Context-free Languages and Pushdown Automata
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1 Context-free Languages and Pushdown Automata Finite Automata vs CFLs E.g., {a n b n } CFLs Regular From earlier results: Languages every regular language is a CFL but there are CFLs that are not regular Can we extend Finite Automata to eual CFLs? I.e., get a machine-like characterization of CFLs? A key feature: recursion Recursion s twin: a pushdown stack Pushdown sufficient? intuitively, yes: Example
2 Alternate way to define this: A PDA Configuration (stack top on left): state, stack, input A PDA Move: p, at, wx, bt, x if p, Q, a Γ {ε}, t Γ*, w Σ {ε}, x Σ* s.t. (,b) δ(p,w,a) Multiple moves: k : exactly k steps * : 0 or more steps M can reach with γ Γ* on its stack after reading w Σ* if Which move top of right 0, ε, w *,γ, ε E.g., M can reach with $ on its stack after reading ab, and M can reach with ε on stack reading ab and M accepts ab. M accepts w if above, and F L(M) = { w Σ* M accepts w } Every CFL is accepted by some PDA PDAs accept all CFLs Top-Down For any CFG G=(V, Σ, R, ), build PDA M = (Q, Σ, Γ, δ, 0, F), where Every regular language is accepted by some PDA (basically, just ignore the stack...) Q = {0,, accept} Above examples show that PDAs are sufficiently powerful to accept some context-free but non-regular languages, too Γ = V Σ {$} ($ V Σ) F = {accept}, and In fact, they can accept every CFL: δ is defined by the diagram Proof : the book s top down parser (next) 0 ε, ε $ a, a ε a Σ ε, A α for all rules A α in R ε, $ ε a Idea: on input w, M nondeterministically picks a leftmost derivation of w from. tack holds intermediate strings in derivation (left end at top); letters in Σ on top of stack matched against input. Proof : bottom up, (aka shift-reduce ) parser (later) matched on input stack 8
3 ab ε Top-Down Parser 9 0 a ε, ε $ a, a ε b, b ε ε, ab ε, ε ε, $ ε L ab L aabb L aabb L aabb L aabbab L aabbab L aabbab a a ε b ε b a ε b a a b b a b ε top bot tate tack Input Deriv 0 ε aabbab $ aabbab ab$ aabbab b$ abbab abb$ abbab bb$ bbab bb$ bbab b$ bab b$ bab $ ab ab$ ab b$ b b$ b $ ε $ ε accept ε ε Input accepted 0 PDAs accept all CFLs Bottom-Up / hift-reduce For any CFG G=(V, Σ, R, ), build PDA M = (Q, Σ, Γ, δ, 0, F), where Q = {0,, accept} Γ = V Σ {$} ($ V Σ) F = {accept}, and δ is defined by the diagram ε,ε $ shift a,ε a, a Σ Idea: on input w, M nondeterministically picks a rightmost derivation backwards, from w to. hift input onto stack or reduce top few symbols at each step. 0 ε,$ ε ε,β A, (A β) R reduce stack unread input a 0 ε, ε $ ε, $ ε a ab ε shift: a, ε a b, ε b reduce: ε, ε ε, ab ab abab abab abab aabbab aabbab aabbab a a ε b ε b a ε b a a b b a b ε hift-reduce Parser bot top tate kcat Input Deriv 0 ε aabbab $ aabbab $a abbab $aa bbab $aa bbab $aab bab $aab bab 8 $a bab 9 $ab ab 0 $aba b $aba b $abab ε $abab ε $ab ε $ ε accept ε ε Input accepted
4 [ 0, ε, [, $, [, $ a, [, $ a if, hift-reduce (Ex ) a if b then a if b then if b then a else a ] a if b then if b then a else a ] if b then if b then a else a ] b then if b then a else a ]. more shifts [, $ a if b then if b then a, [, $ a if b then if b then a, [, $ a if b then if b then, [, $ a if b then, [, $ a if b then else, [, $ a if b then else a, [, $ a if b then else a, [, $ a if b then else, [, $ a, [, $, [ a, ε, if b then else ε ctness Notation: of the ed construction by[state, the following isstack, captured assertion: input] bot top else a ] () else a ] () else a ] () else a ] a ] ε ] () ε ] () ε ] () ε ] () ε ] ε ] R a () R a if b then else () R a if b then else a () R a if b then else a () R a if b then if b then else a () R a if b then if b then a else a () R a if b then if b then a else a () a if b then if b then a ε else a ε Correctness of shiftreduce construction PDA Configurations: [state, stack, input] bot top PDA Moves: [, γα, ay] [, γβ, y] (like slide, except stack reversed) (via w shifts)
5 Notes Both top-down & bottom up PDA s above are nondeterministic. With a carefully designed grammar, and by being able to peek ahead at the next input symbol, it may be possible to tell deterministically which action to take. The CFG's for which this is possible are called LL() (top-down case) or LR() (shift-reduce case) grammars, and are important for programming language design. Every language accepted by a deterministic PDA has an LR () grammar, but not all grammars for a given language are LR(), and for some CFL's no grammar is LR(). ome PDA Facts y, [p, α, x] * [, β, ε] if and only if [p, α, xy] * [, β, y] Why? PDA can t test end of input or peek ahead, so presence/absence of y is invisible. (A bit like the contextfree property in a CFG.) γ, [p, α, x] * [, β, ε] implies [p, γα, x] * [, γβ, ε] Why? γ is buried on bottom of stack, so computation allowed in its absence is still valid in its presence. Note the converse is, in general, false! Computation on right might pop part of γ, then push it back, whereas one at left would block at the attempted pop. Important special case: α = β = ε: p x allowed on empty stack allowed on any stack 8 Q: What L solves this euation? L {a,b}* L = {ε} {a} L {b} Answer: L = { a n b n n 0 } Compare to: ε a b Q: What L solves this euation? L, Σ* ( fixed, e.g. palindromes or odd len ) L = {ε} L L Alt phrasing: the smallest set containing ε and all of and is closed under concatenation? Answer: L = * Compare to: ε grammar_for_ 9 0
6 = the set of input strings x that allow M to go from In English? Lstate to state, starting & ending with empty stack. An euation? PDA to CFG, general construction I. WLOG, assume PDA: a) has only one final state b) accepts only when stack is empty, and c) all transitions either push or pop, never both/neither Goal: Ap gives inputs allowing M to go from state p to state, starting & ending with empty stack. read nothing/no transition p to via any intermediate r. read a + push. go from r to s on empty stack, so re-exposed. read b + pop NB: G can be simplified. E.g., remove A, A & rules using them, since, e.g., x Σ* s.t. A * x. This is just fine in the construction, since there is also no x s.t. [, ε, x] * [, ε, ε]. Easier to construct useless rules locally than to sort out such ramifications globally. Grammar start symbol = Astart-state, final-state
7 I.e., Ap gives set of inputs that allow M to go from state p to state, starting & ending with empty stack. I.e., Ap gives set of inputs that allow M to go from state p to state, starting & ending with empty stack. Case (i): exercise tack (and fact that M s stack is empty when it enters F) Time I.e., Ap gives set of inputs that allow M to go from state p to state, starting & ending with empty stack. ummary: PDA CFG Pushdown stack conveniently allows simulation of recursion in CFG E.g., {anbn} or {wwr} or balanced parens, etc.: push some, match later Nondeterminism sometimes essential - e.g., guess middle ; there is no subset constr for NPDA G M: guess deriv., using stack carefully ( tack L or R) - basis for parsers in most compilers, e.g. M G: Ap = {x go from p to on empty stack} Time 8
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