# Decidable Problems Concerning Regular Languages

Note: there is no requirement that the Turing Machine should halt for strings not in the language. Construct an algorithm recognizing AP as follows: when presented with a string (v,w), check if v is an algorithm for recognizing languages or not. Decidable Languages • We start with problems that are decidable – We first look at problems concerning regular languages and then those for context-free languages • …eventually we will move to problems concerning Turing Machines and show that some problems are not decidable! 10/10/19 Theory of Computation - Fall'19. In Section 2, we recall the de nition of multi-letter QFAs and other related de nitions, and some related results are reviewed. 1 Introduction Languages accepted by multi-tape or multi-head ﬁnite automata were introduced in [36] and [38]. We study the problem of automatic web service composition. Does a TM accept a decidable language? Does a TM accept a regular language? Does a TM accept a context-free language? Does a TM accept a ﬁnite language? Does a TM accept the empty language? Does a TM accept a language that contains all prime numbers? Slides modiﬁed by Benny Chor, based on original slides by Maurice Herlihy, Brown University. CHAPTER 4 Decidability Contents • Decidable Languages • decidable problems concerning regular languages • decidable problems concerning context-free languages • The Halting Problem • The diagonalization method • The halting problem is undecidable • A Turing unrecognizable languages. Show closure or non-closure of languages (regular or CFL) under some operation. Closure Properties of Decidable Languages Decidable languages are closed under ∪, °, *, ∩, and complement Example: Closure under ∪ Need to show that union of 2 decidable L’s is also decidable Let M1 be a decider for L1 and M2 a decider for L2 A decider M for L1 ∪L2: On input w: 1. , under the form of a regular expression on the set {1,,N}). We focus on decidable problems concerning regular languages. Text Books. Also, follow these instructions: 4. Which of the following problems are decidable? 1) Does a given program ever produce an output? 2) If L is context-free language, then is ~L also context free? 3)If L is regular, then is ~L also regular? 4) If L is recursive language, then, is ~:=L also recursive? a) 1,2,3,4 b) 1,2 c) 2,3,4 d) 3,4 e) None of the above. 3) Complement of Regular languages is also regular. describing decidable languages that involve language encodings. A decision problem P is decidable if the language L of all yes instances to P is decidable. we show that it is decidable whether a given regular language belongs to AC’. There exists a one-to-one (or injective) function of the form g : A !N. Every Regular Language is Decidable Recognizable Language. Decidable problems concerning regular languages A DFA = hB;wi B is a DFA that accepts input string w: Theorem A DFA is a decidable language. An inputed language is accepted by a computational model if it runs through the model and ends in an accepting final state. Cleeremans & McClelland, 1991) or, more simply, a repeating. Jkdirectory Page | 1 JKD Syllabus R09 Regulation UNIT-VIII COMPUTABILITY THEORY CONTEXT SENSITIVE LANGUAGE 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. All kinds of aspects of the regular languages have been studied over the past 50 years. 4 DFA can accept only regular language. We also provide a decidable logic for data trees along the same lines. Formulate this problem as a language and show that it is decidable. Zuzana Bednárov. T decides a language L if T recognizes L, and halts in all inputs. All questions are undecidable for languages recognized by general Turing machines (Rice's theorem). Nakhleh NOTES: 1. UNIT - VI COMPUTATIONAL COMPLEXITY 08 Hours Decidability: Decidable problems concerning regular languages, Decidable problems concerning context-free languages, Un-decidability, Halting Problem of TM, A Turing-unrecognizable language. Wewillthenreturntojustify. Proof: Upon halting, simply exchange the verdicts accept and reject. The problem is known to be undecidable (but semidecidable). (b) An inﬁnite regular language. We focus on decidable problems concerning regular languages. Short answers: (a) De ne the following terms and concepts: i. Gave an example of using closure properties of regular languages to prove that a language is not regular. , obtaining factorial value without recursion method. (a) L = a ncb. we’ll define a language HALTTM that’s in RE— Dec. 2-190, TR 11-12. Vector Addition Systems De nition A d-dimensional VAS is a nite set of vectors A Zd. Decidable Languages A language L is called decidable iff there is a decider M such that (ℒ M) = L. If one of these two TM languages happens to be empty, then we are back to EMPTYTM. In the present paper, we tour a fragment of this literature. Decidable and Undecidable Problems Table | TOC January 29, 2018 Anup Patel Resources , TOC Table to check Decidable and Undecidable property of all Grammar (Regular, CFL, DCFL, CSL, Recursive, Recursive Enumerable). Lecture11: Feb. To start a formal investigation of such concepts, Barcel o et al. 1 Introduction The decidability and complexity of. Context-free languages (CFLs) are generated by context-free grammars. We present a TM N that decides 𝐴. For every rule α → β, introduce a new non-terminal symbol A and replace this rule by two rules:. For v;v0: Nd it has a step v !a v0if v0= v + a. [Zai05], [CFGG04]) can be reformulated in the theory of languages. Formulate this problem as a language and show that it is decidable. The set of all context-free languages is identical to the set of languages accepted by pushdown automata, and the set of regular languages is a subset of context-free languages. Time and space measures on computation, completeness, hierarchy theorems, inherently complex problems, oracles, probabilistic computation, and interactive proof systems. [Zai05], [CFGG04]) can be reformulated in the theory of languages. The set R is the set of all decidable languages. N: On input , , where is an NFA and a string, (1) Convert NFA to an equivalent DFA (2) Run TM M from the previous Theorem on input , (3) If M accepts, accept; otherwise reject. Let's lift this to words over A:! " def= Id Nd!aw def=!w!a! L def= w2L! w where "is the empty word, a 2A, w 2A. So, it is natural to expect that A TM is NOT decidable. theory of regular languages. *Note that one can similarly show that the following language is decidable. L is said to beTuring-decidable(Recursiveor simply decidable) if there exists a TM M which decides L. O N SOME DECIDABILITY PROBLEMS CONCERNING DEVELOPMENTAL LANGUAGES Art0 SALOMAA University of Turku, Turku, Finland 1. • AREX = { R,w ⃒R is a regular expression that generates w} ‣By reduction to ANFA. Decidable problems concerning regular language Accepting problem for DFAs: testing if a given DFA accepts a given string A DFA = fhB;wijB is a DFA that accepts string wg Theorem 4. Reduce grammar to CNF. If α is a regular expression, then so is α*. INTRODUCTION TO THE THEORY OF COMPUTATION, SECOND EDITION MICHAEL SIPSER Massachusetts Institute of Technology THOMSON COURSE TECHNOLOGY Australia * Canada * Mexico * Singapore * Spain * United Kingdom * United States. Decidable Problems Concerning Regular Languages Theorems: •Let A DFA = { | B is a DFA that accepts input string w}, then A DFA is a decidable language •Let A NFA = { | B is a NFA that accepts input string w}, then A NFA is a decidable language •Let A REX = { | R is a regular expression that generates string w}, then A REX. Decidable problems concerning regular languages A DFA = hB;wi B is a DFA that accepts input string w: Theorem A DFA is a decidable language. For many operations, the problem turns out to be undecidable for given context-free L and regular R. Moreover, we. 4 DFA can accept only regular language. If we go beyond trees, however, undecidability comes immediately. • the copy language, i. In case the string does not. It established its roots during the 20th Century, as mathematicians began developing - both theoretically and literally - machines which imitated certain features of man, completing calculations more quickly and reliably. 2 Decidable Propertiesof Regular Languages We begin with certain computational problems concerning ﬁnite automata. Decidable and undecidable problems on context free grammars. Text Books. 3 Decidable Languages We have seen in the last chapter an undecidable problem, the 10th problem of Helbert. [ 2 ] Se uma linguagem é não regular, ela requer uma máquina que no mínimo Ω (log log n ) de espaço para reconhecê-la (onde n é o tamaho da entrada). Give a speciﬁc example of a language over the binary alphabet S = f0,1gthat falls into each of the following categories: (a) A ﬁnite language. That is, a decider T is guaranteed to either accept, or reject, and never fall into an infinite loop. 4 Diagonalization Method. The class P. \textbf { Solution: } Let $ U_{PDA} $ = \{ \angles {P} $ | $ $ P $ is a PDA that has useless states \}. UNDECIDABLE. 1 Sets A set is a collection of elements. Finally, a \rule of thumb" concerning decidability questions for the main computation models: \All" questions are decidable for regular languages { the known counter-examples are somewhat \arti cial" problems. ) Interesting problems regarding regular languages are generally decidable. Lecture 8 (2/17) Wrapped up discussion on regular languages and proving non-regularity (including review of complexity of transformations among DFA/NFA/RE, in particular linearity of RE to NFA transformation, and another. Rabin [13] proved that the monadic second-order theory of trees is decidable, although the complexity of the decision procedure is not elementary. 3 Halting Problem. Closure Properties of Decidable Languages Decidable languages are closed under ∪, °, *, ∩, and complement Example: Closure under ∪ Need to show that union of 2 decidable L's is also decidable Let M1 be a decider for L1 and M2 a decider for L2 A decider M for L1 ∪L2: On input w: 1. For Each Of The Following Languages, Indicate If It Is (i) Regular (ii) Context-free But Not Regular (iii) Decidable But Not Context-free (iv) Turing-recognizable But Not Decidable (v) Not Turing-recognizable Prove Your Answers. For an undecidable language, there is no Turing Machine which accepts the language and makes a decision for every input string w (TM can make decision for some input string though). A problem is decidable if there exists at least one Turing machine that halts (concludes true or false) on every possible input. The set of all context-free languages is identical to the set of languages accepted by pushdown automata, and the set of regular languages is a subset of context-free languages. For each of the following languages, specify which type it is. The weak and full pumping lemmas are not a sufficient condition of regular languages. The next proposition follows from Proposition 1. Subsection "Decidable Problems Concerning Regular Languages" of section 4. regular languages starfree languages tree automata Ehrenfeuc h tF decidable theories. L ∈ R iff L is decidable. Simulate M1 on w. Give a speciﬁc example of a language over the binary alphabet S = f0,1gthat falls into each of the following categories: (a) A ﬁnite language. Con ten ts In tro duction Mo dels and F orm ulas W ords T rees and Graphs as Mo dels FirstOrder Logic hnique and some of its applications concerning rstorder logic in formal language theory Th us some complemen tary material to the related. The fundamental result due to Nelson and Oppen yields a decision procedure for the satis ability (or dually, validity) problem concerning quanti er-free formulas in the union of the languages of two decidable theories,. Automata models for data languages, even models beyond register automata such as fresh-register automata [33] and history-register automata [15], often have decidable emptyness problems, and their (less expressive) deterministic restrictions then have decidable inclusion problems. The following theorems summarize the important results concerning reversal-bounded counter machines which we will need in the paper. Regular and recursive languages are closed under complementation. (Information Technology) Syllabus 2012 Course. In particular, recognizable languages would coincide with decidable languages (which is verycoincide with decidable languages (which is very unrealistic!). The problem whether, for a given pair of languages, the ﬁrst language has a relative density in the second one is decidable for the regular lan-guages ([Koz05]). 2: Decidable problems concerning context-free languages, pp. the set of all strings over Σ that consist of two identical halfs, • anbmcndm, • a nbnc , and • a nb cnenf 2. Show closure or non-closure of languages (regular or CFL) under some operation. Proof: For any. L1-L2 is the same as the intersection of L1 and the complement of L2. Since we can encode the DFA as a string, the acceptance problem can be seen as. 3 Halting Problem. A DFA = { | B is the DFA that accepts input string w } Testing if w is in B is the same as testing whether is in A DFA. Use language to represent various computational problems bcs we have terminology to deal with languages. To say L(G) is empty is equivalent to saying D is empty, or that D = Σ*. Decidable Problems Interesting problems regarding regular languages are generally decidable. Let L = fx j x begins with one ag. There are two important, robust classes of !-languages that are definable by fully decidable formalisms. Cleeremans & McClelland, 1991) or, more simply, a repeating. 5 Post’s Correspondence Problem. controversy and by a data problem concerning how to obtain reliable and valid evidence of learners' linguistic knowledge' (Ellis, 2005, p. Write this problem as a language. Decidable Problems Concerning Regular Languages Theorems: •Let A DFA = { | B is a DFA that accepts input string w}, then A DFA is a decidable language •Let A NFA = { | B is a NFA that accepts input string w}, then A NFA is a decidable language •Let A REX = { | R is a regular expression that generates string w}, then A REX. Reading: 1. If α , β are regular expressions, then so is α∪β. COMP481 Review Problems Turing Machines and (Un)Decidability Luay K. show that the inclusion problem for monotone AC-tree automata is not decidable. Decidable and Undecidable Problems Table | TOC January 29, 2018 Anup Patel Resources , TOC Table to check Decidable and Undecidable property of all Grammar (Regular, CFL, DCFL, CSL, Recursive, Recursive Enumerable). The problem of combining decidable rst-order theories has been widely studied (e. Decidable Languages, Decidable Problems Concerning Regular Languages, Decidable Problems Concerning Context-Free Languages. we’ll define a language HALTTM that’s in RE— Dec. Undecidability of finite convergence for concatenation, insertion and bounded shuffle operators will show the undecidability of a number of problems concerning the interaction of regular and context This follows from the fact that the membership problem for context sensitive languages is decidable and the problem to decide if L =. • Because of the decidability of the acceptance problem, all context-free languages are decidable • Hence, regular languages are a proper subset of context-free languages, which are decidable • Furthermore, decidable languages are a proper subset of Turing-recognizable languages Department of Software Systems 171. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): commonly accepted control theory for discrete event systems, due to Ramadge and Wonham [13], followed by several other [17,4], has been more recently extended to temporal logic specifications [8,2,14]. Regular Expressions [11] Regular Languages and Regular Expressions Theorem: If L is a regular language there exists a regular expression E such that L = L(E). Introduction Developmental languages were defined by Lindenmayer [7] in connection with a theory proposed to model the development of filamentous organisms. For many operations, the problem turns out to be undecidable for given context-free L and regular R. Language Decidability A language is called Decidable or Recursive if there is a Turing machine which accepts and halts on every input string w. Every decidable language is Turing-Acceptable. Proof: That AP is recognizable is shown in the same way as for the language M above. Every regular language is decidable. at the same time. Time and space measures on computation, completeness, hierarchy theorems, inherently complex problems, oracles, probabilistic computation, and interactive proof systems. Context-free languages (CFLs) are generated by context-free grammars. L ∈ R iff L is decidable. We also deﬁne a logic over proﬁnite words, called MSO+inf and show that the satisﬁability problem of MSO+B reduces to the satisﬁability problem of our logic. (b) An inﬁnite regular language. L1 TL2 = R”is decidable for a context-free language L1 and regular languages L2, R if and only if the set 0 (T)is ﬁnite. For an undecidable language, there is no Turing Machine which accepts the language and makes a decision for every input string w (TM can make decision for some input string though). Text Books. So in order to prove that I am trying to show that I can move from deterministic finite automaton (DFA) to a Turing decidable machine. , the small model property and the tree model property. (e) An undecidable, semidecidable language. 9,2019 11-3 11. In particular, CSE 105 Sp04, Problem Set 3 Solutions 5 halt. Run M on w. *Note that one can similarly show that the following language is decidable. In this paper we prove that it is decidable whether the set pow(L), which we get by taking all the powers of all the words in some regular language L, is regular or not. There exists a one-to-one (or injective) function of the form g : A !N. Undecidability of finite convergence for concatenation, insertion and bounded shuffle operators will show the undecidability of a number of problems concerning the interaction of regular and context This follows from the fact that the membership problem for context sensitive languages is decidable and the problem to decide if L =. The next proposition follows from Proposition 1. It only takes a minute to sign up. Assume, for a contradiction, that TM T decides the language Proof: We use T to decide A. The Regular Post Embedding Problem is still decidable but, because of the added. we'll argue that there are languages that aren't even in RE! Decidable and Undecidable Languages 32-5 Language Encodings. In a similar way we'll talk about other decision problems, ultimately talking about some underlying language. Introduction Developmental languages were defined by Lindenmayer [7] in connection with a theory proposed to model the development of filamentous organisms. Non-recursive should not be taken as simpler version of computation, i. We also provide a decidable logic for data trees along the same lines. On input x, M simulates (N, y) for length(x) steps. Consider the set of strings on {0,1} in which, every substring of 3 symbols has at most two zeros. The essence of "reducing one problem to another" is the existence of a function from one. As a tool, Parikh simplifying mappings are defined and studied. 2 Decidable Problems Concerning Context Free Languages; 7. Text Books. Or, a recursive language is a recursive subset in the set of all possible words over alphabet Σ of that language. More interesting is the Regular Post Embedding Problem, a further variant where one looks for solutions that belong to a given regular language (submitted, e. whether a ﬁnite automaton accepts a string, b. 1 A DFA is a decidable language. If M tries to move to the marked cell, N moves the head back to the right. Decidable problems concerning regular languages ADFA = fhB;wijBis a DFA that accepts input string wg: That is, for every w2 and DFA B w2L(B) ()hB;wi2ADFA: Theorem ADFA is a decidable language. TM were decidable, the language of any TM would be decidable. (Problem 3. Concerning Regular Languages Yes, you have to remember what they are! Acceptance problem: Testing whether a particular DFA accepts a string. To start a formal investigation of such concepts, Barcel o et al. Recall that in the tutorial, we have proved that the intersection of two regular languages is regular, so the language L0 = F \L is regular. So in order to prove that I am trying to show that I can move from deterministic finite automaton (DFA) to a Turing decidable machine. The problem whether, for a given pair of languages, the ﬁrst language has a relative density in the second one is decidable for the regular lan-guages ([Koz05]). at the same time. Regular language iv. thermore, we prove that the state minimization problem concerning multi-letter QFAs is decidable in EXPSPACE. If L is regular, L passes the conditions of the pumping lemma. All kinds of aspects of the regular languages have been studied over the past 50 years. The Regular Post Embedding Problem is still decidable but, because of the added. A ýB, and 3. For some NP-complete problems, there's a SUCCINCT variant that's NEXP-complete. Kleene's theorem v. To indicate that x is an element of the set S, we write x 2 S. [4] introduced the class of extended conjunctive regular path queries (ECRPQs), allowing to use not only regular languages to express properties of individual paths, but. Claim: There is an algorithm to decide whether a given CFL is empty. (b) An inﬁnite regular language. Undecidability of finite convergence for concatenation, insertion and bounded shuffle operators Charles E. Regular and context-free languages. Reduce grammar to CNF. Since the set of regular languages is closed under each of these operations, L1-L2 must be regular. Prove that the following languages are regular, either by exhibiting a regular expression representing the language, or a DFA/NFA that recognizes the language: [10 x 3 = 30 points] (a) all strings that do not contain the substring aba, for Σ = {a,b} (for instance, aabaa. It is context-free, but not regular. Showing that the language is decidable is the same as showing that. 6 190 Department of Software Systems OHJ-2306 Introduction to Theoretical Computer Science, Fall 2009 5. One may be led to wonder what the results will look like for infinite-state systems. Decidable problems concerning regular languages A DFA = hB;wi B is a DFA that accepts input string w: Theorem A DFA is a decidable language. We consider a formal framework where web service business protocols are described by means of Finite State Machines (FSM) and focus on the protocol synthesis problem. The word automaton itself, closely related to the. If α is a regular expression, then so is α*. This language D, as you may expect, is rather convoluted. Subsection "Decidable Problems Concerning Regular Languages" of section 4. 3 Halting Problem. Consider the set of strings on {0,1} in which, every substring of 3 symbols has at most two zeros. The closure properties and the decid-ability of equational tree automata are also summarized. Regular languages ⊆ context free languages ⊆ context sensitive. Regular languages are useful for many practical applications due to the fact that \all natural" questions concerning regular languages are decidable. Regular and context-free languages. So in order to prove that I am trying to show that I can move from deterministic finite automaton (DFA) to a Turing decidable machine. if L is decidable, then so is ATM We reduceATM to the language L ATM ≤ L We showed:ATM ≤ HALT TM Mapping Reductions f : Σ* →→→Σ* is a computable functionif there is a Turing machine M that halts with just f(w) written on its tape, for every input w. Construct an algorithm recognizing AP as follows: when presented with a string (v,w), check if v is an algorithm for recognizing languages or not. Write this problem as a language. Decidable Languages. Show that single-tape TMs that cannot write on the portion of the tape containing the input string recognize only regular languages. Identify the complexities of various decision problems concerning regular and context-free languages. As a corollary, the decidability of the generalized word problem is deduced. Hughes School of Computer Science, University of Central Florida, Orlando, FL 32816 USA Abstract The k-insertion and shuffle operations on formal languages have been extensively studied in the computer science and control systems literature. A context-free grammar for the language consisting of all strings over {a,b} which contain a different number of a's to b's is The intersection of a context-free language and a regular language is always context-free. Turing Machine FINITE STATE CONTROL INFINITE TAPE I N P U T q10 A 0 → 0, R read write move → , R qaccept A language is called decidable (recursive) if some TM decides it decidable languages recognizable languages A DFA is not regular. For example, 001110 and 011001 are in the language, but 100010 is not. All strings of length less than 3 are also in the language. 2009 • Thus, Mw either accepts the regular language { 0, 1 }* or non- regular { 0 n1 | n ·0 } • Accepting/rejecting the string w on M reduces to the question of the regularity of the language of the TM Mw • Let M ENC be a TM, which • inputs the concatenation of the code ¢M²for. 4) Recursvie language are closed under complement,so it is decidable. , [9,10,13]). Proof: Given a CFG, we can decide whether the start symbol is useful. I and the decidability of similar problems for one-way stack automata [2]. All regular languages { w | count(w, a) = count(w, b) } … All problems that can be solved by correct, terminating programs. Decidable Problems for CFLs Note: Conversions between grammars and PDAs are eﬁective. 1Introduction This paper is an attempt to establish a natural framework for problems related to the limitedness problem. The !-regular languages are definable by finite automata, or equivalently, by the Sequential Calculus. Decidable problems concerning regular language Accepting problem for DFAs: testing if a given DFA accepts a given string A DFA = fhB;wijB is a DFA that accepts string wg Theorem 4. A language A is mapping reducible to language B, written A ≤≤≤≤mB, if there is a. Although language equivalence is decidable. 18) Show that a language is decidable iﬀ some enumerator enumerates the language in lexicographic order. Decidable and Undecidable Problems Table | TOC January 29, 2018 Anup Patel Resources , TOC Table to check Decidable and Undecidable property of all Grammar (Regular, CFL, DCFL, CSL, Recursive, Recursive Enumerable). Time and space measures on computation, completeness, hierarchy theorems, inherently complex problems, oracles, probabilistic computation, and interactive proof systems. The emptiness problem is decidable for the following classes: (a) 2DCM(1) [6]. Implicit Learning in Second Language Acquisition 321. Give a speciﬁc example of a language over the binary alphabet S = f0,1gthat falls into each of the following categories: (a) A ﬁnite language. The statement that x is not in S is written as x 2= S. The idea is that its complement, D, somehow represents derivations in G. Simulate M1 on w. invariance under permutation makes any property concerning the data values, other than equality, irrelevant. Start studying CIS 262 Important Terms. Decidable Problems Any problem can be cast as a language membership problem ¼Does DFA D accept input w? Equivalent to: Is in A DFA = { | D is a DFA that accepts input w}? Decidable problems concerning languages and machines: (see textbook for proofs to some of these) ¼A DFA = { | D is a DFA that accepts input w} ¼A NFA. For example, the acceptance problem for DFAs is whether, given a DFA D and a string w, D accepts input w. For example, language equivalence and inclusion are decidable (see [9]), and for many subclasses of the regular languages it is decidable whether a given automaton accepts a lan-guage inside this subclass (see [19] for some results of this kind). Context-free languages (CFLs) are generated by context-free grammars. length from L belongs to S. at the same time. the set of all strings over Σ that consist of two identical halfs, • anbmcndm, • a nbnc , and • a nb cnenf 2. Gave an example of using closure properties of regular languages to prove that a language is not regular. 2 is dedicated to the control synthesis for the. , their lengths, labels, or similarity. As a tool, Parikh simplifying mappings are defined and studied. For example, 001110 and 011001 are in the language, but 100010 is not. Kleene's theorem v. martin, "Introduction to Language and Theory of Computation", TMH. Regarding the star height problem, we identify a subclass of the regular languages for which we can precisely determine the computational complexity of the star height problem. Eilenberg proposed as a framework for this classi cation the so-called varieties of languages and showed how they are in natural one-to-one correspondence with pseudovarieties of semigroups. For question 29 is 7 and not 5; RE: Theory of Computation questions and answers -Preethi (02/12/15) i think there is a mistake in question29. Some problems are the recognition of languages, and this is how we relate problem classes to language classes. Prove that if L is regular then Prefix(L) is regular. Regular and recursive languages are closed under complementation. 5,page148ofthetext. we'll argue that there are languages that aren't even in RE! Decidable and Undecidable Languages 32-5 Language Encodings. Decidable Languages. Proofs for this result usually rely on model-theoretic properties of PDL, e. The following are the differences between Decidable and Undecidable Problems: Decidable Problems: A decidable problems are the problem for which we are able to built an algorithm that give the answer view the full answer. Concerning the nature of the supervision, it is natural and standard to suppose a partial observation of the plant, as information on it moves and states is incomplete; we then talk about control under partial observation (see [11. we show that it is decidable whether a given regular language belongs to AC'. Otherwise, it isn't in the resulting language. An example is SUCCINCT HAMILTON PATH: A Boolean circuit with 2n inputs and one output represents a graph on 2 n vertices. As a corollary, the decidability of the generalized word problem is deduced. Prove that this problem is decidable. Sound and complete equational theories exist for the various known equivalences, an elegant example is [18]. The statement that x is not in S is written as x 2= S. Prove that if L1 is regular and L2 is regular then so is L1-L2 (the set of all strings in L1 but not in L2). Show closure or non-closure of languages (regular or CFL) under some operation. Let M A be the total TM recognizing language A. How to Understand and Create Mapping Reductions What is a mapping reduction? A mapping reduction A m B(or A P B) is an algorithm (respectively, polytime algorithm) that can transform any instance of decision problem Ainto an instance of decision problem B, in such a way that the answer correspondence property holds. More interesting is the Regular Post Embedding Problem, a further variant where one looks for solutions that belong to a given regular language (submitted, e. Concerning the nature of the supervision, it is natural and standard to suppose a partial observation of the plant, as information on it moves and states is incomplete; we then talk about control under partial observation (see [11. Convert grammar to PDA. Consider the set of strings on {0,1} in which, every substring of 3 symbols has at most two zeros. IScan the input string repeatedly. Decidable and Undecidable Problems Table | TOC January 29, 2018 Anup Patel Resources , TOC Table to check Decidable and Undecidable property of all Grammar (Regular, CFL, DCFL, CSL, Recursive, Recursive Enumerable). Practice Problems for the Final Exam 1. In the present paper, we tour a fragment of this literature. A decision problem P is called “undecidable” if the language L of all yes instances to P is not decidable. Non-recursive should not be taken as simpler version of computation, i. Since K and L are decidable languages, it follows that there exist turing. Showing that the language is decidable is the same as showing that. Decidable problems concerning regular language Accepting problem for DFAs: testing if a given DFA accepts a given string A DFA = fhB;wijB is a DFA that accepts string wg Theorem 4. A problem is decidable if there exists at least one Turing machine that halts (concludes true or false) on every possible input. 8 to give another proof that every regular language is context free, by showing how to convert a regular expression directly to an equivalent co n-text free grammar. On Rice's theorem 3 The class of regular languages The class of Turing-decidable (i. In a slight abuse of terminology, a language is called decidable if it has a decidable grammar. 1 Introduction The decidability and complexity of. A context-free grammar for the language consisting of all strings over {a,b} which contain a different number of a's to b's is The intersection of a context-free language and a regular language is always context-free. Hence it is difficult to generalize to other classical problems concerning certain classes of regular languages. Basic Properties of Turing-recognizable Languages Theorem A Let A, B Y -* be Turing-decidable languages. In these cases, the proofs are based on Theorems 5 and 6, and on the (eﬀective) closure of the family of regular languages under the considered operations. Thus all standard decision questions concerning transductions of 0-FST’s are decidable. A is countable. The closure properties and the decid-ability of equational tree automata are also summarized. If one of these two TM languages happens to be empty, then we are back to EMPTYTM. In fact, the following proof effectively shows that the complement of a context-free language may not even be recursive (i. A language is Recognizable iff there is a Turing Machine which will halt and accept only the strings in that language and for strings not in the language, the TM either rejects, or does not halt at all. • the copy language, i. When proving closure of the class of decidable languages under a given operation the obvious choice is an assumed decider for a given decidable language. , the set of strings generated by regular expression R) is equal to the language defined by T. Turing Machine FINITE STATE A language is called decidable (recursive) if That is, A DFA is not regular. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): commonly accepted control theory for discrete event systems, due to Ramadge and Wonham [13], followed by several other [17,4], has been more recently extended to temporal logic specifications [8,2,14]. We shall represent computational decision problems as problems about languages. All strings of length less than 3 are also in the language. Decidable problems concerning regular languages ADFA = fhB;wijBis a DFA that accepts input string wg: That is, for every w2 and DFA B w2L(B) ()hB;wi2ADFA: Theorem ADFA is a decidable language. We consider a formal framework where web service business protocols are described by means of Finite State Machines (FSM) and focus on the protocol synthesis problem. algorithmic properties. Fi ll Reg = Regular Languages a*b* (a+b)*bbb(a+b)* Finally: 3. For many operations, the problem turns out to be undecidable for given context-free L and regular R. Reducibility: Un-decidable Problems from Language Theory, A Simple Un-decidable Problem PCP, Mapping Reducibility. Although it might take a staggeringly long time, M will eventually accept or reject w. The same methods are used in Section 6 to characterize the regular languages that are AC’-reducible to addition modulo p, where p is prime (that is, the regular languages in the circuit complexity class ACCCp)). IEvery nite language is decidable: For example, by a TM that has all the strings in the language \hard-coded" into it IWe just saw some example algorithms all of which terminate in a nite number of steps, and output yes or no (accept or reject. If M tries to move to the marked cell, N moves the head back to the right. Decidability of inclusion. 4 Diagonalization Method. This is decidable. The following are the differences between Decidable and Undecidable Problems: Decidable Problems: A decidable problems are the problem for which we are able to built an algorithm that give the answer view the full answer. Hence it is difficult to generalize to other classical problems concerning certain classes of regular languages. Vector Addition Systems De nition A d-dimensional VAS is a nite set of vectors A Zd. But ﬁrst, here is so me notation and terminology to do with character strings that we will be using throughout the course. 11/1: Polynomial-time computation. thermore, we prove that the state minimization problem concerning multi-letter QFAs is decidable in EXPSPACE. Consider the set of strings on {0,1} in which, every substring of 3 symbols has at most two zeros. A recognizer of a language is a machine that recognizes that language; A decider of a language is a machine that decides that language; Both types of machine halt in the Accept state on strings that are in the language ; A Decider also halts if the string is not in the language ; A Recogizer MAY or MAY NOT halt on strings that are not in the. c) If r1 is a regular expression associated with L1 and r2 is a regular expression associated with L1: i) (r1)(r2) = L1L2 ii) r1 + r2 = L1 + L2 iii) (r1)* = L1*. Reducibility: Un-decidable Problems from Language Theory, A Simple Un-decidable Problem PCP, Mapping Reducibility Time Complexity: Measuring Complexity, The. In this handout, I regularly make use of two problems, namely † The Halting Problem, denoted by HP, and dened as HP = fhM;wijM is a TM and it halts on string wg. 3 studies algorithmically decidable problems concerning context-free grammars. Since K and L are decidable languages, it follows that there exist turing. Prove that every regular language is decidable. Regular Expressions [11] Regular Languages and Regular Expressions Theorem: If L is a regular language there exists a regular expression E such that L = L(E). Although it might take a staggeringly long time, M will eventually accept or reject w. Simulate M1 on w. Although language equivalence is decidable. Run M on w. A Note on Decidable Separability by Piecewise Testable Languages. The set R is the set of all decidable languages. Cleeremans & McClelland, 1991) or, more simply, a repeating. T's states will be similar to D's. 8 to give another proof that every regular language is context free, by showing how to convert a regular expression directly to an equivalent co n-text free grammar. 3 Prove that the following computational problems concerning ﬁnite automata are decidable: (3 pts) a. A language L is called “decidable” if there exists a halting Turing machine M such that L(M) = L. 5,page148ofthetext. As shown in [5], satisﬁability is decidable for PDL. Thus all standard decision questions concerning transductions of 0-FST's are decidable. Either A is ﬁnite or there exists a one-to-one and onto (or bijective) function of the form h : N!A. To determine if there is an edge between vertices i and j, encode i and j in n bits each, and feed their concatenation to the circuit: there is an edge between these vertices iff the output of. , obtaining factorial value without recursion method. Otherwise, it isn't in the resulting language. Regular and context-free languages. Decidability of inclusion. Examples: PALINDROMES = { w | w is a palindrome } is decidable. A Decidable Class of Problems for Control under Partial Observation temporal logic definable behaviors, etc. 4 Diagonalization Method. A DFA = { | B is the DFA that accepts input string w } Testing if w is in B is the same as testing whether is in A DFA. You may be expected to gure this out for yourself or a hint may be provided. Consider first the problem of deciding whether a particular finite automaton accepts a given string. 1 A DFA is a decidable language. (e) An undecidable, semidecidable language. This language D, as you may expect, is rather convoluted. Since the set of regular languages is closed under each of these operations, L1-L2 must be regular. Equivalence and rationality problems are shown to be decidable for algebraic series with noncommuting variables having bounded supports. Let be a decidable language. A partially completed DFA that accepts this language is shown below. To prove that a language such as this is regular, one uses the Myhill-Nerode theorem or the pumping lemma. Decision Problems for the Veriﬁcation of Real-Time Software 203 φ and the resulting clock valuation ν = ν ↓ λ. 1 introduces the quantified mu-calculus and show its adequacy to control specification; Sec. Context-free languages (CFLs) are generated by context-free grammars. Regular and context-free languages. 15) Show that the collection of decidable languages is closed under (a) Union: (in the textbook). Run M on w. recursive) languages 2 2 Rice’s theorem Rice‘s theorem was originally shown in [4]. As a consequence all problems are decidable for context-free languages. Does a TM accept a decidable language? Does a TM accept a regular language? Does a TM accept a context-free language? Does a TM accept a ﬁnite language? Does a TM accept the empty language? Does a TM accept a language that contains all prime numbers? Slides modiﬁed by Benny Chor, based on original slides by Maurice Herlihy, Brown University. As a corollary, the decidability of the generalized word problem is deduced. The search for a more precise divid- ing line is still active, with the most outstanding open problem concerning the decidability of language equivalence between deterministic push-down automata. The converse is not true: for example the language consisting of all strings having the same number of a's as b's is context-free but not regular. 5 Post's Correspondence Problem. As a tool, Parikh simplifying mappings are defined and studied. By exchanging the accepting and rejecting final state of M A with each other, we. 5,page148ofthetext. Proof: Upon halting, simply exchange the verdicts accept and reject. 1 Decidable Problems Concerning Regular Languages; 7. 8 to give another proof that every regular language is context free, by showing how to convert a regular expression directly to an equivalent co n-text free grammar. 9,2019 11-3 11. PROPOSITION 1. If not, then reject. If you have just a thousand classes, ﬁnding the best schedule may require centuries, even with a supercomputer. We will design a TM $ T $ that decides $ U_{PDA} $. In this paper we discuss decision problems concerning these four properties. (Problem 3. A language is Recognizable iff there is a Turing Machine which will halt and accept only the strings in that language and for strings not in the language, the TM either rejects, or does not halt at all. Convert to a CFG G. ular languages are closed under a wide range of operations and, even more importantly, almost any interesting problem concerning regular languages is decidable. Regular and recursive languages are closed under complementation. Decidable Languages • We start with problems that are decidable – We first look at problems concerning regular languages and then those for context-free languages • …eventually we will move to problems concerning Turing Machines and show that some problems are not decidable! 10/10/19 Theory of Computation - Fall'19. (b) Concatenation: Let K,L be decidable languages. instead is S it should be either 0 or 1 according to the given diagram. Decision Problems for the Veriﬁcation of Real-Time Software 203 φ and the resulting clock valuation ν = ν ↓ λ. We give algo-rithms for testing, a. Either A is ﬁnite or there exists a one-to-one and onto (or bijective) function of the form h : N!A. Let's lift this to words over A:! " def= Id Nd!aw def=!w!a! L def= w2L! w where "is the empty word, a 2A, w 2A. Prove that M recognizes a regular language. , obtaining factorial value without recursion method. Decidability of inclusion. All strings of length less than 3 are also in the language. (a) Suppose on the contrary that F is regular. Answer: The key idea is to design three states q0;q1;q2, where q0 speciﬂes the input string does not end with 0, q1 speciﬂes the input string ends with exactly one 0, and q2 speciﬂes the input string ends with at least. Regular languages ⊆ context free languages ⊆ context sensitive. Remarkably, we don't know. Since the set of regular languages is closed under each of these operations, L1-L2 must be regular. This can be seen by observing the DFA -- if DFA contains a state which contains a loop and that state is reachable from the start state and that state is either a final state or leading to final state, then the language will be infinite. Showing that the language is decidable is the same as showing that. This is the basis of a new automated termination criterion. 11/1: Polynomial-time computation. INTRODUCTION TO THE THEORY OF COMPUTATION, SECOND EDITION MICHAEL SIPSER Massachusetts Institute of Technology THOMSON COURSE TECHNOLOGY Australia * Canada * Mexico * Singapore * Spain * United Kingdom * United States. recognize the same language). Convert grammar to PDA. we'll argue that there are languages that aren't even in RE! Decidable and Undecidable Languages 32-5 Language Encodings. Recap A decision problem is decidable (solvable, recursive) if ∃a TM that a) halts on every input, and b) always gives the right answer A decision problem is partially decidable (partially solvable, recursively enumerable) Regular Languages yExample: strings with no b after an a. The set of all context-free languages is identical to the set of languages accepted by pushdown automata, and the set of regular languages is a subset of context-free languages. Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics : Symbolic Combinatorics. Examples: PALINDROMES = { w | w is a palindrome } is decidable. The class NP. 3 Decidable Languages We have seen in the last chapter an undecidable problem, the 10th problem of Helbert. On Rice‘s theorem 3 The class of regular languages The class of Turing-decidable (i. The concatenation of languages K and L is the language KL = {xy|x ∈ K and y ∈ L}. 11/3: Nondeterministic polynomial-time computation. For example, it is good to have two or three different succinct versions of the same NP-complete problem as examples, if the succinct encodings come in slightly. Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics : Symbolic Combinatorics. $\endgroup$ – fade2black Sep 22 '17 at 17:45. Decidable Problems Interesting problems regarding regular languages are generally decidable. If α is a regular expression, then so is α*. Languages recognized by a TM are called recognizable. Decidable Problems for CFLs Note: Conversions between grammars and PDAs are eﬁective. For example, it is good to have two or three different succinct versions of the same NP-complete problem as examples, if the succinct encodings come in slightly. The following theorems summarize the important results concerning reversal-bounded counter machines which we will need in the paper. Identify the complexities of various decision problems concerning regular and context-free languages. Hughes School of Computer Science, University of Central Florida, Orlando, FL 32816 USA Abstract The k-insertion and shuffle operations on formal languages have been extensively studied in the computer science and control systems literature. Languages decided by a TM are called decidable. 5 Post's Correspondence Problem. Show closure or non-closure of languages (regular or CFL) under some operation. 6 190 Department of Software Systems OHJ-2306 Introduction to Theoretical Computer Science, Fall 2009 5. non-recursive trade-offs and, thus, the borderline between decidable and undecidable problems. Non-recursive should not be taken as simpler version of computation, i. Decidable Languages, Decidable Problems Concerning Regular Languages, Decidable Problems Concerning Context-Free Languages. 1 Claim: There is an algorithm to decide whether a given CFL contains a given string w. If not, then reject. Decidable problems concerning context-free languages A CFG = hG;wi G is a CFG that generates w: Theorem A CFG is a decidable language. a context-free language. The statement that x is not in S is written as x 2= S. Regular and context-free languages. Decidable Problems Interesting problems regarding regular languages are generally decidable. Hughes School of Computer Science, University of Central Florida, Orlando, FL 32816 USA Abstract The k-insertion and shuffle operations on formal languages have been extensively studied in the computer science and control systems literature. Regular Expressions [11] Regular Languages and Regular Expressions Theorem: If L is a regular language there exists a regular expression E such that L = L(E). If M1 accepts, then ACCEPT w. CS5371 Theory of Computation Homework 1 (Solution) 1. Vector Addition Systems De nition A d-dimensional VAS is a nite set of vectors A Zd. Decidable Languages A language L is called decidable iff there is a decider M such that (ℒ M) = L. Instead,weclaimthefollowing,anduseitintheconstruction. A decision problem P is called "undecidable" if the language L of all yes instances to P is not decidable. For an undecidable language, there is no Turing Machine which accepts the language and makes a decision for every input string w (TM can make decision for some input string though). (Problem 3. 1 The downside is that the family of regular languages is quite small. One can often show that a language Lis undecidableby showing that if L is decidable, then so is ATM We reduceATM to the language L ATM ≤ L We showed:ATM ≤ HALT TM Mapping Reductions f : Σ* →→→Σ* is a computable functionif there is a Turing machine M that halts with just f(w) written on its tape, for every input w. The point is that, miraculously, D is context-free! The language D. I understand the definition of decidable is basically if we have an algorithm that decides every instance of the problem (3-SAT in this case). A partially completed DFA that accepts this language is shown below. 1 (Rice‘s theorem) If Sis a non-trivial property of Turing-acceptable languages, then the problem ‘Does L(M) have the property S?’ is undecidable. \textbf { Solution: } Let $ U_{PDA} $ = \{ \angles {P} $ | $ $ P $ is a PDA that has useless states \}. Run M on w. IScan the input string repeatedly. Para provar que uma linguagem como essa não é regular, usamos o Teorema de Myhill-Nerode. Consider the set of strings on {0,1} in which, every substring of 3 symbols has at most two zeros. By exchanging the accepting and rejecting final state of M A with each other, we. If not, then reject. Only the House has the power to impeach members of the executive and judicial branches of government. , [9,10,13]). For many operations, the problem turns out to be undecidable for given context-free L and regular R. A grammar G is called decidable if the membership problem is decidable for every string of terminals of that grammar. An inputed language is accepted by a computational model if it runs through the model and ends in an accepting final state. 1 Decidable Problems Concerning Regular Languages; 7. recursive) languages 2 2 Rice's theorem Rice's theorem was originally shown in [4]. if L is decidable, then so is ATM We reduceATM to the language L ATM ≤ L We showed:ATM ≤ HALT TM Mapping Reductions f : Σ* →→→Σ* is a computable functionif there is a Turing machine M that halts with just f(w) written on its tape, for every input w. 1 Introduction The decidability and complexity of. Recall the inductive definition of regular expressions that was given in class : 1. Since the set of regular languages is closed under each of these operations, L1-L2 must be regular. CS 341 Homework 16 Languages that Are and Are Not Context-Free 1. Let's lift this to words over A:! " def= Id Nd!aw def=!w!a! L def= w2L! w where "is the empty word, a 2A, w 2A. Show that the compliment of regular language is also regular. Turakainen Department of Mathematical Sciences, University of Oulu, SF-90570 Oulu, Finland Received May 1996; revised August 1996 Communicated by A. 4 Diagonalization Method. To indicate that x is an element of the set S, we write x 2 S. A major theme in relational database theory is navigating the tradeoff between expressiveness and tractability for query languages, where the query-containment problem is considered a benchmark of tractability. Also, follow these instructions: 4. The set R is the set of all decidable languages. In a slight abuse of terminology, a language is called decidable if it has a decidable grammar. Turakainen Department of Mathematical Sciences, University of Oulu, SF-90570 Oulu, Finland Received May 1996; revised August 1996 Communicated by A. A partially completed DFA that accepts this language is shown below. For the halting problem instance (N, y), create a new machine M for the L problem. We focus on decidable problems concerning regular languages. \textbf { Solution: } Let $ U_{PDA} $ = \{ \angles {P} $ | $ $ P $ is a PDA that has useless states \}. 1 Decidable Problems Concerning Regular Languages; 7. Concerning Regular Languages Yes, you have to remember what they are! Acceptance problem: Testing whether a particular DFA accepts a string. For question 29 is 7 and not 5; RE: Theory of Computation questions and answers -Preethi (02/12/15) i think there is a mistake in question29. Regular language iv. 2 Decidability and Decidable Languages. It is shown that the cardinality equivalence problem is undecidable for e-free finite substitutions. A language L is called "decidable" if there exists a halting Turing machine M such that L(M) = L. Let R be any set of regular languages is U Ri regular?Prove it. CS5371 Theory of Computation Homework 1 (Solution) 1. It is context-free, but not regular. 3 studies algorithmically decidable problems concerning context-free grammars. , their lengths, labels, or similarity. Languages recognized by a TM are called recognizable. Acall transition (q,a,φ,q,γ,λ) is a move on the (call) input symbol a from q to q where ν satisﬁes φ,theclock valuation is updated from ν to ν ↓ λ,andγ is pushed on the stack. The problem of combining decidable rst-order theories has been widely studied (e. 2-190, TR 11-12. Proofs for this result usually rely on model-theoretic properties of PDL, e. As a corollary, the decidability of the generalized word problem is deduced. Then also languages 1. You can do this by writing a context free grammar or a PDA, or you can use the closure theorems for context-free languages. As a consequence all problems are decidable for context-free languages. There are many automatic verification tools for their analysis which incorporate equivalence checking. Prove that if L1 is regular and L2 is regular then so is L1-L2 (the set of all strings in L1 but not in L2). In Section 2, we recall the de nition of multi-letter QFAs and other related de nitions, and some related results are reviewed. • Now we will show some other problems and consider whether they are decidable by algorithm or not. The representations of regular languages range from various forms of automata (deterministic, nondeterministic, one-way, two-way) to regular expressions and formulas in monadic second-order logic. Formulate this problem as a language and show that it is decidable. Thus all standard decision questions concerning transductions of 0-FST’s are decidable. Supervisory control of deterministic Petri nets with regular specification languages. Namely, the star height problem for bideterministic languages is NP-complete, and this holds already for binary alphabets. Does a TM accept a decidable language? Does a TM accept a regular language? Does a TM accept a context-free language? Does a TM accept a ﬁnite language? Does a TM accept the empty language? Does a TM accept a language that contains all prime numbers? Slides modiﬁed by Benny Chor, based on original slides by Maurice Herlihy, Brown University. IScan the input string hB;wi, determining whether the input constitutes a valid DFA and string w. There is a ﬁxed such that the emptiness problem for 2DCM(2, ) over bounded languages is undecidable [5]. Prove that if L is regular then Prefix(L) is regular. Theoretical Computer Science ELSEVIER Theoretical Computer Science 174 (1997) 269-274 Note The undecidability of some equivalence problems concerning ngsm's and finite substitutions P. Recall the inductive definition of regular expressions that was given in class : 1. The Church-Turing Thesis. Undecidable languages are not recursive languages, but sometimes, they may be recursively enumerable. This can be seen by observing the DFA -- if DFA contains a state which contains a loop and that state is reachable from the start state and that state is either a final state or leading to final state, then the language will be infinite. Learn vocabulary, terms, and more with flashcards, games, and other study tools. The rest of the paper is organized as follows. L is said to beTuring-decidable(Recursiveor simply decidable) if there exists a TM M which decides L. Problem Set 3 Solutions Problem 1 Exercise3. For each of the following languages, specify which type it is. Start studying CIS 262 Important Terms. Termination Proofs for String Rewriting Systems via Inverse Match-Bounds Alfons Geser National Institute of Aerospace, Hampton, Virginia TERMINATION PROOFS FOR STRING REWRITING rewriting systems where this set is eﬀectively a regular language, the two problems are decidable. Now we will show some other problems and consider whether they are decidable by algorithm or not. A partially completed DFA that accepts this language is shown below. So in order to prove that I am trying to show that I can move from deterministic finite automaton (DFA) to a Turing decidable machine. 15) Show that the collection of decidable languages is closed under (a) Union: (in the textbook). Use language to represent various computational problems bcs we have terminology to deal with languages. Every regular language is decidable. Hence it is difficult to generalize to other classical problems concerning certain classes of regular languages. 1 Introduction Languages accepted by multi-tape or multi-head ﬁnite automata were introduced in [36] and [38]. To start a formal investigation of such concepts, Barcel o et al. Closure Problems. Decidable Problems With The TM, Turing Reducibility. Thus, it is sufficient to execute a graph traversal on the graph representation of the FSA from the input string,. 1: Decidable Languages, pp. DECIDABLE PROBLEMS CONCERNING REGULAR LANGUAGES (cont. (Sipser, Problem 3. A is countable. An inputed language is accepted by a computational model if it runs through the model and ends in an accepting final state. If α , β are regular expressions, then so is α∪β. Decidable and Undecidable Problems Table | TOC January 29, 2018 Anup Patel Resources , TOC Table to check Decidable and Undecidable property of all Grammar (Regular, CFL, DCFL, CSL, Recursive, Recursive Enumerable). The statement that x is not in S is written as x 2= S. b) The language associated with ∂ is {∂}, a one-word language. Consider the set of strings on {0,1} in which, every substring of 3 symbols has at most two zeros. The next proposition follows from Proposition 1. We can prove the language EQDFA−REX is decidable by constructing a TM P that decides it as follows: P =" On Input < M,r >:. 1 Sets A set is a collection of elements.