Outline. CS21 Decidability and Tractability. Machine view of FA. Machine view of FA. Machine view of FA. Machine view of FA.

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1 Outline CS21 Decidability and Tractability Lecture 5 January 16, 219 and Languages equivalence of NPDAs and CFGs non context-free languages January 16, 219 CS21 Lecture 5 1 January 16, 219 CS21 Lecture 5 2 Machine view of FA Machine view of FA q q 3 January 16, 219 CS21 Lecture 5 3 January 16, 219 CS21 Lecture 5 4 Machine view of FA Machine view of FA q 1 q 2 etc January 16, 219 CS21 Lecture 5 5 January 16, 219 CS21 Lecture 5 6 1

2 A more powerful machine limitation of FA related to fact that they can only remember a bounded amount of information What is the simplest alteration that adds unbounded memory to our machine? Should be able to recognize, e.g., { n 1 n n } q (in) 1 1 New capabilities can push symbol onto can pop symbol off of January 16, 219 CS21 Lecture 5 7 January 16, 219 CS21 Lecture q q 1 (in) $ (in) $ January 16, 219 CS21 Lecture 5 9 January 16, 219 CS21 Lecture q 1 q 2 (in) $ (in) $ January 16, 219 CS21 Lecture 5 11 January 16, 219 CS21 Lecture

3 (PDA) q (in) $ Note often start by pushing $ marker onto so that we can detect empty We will define nondeterministic pushdown automata immediately potentially several choices of next step Deterministic PDA defined later weaker than NPDA Two ways to describe NPDA diagram formal definition January 16, 219 CS21 Lecture 5 13 January 16, 219 CS21 Lecture 5 14 NPDA diagram NPDA operation tape alphabet Σ alphabet Γ start state states accept states transition label (tape symbol read, symbol popped symbol pushed) transitions, ε 1, ε 1, ε Taking a transition labeled a, b c a (Σ {ε}) b,c (Γ {ε}) read a from tape, or don t read from tape if a = ε pop b from, or don t pop from if b = ε push c onto, or don t push onto if c = ε January 16, 219 CS21 Lecture 5 15 January 16, 219 CS21 Lecture 5 16 Σ = {, 1} Σ = {, 1} Γ = {, 1, $}, ε Γ = {, 1, $}, ε 1, ε 1, ε 1, ε 1, ε tape 1 1 Stack contents $ tape 1 1 Stack contents $ January 16, 219 CS21 Lecture 5 17 January 16, 219 CS21 Lecture

4 Σ = {, 1} Σ = {, 1} Γ = {, 1, $}, ε Γ = {, 1, $}, ε 1, ε 1, ε 1, ε 1, ε tape 1 1 Stack contents $ tape 1 1 Stack contents $ January 16, 219 CS21 Lecture 5 19 January 16, 219 CS21 Lecture 5 2 Σ = {, 1} Σ = {, 1} Γ = {, 1, $}, ε Γ = {, 1, $}, ε 1, ε 1, ε 1, ε 1, ε tape 1 1 Stack contents $ January 16, 219 CS21 Lecture 5 21 tape 1 1 Stack contents $ accepted January 16, 219 CS21 Lecture 5 22 Σ = {, 1} Σ = {, 1} Γ = {, 1, $}, ε Γ = {, 1, $}, ε 1, ε 1, ε 1, ε 1, ε tape 1 Stack contents $ tape 1 Stack contents $ January 16, 219 CS21 Lecture 5 23 January 16, 219 CS21 Lecture

5 Σ = {, 1} Σ = {, 1} Γ = {, 1, $}, ε Γ = {, 1, $}, ε 1, ε 1, ε 1, ε 1, ε tape 1 Stack contents $ tape 1 Stack contents $ January 16, 219 CS21 Lecture 5 25 January 16, 219 CS21 Lecture 5 26 Σ = {, 1} Σ = {, 1} Γ = {, 1, $}, ε Γ = {, 1, $}, ε 1, ε 1, ε 1, ε 1, ε tape 1 Stack contents $ not accepted January 16, 219 CS21 Lecture 5 27 What language does this NPDA accept? January 16, 219 CS21 Lecture 5 28 Formal definition of NPDA A NPDA is a 6-tuple (Q, Σ, Γ, δ, q, F) where Q is a set called the states Σ is a set called the tape alphabet Γ is a set called the alphabet δq x (Σ {ε}) x (Γ {ε}) P(Q x (Γ {ε})) is a function called the transition function q is an element of Q called the start state F is a subset of Q called the accept states Formal definition of NPDA NPDA M = (Q, Σ, Γ, δ, q, F) accepts string w Σ* if w can be written as w 1 w 2 w 3 w m (Σ {ε})*, and there exist states r, r 1, r 2,, r m, and there exist strings s, s 1,, s m in (Γ {ε})* r = q and s = ε (r i+1, b) δ(r i, w i+1, a), where s i = at, s i+1 = bt for some t Γ r m F January 16, 219 CS21 Lecture 5 29 January 16, 219 CS21 Lecture 5 3 5

6 Example of formal definition Q = {q, q 1, q 2, q 3 } Σ = {,1} Γ = {, 1, $} F = {q, q 3 } q q 1 q 3 q 2, ε 1, ε δ(q, ε, ε) = {(q 1, $)} δ(q 1,, ε) = {(q 1, )} δ(q 1, 1, ) = {(q 2, ε)} δ(q 2, 1, ) = {(q 2, ε)} δ(q 2, ε, $) = {(q 3, ε)} 1, ε other values of δ(,, ) equal {} January 16, 219 CS21 Lecture 5 31 Exercise Design a NPDA for the language {a i b j c k i, j, k and i = j or i = k} January 16, 219 CS21 Lecture 5 32 Context-free grammars and languages languages recognized by a (N)FA are exactly the languages described by regular expressions, and they are called the regular languages languages recognized by a NPDA are exactly the languages described by context-free grammars, and they are called the context-free languages start symbol production A A1 A B B # terminal symbols non-terminal symbols January 16, 219 CS21 Lecture 5 33 January 16, 219 CS21 Lecture 5 34 generate strings by repeated replacement of non-terminals with string of terminals and non-terminals write down start symbol (non-terminal) replace a non-terminal with the right-handside of a rule that has that non-terminal as its left-hand-side. repeat above until no more non-terminals Example A A1 A A1 A11 A B A111 B111 B # #111 a derivation of the string #111 set of all strings generated in this way is the language of the grammar L(G) called a Context-Free Language January 16, 219 CS21 Lecture 5 35 January 16, 219 CS21 Lecture

7 Natural languages (e.g. English) have shorthand this sort for of structure multiple rules <sentence> <noun-phrase><verb-phrase> with same lhs <noun-phrase> <cpx-noun> <cpx-noun><prep-phrase> <verb-phrase> <cpx-verb> <cpx-verb><prep-phrase> <prep-phrase> <prep><cpx-noun> <cpx-noun> <article><noun> <cpx-verb> <verb> <verb><noun-phrase> <article> a the <noun> dog cat flower <verb> eats sees <prep> with Generate a string in the language of this grammar. January 16, 219 CS21 Lecture 5 37 CFGs don t capture natural languages completely computer languages often defined by CFG hierarchical structure slightly different notation often used Backus- Naur form see next slide for example January 16, 219 CS21 Lecture 5 38 Example CFG <stmt> <if-stmt> <while-stmt> <begin-stmt> <asgn-stmt> <if-stmt> IF <bool-expr> THEN <stmt> ELSE <stmt> <while-stmt> WHILE <bool-expr> DO <stmt> <begin-stmt> BEGIN <stmt-list> END <stmt-list> <stmt> <stmt>; <stmt-list> <asgn-stmt> <var> = <arith-expr> <bool-expr> <arith-expr><compare-op><arith-expr> <compare-op> < > = <arith-expr> <var> <const> (<arith-expr><arith-op><arith-expr>) <arith-op> + - * / <const> <var> a b c x y z CFG formal definition A context-free grammar is a 4-tuple (V, Σ, R, S) where V is a set called the non-terminals Σ is a set (disjoint from V) called the terminals R is a set of productions where each production is a non-terminal and a string of terminals and nonterminals. S V is the start variable (or start non-terminal) January 16, 219 CS21 Lecture 5 39 January 16, 219 CS21 Lecture 5 4 CFG formal definition u, v, w are strings of non-terminals and terminals, and A w is a production uav yields uwv notation uav uwv also yields in 1 step notation uav + uwv in general yields in k steps notation u, v meaning there exists strings u 1,u 2, u k-1 for which u + u 1 + u u k-1 + v CFG formal definition notation u v meaning k and strings u 1,,u k-1 for which u + u 1 + u u k-1 + v if u = start symbol, this is a derivation of v The language of G, denoted L(G) is {w Σ* S w} January 16, 219 CS21 Lecture 5 41 January 16, 219 CS21 Lecture

8 CFG example Balanced parentheses ( ) ( ( ) ( ( ( ) ( ) ) ) ) a string w in Σ* = { (, ) }* is balanced iff # ( s equals # ) s, and for any prefix of w, # ( s # ) s Exercise design a CFG for balanced parentheses. January 16, 219 CS21 Lecture