# Turing machines and linear bounded automata

Size: px
Start display at page:

## Transcription

1 and linear bounded automata Informatics 2A: Lecture 29 John Longley School of Informatics University of Edinburgh 27 November / 15

2 The Chomsky hierarchy: summary Level Language type Grammars Accepting machines 3 Regular X ɛ, X Y, NFAs (or DFAs) X ay (regular) 2 Context-free X β NPDAs (context-free) 1 Context-sensitive α β Nondet. linear with α β bounded automata (noncontracting) 0 Recursively α β enumerable (unrestricted) The material in red will be introduced today. 2 / 15

3 The length restriction in noncontracting grammars What s the effect of the restriction α β in noncontracting grammar rules? Idea: in a noncontracting derivation S s of a nonempty string s, all the sentential forms are of length at most s. This means that if L is context-sensitive, and we re trying to decide whether s L, we only need to consider possible sentential forms of length s. So intuitively, we have the problem under control, at least in principle. By contrast, without the length restriction, there s no upper limit on the length of intermediate forms that might appear in a derivation of s. So if we re searching for a derivation for s, how do we know when to stop looking? Intuitively, the problem here is wild and out of control. (This will be made more precise next lecture.) 3 / 15

4 Alan Turing ( ) 4 / 15

5 Recall that NFAs are essentially memoryless, whilst NPDAs are equipped with memory in the form of a stack. To find the right kinds of machines for the top two Chomsky levels, we need to allow more general manipulation of memory. A Turing machine consists of a finite-state control unit, equipped with a memory tape, infinite in both directions. Each cell on the tape contains a symbol drawn from a finite alphabet Γ.... : a c 3 a b 5 \$... read, write, move L/R finite control 5 / 15

6 , continued... : a c 3 a b 5 \$... read, write, move L/R finite control At each step, the behaviour of the machine can depend on the current state of the control unit, the tape symbol at the current read position. Depending on these things, the machine may then overwrite the current tape symbol with a new symbol, shift the tape left or right by one cell, jump to a new control state. This happens repeatedly until (if ever) the control unit enters some identified final state. 6 / 15

7 , formally A Turing machine T consists of: A set Q of control states An initial state i Q A final (accepting) state f Q A tape alphabet Γ An input alphabet Σ Γ A blank symbol Γ Σ A transition function δ : Q Γ Q Γ {L, R}. A nondeterministic Turing machine replaces the transition function δ with a transition relation (Q Γ) (Q Γ {L, R}). (Numerous variant definitions of Turing machine are possible. All lead to notions of TM of equivalent power.) 7 / 15

8 as acceptors To use a Turing machine T as an acceptor for a language over Σ, assume Σ Γ, and set up the tape with the test string s Σ written left-to-right starting at the read position, and with blank symbols everywhere else. Then let the machine run (maybe overwriting s), and if it enters the final state, declare that the original string s is accepted. The language accepted by T (written L(T )) consists of all strings s that are accepted in this way. Theorem: A set L Σ is generated by some unrestricted (Type 0) grammar if and only if L = L(T ) for some Turing machine T. So both Type 0 grammars and lead to the same class of recursively enumerable languages. 8 / 15

9 Questions Q1. Which is the most powerful form of language acceptor (i.e., accepts the widest class of languages)? 1 DFAs 2 NPDAs 3 4 Your laptop 9 / 15

10 Questions Q1. Which is the most powerful form of language acceptor (i.e., accepts the widest class of languages)? Q2. Which is the least powerful form of language acceptor (i.e., accepts the narrowest class of languages)? 1 DFAs 2 NPDAs 3 4 Your laptop 9 / 15

11 Suppose we modify our model to allow just a finite tape, initially containing just the test string s with endmarkers on either side: h e y m a n The machine therefore has just a finite amount of memory, determined by the length of the input string. We call this a linear bounded automaton. (LBAs are sometimes defined as having tape length bounded by a constant multiple of length of input string. In fact, this doesn t make any difference.) Theorem: A language L Σ is context-sensitive if and only if L = L(T ) for some non-deterministic linear bounded automaton T. Rough idea: we can guess at a derivation for s. We can check each step since each sentential form fits within the tape. 10 / 15

12 Example: a non-context-sensitive language Sentences of first-order predicate logic can be regarded as strings over a certain alphabet, with symbols like,, x, =, (, ) etc. Let P be the language consisting of all such sentences that are provable using the rules of FOPL. E.g. the sentence ( x. A(x)) ( x. B(x)) ( x. A(x) B(x)) is in P. But the sentence ( x. A(x)) ( x. B(x)) ( x. A(x) B(x)) is not in P, even though it s syntactically well-formed. Theorem: P is recursively enumerable, but not context-sensitive. Intuition: to show s P, we d in effect have to construct a proof of s. But in general, a proof of s might involve formulae much, much longer than s itself, which wouldn t fit on the LBA tape. Put another way, mathematics itself is wild and out of control! 11 / 15

13 Determinism vs. non-determinism: a curiosity At the bottom level of the Chomsky hierarchy, it makes no difference: every NFA can be simulated by a DFA. At the top level, the same happens. Any nondeterministic Turing machine can be simulated by a deterministic one. At the context-free level, there is a difference: we need NPDAs to account for all context-free languages. (Example: Σ {ss s Σ } is a context-free language whose complement isn t context-free, see last lecture. However, if L is accepted by a DPDA then so is its complement.) What about the context-sensitive level? Are NLBAs strictly more powerful than DLBAs? Asked in 1964, and still open!! (Can t use the context-free argument because CSLs are closed under complementation shown in 1988.) 12 / 15

14 Detecting non-acceptance: LBAs versus TMs Suppose T is an LBA. How might we detect that s is not in L(T )? Clearly, if there s an accepting computation for s, there s one that doesn t pass through exactly the same machine configuration twice (if it did, we could shorten it). Since the tape is finite, the total number of machine configurations is finite (though large). So in theory, if T runs for long enough without reaching the final state, it will enter the same configuration twice, and we may as well abort. Note that on this view, repeated configurations would be spotted not by T itself, but by us watching, or perhaps by some super-machine spying on T. For with unlimited tape space, this reasoning doesn t work. Is there some general way of spotting that a computation isn t going to terminate?? See next lecture / 15

15 Wider significance of are important because (it s generally believed that) any symbolic computation that can be done by any mechanical procedure or algorithm can in principle be done by a Turing machine. This is called the Church-Turing Thesis. E.g.: Any language L Σ that can be recognized by some mechanical procedure can be recognized by a TM. Any mathematical function f : N N that can be computed by a mechanical procedure can be computed by a TM (e.g. representing integers in binary, and requiring the TM to write the result onto the tape.) 14 / 15

16 Status of Church-Turing Thesis The CT Thesis is a somewhat informal statement insofar as the general notion of a mechanical procedure isn t formally defined (although we have a pretty good idea of what we mean by it). Although a certain amount of philosophical hair-splitting is possible, the broad idea behind CTT is generally accepted. At any rate, anything that can be done on any present-day computer (even disregarding time/memory limitations) can in principle be done on a TM. So if we buy into CTT, theorems about what TMs can/can t do can be interpreted as fundamental statements about what can/can t be accomplished by mechanical computation in general. We ll see some examples of such theorems next time. 15 / 15

### Turing machines and linear bounded automata

and linear bounded automata Informatics 2A: Lecture 30 John Longley School of Informatics University of Edinburgh jrl@inf.ed.ac.uk 25 November 2016 1 / 17 The Chomsky hierarchy: summary Level Language

### Turing machines and linear bounded automata

and linear bounded automata Informatics 2A: Lecture 29 John Longley School of Informatics University of Edinburgh jrl@inf.ed.ac.uk 25 November, 2011 1 / 13 1 The Chomsky hierarchy: summary 2 3 4 2 / 13

### The Pumping Lemma. for all n 0, u 1 v n u 2 L (i.e. u 1 u 2 L, u 1 vu 2 L [but we knew that anyway], u 1 vvu 2 L, u 1 vvvu 2 L, etc.

The Pumping Lemma For every regular language L, there is a number l 1 satisfying the pumping lemma property: All w L with w l can be expressed as a concatenation of three strings, w = u 1 vu 2, where u

### FORMAL LANGUAGES, AUTOMATA AND COMPUTATION

FORMAL LANGUAGES, AUTOMATA AND COMPUTATION DECIDABILITY ( LECTURE 15) SLIDES FOR 15-453 SPRING 2011 1 / 34 TURING MACHINES-SYNOPSIS The most general model of computation Computations of a TM are described

### Recap DFA,NFA, DTM. Slides by Prof. Debasis Mitra, FIT.

Recap DFA,NFA, DTM Slides by Prof. Debasis Mitra, FIT. 1 Formal Language Finite set of alphabets Σ: e.g., {0, 1}, {a, b, c}, { {, } } Language L is a subset of strings on Σ, e.g., {00, 110, 01} a finite

### CS21 Decidability and Tractability

CS21 Decidability and Tractability Lecture 8 January 24, 2018 Outline Turing Machines and variants multitape TMs nondeterministic TMs Church-Turing Thesis So far several models of computation finite automata

### TURING MAHINES

15-453 TURING MAHINES TURING MACHINE FINITE STATE q 10 CONTROL AI N P U T INFINITE TAPE read write move 0 0, R, R q accept, R q reject 0 0, R 0 0, R, L read write move 0 0, R, R q accept, R 0 0, R 0 0,

### FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY

15-453 FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY Chomsky Normal Form and TURING MACHINES TUESDAY Feb 4 CHOMSKY NORMAL FORM A context-free grammar is in Chomsky normal form if every rule is of the form:

### CSCE 551: Chin-Tser Huang. University of South Carolina

CSCE 551: Theory of Computation Chin-Tser Huang huangct@cse.sc.edu University of South Carolina Church-Turing Thesis The definition of the algorithm came in the 1936 papers of Alonzo Church h and Alan

### Theory of Computation (IX) Yijia Chen Fudan University

Theory of Computation (IX) Yijia Chen Fudan University Review The Definition of Algorithm Polynomials and their roots A polynomial is a sum of terms, where each term is a product of certain variables and

### Equivalence of TMs and Multitape TMs. Theorem 3.13 and Corollary 3.15 By: Joseph Lauman

Equivalence of TMs and Multitape TMs Theorem 3.13 and Corollary 3.15 By: Joseph Lauman Turing Machines First proposed by Alan Turing in 1936 Similar to finite automaton, but with an unlimited and unrestricted

### Languages, regular languages, finite automata

Notes on Computer Theory Last updated: January, 2018 Languages, regular languages, finite automata Content largely taken from Richards [1] and Sipser [2] 1 Languages An alphabet is a finite set of characters,

### Turing Machines A Turing Machine is a 7-tuple, (Q, Σ, Γ, δ, q0, qaccept, qreject), where Q, Σ, Γ are all finite

The Church-Turing Thesis CS60001: Foundations of Computing Science Professor, Dept. of Computer Sc. & Engg., Turing Machines A Turing Machine is a 7-tuple, (Q, Σ, Γ, δ, q 0, q accept, q reject ), where

### CS 301. Lecture 18 Decidable languages. Stephen Checkoway. April 2, 2018

CS 301 Lecture 18 Decidable languages Stephen Checkoway April 2, 2018 1 / 26 Decidable language Recall, a language A is decidable if there is some TM M that 1 recognizes A (i.e., L(M) = A), and 2 halts

### Introduction to Turing Machines. Reading: Chapters 8 & 9

Introduction to Turing Machines Reading: Chapters 8 & 9 1 Turing Machines (TM) Generalize the class of CFLs: Recursively Enumerable Languages Recursive Languages Context-Free Languages Regular Languages

### Chomsky Normal Form and TURING MACHINES. TUESDAY Feb 4

Chomsky Normal Form and TURING MACHINES TUESDAY Feb 4 CHOMSKY NORMAL FORM A context-free grammar is in Chomsky normal form if every rule is of the form: A BC A a S ε B and C aren t start variables a is

### Turing Machine Variants

CS311 Computational Structures Turing Machine Variants Lecture 12 Andrew Black Andrew Tolmach 1 The Church-Turing Thesis The problems that can be decided by an algorithm are exactly those that can be decided

Advanced topic: Space complexity CSCI 3130 Formal Languages and Automata Theory Siu On CHAN Chinese University of Hong Kong Fall 2016 1/28 Review: time complexity We have looked at how long it takes to

### Undecidable Problems and Reducibility

University of Georgia Fall 2014 Reducibility We show a problem decidable/undecidable by reducing it to another problem. One type of reduction: mapping reduction. Definition Let A, B be languages over Σ.

### CSE355 SUMMER 2018 LECTURES TURING MACHINES AND (UN)DECIDABILITY

CSE355 SUMMER 2018 LECTURES TURING MACHINES AND (UN)DECIDABILITY RYAN DOUGHERTY If we want to talk about a program running on a real computer, consider the following: when a program reads an instruction,

### Finite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018

Finite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018 Lecture 15 Ana Bove May 17th 2018 Recap: Context-free Languages Chomsky hierarchy: Regular languages are also context-free; Pumping lemma

### CS5371 Theory of Computation. Lecture 10: Computability Theory I (Turing Machine)

CS537 Theory of Computation Lecture : Computability Theory I (Turing Machine) Objectives Introduce the Turing Machine (TM) Proposed by Alan Turing in 936 finite-state control + infinitely long tape A stronger

### Midterm Exam 2 CS 341: Foundations of Computer Science II Fall 2018, face-to-face day section Prof. Marvin K. Nakayama

Midterm Exam 2 CS 341: Foundations of Computer Science II Fall 2018, face-to-face day section Prof. Marvin K. Nakayama Print family (or last) name: Print given (or first) name: I have read and understand

### Automata & languages. A primer on the Theory of Computation. Laurent Vanbever. ETH Zürich (D-ITET) October,

Automata & languages A primer on the Theory of Computation Laurent Vanbever www.vanbever.eu ETH Zürich (D-ITET) October, 19 2017 Part 5 out of 5 Last week was all about Context-Free Languages Context-Free

### Undecidability COMS Ashley Montanaro 4 April Department of Computer Science, University of Bristol Bristol, UK

COMS11700 Undecidability Department of Computer Science, University of Bristol Bristol, UK 4 April 2014 COMS11700: Undecidability Slide 1/29 Decidability We are particularly interested in Turing machines

### UNIT-VIII COMPUTABILITY THEORY

CONTEXT SENSITIVE LANGUAGE UNIT-VIII COMPUTABILITY THEORY A Context Sensitive Grammar is a 4-tuple, G = (N, Σ P, S) where: N Set of non terminal symbols Σ Set of terminal symbols S Start symbol of the

### Computability and Complexity

Computability and Complexity Lecture 5 Reductions Undecidable problems from language theory Linear bounded automata given by Jiri Srba Lecture 5 Computability and Complexity 1/14 Reduction Informal Definition

### Homework 8. a b b a b a b. two-way, read/write

Homework 8 309 Homework 8 1. Describe a TM that accepts the set {a n n is a power of 2}. Your description should be at the level of the descriptions in Lecture 29 of the TM that accepts {ww w Σ } and the

### cse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska

cse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska LECTURE 13 CHAPTER 4 TURING MACHINES 1. The definition of Turing machine 2. Computing with Turing machines 3. Extensions of Turing

### Midterm Exam 2 CS 341: Foundations of Computer Science II Fall 2016, face-to-face day section Prof. Marvin K. Nakayama

Midterm Exam 2 CS 341: Foundations of Computer Science II Fall 2016, face-to-face day section Prof. Marvin K. Nakayama Print family (or last) name: Print given (or first) name: I have read and understand

### The Turing machine model of computation

The Turing machine model of computation For most of the remainder of the course we will study the Turing machine model of computation, named after Alan Turing (1912 1954) who proposed the model in 1936.

### Decidability (What, stuff is unsolvable?)

University of Georgia Fall 2014 Outline Decidability Decidable Problems for Regular Languages Decidable Problems for Context Free Languages The Halting Problem Countable and Uncountable Sets Diagonalization

### CS500 Homework #2 Solutions

CS500 Homework #2 Solutions 1. Consider the two languages Show that L 1 is context-free but L 2 is not. L 1 = {a i b j c k d l i = j k = l} L 2 = {a i b j c k d l i = k j = l} Answer. L 1 is the concatenation

### Rumination on the Formal Definition of DPDA

Rumination on the Formal Definition of DPDA In the definition of DPDA, there are some parts that do not agree with our intuition. Let M = (Q, Σ, Γ, δ, q 0, Z 0, F ) be a DPDA. According to the definition,

### Turing Machines (TM) The Turing machine is the ultimate model of computation.

TURING MACHINES Turing Machines (TM) The Turing machine is the ultimate model of computation. Alan Turing (92 954), British mathematician/engineer and one of the most influential scientists of the last

### Turing Machines (TM) Deterministic Turing Machine (DTM) Nondeterministic Turing Machine (NDTM)

Turing Machines (TM) Deterministic Turing Machine (DTM) Nondeterministic Turing Machine (NDTM) 1 Deterministic Turing Machine (DTM).. B B 0 1 1 0 0 B B.. Finite Control Two-way, infinite tape, broken into

### Decidability. Human-aware Robotics. 2017/10/31 Chapter 4.1 in Sipser Ø Announcement:

Decidability 2017/10/31 Chapter 4.1 in Sipser Ø Announcement: q q q Slides for this lecture are here: http://www.public.asu.edu/~yzhan442/teaching/cse355/lectures/decidability.pdf Happy Hollaween! Delayed

### Chapter 8. Turing Machine (TMs)

Chapter 8 Turing Machine (TMs) Turing Machines (TMs) Accepts the languages that can be generated by unrestricted (phrase-structured) grammars No computational machine (i.e., computational language recognition

### Context Sensitive Grammar

Context Sensitive Grammar Aparna S Vijayan Department of Computer Science and Automation December 2, 2011 Aparna S Vijayan (CSA) CSG December 2, 2011 1 / 12 Contents Aparna S Vijayan (CSA) CSG December

### CS5371 Theory of Computation. Lecture 10: Computability Theory I (Turing Machine)

CS537 Theory of Computation Lecture : Computability Theory I (Turing Machine) Objectives Introduce the Turing Machine (TM)? Proposed by Alan Turing in 936 finite-state control + infinitely long tape A

### What languages are Turing-decidable? What languages are not Turing-decidable? Is there a language that isn t even Turingrecognizable?

} We ll now take a look at Turing Machines at a high level and consider what types of problems can be solved algorithmically and what types can t: What languages are Turing-decidable? What languages are

### Variants of Turing Machine (intro)

CHAPTER 3 The Church-Turing Thesis Contents Turing Machines definitions, examples, Turing-recognizable and Turing-decidable languages Variants of Turing Machine Multi-tape Turing machines, non-deterministic

### COMP/MATH 300 Topics for Spring 2017 June 5, Review and Regular Languages

COMP/MATH 300 Topics for Spring 2017 June 5, 2017 Review and Regular Languages Exam I I. Introductory and review information from Chapter 0 II. Problems and Languages A. Computable problems can be expressed

### V Honors Theory of Computation

V22.0453-001 Honors Theory of Computation Problem Set 3 Solutions Problem 1 Solution: The class of languages recognized by these machines is the exactly the class of regular languages, thus this TM variant

### Computability Theory. CS215, Lecture 6,

Computability Theory CS215, Lecture 6, 2000 1 The Birth of Turing Machines At the end of the 19th century, Gottlob Frege conjectured that mathematics could be built from fundamental logic In 1900 David

### Turing Machines. Wolfgang Schreiner

Turing Machines Wolfgang Schreiner Wolfgang.Schreiner@risc.jku.at Research Institute for Symbolic Computation (RISC) Johannes Kepler University, Linz, Austria http://www.risc.jku.at Wolfgang Schreiner

### Turing Machines Part III

Turing Machines Part III Announcements Problem Set 6 due now. Problem Set 7 out, due Monday, March 4. Play around with Turing machines, their powers, and their limits. Some problems require Wednesday's

### Introduction to Turing Machines

Introduction to Turing Machines Deepak D Souza Department of Computer Science and Automation Indian Institute of Science, Bangalore. 12 November 2015 Outline 1 Turing Machines 2 Formal definitions 3 Computability

### UNRESTRICTED GRAMMARS

136 UNRESTRICTED GRAMMARS Context-free grammar allows to substitute only variables with strings In an unrestricted grammar (or a rewriting system) one may substitute any non-empty string (containing variables

### CS 154, Lecture 2: Finite Automata, Closure Properties Nondeterminism,

CS 54, Lecture 2: Finite Automata, Closure Properties Nondeterminism, Why so Many Models? Streaming Algorithms 0 42 Deterministic Finite Automata Anatomy of Deterministic Finite Automata transition: for

### CS Lecture 29 P, NP, and NP-Completeness. k ) for all k. Fall The class P. The class NP

CS 301 - Lecture 29 P, NP, and NP-Completeness Fall 2008 Review Languages and Grammars Alphabets, strings, languages Regular Languages Deterministic Finite and Nondeterministic Automata Equivalence of

### Space Complexity. The space complexity of a program is how much memory it uses.

Space Complexity The space complexity of a program is how much memory it uses. Measuring Space When we compute the space used by a TM, we do not count the input (think of input as readonly). We say that

### Accept or reject. Stack

Pushdown Automata CS351 Just as a DFA was equivalent to a regular expression, we have a similar analogy for the context-free grammar. A pushdown automata (PDA) is equivalent in power to contextfree grammars.

### DM17. Beregnelighed. Jacob Aae Mikkelsen

DM17 Beregnelighed Jacob Aae Mikkelsen January 12, 2007 CONTENTS Contents 1 Introduction 2 1.1 Operations with languages...................... 2 2 Finite Automata 3 2.1 Regular expressions/languages....................

### The Church-Turing Thesis

The Church-Turing Thesis Huan Long Shanghai Jiao Tong University Acknowledgements Part of the slides comes from a similar course in Fudan University given by Prof. Yijia Chen. http://basics.sjtu.edu.cn/

### Foundations of

91.304 Foundations of (Theoretical) Computer Science Chapter 3 Lecture Notes (Section 3.2: Variants of Turing Machines) David Martin dm@cs.uml.edu With some modifications by Prof. Karen Daniels, Fall 2012

### Lecture notes on Turing machines

Lecture notes on Turing machines Ivano Ciardelli 1 Introduction Turing machines, introduced by Alan Turing in 1936, are one of the earliest and perhaps the best known model of computation. The importance

### 1 Showing Recognizability

CSCC63 Worksheet Recognizability and Decidability 1 1 Showing Recognizability 1.1 An Example - take 1 Let Σ be an alphabet. L = { M M is a T M and L(M) }, i.e., that M accepts some string from Σ. Prove

### CS4026 Formal Models of Computation

CS4026 Formal Models of Computation Turing Machines Turing Machines Abstract but accurate model of computers Proposed by Alan Turing in 1936 There weren t computers back then! Turing s motivation: find

### CSCE 551 Final Exam, Spring 2004 Answer Key

CSCE 551 Final Exam, Spring 2004 Answer Key 1. (10 points) Using any method you like (including intuition), give the unique minimal DFA equivalent to the following NFA: 0 1 2 0 5 1 3 4 If your answer is

### Harvard CS 121 and CSCI E-121 Lecture 14: Turing Machines and the Church Turing Thesis

Harvard CS 121 and CSCI E-121 Lecture 14: Turing Machines and the Church Turing Thesis Harry Lewis October 22, 2013 Reading: Sipser, 3.2, 3.3. The Basic Turing Machine The Basic Turing Machine a a b a

### Most General computer?

Turing Machines Most General computer? DFAs are simple model of computation. Accept only the regular languages. Is there a kind of computer that can accept any language, or compute any function? Recall

### FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY

15-453 FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY REVIEW for MIDTERM 1 THURSDAY Feb 6 Midterm 1 will cover everything we have seen so far The PROBLEMS will be from Sipser, Chapters 1, 2, 3 It will be

### CSE 105 THEORY OF COMPUTATION

CSE 105 THEORY OF COMPUTATION Spring 2016 http://cseweb.ucsd.edu/classes/sp16/cse105-ab/ Today's learning goals Sipser Ch 2 Design a PDA and a CFG for a given language Give informal description for a PDA,

### Final exam study sheet for CS3719 Turing machines and decidability.

Final exam study sheet for CS3719 Turing machines and decidability. A Turing machine is a finite automaton with an infinite memory (tape). Formally, a Turing machine is a 6-tuple M = (Q, Σ, Γ, δ, q 0,

### Decidability (intro.)

CHAPTER 4 Decidability Contents Decidable Languages decidable problems concerning regular languages decidable problems concerning context-free languages The Halting Problem The diagonalization method The

### ACS2: Decidability Decidability

Decidability Bernhard Nebel and Christian Becker-Asano 1 Overview An investigation into the solvable/decidable Decidable languages The halting problem (undecidable) 2 Decidable problems? Acceptance problem

### Introduction to Languages and Computation

Introduction to Languages and Computation George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 400 George Voutsadakis (LSSU) Languages and Computation July 2014

### Further discussion of Turing machines

Further discussion of Turing machines In this lecture we will discuss various aspects of decidable and Turing-recognizable languages that were not mentioned in previous lectures. In particular, we will

### SYLLABUS. Introduction to Finite Automata, Central Concepts of Automata Theory. CHAPTER - 3 : REGULAR EXPRESSIONS AND LANGUAGES

Contents i SYLLABUS UNIT - I CHAPTER - 1 : AUT UTOMA OMATA Introduction to Finite Automata, Central Concepts of Automata Theory. CHAPTER - 2 : FINITE AUT UTOMA OMATA An Informal Picture of Finite Automata,

### Turing machines COMS Ashley Montanaro 21 March Department of Computer Science, University of Bristol Bristol, UK

COMS11700 Turing machines Department of Computer Science, University of Bristol Bristol, UK 21 March 2014 COMS11700: Turing machines Slide 1/15 Introduction We have seen two models of computation: finite

### Turing Machines Part II

Turing Machines Part II Hello Hello Condensed Slide Slide Readers! Readers! This This lecture lecture is is almost almost entirely entirely animations that that show show how how each each Turing Turing

### T (s, xa) = T (T (s, x), a). The language recognized by M, denoted L(M), is the set of strings accepted by M. That is,

Recall A deterministic finite automaton is a five-tuple where S is a finite set of states, M = (S, Σ, T, s 0, F ) Σ is an alphabet the input alphabet, T : S Σ S is the transition function, s 0 S is the

### MA/CSSE 474 Theory of Computation

MA/CSSE 474 Theory of Computation CFL Hierarchy CFL Decision Problems Your Questions? Previous class days' material Reading Assignments HW 12 or 13 problems Anything else I have included some slides online

### ECS 120 Lesson 15 Turing Machines, Pt. 1

ECS 120 Lesson 15 Turing Machines, Pt. 1 Oliver Kreylos Wednesday, May 2nd, 2001 Before we can start investigating the really interesting problems in theoretical computer science, we have to introduce

### Automata Theory (2A) Young Won Lim 5/31/18

Automata Theory (2A) Copyright (c) 2018 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later

### CS Lecture 28 P, NP, and NP-Completeness. Fall 2008

CS 301 - Lecture 28 P, NP, and NP-Completeness Fall 2008 Review Languages and Grammars Alphabets, strings, languages Regular Languages Deterministic Finite and Nondeterministic Automata Equivalence of

### (a) Definition of TMs. First Problem of URMs

Sec. 4: Turing Machines First Problem of URMs (a) Definition of the Turing Machine. (b) URM computable functions are Turing computable. (c) Undecidability of the Turing Halting Problem That incrementing

### The Pumping Lemma: limitations of regular languages

The Pumping Lemma: limitations of regular languages Informatics 2A: Lecture 8 John Longley School of Informatics University of Edinburgh jrl@inf.ed.ac.uk 6 October, 2016 1 / 16 Recap of Lecture 7 Showing

### CSE 105 Theory of Computation

CSE 105 Theory of Computation http://www.jflap.org/jflaptmp/ Professor Jeanne Ferrante 1 Today s Agenda Quick Review of CFG s and PDA s Introduction to Turing Machines and their Languages Reminders and

### Chapter 3: The Church-Turing Thesis

Chapter 3: The Church-Turing Thesis 1 Turing Machine (TM) Control... Bi-direction Read/Write Turing machine is a much more powerful model, proposed by Alan Turing in 1936. 2 Church/Turing Thesis Anything

### CMSC 330: Organization of Programming Languages. Pushdown Automata Parsing

CMSC 330: Organization of Programming Languages Pushdown Automata Parsing Chomsky Hierarchy Categorization of various languages and grammars Each is strictly more restrictive than the previous First described

### Theory of Computation

Theory of Computation Lecture #10 Sarmad Abbasi Virtual University Sarmad Abbasi (Virtual University) Theory of Computation 1 / 43 Lecture 10: Overview Linear Bounded Automata Acceptance Problem for LBAs

### NODIA AND COMPANY. GATE SOLVED PAPER Computer Science Engineering Theory of Computation. Copyright By NODIA & COMPANY

No part of this publication may be reproduced or distributed in any form or any means, electronic, mechanical, photocopying, or otherwise without the prior permission of the author. GATE SOLVED PAPER Computer

### Closure under the Regular Operations

September 7, 2013 Application of NFA Now we use the NFA to show that collection of regular languages is closed under regular operations union, concatenation, and star Earlier we have shown this closure

### More Turing Machines. CS154 Chris Pollett Mar 15, 2006.

More Turing Machines CS154 Chris Pollett Mar 15, 2006. Outline Multitape Turing Machines Nondeterministic Turing Machines Enumerators Introduction There have been many different proposals for what it means

### Finite Automata. Mahesh Viswanathan

Finite Automata Mahesh Viswanathan In this lecture, we will consider different models of finite state machines and study their relative power. These notes assume that the reader is familiar with DFAs,

### Theory Bridge Exam Example Questions

Theory Bridge Exam Example Questions Annotated version with some (sometimes rather sketchy) answers and notes. This is a collection of sample theory bridge exam questions. This is just to get some idea

### Finite Automata. Finite Automata

Finite Automata Finite Automata Formal Specification of Languages Generators Grammars Context-free Regular Regular Expressions Recognizers Parsers, Push-down Automata Context Free Grammar Finite State

Introduction: Computer Science is a cluster of related scientific and engineering disciplines concerned with the study and application of computations. These disciplines range from the pure and basic scientific

### 6.045: Automata, Computability, and Complexity Or, Great Ideas in Theoretical Computer Science Spring, Class 8 Nancy Lynch

6.045: Automata, Computability, and Complexity Or, Great Ideas in Theoretical Computer Science Spring, 2010 Class 8 Nancy Lynch Today More undecidable problems: About Turing machines: Emptiness, etc. About

### CPSC 421: Tutorial #1

CPSC 421: Tutorial #1 October 14, 2016 Set Theory. 1. Let A be an arbitrary set, and let B = {x A : x / x}. That is, B contains all sets in A that do not contain themselves: For all y, ( ) y B if and only

### Turing machine. Turing Machine Model. Turing Machines. 1. Turing Machines. Wolfgang Schreiner 2. Recognizing Languages

Turing achines Wolfgang Schreiner Wolfgang.Schreiner@risc.jku.at Research Institute for Symbolic Computation (RISC) Johannes Kepler University, Linz, Austria http://www.risc.jku.at 1. Turing achines Wolfgang

### Theory of computation: initial remarks (Chapter 11)

Theory of computation: initial remarks (Chapter 11) For many purposes, computation is elegantly modeled with simple mathematical objects: Turing machines, finite automata, pushdown automata, and such.

### Turing Machines Part II

Turing Machines Part II Problem Set Set Five Five due due in in the the box box up up front front using using a late late day. day. Hello Hello Condensed Slide Slide Readers! Readers! This This lecture

### COMPARATIVE ANALYSIS ON TURING MACHINE AND QUANTUM TURING MACHINE

Volume 3, No. 5, May 2012 Journal of Global Research in Computer Science REVIEW ARTICLE Available Online at www.jgrcs.info COMPARATIVE ANALYSIS ON TURING MACHINE AND QUANTUM TURING MACHINE Tirtharaj Dash

### The Unsolvability of the Halting Problem. Chapter 19

The Unsolvability of the Halting Problem Chapter 19 Languages and Machines SD D Context-Free Languages Regular Languages reg exps FSMs cfgs PDAs unrestricted grammars Turing Machines D and SD A TM M with