Introduction Pushdown Automata Radboud University Nijmegen CFLs and PDAs Closure Properties and Concluding Remarks Formal Languages, Grammars and Automata Lecture 6 Helle Hvid Hansen helle@cs.ru.nl http://www.cs.ru.nl/~helle/ Foundations Group – Intelligent Systems Section Institute for Computing and Information Sciences Radboud University Nijmegen 13 June 2014 Helle Hvid Hansen 13 June 2014 FLGA 1 / 24
Introduction Pushdown Automata Radboud University Nijmegen CFLs and PDAs Closure Properties and Concluding Remarks Overview Last lecture: • Context-free grammars and context-free languages. • Regular grammars. • CYK-algorithm (parsing: w ∈ L ?) Helle Hvid Hansen 13 June 2014 FLGA 2 / 24
Introduction Pushdown Automata Radboud University Nijmegen CFLs and PDAs Closure Properties and Concluding Remarks Overview Last lecture: • Context-free grammars and context-free languages. • Regular grammars. • CYK-algorithm (parsing: w ∈ L ?) Today: • Pushdown automata (NL: stapelautomaten) • Closure properties of context-free languages • Beyond context-free languages Helle Hvid Hansen 13 June 2014 FLGA 2 / 24
Introduction Pushdown Automata Radboud University Nijmegen CFLs and PDAs Closure Properties and Concluding Remarks Automata for Context-Free Languages Language class Syntax/Grammar Automata Context-free context-free grammar ? Regular regular expressions, DFA, NFA, NFA- λ regular grammar Helle Hvid Hansen 13 June 2014 FLGA 3 / 24
Introduction Pushdown Automata Radboud University Nijmegen CFLs and PDAs Closure Properties and Concluding Remarks Automata for Context-Free Languages Language class Syntax/Grammar Automata Context-free context-free grammar ? Regular regular expressions, DFA, NFA, NFA- λ regular grammar • DFA, NFA, NFA- λ : finite memory (states can encode only finite amount of information), for example (for Σ = { a , b } ), • even or odd number of a ’s read: two states even , odd • the last 2 letters read: four states for aa , ab , ba , bb . Helle Hvid Hansen 13 June 2014 FLGA 3 / 24
Introduction Pushdown Automata Radboud University Nijmegen CFLs and PDAs Closure Properties and Concluding Remarks Automata for Context-Free Languages Language class Syntax/Grammar Automata Context-free context-free grammar ? Regular regular expressions, DFA, NFA, NFA- λ regular grammar • DFA, NFA, NFA- λ : finite memory (states can encode only finite amount of information), for example (for Σ = { a , b } ), • even or odd number of a ’s read: two states even , odd • the last 2 letters read: four states for aa , ab , ba , bb . • For example: { a n b n | n ≥ 0 } not regular. We would need to remember how many a ’s we have seen. A DFA with k states can only “count to k ”. Helle Hvid Hansen 13 June 2014 FLGA 3 / 24
Introduction Pushdown Automata Radboud University Nijmegen CFLs and PDAs Closure Properties and Concluding Remarks Automata for Context-Free Languages Language class Syntax/Grammar Automata Context-free context-free grammar ? Regular regular expressions, DFA, NFA, NFA- λ regular grammar • DFA, NFA, NFA- λ : finite memory (states can encode only finite amount of information), for example (for Σ = { a , b } ), • even or odd number of a ’s read: two states even , odd • the last 2 letters read: four states for aa , ab , ba , bb . • For example: { a n b n | n ≥ 0 } not regular. We would need to remember how many a ’s we have seen. A DFA with k states can only “count to k ”. We need some form of unbounded memory. Helle Hvid Hansen 13 June 2014 FLGA 3 / 24
Introduction Pushdown Automata Radboud University Nijmegen CFLs and PDAs Closure Properties and Concluding Remarks Stack Memory Stack of data elements (“stapel”) Note: only top of stack is visible. Helle Hvid Hansen 13 June 2014 FLGA 4 / 24
Introduction Pushdown Automata Radboud University Nijmegen CFLs and PDAs Closure Properties and Concluding Remarks Stack Memory Stack of data elements Stack operations: (“stapel”) push a new pop the top element element onto stack from stack Note: Stack is a Last-In-First-Out (LIFO) only top of stack is visible. data structure. Helle Hvid Hansen 13 June 2014 FLGA 4 / 24
Introduction Pushdown Automata Radboud University Nijmegen CFLs and PDAs Closure Properties and Concluding Remarks Stack Memory: Formal Notation • Stack content described as word over a stack alphabet A Γ = { A , B , C , ..., X , Y , ... } . Last-in = Left-most: B = ABC C • The empty stack is λ ∈ Γ ∗ . Helle Hvid Hansen 13 June 2014 FLGA 5 / 24
Introduction Pushdown Automata Radboud University Nijmegen CFLs and PDAs Closure Properties and Concluding Remarks Stack Memory: Formal Notation • Stack content described as word over a stack alphabet A Γ = { A , B , C , ..., X , Y , ... } . Last-in = Left-most: B = ABC C • The empty stack is λ ∈ Γ ∗ . • Stack operations: X A A A ) = B push ( X , B ) = pop ( B B C C C C push ( X , ABC ) = XABC pop ( ABC ) = BC • Note: The empty stack has no top. Helle Hvid Hansen 13 June 2014 FLGA 5 / 24
Introduction Pushdown Automata Radboud University Nijmegen CFLs and PDAs Closure Properties and Concluding Remarks Pushdown Automaton A pushdown automaton is an NFA- λ equipped with a stack. Helle Hvid Hansen 13 June 2014 FLGA 6 / 24
Introduction Pushdown Automata Radboud University Nijmegen CFLs and PDAs Closure Properties and Concluding Remarks Pushdown Automaton A pushdown automaton is an NFA- λ equipped with a stack. Def. A pushdown automaton (PDA) M = ( Q , Σ , Γ , δ, q 0 , F ) consists of • a finite set of (control) states Q • an input alphabet Σ • an initial state q 0 ∈ Q • a set of final states F ⊆ Q Helle Hvid Hansen 13 June 2014 FLGA 6 / 24
Introduction Pushdown Automata Radboud University Nijmegen CFLs and PDAs Closure Properties and Concluding Remarks Pushdown Automaton A pushdown automaton is an NFA- λ equipped with a stack. Def. A pushdown automaton (PDA) M = ( Q , Σ , Γ , δ, q 0 , F ) consists of • a finite set of (control) states Q • an input alphabet Σ • an initial state q 0 ∈ Q • a set of final states F ⊆ Q • a stack alphabet Γ • a transition function δ : Q × Σ λ × Γ λ → P ( Q × Γ λ ) where Σ λ = Σ ∪ { λ } and Γ λ = Γ ∪ { λ } . That is, for p ∈ Q , a ∈ Σ λ , A ∈ Γ λ , δ ( p , a , A ) is a set of pairs ( q , B ) where q ∈ Q , B ∈ Γ λ (nondeterministic). Helle Hvid Hansen 13 June 2014 FLGA 6 / 24
Introduction Pushdown Automata Radboud University Nijmegen CFLs and PDAs Closure Properties and Concluding Remarks Pushdown Automaton Computation A computation of a PDA M on input word w : Helle Hvid Hansen 13 June 2014 FLGA 7 / 24
Introduction Pushdown Automata Radboud University Nijmegen CFLs and PDAs Closure Properties and Concluding Remarks Pushdown Automaton Computation A computation of a PDA M on input word w : • Start configuration ( q 0 , w , λ ) (start in initial state with empty stack) Helle Hvid Hansen 13 June 2014 FLGA 7 / 24
Introduction Pushdown Automata Radboud University Nijmegen CFLs and PDAs Closure Properties and Concluding Remarks Pushdown Automaton Computation A computation of a PDA M on input word w : • Start configuration ( q 0 , w , λ ) (start in initial state with empty stack) • Transitions are taken (nondeterministically) depending on – the next input symbol (as in DFA, NFA) and – the stack top symbol. (Details next slide.) Changes configuration ( p , u , α ) → ( q , v , β ) according to transition function (where p , q ∈ Q , ; u , v ∈ Σ ∗ , α, β ∈ Γ ∗ ). Helle Hvid Hansen 13 June 2014 FLGA 7 / 24
Introduction Pushdown Automata Radboud University Nijmegen CFLs and PDAs Closure Properties and Concluding Remarks Pushdown Automaton Computation A computation of a PDA M on input word w : • Start configuration ( q 0 , w , λ ) (start in initial state with empty stack) • Transitions are taken (nondeterministically) depending on – the next input symbol (as in DFA, NFA) and – the stack top symbol. (Details next slide.) Changes configuration ( p , u , α ) → ( q , v , β ) according to transition function (where p , q ∈ Q , ; u , v ∈ Σ ∗ , α, β ∈ Γ ∗ ). • Computation is successful if it ends in a configuration ( q , λ, λ ) where q ∈ F . (Acceptance by final state and empty stack) Helle Hvid Hansen 13 June 2014 FLGA 7 / 24
Introduction Pushdown Automata Radboud University Nijmegen CFLs and PDAs Closure Properties and Concluding Remarks Pushdown Automaton Transitions a , A / B � q : p ( q , B ) ∈ δ ( p , a , A ) i.e. • can be taken if: next input symbol is a , stack top is A • actions: read/consume a , pop A from stack, push B . Helle Hvid Hansen 13 June 2014 FLGA 8 / 24
Introduction Pushdown Automata Radboud University Nijmegen CFLs and PDAs Closure Properties and Concluding Remarks Pushdown Automaton Transitions a , A / B � q : p ( q , B ) ∈ δ ( p , a , A ) i.e. • can be taken if: next input symbol is a , stack top is A • actions: read/consume a , pop A from stack, push B . λ, A / B � q p ( q , B ) ∈ δ ( p , λ, A ) i.e. (input λ -transition): • can be taken if: stack top is A (regardless of next input) • actions: pop A from stack, push B . Helle Hvid Hansen 13 June 2014 FLGA 8 / 24
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