CSCE 551: Chin-Tser Huang. University of South Carolina
|
|
- Shonda Lewis
- 5 years ago
- Views:
Transcription
1 CSCE 551: Theory of Computation Chin-Tser Huang University of South Carolina
2 Computation History A computation history of a TM M is a sequence of its configurations C 1, C 2,, C l such hthat tc i+1 legally ll follows from C i according to the rules of M. A computation history is accepting/rejecting if C l is accepting or rejecting configuration respectively. In general, for a TM there may exist an infinite number of configurations i (because of an infinite i tape). However, for TMs that use only limited amount of the tape, the number of configurations is finite. A linear bounded automaton (LBA) is a TM that is permitted to use only the portion of the tape containing the input (i.e., cannot move the head beyond the input). 03/22/2010 2
3 A LBA is Decidable Lemma Let M be an LBA with q states and g symbols in the tape alphabet. Then there are exactly qng n distinct configurations of M for the tape of length n. Lemma Let M be an LBA with q states and g symbols in the tape alphabet. If M on input of length n does not produce an output within first qng n steps, it loops. Theorem A LBA = { M, w M is an LBA and M accepts w} is decidable. 03/22/2010 3
4 E LBA is Undecidable E LBA = { M M is an LBA and L(M) = } We will prove that E LBA is undecidable by reduction from A TM. Namely, we show that if E LBA were decidable, A TM would also be. A TM = { M, w M is a TM and M accepts w} Note that Given a TM M and a string w, w simulating M on input w cannot be done on an LBA (since M may use arbitrarily large portion of its tape, while LBA is limited in tape usage). However, given a sequence of configurations of a fixed TM M, M it can be verified on an LBA whether this sequence represents a valid computation history of M on a fixed input w (since the tape content of M, even if arbitrarily large, is embedded in the given configurations and does not require any additional storage from the LBA verifier). 03/22/2010 4
5 E LBA is Undecidable For a TM M and a string w, M accepts w if and only if the following language is non-empty: L M,w = {#C 1 #C 2 #...#C l # C 1, C 2,, C l is accepting computation history of M on w} Q: How does this help us? 03/22/2010 5
6 ALL CFG is Undecidable ALL CFG = { G G is a CFG and L(G) = Σ * } Again, we will prove that ALL CFG is undecidable by reduction from A TM. We show that, if ALL CFG were decidable, A TM would also be. Idea is for given TM M and string w, construct a PDA (which we know how to convert to a CFG) that generates the following language: Z M,w = {#C 1 #C 2R #C 3 #C 4R #...#C l # C 1, C 2,, C l is NOT accepting computation history of M on w} Note that M accepts w if and only if Z M,w Σ *. Q: How does this help us? 03/22/2010 6
7 ALL CFG is Undecidable For given TM M and string w, we need construct a PDA that generates the following language: Z M,w = {#C 1 #C 2R #C 3 #C 4R #...#C l # C 1, C 2,, C l is NOT accepting computation history of M on w} This PDA verifies a number of conditions (each in its own nondeterministic branch): input string starts and ends with #; each C i is a valid configuration of M ; C 1 is a starting configuration for M on w ; C l is an accepting configuration for M ; C i+1 properly follows from C i. If any of these conditions fails, the PDA accepts. Q: Why do we need to reverse every other configuration in the computation history? 03/22/2010 7
8 Computable Functions A function f : Σ * Σ * is called a computable function if some Turing machine on every input w halts with just f (w) on its tape. Arithmetic functions are computable: m, n m + n; m, n m n; m, n m n; m, n m/n, n 0; m, n m mod n, n 0. 03/22/2010 8
9 Computable Transformations Computable functions may be transformations of TM descriptions: a computable function f may take as an input M and return M where M and M are TMs. Example: M is a TM that recognizes the same language M but never attempts to move its head off the left-hand end of its tape. If input to f is not a legal encoding of a TM, it reutrns ε. 03/22/2010 9
10 Mapping Reducibility Definition Language A is mapping reducible to language B, written A m B, if there is a computable function f : Σ * Σ * such that for every w w A f (w) B The function f is called the reduction of A to B. 03/22/
11 Mapping Reducibility Mapping reduction provides a way to convert questions about membership testing in A to membership testing in B. Instead of testing whether w A, one can compute f (w) and test whether f (w) B. If one problem is mapping reducible to another, previously solved problem, one can thereby obtain a solution to the original problem. 03/22/
12 Theorems Theorem A m B if and only if A m B. Theorem If A m B and B is decidable, then A is decidable. Theorem If A m B and B is Turing-recognizable, then A is Turingrecognizable. Q: How to prove these theorems? 03/22/
13 Corollaries Corollary If A m B and A is undecidable, then B is undecidable. Corollary If A m B and A is not Turing-recognizable, i then B is not Turing-recognizable. 03/22/
14 A TM m HALT TM Corollary If A m B and A is undecidable, then B is undecidable. Let's apply this Corollary to prove that HALT TM is undecidable by showing that A TM m HALT TM. Since A TM is undecidable, so would be HALT TM. We design a TM F that computes a reduction from A TM to HALT TM : F = On input M, w does the following: 1. Construct a TM M that for a given input x : 1.1 Run M on x; 1.2 If M accepts, then accept; otherwise enter a loop. 2. Output M, w. It is easy to see that M, w A TM iff M, w HALT TM. Therefore, F indeed computes a reduction from A TM to HALT TM. 03/22/
15 E TM m EQ TM E TM = { M M is a TM and L(M) = } EQ TM = { M 1, M 2 M 1, M 2 are TMs and L(M 1 ) = L(M 2 )} Q: How to prove that E TM m EQ TM? Let for a given TM M return a pair M, M where M rejects any input. If input is not encoding of a TM, f returns ε. Then M E TM M, M EQ TM So, f is a reduction of E TM to EQ TM. 03/22/
16 E TM is Undecidable Previously we proved that E TM is undecidable by reducing A TM to it. Let s review: A TM = { M, w M is a TM and M accepts w} E TM ={ M M is a TM and L(M) = } We assume that we have a decider D E for E TM, and we need to construct a decider D A for A TM. Note that D A gets as an input a pair M, w that needs to be somehow transformed into an input M 1 for D E such that the output of D E on M 1 would help us produce the required output D A on M, w. 03/22/
17 E TM is Undecidable Basically, we want to have L(M 1 ) if and only if M accepts w. w Then the output of D A on M, w can simply output the opposite of the output of D E on M 1. Let M 1 rejects everything but w on which it runs M and accepts iff M accepts w. Then L(M 1 ) = if M does not accept w; and L(M 1 ) = {w} if M accepts w. D A can be defined as follows: D A = On input M, w, where M is a TM and w is a string: 1. Construct the following M 1. M 1 = On input x: 1. If x w, reject. 2. If x = w, run M on input w and accept if M does. 2. Run D E on input M If D E accepts, reject; if D E rejects, accept. 03/22/
18 A TM m E TM A TM = { M, w M is a TM and M accepts w} E TM = { M M is a TM and L(M) = } Let a TM M rejects everything but w on which it runs M and accepts iff M accepts w. w Then L(M )= if M does not accept w; and L(M ) = {w} if M accepts w. In other words, M, w A TM M E TM. So, computing a TM M for a given pair M, w simply means a reduction of A TM to E TM. Since A TM is undecidable, so is E TM and thus E TM (since decidability is not affected by complementation). 03/22/
19 Not Turing-recognizable recognizable Languages Corollary If A m B and A is not Turing-recognizable, then B is not Turing-recognizable. Q: How to make use of this corollary? In a typical application of this corollary, we let A = A TM which is not Turing-recognizable. To prove that B is not Turing-recognizable we will show that A TM m B Alternatively, ti l we can show that t A TM m B 03/22/
20 EQ TM EQ TM = { M 1, M 2 M 1, M 2 are TMs and L(M 1 ) = L(M 2 )} Theorem EQ TM is neither Turing-recognizable nor co-turing- recognizable. Q: What would be a high-level outline of the proof? To prove that EQ TM is not Turing-recognizable, recognizable, we will show that A TM m EQ TM (alternative of A TM m EQ TM ). To prove that EQ TM is not Turing-recognizable, we will show that t A TM m EQ TM (alternative ti of A TM m EQ TM ). 03/22/
21 A TM m EQ TM To show that A TM m EQ TM we need to construct a computable mapping of M, M w to MM 1, M 2 such hthatt Define a TM F as follows: M, w A TM M 1, M 2 EQ TM. F = On input M, w does the following: 1. Construct the following TMs M 1 and M 2 : M 1 rejects any input. M 2 ignores its input and run M on w. If it accepts, accept. 2. Output M 1, M 2. 03/22/
22 A TM m EQ TM To show that A TM m EQ TM we need to construct a computable mapping of M, M w to MM 1, M 2 such hthatt Define a TM F as follows: M, w A TM M 1, M 2 EQ TM. F = On input M, w does the following: 1. Construct the following TMs M 1 and M 2 : M 1 accepts any input. M 2 ignores its input and run M on w. If it accepts, accept. 2. Output M 1, M 2. 03/22/
23 Announcement Homework 3 is due TODAY in class Do the following problems for your practice: 5.1, 5.9, 5.22, 5.34 We will not cover Chapter 6 Reading assignment: Ch /22/
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
More informationReducability. Sipser, pages
Reducability Sipser, pages 187-214 Reduction Reduction encodes (transforms) one problem as a second problem. A solution to the second, can be transformed into a solution to the first. We expect both transformations
More informationCS5371 Theory of Computation. Lecture 14: Computability V (Prove by Reduction)
CS5371 Theory of Computation Lecture 14: Computability V (Prove by Reduction) Objectives This lecture shows more undecidable languages Our proof is not based on diagonalization Instead, we reduce the problem
More informationCSE 105 THEORY OF COMPUTATION
CSE 105 THEORY OF COMPUTATION Spring 2018 http://cseweb.ucsd.edu/classes/sp18/cse105-ab/ Today's learning goals Sipser Ch 5.1, 5.3 Define and explain core examples of computational problems, including
More informationCSCE 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
More informationUndecidable 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.
More informationComputation Histories
208 Computation Histories The computation history for a Turing machine on an input is simply the sequence of configurations that the machine goes through as it processes the input. An accepting computation
More informationNon-emptiness Testing for TMs
180 5. Reducibility The proof of unsolvability of the halting problem is an example of a reduction: a way of converting problem A to problem B in such a way that a solution to problem B can be used to
More informationV 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
More informationIntroduction 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
More informationCSE 105 THEORY OF COMPUTATION
CSE 105 THEORY OF COMPUTATION "Winter" 2018 http://cseweb.ucsd.edu/classes/wi18/cse105-ab/ Today's learning goals Sipser Ch 4.2 Trace high-level descriptions of algorithms for computational problems. Use
More informationWhat 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
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 4.1, 5.1 Define reductions from one problem to another. Use reductions to prove
More informationACS2: 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
More informationHomework Assignment 6 Answers
Homework Assignment 6 Answers CSCI 2670 Introduction to Theory of Computing, Fall 2016 December 2, 2016 This homework assignment is about Turing machines, decidable languages, Turing recognizable languages,
More informationMapping Reducibility. Human-aware Robotics. 2017/11/16 Chapter 5.3 in Sipser Ø Announcement:
Mapping Reducibility 2017/11/16 Chapter 5.3 in Sipser Ø Announcement: q Slides for this lecture are here: http://www.public.asu.edu/~yzhan442/teaching/cse355/lectures/mapping.pdf 1 Last time Reducibility
More information1 Reducability. CSCC63 Worksheet Reducability. For your reference, A T M is defined to be the language { M, w M accepts w}. Theorem 5.
CSCC63 Worksheet Reducability For your reference, A T M is defined to be the language { M, w M accepts w}. 1 Reducability Theorem 5.1 HALT TM = { M, w M is a T M that halts on input w} is undecidable.
More informationChap. 4,5 Review. Algorithms created in proofs from prior chapters
Chap. 4,5 Review Algorithms created in proofs from prior chapters (p. 55) Theorem 1.39: NFA to DFA (p. 67) Lemma 1.55: Regex to NFA (p. 69) Lemma 1.60: DFA to regex (through GNFA) (p. 112) Lemma 2.21:
More informationTheory 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
More informationCS 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
More information1 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
More informationLecture 23: Rice Theorem and Turing machine behavior properties 21 April 2009
CS 373: Theory of Computation Sariel Har-Peled and Madhusudan Parthasarathy Lecture 23: Rice Theorem and Turing machine behavior properties 21 April 2009 This lecture covers Rice s theorem, as well as
More informationFORMAL 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
More informationTheory 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
More informationLecture 12: Mapping Reductions
Lecture 12: Mapping Reductions October 18, 2016 CS 1010 Theory of Computation Topics Covered 1. The Language EQ T M 2. Mapping Reducibility 3. The Post Correspondence Problem 1 The Language EQ T M The
More informationCS5371 Theory of Computation. Lecture 12: Computability III (Decidable Languages relating to DFA, NFA, and CFG)
CS5371 Theory of Computation Lecture 12: Computability III (Decidable Languages relating to DFA, NFA, and CFG) Objectives Recall that decidable languages are languages that can be decided by TM (that means,
More information6.045: 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
More informationCS20a: Turing Machines (Oct 29, 2002)
CS20a: Turing Machines (Oct 29, 2002) So far: DFA = regular languages PDA = context-free languages Today: Computability 1 Church s thesis The computable functions are the same as the partial recursive
More information} Some languages are Turing-decidable A Turing Machine will halt on all inputs (either accepting or rejecting). No infinite loops.
and their languages } Some languages are Turing-decidable A Turing Machine will halt on all inputs (either accepting or rejecting). No infinite loops. } Some languages are Turing-recognizable, but not
More informationTheory of Computation p.1/?? Theory of Computation p.2/?? We develop examples of languages that are decidable
Decidable Languages We use languages to represent various computational problems because we have a terminology for dealing with languages Definition: A language is decidable if there is an algorithm (i.e.
More informationDecidability and Undecidability
Decidability and Undecidability Major Ideas from Last Time Every TM can be converted into a string representation of itself. The encoding of M is denoted M. The universal Turing machine U TM accepts an
More informationIntro to Theory of Computation
Intro to Theory of Computation LECTURE 14 Last time Turing Machine Variants Church-Turing Thesis Today Universal TM Decidable languages Designing deciders Sofya Raskhodnikova 3/1/2016 Sofya Raskhodnikova;
More informationTuring 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
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 3.3, 4.1 State and use the Church-Turing thesis. Give examples of decidable problems.
More informationCSE 105 THEORY OF COMPUTATION
CSE 105 THEORY OF COMPUTATION "Winter" 2018 http://cseweb.ucsd.edu/classes/wi18/cse105-ab/ Today's learning goals Sipser Section 4.1 Explain what it means for a problem to be decidable. Justify the use
More informationCpSc 421 Homework 9 Solution
CpSc 421 Homework 9 Solution Attempt any three of the six problems below. The homework is graded on a scale of 100 points, even though you can attempt fewer or more points than that. Your recorded grade
More informationCS154, Lecture 10: Rice s Theorem, Oracle Machines
CS154, Lecture 10: Rice s Theorem, Oracle Machines Moral: Analyzing Programs is Really, Really Hard But can we more easily tell when some program analysis problem is undecidable? Problem 1 Undecidable
More informationFurther 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
More informationMidterm 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
More informationproblem X reduces to Problem Y solving X would be easy, if we knew how to solve Y
CPS220 Reducibility A reduction is a procedure to convert one problem to another problem, in such a way that a solution to the second problem can be used to solve the first problem. The conversion itself
More informationCSE 105 THEORY OF COMPUTATION. Spring 2018 review class
CSE 105 THEORY OF COMPUTATION Spring 2018 review class Today's learning goals Summarize key concepts, ideas, themes from CSE 105. Approach your final exam studying with confidence. Identify areas to focus
More informationCSE 105 Theory of Computation
CSE 105 Theory of Computation http://www.jflap.org/jflaptmp/ Professor Jeanne Ferrante 1 Undecidability Today s Agenda Review and More Problems A Non-TR Language Reminders and announcements: HW 7 (Last!!)
More informationCS5371 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
More information4.2 The Halting Problem
172 4.2 The Halting Problem The technique of diagonalization was discovered in 1873 by Georg Cantor who was concerned with the problem of measuring the sizes of infinite sets For finite sets we can simply
More informationCS21 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
More informationDecidability (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
More informationCS5371 Theory of Computation. Lecture 15: Computability VI (Post s Problem, Reducibility)
CS5371 Theory of Computation Lecture 15: Computability VI (Post s Problem, Reducibility) Objectives In this lecture, we introduce Post s correspondence problem (playing with a special type of domino) We
More informationCS5371 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
More informationMapping Reductions. Mapping Reductions. Introduction to Computability. Theory. Mapping Reductions. Lecture13: Mapping. Reductions
Introduction to Computability Theory Lecture13: apping Reductions Prof. Amos Israeli 1 apping Reductions apping reductionsconstitute the generalization we are looking for. They are always in a specified
More informationThe 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
More informationCSE 105 THEORY OF COMPUTATION
CSE 105 THEORY OF COMPUTATION Spring 2017 http://cseweb.ucsd.edu/classes/sp17/cse105-ab/ Today's learning goals Summarize key concepts, ideas, themes from CSE 105. Approach your final exam studying with
More informationacs-07: Decidability Decidability Andreas Karwath und Malte Helmert Informatik Theorie II (A) WS2009/10
Decidability Andreas Karwath und Malte Helmert 1 Overview An investigation into the solvable/decidable Decidable languages The halting problem (undecidable) 2 Decidable problems? Acceptance problem : decide
More informationComputational Models Lecture 8 1
Computational Models Lecture 8 1 Handout Mode Nachum Dershowitz & Yishay Mansour. Tel Aviv University. May 17 22, 2017 1 Based on frames by Benny Chor, Tel Aviv University, modifying frames by Maurice
More informationDecidability. 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
More informationComputability and Complexity
Computability and Complexity Decidability, Undecidability and Reducibility; Codes, Algorithms and Languages CAS 705 Ryszard Janicki Department of Computing and Software McMaster University Hamilton, Ontario,
More informationEquivalence 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
More informationComputational Models Lecture 8 1
Computational Models Lecture 8 1 Handout Mode Ronitt Rubinfeld and Iftach Haitner. Tel Aviv University. May 11/13, 2015 1 Based on frames by Benny Chor, Tel Aviv University, modifying frames by Maurice
More informationExam Computability and Complexity
Total number of points:... Number of extra sheets of paper:... Exam Computability and Complexity by Jiri Srba, January 2009 Student s full name CPR number Study number Before you start, fill in the three
More informationCSE 105 Theory of Computation
CSE 105 Theory of Computation http://www.jflap.org/jflaptmp/ Professor Jeanne Ferrante 1 Undecidability Today s Agenda Review: The TM Acceptance problem, A TM The Halting Problem for TM s Other problems
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 2 Design a PDA and a CFG for a given language Give informal description for a PDA,
More informationCS 361 Meeting 28 11/14/18
CS 361 Meeting 28 11/14/18 Announcements 1. Homework 9 due Friday Computation Histories 1. Some very interesting proos o undecidability rely on the technique o constructing a language that describes the
More informationIntroduction 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
More informationSCHEME FOR INTERNAL ASSESSMENT TEST 3
SCHEME FOR INTERNAL ASSESSMENT TEST 3 Max Marks: 40 Subject& Code: Automata Theory & Computability (15CS54) Sem: V ISE (A & B) Note: Answer any FIVE full questions, choosing one full question from each
More informationDecidability (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
More informationHomework 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
More informationUndecidability. We are not so much concerned if you are slow as when you come to a halt. (Chinese Proverb)
We are not so much concerned if you are slow as when you come to a halt. (Chinese Proverb) CS /55 Theory of Computation The is A TM = { M,w M is a TM and w L(M)} A TM is Turing-recognizable. Proof Sketch:
More informationTuring Machines. Chapter 17
Turing Machines Chapter 17 Languages and Machines SD D Context-Free Languages Regular Languages reg exps FSMs cfgs PDAs unrestricted grammars Turing Machines Grammars, SD Languages, and Turing Machines
More informationUNIT-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
More informationPart I: Definitions and Properties
Turing Machines Part I: Definitions and Properties Finite State Automata Deterministic Automata (DFSA) M = {Q, Σ, δ, q 0, F} -- Σ = Symbols -- Q = States -- q 0 = Initial State -- F = Accepting States
More informationComputational Models Lecture 9, Spring 2009
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. p. 1 Computational Models Lecture 9, Spring 2009 Reducibility among languages Mapping reductions More undecidable
More informationCPSC 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
More informationAn example of a decidable language that is not a CFL Implementation-level description of a TM State diagram of TM
Turing Machines Review An example of a decidable language that is not a CFL Implementation-level description of a TM State diagram of TM Varieties of TMs Multi-Tape TMs Nondeterministic TMs String Enumerators
More informationDecidable Languages - relationship with other classes.
CSE2001, Fall 2006 1 Last time we saw some examples of decidable languages (or, solvable problems). Today we will start by looking at the relationship between the decidable languages, and the regular and
More informationTHEORY OF COMPUTATION (AUBER) EXAM CRIB SHEET
THEORY OF COMPUTATION (AUBER) EXAM CRIB SHEET Regular Languages and FA A language is a set of strings over a finite alphabet Σ. All languages are finite or countably infinite. The set of all languages
More informationCSE355 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,
More informationAdvanced Undecidability Proofs
17 Advanced Undecidability Proofs In this chapter, we will discuss Rice s Theorem in Section 17.1, and the computational history method in Section 17.3. As discussed in Chapter 16, these are two additional
More informationComputational Models Lecture 8 1
Computational Models Lecture 8 1 Handout Mode Ronitt Rubinfeld and Iftach Haitner. Tel Aviv University. April 18/ May 2, 2016 1 Based on frames by Benny Chor, Tel Aviv University, modifying frames by Maurice
More informationMore 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
More informationDecidability. Overview. Preliminaries to Halting Problem. Decidable Problems of Regular Languages. Decidable Problems of Context-Free Languages
CS533 Class 04a: 1 c P. Heeman, 2017 Introduction: Decidability Overview Decidable Problems of Regular Languages Decidable Problems of Context-Free Languages Preliminaries to Halting Problem CS533 Class
More informationDecidability. Linz 6 th, Chapter 12: Limits of Algorithmic Computation, page 309ff
Decidability Linz 6 th, Chapter 12: Limits of Algorithmic Computation, page 309ff 1 A property P of strings is said to be decidable if the set of all strings having property P is a recursive set; that
More informationLecture Notes: The Halting Problem; Reductions
Lecture Notes: The Halting Problem; Reductions COMS W3261 Columbia University 20 Mar 2012 1 Review Key point. Turing machines can be encoded as strings, and other Turing machines can read those strings
More informationMidterm 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
More informationCS481F01 Solutions 8
CS481F01 Solutions 8 A. Demers 7 Dec 2001 1. Prob. 111 from p. 344 of the text. One of the following sets is r.e. and the other is not. Which is which? (a) { i L(M i ) contains at least 481 elements }
More informationDM17. 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....................
More informationCS20a: Turing Machines (Oct 29, 2002)
CS20a: Turing Machines (Oct 29, 2002) So far: DFA = regular languages PDA = context-free languages Today: Computability 1 Handicapped machines DFA limitations Tape head moves only one direction 2-way DFA
More informationTheory of Computation
Theory of Computation Lecture #10 Sarmad Abbasi Virtual University Sarmad Abbasi (Virtual University) Theory of Computation 1 / 35 Lecture 10: Overview Undecidability of the Halting Problem Russell s Paradox
More informationCSCC63 Worksheet Turing Machines
1 An Example CSCC63 Worksheet Turing Machines Goal. Design a turing machine, M that accepts only strings of the form {w#w w {0, 1} }. Idea. Describe in words how the machine would work. Read first symbol
More informationFoundations 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
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 Ch 3 Trace the computation of a Turing machine using its transition function and configurations.
More informationChapter 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
More informationComputability Theory
CS:4330 Theory of Computation Spring 2018 Computability Theory Decidable Languages Haniel Barbosa Readings for this lecture Chapter 4 of [Sipser 1996], 3rd edition. Section 4.1. Decidable Languages We
More information6.045J/18.400J: Automata, Computability and Complexity. Quiz 2. March 30, Please write your name in the upper corner of each page.
6.045J/18.400J: Automata, Computability and Complexity March 30, 2005 Quiz 2 Prof. Nancy Lynch Please write your name in the upper corner of each page. Problem Score 1 2 3 4 5 6 Total Q2-1 Problem 1: True
More informationDecision Problems with TM s. Lecture 31: Halting Problem. Universe of discourse. Semi-decidable. Look at following sets: CSCI 81 Spring, 2012
Decision Problems with TM s Look at following sets: Lecture 31: Halting Problem CSCI 81 Spring, 2012 Kim Bruce A TM = { M,w M is a TM and w L(M)} H TM = { M,w M is a TM which halts on input w} TOTAL TM
More informationDecidable and undecidable languages
The Chinese University of Hong Kong Fall 2011 CSCI 3130: Formal languages and automata theory Decidable and undecidable languages Andrej Bogdanov http://www.cse.cuhk.edu.hk/~andrejb/csc3130 Problems about
More informationTURING 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,
More information6.045: Automata, Computability, and Complexity Or, Great Ideas in Theoretical Computer Science Spring, Class 10 Nancy Lynch
6.045: Automata, Computability, and Complexity Or, Great Ideas in Theoretical Computer Science Spring, 2010 Class 10 Nancy Lynch Today Final topic in computability theory: Self-Reference and the Recursion
More informationCSCE 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
More informationCpSc 421 Final Exam December 15, 2006
CpSc 421 Final Exam December 15, 2006 Do problem zero and six of problems 1 through 9. If you write down solutions for more that six problems, clearly indicate those that you want graded. Note that problems
More informationIntroduction 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
More informationTuring Machines. Reading Assignment: Sipser Chapter 3.1, 4.2
Reading Assignment: Sipser Chapter 31, 42 Turing Machines 41 covers algorithms for decidable problems about DFA, NFA, RegExp, CFG, and PDAs, eg slides 17 & 18 below I ve talked about most of this in class
More information