Automata Theory Nested Word Automata Christian Schilling June 4th, 2012 Nested Word Automata Overview Motivation and background Nested words and their acceptors Determinization proof Conclusion 1 / 30 Nested Word Automata Motivation and background Overview Motivation and background Common languages Visibly pushdown languages Nested words and their acceptors Determinization proof Conclusion 1 / 30 q0 c q q 1 r 2 Nested Word Automata Motivation and background Common languages Regular language 1 procedure foo () § ¤ 2 { 3 r e t u r n ; 4 } ¦ L1 = fc rg ¥ 2 / 30 Nested Word Automata Motivation and background Common languages Regular language 1 procedure foo () § ¤ 2 { q0 3 r e t u r n ; 4 } c ¦ L1 = fc rg ¥ q q 1 r 2 2 / 30 q0 q3 ?, c, ?A A, r, " A, r, " q1 q2 B, r, " A, c, AB A, r, " B, c, BB Nested Word Automata Motivation and background Common languages (det.) Context-free language 1 procedure bar () § ¤ 2 { 3 i f (*) 4 c a l l bar (); 5 r e t u r n ; 6 } n n ¦ L2 = fc r j n > 0g ¥ 3 / 30 Nested Word Automata Motivation and background Common languages (det.) Context-free language q0 q3 1 procedure bar () § ¤ 2 { 3 i f (*) ?, c, ?A A, r, " A, r, " 4 c a l l bar (); 5 r e t u r n ; q1 q2 6 } B, r, " L = fcn rn j n > 0g ¦ 2 ¥ A, c, AB A, r, " B, c, BB 3 / 30 / / / all standard properties not under intersection and complementation all standard problems intersection, inclusion, equivalence undecidable powerset construction not possible Question: Is there some class of languages in between that is more expressive than regular languages, but keeps their nice properties? Answer (Alur & Madhusudan 2004): yes, at least in some sense Nested Word Automata Motivation and background Common languages Comparison regular context-free comparison, constants two variables of numbers closure decidability determinize 4 / 30 / / all standard problems intersection, inclusion, equivalence undecidable powerset construction not possible Question: Is there some class of languages in between that is more expressive than regular languages, but keeps their nice properties? Answer (Alur & Madhusudan 2004): yes, at least in some sense Nested Word Automata Motivation and background Common languages Comparison regular context-free comparison, / constants two variables of numbers closure all standard properties not under intersection and complementation decidability determinize 4 / 30 / powerset construction not possible Question: Is there some class of languages in between that is more expressive than regular languages, but keeps their nice properties? Answer (Alur & Madhusudan 2004): yes, at least in some sense Nested Word Automata Motivation and background Common languages Comparison regular context-free comparison, / / constants two variables of numbers closure all standard properties not under intersection and complementation decidability all standard problems intersection, inclusion, equivalence undecidable determinize 4 / 30 Question: Is there some class of languages in between that is more expressive than regular languages, but keeps their nice properties? Answer (Alur & Madhusudan 2004): yes, at least in some sense Nested Word Automata Motivation and background Common languages Comparison regular context-free comparison, / / / constants two variables of numbers closure all standard properties not under intersection and complementation decidability all standard problems intersection, inclusion, equivalence undecidable determinize powerset construction not possible 4 / 30 Answer (Alur & Madhusudan 2004): yes, at least in some sense Nested Word Automata Motivation and background Common languages Comparison regular context-free comparison, / / / constants two variables of numbers closure all standard properties not under intersection and complementation decidability all standard problems intersection, inclusion, equivalence undecidable determinize powerset construction not possible Question: Is there some class of languages in between that is more expressive than regular languages, but keeps their nice properties? 4 / 30 Nested Word Automata Motivation and background Common languages Comparison regular context-free comparison, / / / constants two variables of numbers closure all standard properties not under intersection and complementation decidability all standard problems intersection, inclusion, equivalence undecidable determinize powerset construction not possible Question: Is there some class of languages in between that is more expressive than regular languages, but keeps their nice properties? Answer (Alur & Madhusudan 2004): yes, at least in some sense 4 / 30 • states, initial state, final states, stack alphabet, bottom-of-stack symbol (no change here), • partitioning of the input alphabet: Σ = Σi ] Σc ] Σr , • δ = δi ] δc ] δr , • δi ⊆ Q × Σi ! Q • δc ⊆ Q × Σc ! (Γ n f?g) × Q • δr ⊆ Q × Σr × Γ ! Q Note: pops occur implicitly, ? never popped, no " Nested Word Automata Motivation and background Visibly pushdown languages Visibly pushdown languages (VPLs) A visibly pushdown language (VPL) is the language accepted by a visibly pushdown automaton (VPA). A VPA A = hQ; q0; Qf ; Σ; Γ; ?;δ i is a deterministic PDA with special rules: Determined by the input symbol, only one symbol per push is allowed and reading the stack implies a pop. 5 / 30 • partitioning of the input alphabet: Σ = Σi ] Σc ] Σr , • δ = δi ] δc ] δr , • δi ⊆ Q × Σi ! Q • δc ⊆ Q × Σc ! (Γ n f?g) × Q • δr ⊆ Q × Σr × Γ ! Q Note: pops occur implicitly, ? never popped, no " Nested Word Automata Motivation and background Visibly pushdown languages Visibly pushdown languages (VPLs) A visibly pushdown language (VPL) is the language accepted by a visibly pushdown automaton (VPA). A VPA A = hQ; q0; Qf ; Σ; Γ; ?;δ i is a deterministic PDA with special rules: Determined by the input symbol, only one symbol per push is allowed and reading the stack implies a pop. • states, initial state, final states, stack alphabet, bottom-of-stack symbol (no change here), 5 / 30 • δ = δi ] δc ] δr , • δi ⊆ Q × Σi ! Q • δc ⊆ Q × Σc ! (Γ n f?g) × Q • δr ⊆ Q × Σr × Γ ! Q Note: pops occur implicitly, ? never popped, no " Nested Word Automata Motivation and background Visibly pushdown languages Visibly pushdown languages (VPLs) A visibly pushdown language (VPL) is the language accepted by a visibly pushdown automaton (VPA). A VPA A = hQ; q0; Qf ; Σ; Γ; ?;δ i is a deterministic PDA with special rules: Determined by the input symbol, only one symbol per push is allowed and reading the stack implies a pop. • states, initial state, final states, stack alphabet, bottom-of-stack symbol (no change here), • partitioning of the input alphabet: Σ = Σi ] Σc ] Σr , 5 / 30 Note: pops occur implicitly, ? never popped, no " Nested Word Automata Motivation and background Visibly pushdown languages Visibly pushdown languages (VPLs) A visibly pushdown language (VPL) is the language accepted by a visibly pushdown automaton (VPA). A VPA A = hQ; q0; Qf ; Σ; Γ; ?;δ i is a deterministic PDA with special rules: Determined by the input symbol, only one symbol per push is allowed and reading the stack implies a pop. • states, initial state, final states, stack alphabet, bottom-of-stack symbol (no change here), • partitioning of the input alphabet: Σ = Σi ] Σc ] Σr , • δ = δi ] δc ] δr , • δi ⊆ Q × Σi ! Q • δc ⊆ Q × Σc ! (Γ n f?g) × Q • δr ⊆ Q × Σr × Γ ! Q 5 / 30 Nested Word Automata Motivation and background Visibly pushdown languages Visibly pushdown languages (VPLs) A visibly pushdown language (VPL) is the language accepted by a visibly pushdown automaton (VPA). A VPA A = hQ; q0; Qf ; Σ; Γ; ?;δ i is a deterministic PDA with special rules: Determined by the input symbol, only one symbol per push is allowed and reading the stack implies a pop. • states, initial state, final states, stack alphabet, bottom-of-stack symbol (no change here), • partitioning of the input alphabet: Σ = Σi ] Σc ] Σr , • δ = δi ] δc ] δr , • δi ⊆ Q × Σi ! Q • δc ⊆ Q × Σc ! (Γ n f?g) × Q • δr ⊆ Q × Σr × Γ ! Q Note: pops occur implicitly, ? never popped, no " 5 / 30 We construct a VPA for L2. Partitioning: Σi = ;,Σc = fcg,Σr = frg δc = f (q0; c; A; q1), (q1; c; B; q1) g δr = f (q1; r; A; q3), (q1; r; B; q2), (q2; r; A; q3), (q2; r; B; q2) g Nested Word Automata Motivation and background Visibly pushdown languages L2 as VPL n n Consider again L2 = fc r j n > 0g. q0 q3 ?, c, ?A A, r, " A, r, " q1 q2 B, r, " A, c, AB A, r, " B, c, BB 6 / 30 δc = f (q0; c; A; q1), (q1; c; B; q1) g δr = f (q1; r; A; q3), (q1; r; B; q2), (q2; r; A; q3), (q2; r; B; q2) g Nested Word Automata Motivation and background Visibly pushdown languages L2 as VPL n n Consider again L2 = fc r j n > 0g. We construct a VPA for L2. Partitioning: q q 0 3 Σi = ;,Σc = fcg,Σr = frg ?, c, ?A A, r, " A, r, " q1 q2 B, r, " A, c, AB A, r, " B, c, BB 6 / 30 Nested Word Automata Motivation and background Visibly pushdown languages L2 as VPL n n Consider again L2 = fc r j n > 0g. We construct a VPA for L2. Partitioning: q q 0 3 Σi = ;,Σc = fcg,Σr = frg c, A r, A δc = f (q0; c; A; q1), r, A (q1; c; B; q1) g q1 q2 δr = f (q1; r; A; q3), r, B (q1; r; B; q2), (q2; r; A; q3), c, B r, B (q2; r; B; q2) g 6 / 30 (Alur & Madhusudan 2006): no, with some further treatment of the input ! nested words (NWs) • automaton model: nested word automata (NWAs) • nested word languages (NWLs) and VPLs have same power ! NWAs deterministic PDAs • main idea: call and return symbols are matched in the input Nested Word Automata Motivation and background Visibly pushdown languages From VPAs to NWAs • main differences between VPAs and PDAs: • closed