Department of Computer Science & Automation Indian Institute of Science Nested Word Automata Seminar Presentation 08 Dec 16 Dara Singh Vohra Jibin Surendran Ashok Rawat 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