# Notes on State Minimization

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 U.C. Berkeley CS172: Automata, Computability and Complexity Handout 1 Professor Luca Trevisan 2/3/2015 Notes on State Minimization These notes present a technique to prove a lower bound on the number of states of any DFA that recognizes a given language. The technique can also be used to prove that a language is not regular. (By showing that for every k one needs at least k states to recognize the language.) We also present an efficient algorithm to convert a given DFA into a DFA for the same language and with a minimum number of states. 1 Distinguishable and Indistinguishable States It will be helpful to keep in mind the following two languages over the alphabet Σ = {0, 1} as examples: the language EQ = {0 n 1 n n 1} of strings containing a sequence of zeroes followed by an equally long sequence of ones, and the language A = (0 1) 1 (0 1) of strings containing a 1 in the second-to-last position. We start with the following basic notion. Definition 1 (Distinguishable Strings) Let L be a language over an alphabet Σ. We say that two strings x and y are distinguishable with respect to L if there is a string z such that xz L and yz L, or vice versa. For example the strings x = 0 and y = 00 are distinguishable with respect to EQ, because if we take z = 1 we have xz = 01 EQ and yz = 001 EQ. Also, the strings x = 00 and y=01 are distinguishable with respect to A as can be seen by taking z = 0. On the other hand, the strings x = 0110 and y = 10 are not distinguishable with respect to EQ because for every z we have xz EQ and yz EQ. Exercise 2 Find two strings that are not distinguishable with respect to A. The intuition behind Definition 1 is captured by the following simple fact. Lemma 3 Let L be a language, M be a DFA that decides L, and x and y be distinguishable strings with respect to L. Then the state reached by M on input x is different from the state reached by M on input y. Proof: Suppose by contradiction that M reaches the same state q on input x and on input y. Let z be the string such that xz L and yz L (or vice versa). Let us call q the state reached by M on input xz. Note that q is the state reached by M starting from q and given the string z. But also, on input yz, M must reach the same state q, because M reaches state q given y, and then goes from q to q given z. This means that M either accepts both xz and yz, or it rejects both. In either case, M is incorrect and we reach a contradiction. Consider now the following generalization of the notion of distinguishability. Definition 4 (Distinguishable Set of Strings) Let L be a language. A set of strings {x 1,..., x k } is distinguishable if for every two distinct strings x i, x j we have that x i is distinguishable from x j. 1

2 For example one can verify that {0, 00, 000} are distinguishable with respect to EQ and that {00, 01, 10, 11} are distinguishable with respect to A. We now prove the main result of this section. Lemma 5 Let L be a language, and suppose there is a set of k distinguishable strings with respect to L. Then every DFA for L has at least k states. Proof: If L is not regular, then there is no DFA for L, much less a DFA with less than k states. If L is regular, let M be a DFA for L, let x 1,..., x k be the distinguishable strings, and let q i be the state reached by M after reading x i. For every i j, we have that x i and x j are distinguishable, and so q i q j because of Lemma 3. So we have k different states q 1,..., q k in M, and so M has at least k states. Using Lemma 5 and the fact that the strings {00, 01, 10, 11} are distinguishable with respect to A we conclude that every DFA for A has at least 4 states. For every k 1, consider the set {0, 00,..., 0 k } of strings made of k or fewer zeroes. It is easy to see that this is a set of distinguishable strings with respect to EQ. This means that there cannot be a DFA for EQ, because, if there were one, it would have to have at least k states for every k, which is clearly impossible. 2 State Minimization Let L be a language over an alphabet Σ. We have seen in the previous section the definition of distinguishable strings with respect to L. We say that two strings x and y are indistinguishable, and we write it x L y if they are not distinguishable. That is, x L y means that, for every string z, the string xz belongs to L if and only if the string yz does. By definition, x L y if and only if y L x, and we always have x L x. It is also easy to see that if x L y and y L w then we must have x L w. In other words: Fact 6 The relation L is an equivalence relation over the strings in Σ. As you may remember from such classes as Math55 or CS70, when you define an equivalence relation over a set you also define a way to partition the set into a collection of subsets, called equivalence class. An equivalence class in Σ with respect to L is a set of strings that are all indistinguishable from one another, and that are all distinguishable from all the others not in the set. We denote by [x] the equivalence class that contains the string x. A fancy way of stating Lemma 5 is to say that every DFA for L must have at least as many states as the number of equivalence class in Σ with respect to L. Perhaps surprisingly, the converse is also true: there is always a DFA that has precisely as many states as the number of equivalence classes. Theorem 7 (Myhill-Nerode) Let L be a language over Σ. If Σ has infinitely many equivalence classes with respect to L, then L is not regular. Otherwise, L can be decided by a DFA whose number of states is equal to the number of equivalence classes in Σ with respect to L. Proof: If there are infinitely many equivalence classes, then it follows from Lemma 5 that no DFA can decide L, and so L is not regular. 2

3 Suppose then that there is a finite number of equivalence class. We define an automaton that has a state for each equivalence class. The start state is the class [ɛ] and every state of the form [x] for x L is a final state. It remains to describe the transition function. From a state [x], reading the character a, the automaton moves to state [xa]. We need to make sure that this definition makes sense. If x L x, then the state [x] and the state [x ] are the same, so we need to verify that the state [xa] and the state [x a] are also the same. That is, we need to verify that, for every string z, xaz L if and only if x az L; this is clearly true because x and x are indistinguishable and so appending the string az makes xaz an element of the language L if and only if it also makes x az an element of the language. So the automaton is well defined. Let now x = x 1 x 2 x n be an input string for the automaton. The automaton starts in [ɛ], then moves to state [x 1 ], and so on, and at the end is in state [x 1 x n ]; this is an accepting state if and only if x L, and so the automaton works correctly on x. The Myhill-Nerode theorem shows that one can use the distinguishability method to prove optimal lower bounds on the number of states of a DFA for a given language, but it does not give an efficient way to construct an optimal DFA. We will now see a polynomial time algorithm that given a DFA finds an equivalent DFA with a minimum number of states. There is even an O(n log n) algorithm for this problem, where n = Q Σ is the size of the input, but we will not see it. Before describing the algorithm, consider the following definition. Definition 8 (Equivalence of states) Let M = (Q, Σ, δ, q 0, F ) be a DFA. We say that two states p, q Q are equivalent, and we write it p q, if for every string x Σ the state that M reaches from p given x is accepting if and only if the state that M reaches from q given x is accepting. An equivalent way to think of the definition is that if we change M so that p is the start state the language that we recognize is the same as if we change M so that q is the start state. We can verify that is an equivalence relation among the set of states Q, and so partitions Q into a set of equivalence classes. The intution behind this definition is that if p q, then it is redundant to have two different states p and q. We would then like to compute the set of equivalence classes of Q with respect to and then construct a new automaton that has only one state for each equivalence class. The problem here is that it is not clear how to compute the relation, since it looks like we need to test infinitely many cases (one for each string x Σ ) in order to verify that p q. Fortunately, there is a shortcut. To discuss the shortcut, we need one more definition. Definition 9 Let M = (Q, Σ, δ, q 0, F ) be a DFA. We say that two states p, q Q are equivalent for length n, and we write it p n q, if for every string x Σ of length n the state that M reaches from p given x is accepting if and only if the state that M reaches from q given x is accepting. In other words, if we change M so that p becomes the start state, the set of strings of length at most n that we accept is precisely the same as if we had made q be the start state. The equivalence relations n can be computed recursively. First, note that p 0 q if and only if p is accepting and q is not (or vice versa). That is, the equivalence classes for 0 are the set F and the set Q F. 3

4 Suppose we have computed n 1. Now we have that p n q if and only if p n 1 q and for every a Σ we have δ(p, a) n 1 δ(q, a). The second part requires some justification, and we prove it as a lemma below. Lemma 10 Let M = (Q, Σ, δ, q 0, F ) be a DFA. For any two states p, q and integer n 1, we have that p n q if and only if p n 1 q or there is an a Σ such that δ(p, a) n 1 δ(q, a). Proof: If p n q, then there is a string x such that M accepts when starting from p and given x, but it rejects when starting from q and given x. If the length of x is n 1, then we have p n 1 q. Otherwise, let us write x = ax, where a is the first character of x, and let us call p = δ(p, a) and q = δ(q, a). Then the string x has length n 1 and it shows that p n 1 q. For the other direction, if p n 1 q then clearly also p n q. Otherwise, if there is an a such that δ(p, a) n 1 δ(q, a), then let x be the string of length n 1 that shows that δ(p, a) n 1 δ(q, a). Then the string ax of length n shows that p n q. This gives an O( Σ Q ) time algorithm to compute n given n 1, and so an O(n Σ Q ) time algorithm to compute n from scratch. Now we come to an important observation: If, for some k, we have that the relation k is the same as the relation k+1, then also k is the same as n for all n > k. To see that it is true, note that the process that we use to go from k to k+1 is independent of k, and so if it does not change the relation when applied once, it will not change the relation if applied an arbitrary number of times. This shows that the algorithm converges to a fixed partition in Q 1 steps, because it starts with two equivalence classes, it cannot create more than Q equivalence classes, and at each step it must either increase the number of equivalence classes or reach the final partition. In particular, if we let k = Q 1, then we have that k is the same as n for every n k, and it can be computed in time at most O( Q 2 Σ ). We can also see that k is the same relation as. This is true because if p q then certainly p k q. But also, if p q, then there is a string x that shows this is the case, and if we call n the length of x we have that p n q. If n k, then certainly this also means that p k q; if n > k we can still say that p k q because we have observed above that n and k are the same for n > k. We can now finally describe our state minimization algorithm. Given M = (Q, Σ, δ, q 0, F ). Let k = Q 1 and compute the equivalence classes of Q with respect to k. Define a new automaton M = (Q, Σ, δ, q 0, F ) as follows. There is a state in Q for each equivalence class. The initial state q 0 is the equivalence class [q 0]. The set F contain all the equivalence classes that contain final states in F. 1 Define δ ([q], a) = [δ(q, a)]. Remove from Q all the states that are not reachable from q 0. Such states can be easily found doing a depth-first search of the graph of the automaton starting from q 0. The removal of these states does not change the language accepted by automaton because they never occur in a computation. Let M = (Q, Σ, δ, q 0, F ) be the new automaton. 1 Note that an equivalence class contains either only states in F or only states not in F. 4

5 The reader should verify that our definition of δ makes sense and that the automaton M decides the same language as M. To see that the algorithm constructs an optimal automaton, let t = Q be number of states of the automaton. Every state [q] of the automaton is reachable from [q 0 ], so for every state [q] there is at least a string x [q] such that M reaches [q] when given in input x [q]. Consider now two different states [p] [q]. We have p q in M, and so there must be some string y such that starting from p in M we accept when given y, but starting from q we reject (or vice versa). Also, in M, starting from [p] we accept when given y but starting from [q] we reject. This means that y shows that the strings x [q] and x [p] are distinguishable, and so we have a set of t distinguishable strings, which implies that it is impossible to decide the same language with fewer than t states. 5

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

cse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska LECTURE 6 CHAPTER 2 FINITE AUTOMATA 2. Nondeterministic Finite Automata NFA 3. Finite Automata and Regular Expressions 4. Languages

### 1 More finite deterministic automata

CS 125 Section #6 Finite automata October 18, 2016 1 More finite deterministic automata Exercise. Consider the following game with two players: Repeatedly flip a coin. On heads, player 1 gets a point.

### UNIT-VI PUSHDOWN AUTOMATA

Syllabus R09 Regulation UNIT-VI PUSHDOWN AUTOMATA The context free languages have a type of automaton that defined them. This automaton, called a pushdown automaton, is an extension of the nondeterministic

### 3515ICT: Theory of Computation. Regular languages

3515ICT: Theory of Computation Regular languages Notation and concepts concerning alphabets, strings and languages, and identification of languages with problems (H, 1.5). Regular expressions (H, 3.1,

### Fooling Sets and. Lecture 5

Fooling Sets and Introduction to Nondeterministic Finite Automata Lecture 5 Proving that a language is not regular Given a language, we saw how to prove it is regular (union, intersection, concatenation,

### Equivalence of DFAs and NFAs

CS 172: Computability and Complexity Equivalence of DFAs and NFAs It s a tie! DFA NFA Sanjit A. Seshia EECS, UC Berkeley Acknowledgments: L.von Ahn, L. Blum, M. Blum What we ll do today Prove that DFAs

### Before we show how languages can be proven not regular, first, how would we show a language is regular?

CS35 Proving Languages not to be Regular Before we show how languages can be proven not regular, first, how would we show a language is regular? Although regular languages and automata are quite powerful

### Chapter 2: Finite Automata

Chapter 2: Finite Automata Peter Cappello Department of Computer Science University of California, Santa Barbara Santa Barbara, CA 93106 cappello@cs.ucsb.edu Please read the corresponding chapter before

### Nondeterministic finite automata

Lecture 3 Nondeterministic finite automata This lecture is focused on the nondeterministic finite automata (NFA) model and its relationship to the DFA model. Nondeterminism is an important concept in the

### Nondeterministic Finite Automata

Nondeterministic Finite Automata Mahesh Viswanathan Introducing Nondeterminism Consider the machine shown in Figure. Like a DFA it has finitely many states and transitions labeled by symbols from an input

### Formal Definition of Computation. August 28, 2013

August 28, 2013 Computation model The model of computation considered so far is the work performed by a finite automaton Finite automata were described informally, using state diagrams, and formally, as

### Theory of Computation

Thomas Zeugmann Hokkaido University Laboratory for Algorithmics http://www-alg.ist.hokudai.ac.jp/ thomas/toc/ Lecture 3: Finite State Automata Motivation In the previous lecture we learned how to formalize

### 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....................

### Lecture 3: Nondeterministic Finite Automata

Lecture 3: Nondeterministic Finite Automata September 5, 206 CS 00 Theory of Computation As a recap of last lecture, recall that a deterministic finite automaton (DFA) consists of (Q, Σ, δ, q 0, F ) where

### Automata and Formal Languages - CM0081 Non-Deterministic Finite Automata

Automata and Formal Languages - CM81 Non-Deterministic Finite Automata Andrés Sicard-Ramírez Universidad EAFIT Semester 217-2 Non-Deterministic Finite Automata (NFA) Introduction q i a a q j a q k The

### Extended transition function of a DFA

Extended transition function of a DFA The next two pages describe the extended transition function of a DFA in a more detailed way than Handout 3.. p./43 Formal approach to accepted strings We define the

### September 7, Formal Definition of a Nondeterministic Finite Automaton

Formal Definition of a Nondeterministic Finite Automaton September 7, 2014 A comment first The formal definition of an NFA is similar to that of a DFA. Both have states, an alphabet, transition function,

### Properties of Regular Languages. BBM Automata Theory and Formal Languages 1

Properties of Regular Languages BBM 401 - Automata Theory and Formal Languages 1 Properties of Regular Languages Pumping Lemma: Every regular language satisfies the pumping lemma. A non-regular language

### 6.045 Final Exam Solutions

6.045J/18.400J: Automata, Computability and Complexity Prof. Nancy Lynch, Nati Srebro 6.045 Final Exam Solutions May 18, 2004 Susan Hohenberger Name: Please write your name on each page. This exam is open

### Deterministic Finite Automata. Non deterministic finite automata. Non-Deterministic Finite Automata (NFA) Non-Deterministic Finite Automata (NFA)

Deterministic Finite Automata Non deterministic finite automata Automata we ve been dealing with have been deterministic For every state and every alphabet symbol there is exactly one move that the machine

### CS243, Logic and Computation Nondeterministic finite automata

CS243, Prof. Alvarez NONDETERMINISTIC FINITE AUTOMATA (NFA) Prof. Sergio A. Alvarez http://www.cs.bc.edu/ alvarez/ Maloney Hall, room 569 alvarez@cs.bc.edu Computer Science Department voice: (67) 552-4333

### ECS 120: Theory of Computation UC Davis Phillip Rogaway February 16, Midterm Exam

ECS 120: Theory of Computation Handout MT UC Davis Phillip Rogaway February 16, 2012 Midterm Exam Instructions: The exam has six pages, including this cover page, printed out two-sided (no more wasted

### PS2 - Comments. University of Virginia - cs3102: Theory of Computation Spring 2010

University of Virginia - cs3102: Theory of Computation Spring 2010 PS2 - Comments Average: 77.4 (full credit for each question is 100 points) Distribution (of 54 submissions): 90, 12; 80 89, 11; 70-79,

### Finite Automata and Regular languages

Finite Automata and Regular languages 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/

### 1 Computational Problems

Stanford University CS254: Computational Complexity Handout 2 Luca Trevisan March 31, 2010 Last revised 4/29/2010 In this lecture we define NP, we state the P versus NP problem, we prove that its formulation

### Chapter 2: Finite Automata

Chapter 2: Finite Automata 2.1 States, State Diagrams, and Transitions Finite automaton is the simplest acceptor or recognizer for language specification. It is also the simplest model of a computer. A

### Finite Automata (contd)

Finite Automata (contd) CS 2800: Discrete Structures, Fall 2014 Sid Chaudhuri Recap: Deterministic Finite Automaton A DFA is a 5-tuple M = (Q, Σ, δ, q 0, F) Q is a fnite set of states Σ is a fnite input

### Languages. Non deterministic finite automata with ε transitions. First there was the DFA. Finite Automata. Non-Deterministic Finite Automata (NFA)

Languages Non deterministic finite automata with ε transitions Recall What is a language? What is a class of languages? Finite Automata Consists of A set of states (Q) A start state (q o ) A set of accepting

### Warshall s algorithm

Regular Expressions [1] Warshall s algorithm See Floyd-Warshall algorithm on Wikipedia The Floyd-Warshall algorithm is a graph analysis algorithm for finding shortest paths in a weigthed, directed graph

### Formal Definition of a Finite Automaton. August 26, 2013

August 26, 2013 Why a formal definition? A formal definition is precise: - It resolves any uncertainties about what is allowed in a finite automaton such as the number of accept states and number of transitions

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

### CONCATENATION AND KLEENE STAR ON DETERMINISTIC FINITE AUTOMATA

1 CONCATENATION AND KLEENE STAR ON DETERMINISTIC FINITE AUTOMATA GUO-QIANG ZHANG, XIANGNAN ZHOU, ROBERT FRASER, LICONG CUI Department of Electrical Engineering and Computer Science, Case Western Reserve

### Büchi Automata and their closure properties. - Ajith S and Ankit Kumar

Büchi Automata and their closure properties - Ajith S and Ankit Kumar Motivation Conventional programs accept input, compute, output result, then terminate Reactive program : not expected to terminate

### October 6, Equivalence of Pushdown Automata with Context-Free Gramm

Equivalence of Pushdown Automata with Context-Free Grammar October 6, 2013 Motivation Motivation CFG and PDA are equivalent in power: a CFG generates a context-free language and a PDA recognizes a context-free

### 3 Finite-State Machines

Lecture 3: Finite-State Machines [Fa 4] Caveat lector! This is the first edition of this lecture note. A few topics are missing, and there are almost certainly a few serious errors. Please send bug reports

### The efficiency of identifying timed automata and the power of clocks

The efficiency of identifying timed automata and the power of clocks Sicco Verwer a,b,1,, Mathijs de Weerdt b, Cees Witteveen b a Eindhoven University of Technology, Department of Mathematics and Computer

### 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

### Finite State Automata

Trento 2005 p. 1/4 Finite State Automata Automata: Theory and Practice Paritosh K. Pandya (TIFR, Mumbai, India) Unversity of Trento 10-24 May 2005 Trento 2005 p. 2/4 Finite Word Langauges Alphabet Σ is

### 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

### Notes on Zero Knowledge

U.C. Berkeley CS172: Automata, Computability and Complexity Handout 9 Professor Luca Trevisan 4/21/2015 Notes on Zero Knowledge These notes on zero knowledge protocols for quadratic residuosity are based

### Finite-state Machines: Theory and Applications

Finite-state Machines: Theory and Applications Unweighted Finite-state Automata Thomas Hanneforth Institut für Linguistik Universität Potsdam December 10, 2008 Thomas Hanneforth (Universität Potsdam) Finite-state

### CS154. Non-Regular Languages, Minimizing DFAs

CS54 Non-Regular Languages, Minimizing FAs CS54 Homework is due! Homework 2 will appear this afternoon 2 The Pumping Lemma: Structure in Regular Languages Let L be a regular language Then there is a positive

### 1 Two-Way Deterministic Finite Automata

1 Two-Way Deterministic Finite Automata 1.1 Introduction Hing Leung 1 In 1943, McCulloch and Pitts [4] published a pioneering work on a model for studying the behavior of the nervous systems. Following

### Notes for Lecture 2. Statement of the PCP Theorem and Constraint Satisfaction

U.C. Berkeley Handout N2 CS294: PCP and Hardness of Approximation January 23, 2006 Professor Luca Trevisan Scribe: Luca Trevisan Notes for Lecture 2 These notes are based on my survey paper [5]. L.T. Statement

### Finite Automata Part One

Finite Automata Part One Computability Theory What problems can we solve with a computer? What problems can we solve with a computer? What kind of computer? Computers are Messy http://www.intel.com/design/intarch/prodbref/27273.htm

### CSE 135: Introduction to Theory of Computation Nondeterministic Finite Automata (cont )

CSE 135: Introduction to Theory of Computation Nondeterministic Finite Automata (cont ) Sungjin Im University of California, Merced 2-3-214 Example II A ɛ B ɛ D F C E Example II A ɛ B ɛ D F C E NFA accepting

### Lecture 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

### Fall 1999 Formal Language Theory Dr. R. Boyer. 1. There are other methods of nding a regular expression equivalent to a nite automaton in

Fall 1999 Formal Language Theory Dr. R. Boyer Week Four: Regular Languages; Pumping Lemma 1. There are other methods of nding a regular expression equivalent to a nite automaton in addition to the ones

### Uses of finite automata

Chapter 2 :Finite Automata 2.1 Finite Automata Automata are computational devices to solve language recognition problems. Language recognition problem is to determine whether a word belongs to a language.

### Deterministic (DFA) There is a fixed number of states and we can only be in one state at a time

CS35 - Finite utomata This handout will describe finite automata, a mechanism that can be used to construct regular languages. We ll describe regular languages in an upcoming set of lecture notes. We will

### CS 154, Lecture 4: Limitations on DFAs (I), Pumping Lemma, Minimizing DFAs

CS 154, Lecture 4: Limitations on FAs (I), Pumping Lemma, Minimizing FAs Regular or Not? Non-Regular Languages = { w w has equal number of occurrences of 01 and 10 } REGULAR! C = { w w has equal number

### U.C. Berkeley CS278: Computational Complexity Professor Luca Trevisan 9/6/2004. Notes for Lecture 3

U.C. Berkeley CS278: Computational Complexity Handout N3 Professor Luca Trevisan 9/6/2004 Notes for Lecture 3 Revised 10/6/04 1 Space-Bounded Complexity Classes A machine solves a problem using space s(

### IN THIS paper we investigate the diagnosability of stochastic

476 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 50, NO 4, APRIL 2005 Diagnosability of Stochastic Discrete-Event Systems David Thorsley and Demosthenis Teneketzis, Fellow, IEEE Abstract We investigate

### Lecture 23 : Nondeterministic Finite Automata DRAFT Connection between Regular Expressions and Finite Automata

CS/Math 24: Introduction to Discrete Mathematics 4/2/2 Lecture 23 : Nondeterministic Finite Automata Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last time we designed finite state automata

### Incorrect reasoning about RL. Equivalence of NFA, DFA. Epsilon Closure. Proving equivalence. One direction is easy:

Incorrect reasoning about RL Since L 1 = {w w=a n, n N}, L 2 = {w w = b n, n N} are regular, therefore L 1 L 2 = {w w=a n b n, n N} is regular If L 1 is a regular language, then L 2 = {w R w L 1 } is regular,

### Introduction to Formal Languages, Automata and Computability p.1/51

Introduction to Formal Languages, Automata and Computability Finite State Automata K. Krithivasan and R. Rama Introduction to Formal Languages, Automata and Computability p.1/51 Introduction As another

### arxiv: v2 [cs.fl] 29 Nov 2013

A Survey of Multi-Tape Automata Carlo A. Furia May 2012 arxiv:1205.0178v2 [cs.fl] 29 Nov 2013 Abstract This paper summarizes the fundamental expressiveness, closure, and decidability properties of various

### Computer Sciences Department

1 Reference Book: INTRODUCTION TO THE THEORY OF COMPUTATION, SECOND EDITION, by: MICHAEL SIPSER 3 objectives Finite automaton Infinite automaton Formal definition State diagram Regular and Non-regular

### CpSc 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

### Automata: a short introduction

ILIAS, University of Luxembourg Discrete Mathematics II May 2012 What is a computer? Real computers are complicated; We abstract up to an essential model of computation; We begin with the simplest possible

### Lecture 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

### Nondeterministic Finite Automata

Nondeterministic Finite Automata COMP2600 Formal Methods for Software Engineering Katya Lebedeva Australian National University Semester 2, 206 Slides by Katya Lebedeva. COMP 2600 Nondeterministic Finite

### Nondeterminism and Epsilon Transitions

Nondeterminism and Epsilon Transitions Mridul Aanjaneya Stanford University June 28, 22 Mridul Aanjaneya Automata Theory / 3 Challenge Problem Question Prove that any square with side length a power of

### 1 Functions, relations, and infinite cardinality

Discrete Structures Prelim 2 sample questions CS2800 Questions selected for fall 2017 1 Functions, relations, and infinite cardinality 1. True/false. For each of the following statements, indicate whether

### CSE 105 THEORY OF COMPUTATION

CSE 105 THEORY OF COMPUTATION Spring 2017 http://cseweb.ucsd.edu/classes/sp17/cse105-ab/ Today's learning goals Sipser Ch 1.4 Explain the limits of the class of regular languages Justify why the Pumping

### 3 Probabilistic Turing Machines

CS 252 - Probabilistic Turing Machines (Algorithms) Additional Reading 3 and Homework problems 3 Probabilistic Turing Machines When we talked about computability, we found that nondeterminism was sometimes

### 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

### CSE 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

### Kleene Algebras and Algebraic Path Problems

Kleene Algebras and Algebraic Path Problems Davis Foote May 8, 015 1 Regular Languages 1.1 Deterministic Finite Automata A deterministic finite automaton (DFA) is a model of computation that can simulate

### 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.

### 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

### Theory of Computation p.1/?? Theory of Computation p.2/?? Unknown: Implicitly a Boolean variable: true if a word is

Abstraction of Problems Data: abstracted as a word in a given alphabet. Σ: alphabet, a finite, non-empty set of symbols. Σ : all the words of finite length built up using Σ: Conditions: abstracted as a

### On Rice s theorem. Hans Hüttel. October 2001

On Rice s theorem Hans Hüttel October 2001 We have seen that there exist languages that are Turing-acceptable but not Turing-decidable. An important example of such a language was the language of the Halting

### 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

### CSE 211. Pushdown Automata. CSE 211 (Theory of Computation) Atif Hasan Rahman

CSE 211 Pushdown Automata CSE 211 (Theory of Computation) Atif Hasan Rahman Lecturer Department of Computer Science and Engineering Bangladesh University of Engineering & Technology Adapted from slides

### Reversal of Regular Languages and State Complexity

Reversal of Regular Languages and State Complexity Juraj Šebej Institute of Computer Science, Faculty of Science, P. J. Šafárik University, Jesenná 5, 04001 Košice, Slovakia juraj.sebej@gmail.com Abstract.

### CSC173 Workshop: 13 Sept. Notes

CSC173 Workshop: 13 Sept. Notes Frank Ferraro Department of Computer Science University of Rochester September 14, 2010 1 Regular Languages and Equivalent Forms A language can be thought of a set L of

### 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.

### Automata and Computability. Solutions to Exercises

Automata and Computability Solutions to Exercises Spring 27 Alexis Maciel Department of Computer Science Clarkson University Copyright c 27 Alexis Maciel ii Contents Preface vii Introduction 2 Finite Automata

### CpSc 421 Homework 1 Solutions

CpSc 421 Homework 1 Solutions 1. (15 points) Let Σ = {a, b, c}. Figure 7 depicts two finite state machines that read Let L a and L b denote the languages recognized by DFA (a) and DFA (b) respectively.

### Axioms of Kleene Algebra

Introduction to Kleene Algebra Lecture 2 CS786 Spring 2004 January 28, 2004 Axioms of Kleene Algebra In this lecture we give the formal definition of a Kleene algebra and derive some basic consequences.

### Learning of Bi-ω Languages from Factors

JMLR: Workshop and Conference Proceedings 21:139 144, 2012 The 11th ICGI Learning of Bi-ω Languages from Factors M. Jayasrirani mjayasrirani@yahoo.co.in Arignar Anna Government Arts College, Walajapet

### Announcements. Problem Set 6 due next Monday, February 25, at 12:50PM. Midterm graded, will be returned at end of lecture.

Turing Machines 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 machine

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

cse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska LECTURE 14 SMALL REVIEW FOR FINAL SOME Y/N QUESTIONS Q1 Given Σ =, there is L over Σ Yes: = {e} and L = {e} Σ Q2 There are uncountably

### Constructive Formalization of Regular Languages

Constructive Formalization of Regular Languages Jan-Oliver Kaiser Advisors: Christian Doczkal, Gert Smolka Supervisor: Gert Smolka UdS November 7, 2012 Jan-Oliver Kaiser (UdS) Constr. Formalization of

### Properties of Regular Languages (2015/10/15)

Chapter 4 Properties of Regular Languages (25//5) Pasbag, Turkey Outline 4. Proving Languages Not to e Regular 4.2 Closure Properties of Regular Languages 4.3 Decision Properties of Regular Languages 4.4

### ,

Kolmogorov Complexity Carleton College, CS 254, Fall 2013, Prof. Joshua R. Davis based on Sipser, Introduction to the Theory of Computation 1. Introduction Kolmogorov complexity is a theory of lossless

### Notes on Space-Bounded Complexity

U.C. Berkeley CS172: Automata, Computability and Complexity Handout 7 Professor Luca Trevisan April 14, 2015 Notes on Space-Bounded Complexity These are notes for CS278, Computational Complexity, scribed

### Compositional Reasoning

EECS 219C: Computer-Aided Verification Compositional Reasoning and Learning for Model Generation Sanjit A. Seshia EECS, UC Berkeley Acknowledgments: Avrim Blum Compositional Reasoning S. A. Seshia 2 1

### 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

### Automata Theory, Computability and Complexity

Automata Theory, Computability and Complexity Mridul Aanjaneya Stanford University June 26, 22 Mridul Aanjaneya Automata Theory / 64 Course Staff Instructor: Mridul Aanjaneya Office Hours: 2:PM - 4:PM,

### COMP-330 Theory of Computation. Fall Prof. Claude Crépeau. Lecture 2 : Regular Expressions & DFAs

COMP-330 Theory of Computation Fall 2017 -- Prof. Claude Crépeau Lecture 2 : Regular Expressions & DFAs COMP 330 Fall 2017: Lectures Schedule 1-2. Introduction 1.5. Some basic mathematics 2-3. Deterministic

### Lecture 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

### Going from graphic solutions to algebraic

Going from graphic solutions to algebraic 2 variables: Graph constraints Identify corner points of feasible area Find which corner point has best objective value More variables: Think about constraints

### MINIMIZATION OF DETERMINISTIC FINITE AUTOMATA WITH VAGUE (FINAL) STATES AND INTUITIONISTIC FUZZY (FINAL) STATES

Iranian Journal of Fuzzy Systems Vol. 10, No. 1, (2013) pp. 75-88 75 MINIMIZATION OF DETERMINISTIC FINITE AUTOMATA WITH VAGUE (FINAL) STATES AND INTUITIONISTIC FUZZY (FINAL) STATES A. CHOUBEY AND K. M.

### Deterministic Finite Automata

Deterministic Finite Automata COMP2600 Formal Methods for Software Engineering Katya Lebedeva Australian National University Semester 2, 2016 Slides by Ranald Clouston and Katya Lebedeva. COMP 2600 Deterministic

### FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY

5-453 FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY NON-DETERMINISM and REGULAR OPERATIONS THURSDAY JAN 6 UNION THEOREM The union of two regular languages is also a regular language Regular Languages Are

### 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