Context Free Languages: Decidability of a CFL

Transcription

1 Theorem 14.1 Context Free Languages: Decidability of a CFL Statement: Given a CFL L and string w, there is a decision procedure that determines whether w L. Proof: By construction. 1. Proof using a grammar Proof (a) if L is represented by a PDA M, construct a CFG G such that L(G) = L(M) (b) if w = ɛ and S is nullable, accept; otherwise reject (c) Convert G to CNF (G ) such that L(G ) = L(G) ɛ (d) Try all derivations from G using no more than 2 w 1 steps. If w derived, accept; otherwise reject Analysis Algorithm requires at most 2 w 1 steps because grammar is in CNF Time to build G independent of w, so is constant Let n = w g = search-branching factor of G : maximum number of rules with same LHS Number of derivations of length 2 w 1 g 2n 1 It takes fewer than 2n 1 steps to test each So, run time O(n2 n ) 2. PDA-based proof See below 1

2 Theorem 14.2 Context Free Languages: Decidability of a CFL (2) Statement: Given a CFG G = (V, Σ, R, S), there is a PDA M such that L(M) = L(G) ɛ which contains no transitions of the form ((p, ɛ, α), (q, β)) Proof: By construction (see below). Discussion Algorithm convertcfgtopda-td generates PDA M that simulates derivation of a string from the given CFG M = ({p, q}, Σ, V,, p, {q}), where contains 1. A start-up transition ((p, ɛ, ɛ), (q, s)) whose sole job is to push S 2. Transitions of the form ((p, ɛ, X), (q, s 1 s 2...s n )) for each rule of the form X s 1 s 2...s n These replace X by s 1 s 2...s n in the derivation When n = 0, transition is ((p, ɛ, X), (q, ɛ)) 3. Transitions of the form ((p, c, c), (q, ɛ)) for each c Σ These replace allow consumption of input Transitions of type 1 and type 2 above are ɛ-transitions Consider a CFG in GNF 1. Rules containing S are of form S cs 1 s 2...s n, where s 1, s 2,..., s n V Σ No need to push S Simply read the c and push s 1 s 2...s n 2. Other rules are of form X cs 1 s 2...s n (X S) Rather than (a) Push c, and then (b) Immediately pop c in next transition (ala sequence of rules of type 2 and 3 above) Can simply read the c and push s 1 s 2...s n Algorithm: 1. Convert G to GNF 2. Use the above to build PDA M 2

3 Theorem 14.3 Context Free Languages: Decidability of a CFL (3) Statement: Let M be a PDA that contains no ɛ-transitions. Consider operations of M on w Σ where w = n. 1. M will halt and either accept or reject w. 2. Each computation will halt within n steps. 3. The total number of computations pursued by M is less than or equal to b n, where b is the maximum number of competing transitions from any state of M. 4. The maximum number of steps executed by any computation of m is nb n. Proof: (See p 317) Proof of Theorem 14.1, part 2: Using a PDA Algorithm 1. If L represented by a PDA M, convert to CFG G so that L(G) = L(M) 2. if w = ɛ and S is nullable, accept; otherwise reject 3. Convert G to GNF (G ) such that L(G) = L(G ) ɛ 4. Construct PDA M such that L(M ) = L(G ) and M has no ɛ transitions 5. By theorem 14.3, all paths of M are guaranteed to halt in a finite number of steps. Run M on w Accept if M accepts; otherwise reject Analysis M is constructed in constant time (only done once) Let w = n Time needed to analyze w is time needed to simulate all paths of M on w Total number of steps bounded by nb n Number of steps could grow exponentially, or be less than b Depends on how the simulation is performed: depth-first search is O(b n ) 3

4 Context Free Languages: Decidability of CFL Emptiness and Finiteness Theorem 14.4 Statement: Given CFL L, there is a decision procedure that determines whether 1. L = NULL 2. L is infinite Proof: By construction 1. Emptiness: L(G) is empty if S is unproductive. Boolean decidecflempty (CFG G) { G = removeunproductive(g); if (G.S not in G.V) return TRUE; else return FALSE; } 2. Non-finiteness Let G = (V, Σ, R, S) be CFG that generates L n = V Σ b = branching factor of G Longest string that can be generated without repeating a NT has length b n If G generates no strings longer than this, L is finite If G generates one string longer than this, L is infinite Cannot apply decidecf L(G, w) to all strings longer than b n Σ, as there are an infinite number of them Let t be the shortest string generated by G whose length is b n+1 + b n Pumping Theorem states that t = uvxyz vy > 0 uxz L, and uxz < t Since t is the shortest string whose length is greater than b n+1 + b n uxz b n+1 + b n 4

5 Context Free Languages: Decidability of CFL Emptiness and Finiteness (2) Since the Pumping Theorem states that vxy k (i.e., b n ), no more than b n+1 strings could be pumped out Thus, b n < uxz b n+1 + b n If L has any strings longer than b n, it must have at least one of length b n+1 + b n Algorithm: Boolean decidecflinfinite (CFG G) { lexicographically enumerate all strings W L(G) where b n < w b n+1 + b n ; for (each w) if (return decidecfl(g, w)) return TRUE; return FALSE; } 5

6 Theorem 14.5 Context Free Languages: Decidability of Equivalence Statement: Given 2 deterministic CFLs L 1 and L 2, there is a decision procedure to determine whether L 1 = L 2. Proof: (See p 320) 6

7 Context Free Languages: Questions That Are Not Decidable The following are not decidable: Given CFLs L 1 and L 2 defined over alphabet Σ 1. L 1 = Σ? 2. Is L 1 s complement a CFL? 3. Is L 1 regular? 4. L 1 = L 2? 5. L 1 L 2? 6. L 1 L 2 = NULL? 7. Is L 1 inherently ambiguous? 8. Is CFG G ambiguous? 7