E Cfg Decidable, What can you conclude? C) Finiteness problem

  • E Cfg Decidable, What can you conclude? C) Finiteness problem of CFL (or CFG) decidable, because we can check the existence of loops in the dependency graph over the non terminals. This problem has confused me a lot. A language L is said to be a context-free language (CFL), if there exists a CFG G, such that . Theorem 5. 15 of the third e Acceptance Problem for CFGs is Decidable • Decision problem: Does a CFG G generate a string w? ACFG = { G, w| G is a CFG that generates string w } ⊆ { G, w| G is a CFG and w a string } ≡ Ω. Solution Consider the following Turing machine = On input hG; xi, where G is a CFG Option 2: The completeness problem for Context-free language is not decidable i. Initially Nmarked is the set non-terminals X which have a production with a terminal RHS. Decidable Problems Concerning CFLs Here we describe algorithms to test whether a CFG generates a particular string and to test whether the language of a CFG is empty. Learning Outcomes of Section 4. In this article, we will discuss Decidable and Undecidable problems in detail. In the case of deterministic nite automata, problems like equivalence can be solved even in polynomial time. So instead we use set builder notation. Each language L in L has a characteristic function, F, which is an infinite sequence of 0’s and 1’s (I. Let A be a recognizer for L, and B for its complement. The following \On input hG; ki, where G is a CFG and k is a natural number: Lecture 18 – Decidable languages recognizes A (i. How could I prove that the following language is decidable? $\ {\langle G\rangle \mid G\ \text {is a CFG over}\ \ {0,1\}\ \text {and}\ 1^* \subseteq L (G)\}$ P. Here we show that the equality problem for context-free grammars is undecidable, and its complement is recognizable. there exists a derivation X ) w for some terminal string w. Solution Consider the following Turing machine = On input hG; xi, where G is a CFG Undecidable Problems Showed these problems for FA and CFG’s decidable: Acceptance, Emptiness, Equivalence Qu: What about for TM? Clearly, D is decidable. Turing Machines, Recognizability, Decidability A Turing recognizable (by M1) and A Turing recognizable (by M2) = ) A decidable run M1and M2in lockstep, see which halts rst Enumerating Turing-recognizable , (recursively) enumerable run for k steps on s1;s2;:::sk(dovetailing) Decidable , enumerable in lexicographical order Problems for Recognizers Partially Decidable Languages Undecidable Languages Decidable Language A decision problem PPP is said to be decidable if there exists an algorithm (or Turing Machine) that always halts and correctly determines the answer for every input. Every CFL is decidable. In other words, show that { G ∣ G is a CFG over {0, 1} and 1 ∗ ⊆ L (G)} is a decidable language. Option 4: The problem of whether any language will get accepted by DFA and NFA is Check-in 7. Here we show that the emptiness problem for context-free grammars (CFGs) is decidable. Since the emptiness of CFG is decidable, we can use it to decide whether $\bar {C}$ is empty and therefore whether $C$ is universal. Example E -> E + E | E * E | (E) | id Reductions “If A is decidable then B is decidable” Can be proved using closure properties, or assuming a decider for A Assume B<A and A is decidable. (Bonus Question) Show that the problem of determining whether a CFG generates all string in 1¤ is decidable. Two distinct derivations of a string in a given CFG may sometimes attribute the same parse tree to the string. Some context-free languages generated by ambiguous CFGs are also generated by unambiguous CFGs. Moreover, if a language is decidable, then so is its complement, and hence that complement is recognizable. Now suppose a language L and its complement are recognizable. 2 & 5. Also there are efficient parsing algorithms for context-free grammars. un-decidable. 1 Be able to formally and informally describe the contents of languages such as and including the following: EDFA ECFG EQDFA ADFA, ANFA, AREX ACFG, APDA E DFA E CFG EQ DFA A DFA, A NFA, A REX A CFG, A PDA Prove that the following are Turing Decidable languages: the languages above every Context Free language A the languages above every Context Free language A Machine M is a valid decider for ALLCFG since in step 1, we can easily pick a CFG that accept every string in finite time. You should nd some inspiration for this by reviewing the algorithm for E CFG. 29. Hence M will only loop if L(G1) = L(G2). It's the problem 4. What are Decidable Problems? A problem is said to be Decidable if we can always construct a corresponding algorithm that can answer the problem correctly. Decidable problems from language theory For simple machine models, such as nite automata or pushdown automata, many decision problems are solvable. For sure you can effectively tests if a string codes a CFG (of course, you have to know how this code is built, but you can assume that the grammar is given in some "normal" form). 1 可判定语言(DECIDABLE LANGUAGES)本节将给出一些语言的例子，它们都是算法上可判定的。 与正则语言相关的可判定性问题(DECIDABLE PROBLEMS CONCERNING REGULAR LANGUAGES)DFA的接受问题(acceptance problem fo… decidable problems concerning context-free languages describe algorithms to determine whether a CFG generates a particular string and to determine whether the language of a CFG is empty We have defined the concept of Decidability in Theory of Computation, Decidable Languages and which Languages are decidable and undecidable. 7 on page 172). , see the proof of Theorem 4. Thursday, December 5/6, 2007 Reading: Sipser 4 and 5. Moreover, EQCFG is decidable so step 2 and 3 will terminate in finite time. Express this problem as a language and show that it is decidable. Prove that ADFA = { B, w : B is a DFA that accepts input string w} is a decidable language. Let A = { < G , w > : G is a CFG that generates string w }. I know that generation of a particular string by a given CFG is a decidable problem. [1] J. We have defined the concept of Decidability in Theory of Computation, Decidable Languages and which Languages are decidable and undecidable. It is decidable whether such a language is finite, but not whether it contains every possible string, is regular, is unambiguous, or is equivalent to a language with a different grammar. 3 Revised Thursday, December 6 The general form of the ambiguous grammar is: S -> α S β S γ | α1| α2 | α3 | …. Homework 8 Solutions 1. Thus This is a contradiction and thus our assumption jtj > n1n2 is false. 1007/BF01746517. Given the string encoding of a CFG , a TM can decide whether the language of is empty. Maintain a set of \marked" non-terminals. Is the decision problem "Does a given context free grammar generate an infinite number of strings" decidable? In order to test whether a context free grammar generates an infinite number of strings or not, we can write a program that would test this but would not halt. An extended context-free grammar (ECFG) is a formal language grammar that extends the context-free grammar (CFG) by allowing the use of variables on the right-hand side of production rules. e. Show that AECFG is decidable. • For any specific pair G, w∈ Ω of a CFG G and string w, if G generates w, i. Anyway, the problem of testing whether a CFG generates all the strings of the language $1^∗$ is truly decidable. , w ∈ L(G). If anyone could point me in the right direction, I would appreciate it. Also, x 2 C if and only if there exists y such that hx; yi 2 D. The reason is that there is a procedure that always halts for checking whether a CFG in Chomsky normal form can generate a particular string w or not; e. However, we run into one big problem: the class of context-free languages is not closed under complement! 2) Let AecFG <G> | G is a CFG that generates e . Where α,β and γ are some string. , 3⁄4w " Σ , M either accepts or rejects w) The algorithm you are describing shows that the problem of testing whether a CFG generates some string from $1^∗$ is decidable (e. Thank you. Now we have to show that ACFG A C F G is decidable, or in other words, there exists an algorithm that determines whether w w is generated by G G or not. 5 Decidable CFL Properties Finally, we show that two languages related to properties about context-free grammars are decidable. In other words, show that fhGi j G is a CFG over f0; 1g and 1¤ μ L(G)g is a decidable language. For context-free languages, we know that ECFG is decidable, so we might be tempted to take a similar approach to show that UCFG is decidable. Context-free language The language of a grammar is the set of all terminal-symbol strings derivable from the start symbol. Option 1 is whether a CFG is empty or not, this problem is decidable. Thus To approach the problem of showing that A CFG is decidable, consider the following tips: Understand CFGs: Review the properties of context-free grammars (CFGs) and how they generate strings. Option 4 is whether language generated by DFA and NFA are same is decidable. an infinite binary sequence) Eg consider L = { x | length of x is even } Observe that the checks in step 2. , L M A), and halts on every input (i. Instead, show that Aε CFG is decidable by exhibiting a direct algorithm for it that doesn’t involve converting G to another form. g. Learning Outcomes of Section 4. 1. E. Proof: Let G be a CFG for L. ) You are on the right track in that if you can construct a TM that always halts for a given language L, then L is decidable. ) S = "On input <G,w>, where G is a CFG and w is a string: 3 I am trying to prove the fact that every CFL is decidable, however I can't come to terms with what the statement exactly means. Let G 1 is Context Free Grammar that have S as starting non-terminal for which every follow the rule S → a S and S → ∈ terminal and L G 1 = ∑ * i. The Halting Problem and The Return of Diagonalization Wednesday. can be done algorithmically because ACFG is a decidable language (see Lecture 3 or Theorem 4. Show that the following language is decidable ALLDFA = {< A > |A is a DFA and L (A) = } 31 Problem 4. We utilize the fact that ALL_CFG is unde For a regular language $R$, the problem $P_R$ is decidable iff $R$ is bounded [1]. To prove that ACFG is decidable, we note the following two facts: Every CFG has an equivalent Chomsky Normal Form grammar TM can simulate CFG (2) Theorem 8: Given a CFG G, we can construct a TM that recognizes the same language. This means that we can design a TM that will definitely halt and tell us whether the given string is generated by a CFG. Option 2 is whether a CFG will generate all possible strings (everything or completeness of CFG), this problem is undecidable. Decidable Problems Concerning Context-Free Languages Topics Problem 1: describe algorithms to test whether a CFG generates a particular string Problem 2 describe algorithms to test whether the language generated by a CFG is empty. , see here at page 21). The following language is decidable ACFG = fhG; wijG is a CFG; w 2 L(G)g Proof. Theorem. Whether a string belongs to the language generated by a CNF ? Proof: Surely, a decidable language is recognizable. Regular Languages 1. 1 Be able to formally and informally describe the contents of languages such as and including the following: EDFA ECFG EQDFA ADFA, ANFA, AREX ACFG, APDA E DFA E CFG EQ DFA A DFA, A NFA, A REX A CFG, A PDA Prove that the following are Turing Decidable languages: the languages above every Context Free language A the languages above every Context Free language A Undecidable Problems Showed these problems for FA and CFG’s decidable: Acceptance, Emptiness, Equivalence Qu: What about for TM? Some tips for proofs (Turing machines, decidability, reducibility) Set builder notation Enumerating all elements of a set is very often (almost always) impractical. , language L of Grammar G 1 contain all the strings in ∑. Closure Properties of Decidable and Recognizable Languages Theorem. We can nd out which non-terminals of G can derive a terminal string: i. Consider the decision problem of testing whether a DFA and a regular expression are equivalent. In other words, every CFL is a decidable language (The fact that you have a CFG in this case is irrelevant. So, L (G) = Σ* is un-decidable. Thus, C = fx j 9y(hx; yi 2 D)g. We can intuitively understand Decidable issues by considering a simple example. The class of decidable languages is prove that languages are decidable (“good-news reductions”) prove that languages are not decidable (“bad-news reductions”) “If B is decidable, then A is decidable” Consider the language ACFG A C F G = { <G G, w w> | G G is CFG that generates w w }, where <G G, w w> is a string encoding of G G and w w. I know I am supposed to prove that it is decidable or not L L contains some string of the language 1∗ 1 ∗ and I know that CFLs are not closed under intersection and that ETM E T M is not decidable but I am having trouble constructing the proof knowing this. Proof: Since we have a decider M for INFINITECFG = fhGi j G is a CFG and L(G) is an in nite languageg, we use M to show a decider T for CCFG. the language consisting of every possible string over the alphabet. Here’s how we would write the sets of all even and odd numbers (respectively): Even: {n ∣ n = 2 k, k ∈ Z} {n ∣ n = 2k,k ∈ Z} Odd: {n ∣ n = 2 k + 1, k ∈ Z} {n ∣ n = 2k +1,k ∈ Decidable/undecidable : Does a given context free grammar generate an infinite number of strings? This explained in given link : check if the CFG generates any string of length between $n$ and $2n-1$. Option 3: Whether the language generated by a Turing machine is always un-decidable. Hopcroft: On the equivalence and containment problems for context-free languages, Mathematical Systems Theory 3 (1969), pp. Closure properties of Decidable languages. When there are derivations of some string in a given CFG that attribute different parse trees, the CFG is ambiguous. Option 3 is whether language generated by TM is regular is undecidable. Decidability In formal language theory, questions about regular languages are usually decidable, but ones about context-free languages are often not. . There exists a TM that decides the above language. However, the machine that you describe will actually accept every string, i. Thus decider5-ATM is indeed a decider for ATM, but this is impossible, and we thus conclude that our assumption, that ELBA is decidable, was false, implying the claim. 119–124, doi 10. 7, which shows ACFG is decidable. Let CCFG = fhG; ki j G is a CFG and L(G) contains exactly k strings where k 0 or k = 1 g. Is it decidable whether a given CFG accepts a non-empty language? Yes, it is. 4. ∈ ACFG if G doesn’t generate w, i. Let C = fhG; xijG is a CFG; x is a substring of some y 2 L(G)g. 3 Why can’t we use the same technique we used to show !"DFA is decidable to show that !"CFG is decidable? Problem 15 Show that the problem of determining whether a CFG generates all strings in 1 ∗ is decidable. 1; Kozen 31; Stoughton 5. Design a TM that decides L. Show that C is decidable. Therefore, Is L (M) regular is un-decidable. In other words, the language LLL, consisting of all inputs for which the answer is “yes,” is decidable This is a contradiction and thus our assumption jtj > n1n2 is false. |αn this production has an ambiguity problem because S are present twice on the right hand side of the production. • Is it a good idea to convert G to an equivalent PDA P and have simulate P? Theorem: ACFG is a decidable language proof (cont. Non-deterministic pushdown automata recognize exactly the context-free languages. A variable is "productive" if it can make any string of terminals at a If it were decidable, given the regular expression $ (a \cup b)^* $, we would be able to determine if $ \Sigma ^* \subseteq L (G) \Leftrightarrow L (G) = \Sigma ^* $ which is undecidable. 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