
5.2 primitive recursive functionsSampath Kumar S The document discusses primitive recursive functions and predicates. It defines primitive recursive functions as those that can be constructed from initial functions using only composition and recursion. Some examples of primitive recursive functions given are addition, multiplication, factorial, power, predecessor, absolute value, and bounded quantification. Predicates like equality, less than, negation, conjunction, disjunction, division, and primality are also shown to be primitive recursive. The concept of bounded minimalization is introduced, where a function returns the least value t for which a primitive recursive predicate P(t,x1,...xn) is true.
4.7. chomskian hierarchy of languages
4.7. chomskian hierarchy of languagesSampath Kumar S Noam Chomsky classified languages into four types (Type 0-3) based on the type of grammar and automaton that can describe them. Type 0 languages have the most expressive power and are recursively enumerable, while Type 3 languages have the least expressive power and are regular. The types differ based on the grammar and machines used to generate and recognize the language.
4.6 halting problem
4.6 halting problemSampath Kumar S The document discusses the halting problem and decidability of languages. It defines a decidable language as one where a Turing machine can accept or reject every input string and halt. The halting problem asks whether it is possible to determine if an arbitrary Turing machine will halt on a given input. The document proves the halting problem is undecidable by constructing a machine that contradicts itself, showing no algorithm can correctly determine if all Turing machines halt on all inputs.
4.3 techniques for turing machines construction
4.3 techniques for turing machines constructionSampath Kumar S This document discusses techniques for constructing Turing machines including storing finite control in storage, using multiple tracks, checking off symbols, and implementing subroutines. It was presented by Sampath Kumar S on November 21, 2017 to discuss methods for building Turing machines.
4.2 variantsof turing machines (types of tm)
4.2 variantsof turing machines (types of tm)Sampath Kumar S This document discusses different types of Turing machines. It describes multi-tape/head Turing machines which have multiple tapes that each have their own independent read/write head. It also describes multi-track Turing machines which have a single tape head that can read and write to multiple tracks on a single tape simultaneously. The document is a presentation on Turing machines given by Sampath Kumar S on November 21, 2017.
4.1 turing machines
4.1 turing machinesSampath Kumar S A Turing machine is a mathematical model of computation that consists of an infinite tape divided into cells, a head that reads and writes symbols on the tape, and a finite set of states. Alan Turing invented the Turing machine in 1936 to formalize the idea of algorithmic computation. A Turing machine operates by reading a symbol from the tape according to its transition function, then writing a symbol, changing its state, and moving the head left or right. If the machine reaches an accepting state, the input is accepted, otherwise it is rejected.
3.6 & 7. pumping lemma for cfl & problems based on pl
3.6 & 7. pumping lemma for cfl & problems based on plSampath Kumar S The document discusses the pumping lemma for context-free grammars (CFGs). It defines the pumping lemma, which states that for any CFG there is a pumping length n such that any string in the language of length >= n can be broken into five parts that satisfy certain properties related to pumping. The document gives examples of using the pumping lemma to determine whether languages are context-free or not, such as the language L={0n1n2n | n >= 1}. It concludes by listing additional problems involving using the pumping lemma to prove languages are not context-free.
3.5 equivalence of pushdown automata and cfl
3.5 equivalence of pushdown automata and cflSampath Kumar S The document discusses the equivalence between context-free grammars (CFGs) and pushdown automata (PDAs). It states that for any CFG, an equivalent PDA can be constructed to accept the language generated by the grammar, and vice versa. This allows a programming language to be specified by a CFG and implemented with a PDA in a compiler. The document also provides procedures for converting between CFGs and PDAs, including an example of constructing a PDA from a given CFG.
3.4 deterministic pda
3.4 deterministic pdaSampath Kumar S The document discusses deterministic pushdown automata (DPDA) and provides examples of DPDAs that accept specific languages. It begins by defining DPDA as a PDA that has at most one choice of move in any state, as opposed to a non-deterministic PDA. It then gives an example of a DPDA that accepts the language L={anbn|n>1} and explains its transition function. Finally, it lists several other language recognition problems and invites the reader to design DPDAs to solve them.
3.1,2,3 pushdown automata definition, moves & id
3.1,2,3 pushdown automata definition, moves & idSampath Kumar S A pushdown automaton (PDA) is a type of finite state machine that uses a stack to remember an infinite amount of information. A PDA has three components: an input tape, a control unit, and a stack with infinite size. It can perform two stack operations - push, which adds a new symbol to the top of the stack, and pop, which reads and removes the top symbol. A PDA is formally defined as a 7-tuple that specifies its states, alphabets, transition function, initial state, initial stack symbol, and accepting states. A PDA accepts a language either by entering an accepting state after consuming the input string or by having an empty stack when reading is complete.
2.8 normal forms gnf & problems
2.8 normal forms gnf & problemsSampath Kumar S Greibach Normal Form (GNF) is a type of context-free grammar where the right-hand side of each production rule consists of a single terminal symbol followed by zero or more non-terminal symbols. The document discusses GNF, provides an example grammar in GNF, describes two lemmas used to convert grammars to GNF, and shows the procedure and steps to convert an arbitrary context-free grammar into GNF. It also provides examples of converting several grammars to GNF and solving related problems.
2.7 normal forms cnf & problems
2.7 normal forms cnf & problemsSampath Kumar S The document discusses Chomsky Normal Form (CNF) for context-free grammars. A grammar is in CNF if productions are of two forms: A → BC, with exactly two nonterminals on the right-hand side, or A → a, with a single terminal. The document outlines a procedure to convert any grammar into CNF by introducing new nonterminals to restrict the number and type of symbols on the right-hand side of productions. Several examples of converting grammars to CNF are provided.
2.5 ambiguity in context free grammars
2.5 ambiguity in context free grammarsSampath Kumar S The document discusses ambiguity in context-free grammars. It provides examples of ambiguous grammars, including one where the grammar G with production rules X → X+X | X*X |X|a is shown to be ambiguous because the string "a+a*a" has two different leftmost derivation trees. It then lists four other grammars and briefly states that each one is ambiguous without showing the derivations.
2.4 derivations and languages
2.4 derivations and languagesSampath Kumar S This document discusses derivations and languages in context-free grammars. It covers recursive inference versus derivation to determine if a string belongs to a language. It also discusses leftmost and rightmost derivations, derivation trees, sentential forms, and left and right recursive grammars. Several example problems are provided to demonstrate derivations, parse trees, and generating strings from given grammars.
2.3 context free grammars and languages
2.3 context free grammars and languagesSampath Kumar S The document discusses context-free grammars and languages. It defines a context-free grammar as a quadruple consisting of non-terminal symbols, terminals, production rules, and a start symbol. Examples of context-free grammars are provided. Problems involving constructing context-free grammars for various languages are listed, along with their answers.
2.1 & 2.2 grammar introduction – types of grammar
2.1 & 2.2 grammar introduction – types of grammarSampath Kumar S The document discusses different types of grammars according to Chomsky's classification. It defines a grammar as a 4-tuple consisting of non-terminal symbols, terminal symbols, a start symbol, and production rules. Type-3 grammars generate regular languages using productions with a single non-terminal and terminal. Type-2 grammars are context-free using productions of strings. Type-1 grammars are context-sensitive with restricted productions. Type-0 grammars are unrestricted and generate recursively enumerable languages.
1.10. pumping lemma for regular sets
1.10. pumping lemma for regular setsSampath Kumar S The document discusses the pumping lemma for regular sets. It states that for any regular language L, there exists a constant n such that any string w in L of length at least n can be broken down into sections xyz such that y is not empty, xy is less than or equal to n, and xykz is in L for all k. The pumping lemma can be used to show a language is not regular by finding a string that does not satisfy the lemma conditions. Examples are provided to demonstrate how to use the pumping lemma to prove languages are not regular.
1.9. minimization of dfa
1.9. minimization of dfaSampath Kumar S The document describes the process of minimizing a deterministic finite automaton (DFA) using the Myhill-Nerode theorem. It involves 4 steps: 1) creating a table of all state pairs, 2) marking pairs where one state is final and the other is not, 3) transitively marking additional pairs, 4) combining any remaining unmarked pairs into single states. An example is provided where the given 6-state DFA is minimized to a 3-state DFA using this process. Terminology used in DFA minimization is also defined.
1.8. equivalence of finite automaton and regular expressions
1.8. equivalence of finite automaton and regular expressionsSampath Kumar S This document discusses Thompson's construction algorithm for converting regular expressions to equivalent nondeterministic finite automata (NFAs). It explains the rules for converting the operators in regular expressions, such as empty strings, symbols, union, concatenation, and Kleene star, into NFA transitions and states. Examples of converting regular expressions to NFAs are provided for practice. Thompson's construction allows regular expressions to be matched against strings using the resulting NFA.
1.7. eqivalence of nfa and dfa
1.7. eqivalence of nfa and dfaSampath Kumar S The document discusses the conversion of non-deterministic finite automata (NFA) to deterministic finite automata (DFA). It states that an NFA with epsilon transitions can first be converted to an NFA without epsilon transitions, which can then be converted to an equivalent DFA. The conversion of NFA to DFA involves constructing a DFA with up to 2^n states for an n-state NFA, with the same input alphabet and transition and acceptance functions based on the NFA. Examples are provided of converting epsilon NFAs to NFAs without epsilon transitions and NFAs to equivalent DFAs.
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0.0 Introduction to theory of computation
- 1. Introduction to THEORY OF COMPUTATION -Sampath Kumar S, AP/CSE, SECE
- 2. THEORY OF COMPUTATION In theoretical computer science and mathematics, the theory of computation is the branch that deals with how efficiently problems can be solved on a model of computation, using an algorithm. 21 November 20172
- 3. THEORY OF COMPUTATION Computation: Task performed by ‘calculator’ or ‘computer’ or a ‘machine’. Theory: What is the theory behind about this machine Capabilities of the machine Problems that could be solved by this machine Limitations of this machine 21 November 20173
- 4. Branches of ToC The field is divided into three major branches: 1. Automata theory and language 2. Computability theory 3. Computational complexity theory which are linked by the question: "What are the fundamental capabilities and limitations of computers?“. 21 November 20174
- 5. OBJECTIVES of this course: Understand various Computing models like Finite State Machine, Pushdown Automata, and Turing Machine. Be aware of Decidability and Un-decidability of various problems. Learn types of grammars. 21 November 20175
- 6. UNIT I: FINITE AUTOMATA Introduction- Basic Mathematical Notation and techniques- Finite State systems – Basic Definitions – Finite Automaton – DFA & NDFA – Finite Automaton with €- moves – Regular Languages - Regular Expression – Equivalence of NFA and DFA – Equivalence of NDFA’s with and without €-moves – Equivalence of finite Automaton and regular expressions –Minimization of DFA - Pumping Lemma for Regular sets – Problems based on Pumping Lemma. 21 November 20176
- 7. UNIT II: GRAMMARS Grammar Introduction – Types of Grammar - Context Free Grammars and Languages – Derivations and Languages – Ambiguity- Relationship between derivation and derivation trees – Simplification of CFG – Elimination of Useless symbols - Unit productions - Null productions – Greiback Normal form – Chomsky normal form – Problems related to CNF and GNF. 21 November 20177
- 8. UNIT III: PUSHDOWN AUTOMATA Pushdown Automata - Definitions – Moves – Instantaneous descriptions – Deterministic pushdown automata – Equivalence of Pushdown automata and CFL - pumping lemma for CFL – problems based on pumping Lemma. 21 November 20178
- 9. UNIT IV: TURING MACHINES Definitions of Turing machines – Models – Computable languages and functions – Techniques for Turing machine construction – Multi head and Multi tape Turing Machines - The Halting problem – Partial Solvability – Problems about Turing machine- Chomskian hierarchy of languages. 21 November 20179
- 10. UNIT V: UNSOLVABLE PROBLEMS AND COMPUTABLE FUNCTIONS Unsolvable Problems and Computable Functions – Primitive recursive functions – Recursive and recursively enumerable languages – Universal Turing machine. MEASURING AND CLASSIFYING COMPLEXITY: Tractable and Intractable problems- Tractable and possibly intractable problems – P and NP completeness - Polynomial time reductions. 21 November 201710
- 11. OUTCOMES: Design Finite State Machine, Pushdown Automata, and Turing Machine. Explain the Decidability or Undecidability of various problems 21 November 201711
- 12. Applications of ToC Artificial Intelligence Compiler Design Robotics Circuit Design Natural Language Processing Knowledge based System Quantum Computing Pattern Recognition 21 November 201712