Intro to Theory of Computation

LECTURE 6 • Last time: • Pumping lemma • Proving a language is not regular • Today: • Pushdown automata (PDAs) • Context-free grammars (CFGs) Intro to Theory of Computation CS 332 Sofya Raskhodnikova On Friday: Homework 2 due Homework 3 out SofyaRaskhodnikova; based on slides by Nick Hopper

CS332 so far MODEL OF A PROBLEM: LANGUAGE MODEL OF A PROGRAM: DFA EQUIVALENT MODELS: NFA, REGEXP PROBLEMS THAT A DFA CAN’T SOLVE ARE WE DONE? Sofya Raskhodnikova; based on slides by Nick Hopper

NONE OF THESE ARE REGULAR • Σ = {0, 1}, L = { 0n1n | n ≥ 0 } • Σ = {a, b, c, …, z}, L = { w | w = wR } • Σ = { (, ) }, L = { balanced strings of parens } We can write a Cor JAVA program for any of them! Sofya Raskhodnikova; based on slides by Nick Hopper

PUSHDOWNAUTOMATA (PDA) FINITE STATE CONTROL INPUT STACK (Last in, first out) Sofya Raskhodnikova; based on slides by Nick Hopper

string pop push ε,ε → $ 0,ε→ 0 11 0011 0011 011 1,0 →ε ε,$ → ε 1,0 →ε 1 0 STACK $ 0 $ $

string pop push ε,ε → $ 0,ε→ 0 1 001 001 01 1,0 →ε ε,$ → ε 1,0 →ε 0 STACK $ 0 $ $ PDA to recognize L = { 0n1n | n ≥ 0 }

Formal Definition A PDAis a 6-tuple P = (Q, Σ, Γ, , q0, F) Q is a finite set of states Σ is the alphabet Γis the stack alphabet  : Q ΣεΓε → P(Q Γε) is the transition function q0 Q is the start state F  Q is the set of accept states Σε= Σ {ε}and P(Q Γε) is the set of subsets of Q Γε Note: A PDA is defined to be nondeterministic. Sofya Raskhodnikova; based on slides by Nick Hopper

Formal Definition A PDAis a 6-tuple P = (Q, Σ, Γ, , q0, F) Q is a finite set of states Σ is the alphabet Γis the stack alphabet  : Q ΣεΓε → P(Q Γε) is the transition function q0 Q is the start state F  Q is the set of accept states A PDA starts with an empty stack. Itaccepts a string if at least one of its computational branches reads all the input and gets into an accept state at the end of it. Sofya Raskhodnikova; based on slides by Nick Hopper

An example PDA ε,ε → $ 0,ε→ 0 q0 q1 1,0 →ε ε,$ → ε q3 q2 1,0 →ε Q = {q0, q1, q2, q3} Σ = {0,1} Γ = {$,0}  : Q Σε Γε→ P(Q  Γε) (q1,1,0) = { (q2,ε) } (q2,1,1) = 

PDA: algorithmic description ε,ε → $ 0,ε→ 0 q0 q1 • Place the marker symbol $ onto the stack. • Keep reading a 0 and pushing it onto the stack. If you read a 1, pop a 0 and go to the next step. • Nondeterministically keep reading a 1 and popping a 0 or go to the next step. • If the top of the stack is $, enter the accept state. (Then PDA accepts if the input has been read). 1,0 →ε ε,$ → ε q3 q2 1,0 →ε

Exercise Σ = {a, b, c, …, z} ε,ε → $ ,ε→ q0 q1 ε,ε→ε ε,$ → ε ,→ε q3 q2 What strings are accepted by this PDA? • Only • Palindromes • Even-length palindromes • All strings that start and end with the same letter • None of the above

Give an ALGORITHMIC description Σ = {a, b, c, …, z} ε,ε → $ ,ε→ q0 q1 • Place the marker symbol $ onto the stack. • Nondeterministically keep reading a character and pushing it onto the stack or go to the next step. • Nondeterministically keep reading and popping a matching character or go to the next step. • If the top of the stack is $, enter the accept state. ε,ε→ε ε,$ → ε ,→ε q3 q2

q2 q3 q1 q4 q6 q5 Build a PDA to recognize L = { aibjck | i, j, k ≥ 0 and (i = j or i = k) } c,ε→ε b,a →ε q0 ε,$ → ε ε,ε → ε ε,ε → $ ε,ε → ε ε,ε → ε ε,$ → ε a,ε→ a b,ε→ε c,a →ε

CONTEXT-FREEGRAMMARS (CFGs) production rules start variable A → 0A1 A → B B → # variables terminals A  0A1  00A11  00B11  00#11 Sofya Raskhodnikova; based on slides by Nick Hopper

VALLEY GIRL GRAMMAR <PHRASE> → <FILLER><PHRASE> <PHRASE> → <START><END> <FILLER> → LIKE <FILLER> → UMM <START> → YOU KNOW <START> → ε <END> → GAG ME WITH A SPOON <END> → AS IF <END> → WHATEVER <END> → LOSER Sofya Raskhodnikova; based on slides by Nick Hopper

Formal Definition A CFGis a 4-tuple G = (V, Σ, R, S) Vis a finite set of variables Σ is a finite set of terminals (disjoint from V) R is set of production rules of the form A → W, where A  V and W  (VΣ)* S  V is the start variable L(G) is the set of strings generated by G A language is context-free if it is generated by some CFG. Sofya Raskhodnikova; based on slides by Nick Hopper

Formal Definition A CFGis a 4-tuple G = (V, Σ, R, S) Vis a finite set of variables Σ is a finite set of terminals (disjoint from V) R is set of production rules of the form A → W, where A  V and W  (VΣ)* S  V is the start variable R = { S → 0S1, S → ε } Example: G = {{S}, {0,1}, R, S} L(G) = { 0n1n | n ≥ 0 } Sofya Raskhodnikova; based on slides by Nick Hopper

CFG terminology production rules start variable A → 0A1 A → B B → # variables terminals A  0A1  00A11  00B11  00#11 uVwyields uvw if (V → v)  R. A derives 00#11 in 4 steps. Sofya Raskhodnikova; based on slides by Nick Hopper

Example GIVE A CFG FOR EVEN-LENGTH PALINDROMES S → S for all   Σ S → ε Sofya Raskhodnikova; based on slides by Nick Hopper

Example GIVE A CFG FOR THE EMPTY SET G = { {S}, Σ, , S } Sofya Raskhodnikova; based on slides by Nick Hopper

GIVE A CFG FOR… L3 = { strings of balanced parens } L4 = { aibjck | i, j, k ≥ 0 and (i = j or j = k) } Sofya Raskhodnikova; based on slides by Nick Hopper

CFGs in the real world The syntactic grammar for the Java programming language BasicForStatement: for ( ; ; ) Statement for ( ; ; ForUpdate ) Statement for ( ; Expression ; ) Statement for ( ; Expression ; ForUpdate ) Statement for ( ForInit ; ; ) Statement for ( ForInit ; ; ForUpdate ) Statement for ( ForInit ; Expression ; ) Statement for ( ForInit ; Expression ; ForUpdate ) Statement Sofya Raskhodnikova; based on slides by Nick Hopper

COMPILER MODULES LEXER PARSER SEMANTIC ANALYZER TRANSLATOR/INTERPRETER Sofya Raskhodnikova; based on slides by Nick Hopper

Parse trees A A A B 0 0 # 1 1 A  0A1  00A11  00B11  00#11 Sofya Raskhodnikova; based on slides by Nick Hopper

Equivalence of CFGs & PDAs A language is generated by a CFG  It is recognized by a PDA Sofya Raskhodnikova; based on slides by Nick Hopper

Intro to Theory of Computation