Universal Turing Machine Chomsky Hierarchy Decidability Reducibility Uncomputable Functions Rice s Theorem Decidability Continued
|
|
- Godwin Watkins
- 5 years ago
- Views:
Transcription
1 CD5080 AUBER odels of Computation, anguages and Automata ecture 14 älardalen University Content Universal Turing achine Chomsky Hierarchy Decidability Reducibility Uncomputable Functions Rice s Decidability Continued Universal Turing achine A limitation of Turing achines: Turing achines are hardired they execute only one program Real Computers are re-programmable Solution: Universal Turing achine Characteristics: Reprogrammable machine Simulates any other Turing achine Universal Turing achine simulates any other Turing achine Input of Universal Turing achine Description of transitions of Initial tape contents of Tape 1 Three tapes Universal Turing achine Description of Tape 2 Tape Contents of Tape 1 Description of We describe Turing machine as a string of symbols: Alphabet Encoding Symbols: a b c d l Tape 3 State of We encode as a string of symbols Encoding:
2 State Encoding States: l Transition Encoding achine Encoding Encoding: ove: Head ove Encoding R Transition: δ ( 1, a) = ( 2, b, ) Encoding: separator Transitions: δ ( 1, a) = ( 2, b, Encoding: ) δ ( 2, b) = ( 3, c, R) Encoding: separator 12 Tape 1 contents of Universal Turing achine: encoding of the simulated machine as a binary string of 0 s and 1 s 13 A Turing achine is described ith a binary string of 0 s and 1 s. Therefore: The set of Turing machines forms a language: Each string of the language is the binary encoding of a Turing achine. 14 anguage of Turing achines = { , (Turing achine 1) , (Turing achine 2) , } 15 The Chomsky Hierarchy Unrestricted Grammars String of variables and terminals Productions u v String of variables and terminals Example of unrestricted grammar S abc ab ca Ac d
3 Context-Sensitive Grammars The language n n n { a b c } A language is recursively enumerable if and only if it is generated by an unrestricted grammar. String of variables and terminals Productions u v and u v String of variables and terminals is context-sensitive: S abc aabc Ab ba Ac Bbcc bb Bb ab aa aaa inear Bounded Automaton (BA) A language is context sensitive if and only if it is accepted by a inear-bounded automaton. inear Bounded Automata (BAs) are the same as Turing achines ith one difference: The input string tape space is the only tape space alloed to use. eft-end marker Input string [ a b c d e ] Working space in tape Right-end marker All computation is done beteen end markers. 24 Observation There is a language hich is context-sensitive but not recursive. Non-recursively enumerable Recursively-enumerable Recursive Context-sensitive Context-free Regular Decidability 25 The Chomsky anguage Hierarchy 26 27
4 Consider problems ith anser or. Examples Does achine have three states? Is string a binary number? Does DFA accept any input? A problem is decidable if some Turing machine solves (decides) the problem. Decidable problems: Does achine have three states? Is string a binary number? Does DFA accept any input? The Turing machine that solves a problem ansers or for each instance. Input problem instance Turing achine The machine that decides a problem: If the anser is then halts in a yes state If the anser is then halts in a no state These states may not be final states. Turing achine that decides a problem Difference beteen Recursive anguages and Decidable problems For decidable problems: The states may not be final states. 31 and states are halting states Some problems are undecidable: There is no Turing achine that solves all instances of the problem. A simple undecidable problem: The membership problem The embership Problem Input: Turing achine String Question: Does accept?
5 The membership problem is undecidable. Assume for contradiction that the membership problem is decidable. There exists a Turing achine H that solves the membership problem H accepts rejects et be a recursively enumerable language. et be the Turing achine that accepts. We ill prove that is also recursive: We ill describe a Turing machine that accepts and halts on any input Turing achine that accepts and halts on any input H accepts? accept reject Therefore, is recursive. Since is chosen arbitrarily, e have proven that every recursively enumerable language is also recursive. But there are recursively enumerable languages hich are not recursive. Therefore, the membership problem is undecidable. Contradiction! A famous undecidable problem: The halting problem The Halting Problem Input: Turing achine String Question: Does halt on? The halting problem is undecidable. Assume for contradiction that the halting problem is decidable
6 There exists Turing achine H that solves the halting problem Construction of H H halts on doesn t halt on Input: initial tape contents 0 Encoding String of H y n Construct machine H If H returns then loop forever. If H returns then halt H H y n oop forever a b Construct machine Input: Ĥ (machine ) If halts on input Then loop forever Else halt copy Ĥ H Run machine Ĥ ith input itself Input: If Ĥ H ˆ (machine Ĥ ) halts on input Then loop forever Else halt H ˆ Ĥ on input Ĥ H ˆ If halts then loops forever. Ĥ : If doesn t halt then it halts. CONTRADICTION! This means that The halting problem is undecidable
7 Another proof of the same theorem If the halting problem as decidable then every recursively enumerable language ould be recursive. The halting problem is undecidable. Assume for contradiction that the halting problem is decidable. There exists Turing achine H that solves the halting problem. H halts on doesn t halt on et be a recursively enumerable language. et be the Turing achine that accepts. We ill prove that is also recursive: We ill describe a Turing machine that accepts and halts on any input. 58 Turing achine that accepts and halts on any input H halts on? Run ith input reject accept Halts on final state reject Halts on non-final Therefore is recursive. Since is chosen arbitrarily, e have proven that every recursively enumerable language is also recursive. But there are recursively enumerable languages hich are not recursive. Contradiction! state Problem A is reduced to problem B Therefore, the halting problem is undecidable. Reducibility B If e can solve problem then e can solve problem A. B A
8 Problem A is reduced to problem If B is decidable then A is decidable. If A is undecidable then B is undecidable. B Example the halting problem reduced to the state-entry problem. The state-entry problem Inputs: Question: Turing achine State String Does enter state on input? The state-entry problem is undecidable. Reduce the halting problem to the state-entry problem. Suppose e have an algorithm (Turing achine) that solves the state-entry problem. We ill construct an algorithm that solves the halting problem. Assume e have the state-entry algorithm: state-entry problem enters doesn t enter We ant to design the halting algorithm: Halting problem halts on doesn t halt on odify input machine Add ne state From any halting state add transitions to halts if and only if halts on state halting states Single halt state
9 halting problem Inputs: machine and string Halting problem algorithm We reduced the halting problem to the state-entry problem. 1. Construct machine ith state 2. Run algorithm for state-entry problem ith inputs:,, Generate State-entry algorithm Since the halting problem is undecidable, it must be that the state-entry problem is also undecidable Another example The halting problem reduced to the blank-tape halting problem. The blank-tape halting problem Input: Turing achine Question: Does halt hen started ith a blank tape? The blank-tape halting problem is undecidable. Reduce the halting problem to the blank-tape halting problem Assume e have the blank-tape halting algorithm We ant to design the halting algorithm: Suppose e have an algorithm for the blank-tape halting problem. We ill construct an algorithm for the halting problem. blank-tape halting problem halts on blanks tape doesn t halt on blank tape halting problem halts on doesn t halt on
10 Construct a ne machine On blank tape rites Then continues execution like halts on input string halting problem Inputs: machine and string step 1 if blank tape then rite step2 execute ith input if and only if halts hen started ith blank tape. 1. Construct 2. Run algorithm for blank-tape halting problem ith input Halting problem algorithm Summary of Undecidable Problems Generate Blank-tape halting algorithm We reduced the halting problem to the blank-tape halting problem. Since the halting problem is undecidable, the blank-tape halting problem is also undecidable. Halting Problem Does machine halt on input? embership problem Does machine accept string? (In other ords: Is a string member of a recursively enumerable language?) Blank-tape halting problem Does machine halt hen starting on blank tape? State-entry Problem: Does machine enter state on input? Uncomputable Functions Uncomputable Functions Domain f Range A function is uncomputable if it cannot be computed for all of its domain
11 f (n) = Example An uncomputable function: maximum number of moves until any Turing machine ith n states halts hen started ith the blank tape. 91 Function f (n) is uncomputable. Assume for contradiction that f (n) is computable. Then the blank-tape halting problem is decidable. 92 blank-tape halting problem Input: machine 1. Count states of : 2. Compute f (m) m f (m) 3. Simulate for steps starting ith empty tape If halts then return otherise return 93 Therefore, the blank-tape halting problem must be decidable. Hoever, e kno that the blank-tape halting problem is undecidable. Therefore, function f (n) is uncomputable. Rice s Contradiction! Definition Non-trivial properties of recursively enumerable languages: any property possessed by some (not all) recursively enumerable languages. Some non-trivial properties of recursively enumerable languages is empty is finite contains to different strings of the same length Rice s Any non-trivial property of a recursively enumerable language is undecidable
12 We ill prove some non-trivial properties ithout using Rice s theorem. For any recursively enumerable language it is undecidable hether it is empty. We ill reduce the membership problem to the problem of deciding hether is empty. embership problem: Does machine accept string? Assume e have the empty language algorithm: et be the machine that accepts ( ) = empty language problem ( ) ( ) empty not empty We ill design the membership algorithm: membership problem accepts rejects First construct machine : When enters a final state, compare original input string ith. Accept if original input is the same as membership problem Inputs: machine and string embership algorithm ( ) if and only if is not empty ( ) = { } 1. Construct ( ) 2. Determine if is empty : then ( ) : then ( ) construct Check if ( ) is empty
13 We reduced the empty language problem to the membership problem. Since the membership problem is undecidable, the empty language problem is also undecidable. Decidability continued For a recursively enumerable language it is undecidable to determine hether is finite. We ill reduce the halting problem to the finite language problem Assume e have the finite language algorithm: et be the machine that accepts finite language problem ( ) = ( ) ( ) finite not finite We ill design the halting problem algorithm: Halting problem halts on doesn t halt on First construct machine. Initially, simulates on input. When enters a halt state, accept any input (infinite language). Otherise accept nothing (finite language) halts on halting problem: Inputs: machine and string achine for halting problem ( ) if and only if is not finite. 1. Construct ( ) 2. Determine if is finite : : then doesn t halt on then halts on construct Check if ( ) is finite
14 We reduced the finite language problem to the halting problem. Since the halting problem is undecidable, the finite language problem is also undecidable. 118 For a recursively enumerable language it is undecidable hether contains to different strings of same length. We ill reduce the halting problem to the to strings of eual length- problem. 119 et be the machine that accepts ( ) = Assume e have the to-strings algorithm: to-strings problem ( ) contains doesn t ( ) contain to eual length strings 120 We ill design the halting problem algorithm: Halting problem halts on doesn t halt on First construct machine. Initially, simulates on input. When enters a halt state, accept symbols or. a b (to eual length strings) halts on if and only if accepts a and b (to eual length strings) halting problem Inputs: machine and string 1. Construct 2. Determine if accepts to strings of eual length : : then halts on then doesn t halt on achine for halting problem construct Check if ( ) has to eual length strings We reduced the to strings of eual length - problem to the halting problem. Since the halting problem is undecidable, the to strings of eual length problem is also undecidable
TAFL 1 (ECS-403) Unit- V. 5.1 Turing Machine. 5.2 TM as computer of Integer Function
TAFL 1 (ECS-403) Unit- V 5.1 Turing Machine 5.2 TM as computer of Integer Function 5.2.1 Simulating Turing Machine by Computer 5.2.2 Simulating Computer by Turing Machine 5.3 Universal Turing Machine 5.4
More informationDecidable Problems. We examine the problems for which there is an algorithm.
Decidable Problems We examine the problems for which there is an algorithm. Decidable Problems A problem asks a yes/no question about some input. The problem is decidable if there is a program that always
More informationR10 SET a) Construct a DFA that accepts an identifier of a C programming language. b) Differentiate between NFA and DFA?
R1 SET - 1 1. a) Construct a DFA that accepts an identifier of a C programming language. b) Differentiate between NFA and DFA? 2. a) Design a DFA that accepts the language over = {, 1} of all strings that
More informationRegular Languages (14 points) Solution: Problem 1 (6 points) Minimize the following automaton M. Show that the resulting DFA is minimal.
Regular Languages (14 points) Problem 1 (6 points) inimize the following automaton Show that the resulting DFA is minimal. Solution: We apply the State Reduction by Set Partitioning algorithm (särskiljandealgoritmen)
More informationLimitations of Algorithmic Solvability In this Chapter we investigate the power of algorithms to solve problems Some can be solved algorithmically and
Computer Language Theory Chapter 4: Decidability 1 Limitations of Algorithmic Solvability In this Chapter we investigate the power of algorithms to solve problems Some can be solved algorithmically and
More informationTheory of Computation, Homework 3 Sample Solution
Theory of Computation, Homework 3 Sample Solution 3.8 b.) The following machine M will do: M = "On input string : 1. Scan the tape and mark the first 1 which has not been marked. If no unmarked 1 is found,
More informationSource of Slides: Introduction to Automata Theory, Languages, and Computation By John E. Hopcroft, Rajeev Motwani and Jeffrey D.
Source of Slides: Introduction to Automata Theory, Languages, and Computation By John E. Hopcroft, Rajeev Motwani and Jeffrey D. Ullman And Introduction to Languages and The by J. C. Martin Basic Mathematical
More informationTuring Machine Languages
Turing Machine Languages Based on Chapters 23-24-25 of (Cohen 1997) Introduction A language L over alphabet is called recursively enumerable (r.e.) if there is a Turing Machine T that accepts every word
More informationTheory of Programming Languages COMP360
Theory of Programming Languages COMP360 Sometimes it is the people no one imagines anything of, who do the things that no one can imagine Alan Turing What can be computed? Before people even built computers,
More informationComputer Sciences Department
1 Reference Book: INTRODUCTION TO THE THEORY OF COMPUTATION, SECOND EDITION, by: MICHAEL SIPSER 3 D E C I D A B I L I T Y 4 Objectives 5 Objectives investigate the power of algorithms to solve problems.
More informationRice s Theorem and Enumeration
Rice s Theorem and Enumeration 11/6/17 (Using slides adapted from the book) Administrivia HW on reductions due Wed at beginning of class For Wednesday, read Sections 20.1-20.3 and pp. 331-344 (Sections
More informationOutline. Language Hierarchy
Outline Language Hierarchy Definition of Turing Machine TM Variants and Equivalence Decidability Reducibility Language Hierarchy Regular: finite memory CFG/PDA: infinite memory but in stack space TM: infinite
More information6.045J/18.400J: Automata, Computability and Complexity. Practice Quiz 2
6.045J/18.400J: Automata, omputability and omplexity March 21, 2007 Practice Quiz 2 Prof. Nancy Lynch Elena Grigorescu Please write your name in the upper corner of each page. INFORMATION ABOUT QUIZ 2:
More informationSkyup's Media. PART-B 2) Construct a Mealy machine which is equivalent to the Moore machine given in table.
Code No: XXXXX JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY, HYDERABAD B.Tech II Year I Semester Examinations (Common to CSE and IT) Note: This question paper contains two parts A and B. Part A is compulsory
More informationDecision Properties for Context-free Languages
Previously: Decision Properties for Context-free Languages CMPU 240 Language Theory and Computation Fall 2018 Context-free languages Pumping Lemma for CFLs Closure properties for CFLs Today: Assignment
More information(a) R=01[((10)*+111)*+0]*1 (b) ((01+10)*00)*. [8+8] 4. (a) Find the left most and right most derivations for the word abba in the grammar
Code No: R05310501 Set No. 1 III B.Tech I Semester Regular Examinations, November 2008 FORMAL LANGUAGES AND AUTOMATA THEORY (Computer Science & Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE
More informationActually talking about Turing machines this time
Actually talking about Turing machines this time 10/25/17 (Using slides adapted from the book) Administrivia HW due now (Pumping lemma for context-free languages) HW due Friday (Building TMs) Exam 2 out
More informationCS21 Decidability and Tractability
CS21 Decidability and Tractability Lecture 9 January 26, 2018 Outline Turing Machines and variants multitape TMs nondeterministic TMs Church-Turing Thesis decidable, RE, co-re languages Deciding and Recognizing
More informationwhere is the transition function, Pushdown Automata is the set of accept states. is the start state, and is the set of states, is the input alphabet,
!# #! + Pushdown utomata CS5 ecture Pushdown utomata where pushdown automaton is a 6tuple are finite sets is the set of states is the input alphabet is the stack alphabet is the transition function is
More informationJNTUWORLD. Code No: R
Code No: R09220504 R09 SET-1 B.Tech II Year - II Semester Examinations, April-May, 2012 FORMAL LANGUAGES AND AUTOMATA THEORY (Computer Science and Engineering) Time: 3 hours Max. Marks: 75 Answer any five
More informationMaterial from Recitation 1
Material from Recitation 1 Darcey Riley Frank Ferraro January 18, 2011 1 Introduction In CSC 280 we will be formalizing computation, i.e. we will be creating precise mathematical models for describing
More informationThe Turing Machine. Unsolvable Problems. Undecidability. The Church-Turing Thesis (1936) Decision Problem. Decision Problems
The Turing Machine Unsolvable Problems Motivating idea Build a theoretical a human computer Likened to a human with a paper and pencil that can solve problems in an algorithmic way The theoretical machine
More informationTheory of Languages and Automata
Theory of Languages and Automata Chapter 3- The Church-Turing Thesis Sharif University of Technology Turing Machine O Several models of computing devices Finite automata Pushdown automata O Tasks that
More informationTheory Bridge Exam Example Questions Version of June 6, 2008
Theory Bridge Exam Example Questions Version of June 6, 2008 This is a collection of sample theory bridge exam questions. This is just to get some idea of the format of the bridge exam and the level of
More informationLec-5-HW-1, TM basics
Lec-5-HW-1, TM basics (Problem 0)-------------------- Design a Turing Machine (TM), T_sub, that does unary decrement by one. Assume a legal, initial tape consists of a contiguous set of cells, each containing
More informationRecursively Enumerable Languages, Turing Machines, and Decidability
Recursively Enumerable Languages, Turing Machines, and Decidability 1 Problem Reduction: Basic Concepts and Analogies The concept of problem reduction is simple at a high level. You simply take an algorithm
More informationCT32 COMPUTER NETWORKS DEC 2015
Q.2 a. Using the principle of mathematical induction, prove that (10 (2n-1) +1) is divisible by 11 for all n N (8) Let P(n): (10 (2n-1) +1) is divisible by 11 For n = 1, the given expression becomes (10
More informationTheory of Computations Spring 2016 Practice Final Exam Solutions
1 of 8 Theory of Computations Spring 2016 Practice Final Exam Solutions Name: Directions: Answer the questions as well as you can. Partial credit will be given, so show your work where appropriate. Try
More informationQUESTION BANK. Unit 1. Introduction to Finite Automata
QUESTION BANK Unit 1 Introduction to Finite Automata 1. Obtain DFAs to accept strings of a s and b s having exactly one a.(5m )(Jun-Jul 10) 2. Obtain a DFA to accept strings of a s and b s having even
More informationEnumerations and Turing Machines
Enumerations and Turing Machines Mridul Aanjaneya Stanford University August 07, 2012 Mridul Aanjaneya Automata Theory 1/ 35 Finite Sets Intuitively, a finite set is a set for which there is a particular
More informationThe Big Picture. Chapter 3
The Big Picture Chapter 3 Examining Computational Problems We want to examine a given computational problem and see how difficult it is. Then we need to compare problems Problems appear different We want
More informationUniversity of Nevada, Las Vegas Computer Science 456/656 Fall 2016
University of Nevada, Las Vegas Computer Science 456/656 Fall 2016 The entire examination is 925 points. The real final will be much shorter. Name: No books, notes, scratch paper, or calculators. Use pen
More informationA Characterization of the Chomsky Hierarchy by String Turing Machines
A Characterization of the Chomsky Hierarchy by String Turing Machines Hans W. Lang University of Applied Sciences, Flensburg, Germany Abstract A string Turing machine is a variant of a Turing machine designed
More informationDHANALAKSHMI SRINIVASAN INSTITUTE OF RESEARCH AND TECHNOLOGY SIRUVACHUR, PERAMBALUR DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING
DHANALAKSHMI SRINIVASAN INSTITUTE OF RESEARCH AND TECHNOLOGY SIRUVACHUR, PERAMBALUR-621113 DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING Third Year CSE( Sem:V) CS2303- THEORY OF COMPUTATION PART B-16
More informationVariants of Turing Machines
November 4, 2013 Robustness Robustness Robustness of a mathematical object (such as proof, definition, algorithm, method, etc.) is measured by its invariance to certain changes Robustness Robustness of
More informationVALLIAMMAI ENGNIEERING COLLEGE SRM Nagar, Kattankulathur 603203. DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING Year & Semester : III Year, V Semester Section : CSE - 1 & 2 Subject Code : CS6503 Subject
More informationNormal Forms for CFG s. Eliminating Useless Variables Removing Epsilon Removing Unit Productions Chomsky Normal Form
Normal Forms for CFG s Eliminating Useless Variables Removing Epsilon Removing Unit Productions Chomsky Normal Form 1 Variables That Derive Nothing Consider: S -> AB, A -> aa a, B -> AB Although A derives
More informationWe can create PDAs with multiple stacks. At each step we look at the current state, the current input symbol, and the top of each stack.
Other Automata We can create PDAs with multiple stacks. At each step we look at the current state, the current input symbol, and the top of each stack. From all of this information we decide what state
More informationCpSc 421 Final Solutions
CpSc 421 Final Solutions Do any eight of the ten problems below. If you attempt more than eight problems, please indicate which ones to grade (otherwise we will make a random choice). This allows you to
More informationECS 120 Lesson 16 Turing Machines, Pt. 2
ECS 120 Lesson 16 Turing Machines, Pt. 2 Oliver Kreylos Friday, May 4th, 2001 In the last lesson, we looked at Turing Machines, their differences to finite state machines and pushdown automata, and their
More information1. [5 points each] True or False. If the question is currently open, write O or Open.
University of Nevada, Las Vegas Computer Science 456/656 Spring 2018 Practice for the Final on May 9, 2018 The entire examination is 775 points. The real final will be much shorter. Name: No books, notes,
More informationStructure of a Compiler: Scanner reads a source, character by character, extracting lexemes that are then represented by tokens.
CS 441 Fall 2018 Notes Compiler - software that translates a program written in a source file into a program stored in a target file, reporting errors when found. Source Target program written in a High
More informationFormal Grammars and Abstract Machines. Sahar Al Seesi
Formal Grammars and Abstract Machines Sahar Al Seesi What are Formal Languages Describing the sentence structure of a language in a formal way Used in Natural Language Processing Applications (translators,
More informationLanguages and Automata
Languages and Automata What are the Big Ideas? Tuesday, August 30, 2011 Reading: Sipser 0.1 CS235 Languages and Automata Department of Computer Science Wellesley College Why Take CS235? 1. It s required
More informationProblems, Languages, Machines, Computability, Complexity
CS311 Computational Structures Problems, Languages, Machines, Computability, Complexity Lecture 1 Andrew P. Black Andrew Tolmach 1 The Geography Game Could you write a computer program to play the geography
More informationt!1; R 0! 1; R q HALT
CS 181 FINA EXAM SPRING 2014 NAME UCA ID You have 3 hours to complete this exam. You may assume without proof any statement proved in class. (2 pts) 1 Give a simple verbal description of the function f
More informationFormal Languages and Automata
Mobile Computing and Software Engineering p. 1/3 Formal Languages and Automata Chapter 3 Regular languages and Regular Grammars Chuan-Ming Liu cmliu@csie.ntut.edu.tw Department of Computer Science and
More informationFrom Theorem 8.5, page 223, we have that the intersection of a context-free language with a regular language is context-free. Therefore, the language
CSCI 2400 Models of Computation, Section 3 Solutions to Practice Final Exam Here are solutions to the practice final exam. For some problems some details are missing for brevity. You should write complete
More informationEquivalence of NTMs and TMs
Equivalence of NTMs and TMs What is a Turing Machine? Similar to a finite automaton, but with unlimited and unrestricted memory. It uses an infinitely long tape as its memory which can be read from and
More informationUNIT I PART A PART B
OXFORD ENGINEERING COLLEGE (NAAC ACCREDITED WITH B GRADE) DEPARTMENT OF COMPUTER SCIENCE & ENGINEERING LIST OF QUESTIONS YEAR/SEM: III/V STAFF NAME: Dr. Sangeetha Senthilkumar SUB.CODE: CS6503 SUB.NAME:
More informationQUESTION BANK. Formal Languages and Automata Theory(10CS56)
QUESTION BANK Formal Languages and Automata Theory(10CS56) Chapter 1 1. Define the following terms & explain with examples. i) Grammar ii) Language 2. Mention the difference between DFA, NFA and εnfa.
More informationClosure Properties of CFLs; Introducing TMs. CS154 Chris Pollett Apr 9, 2007.
Closure Properties of CFLs; Introducing TMs CS154 Chris Pollett Apr 9, 2007. Outline Closure Properties of Context Free Languages Algorithms for CFLs Introducing Turing Machines Closure Properties of CFL
More informationTheory of Computation
Theory of Computation For Computer Science & Information Technology By www.thegateacademy.com Syllabus Syllabus for Theory of Computation Regular Expressions and Finite Automata, Context-Free Grammar s
More informationONE-STACK AUTOMATA AS ACCEPTORS OF CONTEXT-FREE LANGUAGES *
ONE-STACK AUTOMATA AS ACCEPTORS OF CONTEXT-FREE LANGUAGES * Pradip Peter Dey, Mohammad Amin, Bhaskar Raj Sinha and Alireza Farahani National University 3678 Aero Court San Diego, CA 92123 {pdey, mamin,
More informationKEY. A 1. The action of a grammar when a derivation can be found for a sentence. Y 2. program written in a High Level Language
1 KEY CS 441G Fall 2018 Exam 1 Matching: match the best term from the following list to its definition by writing the LETTER of the term in the blank to the left of the definition. (1 point each) A Accepts
More informationComputability Summary
Computability Summary Recursive Languages The following are all equivalent: A language B is recursive iff B = L(M) for some total TM M. A language B is (Turing) computable iff some total TM M computes
More informationMidterm Exam II CIS 341: Foundations of Computer Science II Spring 2006, day section Prof. Marvin K. Nakayama
Midterm Exam II CIS 341: Foundations of Computer Science II Spring 2006, day section Prof. Marvin K. Nakayama Print family (or last) name: Print given (or first) name: I have read and understand all of
More informationDoes this program exist? Limits of computability. Does this program exist? A helper function. For example, if the string program is
Does this program exist? For example, if the string program is Limits of computability public void abc(string s) { if( s == "abc" ) { print "Hello world!"; } else { print "Whatever"; }} then test called
More informationAutomata Theory TEST 1 Answers Max points: 156 Grade basis: 150 Median grade: 81%
Automata Theory TEST 1 Answers Max points: 156 Grade basis: 150 Median grade: 81% 1. (2 pts) See text. You can t be sloppy defining terms like this. You must show a bijection between the natural numbers
More information1. Draw the state graphs for the finite automata which accept sets of strings composed of zeros and ones which:
P R O B L E M S Finite Autom ata. Draw the state graphs for the finite automata which accept sets of strings composed of zeros and ones which: a) Are a multiple of three in length. b) End with the string
More informationTheory of Computations Spring 2016 Practice Final
1 of 6 Theory of Computations Spring 2016 Practice Final 1. True/False questions: For each part, circle either True or False. (23 points: 1 points each) a. A TM can compute anything a desktop PC can, although
More informationTuring Machines. A transducer is a finite state machine (FST) whose output is a string and not just accept or reject.
Turing Machines Transducers: A transducer is a finite state machine (FST) whose output is a string and not just accept or reject. Each transition of an FST is labeled with two symbols, one designating
More informationMultiple Choice Questions
Techno India Batanagar Computer Science and Engineering Model Questions Subject Name: Formal Language and Automata Theory Subject Code: CS 402 Multiple Choice Questions 1. The basic limitation of an FSM
More informationAutomata Theory CS S-FR Final Review
Automata Theory CS411-2015S-FR Final Review David Galles Department of Computer Science University of San Francisco FR-0: Sets & Functions Sets Membership: a?{a,b,c} a?{b,c} a?{b,{a,b,c},d} {a,b,c}?{b,{a,b,c},d}
More informationCSE 120. Computer Science Principles
Adam Blank Lecture 17 Winter 2017 CSE 120 Computer Science Principles CSE 120: Computer Science Principles Proofs & Computation e w h e q 0 q 1 q 2 q 3 h,e w,e w,h w,h q garbage w,h,e CSE = Abstraction
More informationTheory of Computation Dr. Weiss Extra Practice Exam Solutions
Name: of 7 Theory of Computation Dr. Weiss Extra Practice Exam Solutions Directions: Answer the questions as well as you can. Partial credit will be given, so show your work where appropriate. Try to be
More informationName: CS 341 Practice Final Exam. 1 a 20 b 20 c 20 d 20 e 20 f 20 g Total 207
Name: 1 a 20 b 20 c 20 d 20 e 20 f 20 g 20 2 10 3 30 4 12 5 15 Total 207 CS 341 Practice Final Exam 1. Please write neatly. You will lose points if we cannot figure out what you are saying. 2. Whenever
More informationFinal Course Review. Reading: Chapters 1-9
Final Course Review Reading: Chapters 1-9 1 Objectives Introduce concepts in automata theory and theory of computation Identify different formal language classes and their relationships Design grammars
More informationFinite Automata. Dr. Nadeem Akhtar. Assistant Professor Department of Computer Science & IT The Islamia University of Bahawalpur
Finite Automata Dr. Nadeem Akhtar Assistant Professor Department of Computer Science & IT The Islamia University of Bahawalpur PhD Laboratory IRISA-UBS University of South Brittany European University
More informationCS 441G Fall 2018 Exam 1 Matching: LETTER
CS 441G Fall 2018 Exam 1 Matching: match the best term from the following list to its definition by writing the LETTER of the term in the blank to the left of the definition. All 31 definitions are given
More informationContext Free Languages and Pushdown Automata
Context Free Languages and Pushdown Automata COMP2600 Formal Methods for Software Engineering Ranald Clouston Australian National University Semester 2, 2013 COMP 2600 Context Free Languages and Pushdown
More informationCSCI 340: Computational Models. Turing Machines. Department of Computer Science
CSCI 340: Computational Models Turing Machines Chapter 19 Department of Computer Science The Turing Machine Regular Expressions Acceptor: FA, TG Nondeterminism equal? Yes Closed Under: L 1 + L 2 L 1 L
More informationT.E. (Computer Engineering) (Semester I) Examination, 2013 THEORY OF COMPUTATION (2008 Course)
*4459255* [4459] 255 Seat No. T.E. (Computer Engineering) (Semester I) Examination, 2013 THEY OF COMPUTATION (2008 Course) Time : 3 Hours Max. Marks : 100 Instructions : 1) Answers to the two Sections
More informationIntroduction to Automata Theory. BİL405 - Automata Theory and Formal Languages 1
Introduction to Automata Theory BİL405 - Automata Theory and Formal Languages 1 Automata, Computability and Complexity Automata, Computability and Complexity are linked by the question: What are the fundamental
More information14.1 Encoding for different models of computation
Lecture 14 Decidable languages In the previous lecture we discussed some examples of encoding schemes, through which various objects can be represented by strings over a given alphabet. We will begin this
More informationLanguages and Compilers
Principles of Software Engineering and Operational Systems Languages and Compilers SDAGE: Level I 2012-13 3. Formal Languages, Grammars and Automata Dr Valery Adzhiev vadzhiev@bournemouth.ac.uk Office:
More informationAUTOMATA THEORY AND COMPUTABILITY
AUTOMATA THEORY AND COMPUTABILITY QUESTION BANK Module 1 : Introduction to theory of computation and FSM Objective: Upon the completion of this chapter you will be able to Define Finite automata, Basic
More informationCSE450. Translation of Programming Languages. Lecture 20: Automata and Regular Expressions
CSE45 Translation of Programming Languages Lecture 2: Automata and Regular Expressions Finite Automata Regular Expression = Specification Finite Automata = Implementation A finite automaton consists of:
More informationRegular Expressions & Automata
Regular Expressions & Automata CMSC 132 Department of Computer Science University of Maryland, College Park Regular expressions Notation Patterns Java support Automata Languages Finite State Machines Turing
More informationTuring Machines, continued
Previously: Turing Machines, continued CMPU 240 Language Theory and Computation Fall 2018 Introduce Turing machines Today: Assignment 5 back TM variants, relation to algorithms, history Later Exam 2 due
More informationCSE 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 3.2, 3.3 Define variants of TMs Enumerators Multi-tape TMs Nondeterministic TMs
More informationAUBER (Models of Computation, Languages and Automata) EXERCISES
AUBER (Models of Computation, Languages and Automata) EXERCISES Xavier Vera, 2002 Languages and alphabets 1.1 Let be an alphabet, and λ the empty string over. (i) Is λ in? (ii) Is it true that λλλ=λ? Is
More informationCS 275 Automata and Formal Language Theory. First Problem of URMs. (a) Definition of the Turing Machine. III.3 (a) Definition of the Turing Machine
CS 275 Automata and Formal Language Theory Course Notes Part III: Limits of Computation Chapt. III.3: Turing Machines Anton Setzer http://www.cs.swan.ac.uk/ csetzer/lectures/ automataformallanguage/13/index.html
More informationA Note on the Succinctness of Descriptions of Deterministic Languages
INFORMATION AND CONTROL 32, 139-145 (1976) A Note on the Succinctness of Descriptions of Deterministic Languages LESLIE G. VALIANT Centre for Computer Studies, University of Leeds, Leeds, United Kingdom
More informationCISC 4090 Theory of Computation
CISC 4090 Theory of Computation Turing machines Professor Daniel Leeds dleeds@fordham.edu JMH 332 Alan Turing (1912-1954) Father of Theoretical Computer Science Key figure in Artificial Intelligence Codebreaker
More informationAmbiguous Grammars and Compactification
Ambiguous Grammars and Compactification Mridul Aanjaneya Stanford University July 17, 2012 Mridul Aanjaneya Automata Theory 1/ 44 Midterm Review Mathematical Induction and Pigeonhole Principle Finite Automata
More informationToday, we will dispel the notion that Java is a magical language that allows us to solve any problem we want if we re smart enough.
CSE 311: Foundations of Computing Lecture 25: Limits of Computation Showing a Language L is not regular 1. Suppose for contradiction that some DFA M accepts L. 2. Consider an INFINITE set of half strings
More informationFinite Automata Theory and Formal Languages TMV027/DIT321 LP4 2016
Finite Automata Theory and Formal Languages TMV027/DIT321 LP4 2016 Lecture 15 Ana Bove May 23rd 2016 More on Turing machines; Summary of the course. Overview of today s lecture: Recap: PDA, TM Push-down
More informationTheory of Computation p.1/?? Theory of Computation p.2/?? A Turing machine can do everything that any computing
Turing Machines A Turing machine is similar to a finite automaton with supply of unlimited memory. A Turing machine can do everything that any computing device can do. There exist problems that even a
More informationReflection in the Chomsky Hierarchy
Reflection in the Chomsky Hierarchy Henk Barendregt Venanzio Capretta Dexter Kozen 1 Introduction We investigate which classes of formal languages in the Chomsky hierarchy are reflexive, that is, contain
More informationPower Set of a set and Relations
Power Set of a set and Relations 1 Power Set (1) Definition: The power set of a set S, denoted P(S), is the set of all subsets of S. Examples Let A={a,b,c}, P(A)={,{a},{b},{c},{a,b},{b,c},{a,c},{a,b,c}}
More informationModels of Computation II: Grammars and Pushdown Automata
Models of Computation II: Grammars and Pushdown Automata COMP1600 / COMP6260 Dirk Pattinson Australian National University Semester 2, 2018 Catch Up / Drop in Lab Session 1 Monday 1100-1200 at Room 2.41
More informationLambda Calculus and Computation
6.037 Structure and Interpretation of Computer Programs Chelsea Voss csvoss@mit.edu Massachusetts Institute of Technology With material from Mike Phillips and Nelson Elhage February 1, 2018 Limits to Computation
More informationIntroduction to Computers & Programming
16.070 Introduction to Computers & Programming Theory of computation 5: Reducibility, Turing machines Prof. Kristina Lundqvist Dept. of Aero/Astro, MIT States and transition function State control A finite
More informationPractice: Large Systems Part 2, Chapter 2
Practice: Large Systems Part 2, Chapter 2 Overvie Introduction Strong Consistency Crash Failures: Primary Copy, Commit Protocols Crash-Recovery Failures: Paxos, Chubby Byzantine Failures: PBFT, Zyzzyva
More informationMIT Specifying Languages with Regular Expressions and Context-Free Grammars. Martin Rinard Massachusetts Institute of Technology
MIT 6.035 Specifying Languages with Regular essions and Context-Free Grammars Martin Rinard Massachusetts Institute of Technology Language Definition Problem How to precisely define language Layered structure
More informationChapter 14: Pushdown Automata
Chapter 14: Pushdown Automata Peter Cappello Department of Computer Science University of California, Santa Barbara Santa Barbara, CA 93106 cappello@cs.ucsb.edu The corresponding textbook chapter should
More informationAutomata & languages. A primer on the Theory of Computation. The imitation game (2014) Benedict Cumberbatch Alan Turing ( ) Laurent Vanbever
Automata & languages A primer on the Theory of Computation The imitation game (24) Benedict Cumberbatch Alan Turing (92-954) Laurent Vanbever www.vanbever.eu ETH Zürich (D-ITET) September, 2 27 Brief CV
More informationLimited Automata and Unary Languages
Limited Automata and Unary Languages Giovanni Pighizzini and Luca Prigioniero Dipartimento di Informatica, Università degli Studi di Milano, Italy {pighizzini,prigioniero}@di.unimi.it Abstract. Limited
More informationCSE 105 THEORY OF COMPUTATION
CSE 105 THEORY OF COMPUTATION Fall 2016 http://cseweb.ucsd.edu/classes/fa16/cse105-abc/ Today's learning goals Sipser sec 3.2 Describe several variants of Turing machines and informally explain why they
More information