Nested Word Automata Automata Theory and Computability S Gaurang, Pranshu Gaba Indian Institute of Science, Bangalore November 16, 2018 Motivation Definitions NWA Restrictions Closure Equivalent Definitions Applications Overview 1 Motivation 2 Definitions 3 NWA 4 Restrictions 5 Closure 6 Equivalent Definitions 7 Applications Proper fragment of deterministic CFL's More expressive than regular languages Nice closure properties Motivation Definitions NWA Restrictions Closure Equivalent Definitions Applications Motivation More expressive than regular languages Nice closure properties Motivation Definitions NWA Restrictions Closure Equivalent Definitions Applications Motivation Proper fragment of deterministic CFL's Nice closure properties Motivation Definitions NWA Restrictions Closure Equivalent Definitions Applications Motivation Proper fragment of deterministic CFL's More expressive than regular languages Motivation Definitions NWA Restrictions Closure Equivalent Definitions Applications Motivation Proper fragment of deterministic CFL's More expressive than regular languages Nice closure properties Motivation Definitions NWA Restrictions Closure Equivalent Definitions Applications Motivation Proper fragment of deterministic CFL's More expressive than regular languages Nice closure properties Motivation Definitions NWA Restrictions Closure Equivalent Definitions Applications Overview 1 Motivation 2 Definitions 3 NWA 4 Restrictions 5 Closure 6 Equivalent Definitions 7 Applications Nested words only go forward: if i j, then i j; No two nested edges share a position: For i , there is at most one position h such that↝h i and< at most one position j such that i j; −∞ < < ∞ A position cannot be both a call and a↝ return. ↝ Nesting edges never cross: if i j, and i j , then it is not the case that i i j j . ′ ′ ′ ′ ↝ ↝ < ≤ < Motivation Definitions NWA Restrictions Closure Equivalent Definitions Applications Matching Relation A matching relation of length ` 0 is a subset of ; 1; 2; 3; : : : ; ` 1; 2; 3; : : : ; `; such that: ↝ ≥ {−∞ } × { +∞} No two nested edges share a position: For i , there is at most one position h such that h i and at most one position j such that i j; −∞ < < ∞ A position cannot be both a call and a↝ return. ↝ Nesting edges never cross: if i j, and i j , then it is not the case that i i j j . ′ ′ ′ ′ ↝ ↝ < ≤ < Motivation Definitions NWA Restrictions Closure Equivalent Definitions Applications Matching Relation A matching relation of length ` 0 is a subset of ; 1; 2; 3; : : : ; ` 1; 2; 3; : : : ; `; such that: Nested words only↝ go forward:≥ if i j, then i j; {−∞ } × { +∞} ↝ < A position cannot be both a call and a return. Nesting edges never cross: if i j, and i j , then it is not the case that i i j j . ′ ′ ′ ′ ↝ ↝ < ≤ < Motivation Definitions NWA Restrictions Closure Equivalent Definitions Applications Matching Relation A matching relation of length ` 0 is a subset of ; 1; 2; 3; : : : ; ` 1; 2; 3; : : : ; `; such that: Nested words only↝ go forward:≥ if i j, then i j; {−∞ } × { +∞} No two nested edges share a position: For i , there is at most one position h such that↝h i and< at most one position j such that i j; −∞ < < ∞ ↝ ↝ Nesting edges never cross: if i j, and i j , then it is not the case that i i j j . ′ ′ ′ ′ ↝ ↝ < ≤ < Motivation Definitions NWA Restrictions Closure Equivalent Definitions Applications Matching Relation A matching relation of length ` 0 is a subset of ; 1; 2; 3; : : : ; ` 1; 2; 3; : : : ; `; such that: Nested words only↝ go forward:≥ if i j, then i j; {−∞ } × { +∞} No two nested edges share a position: For i , there is at most one position h such that↝h i and< at most one position j such that i j; −∞ < < ∞ A position cannot be both a call and a↝ return. ↝ Motivation Definitions NWA Restrictions Closure Equivalent Definitions Applications Matching Relation A matching relation of length ` 0 is a subset of ; 1; 2; 3; : : : ; ` 1; 2; 3; : : : ; `; such that: Nested words only↝ go forward:≥ if i j, then i j; {−∞ } × { +∞} No two nested edges share a position: For i , there is at most one position h such that↝h i and< at most one position j such that i j; −∞ < < ∞ A position cannot be both a call and a↝ return. ↝ Nesting edges never cross: if i j, and i j , then it is not the case that i i j j . ′ ′ ′ ′ ↝ ↝ < ≤ < a call position, if i j for some j, if i , then i is a pending call. else i is a matched↝ call, and j is the return-successor of i. a return↝ position+∞ , if h i, for some h, if i, then i is a pending return. else i is a matched↝ return, and h is the call-predecessor of i. −∞ ↝ an internal position, for all remaining cases. Motivation Definitions NWA Restrictions Closure Equivalent Definitions Applications Matching Relation A position 1 i ` is referred to as ≤ ≤ a return position, if h i, for some h, if i, then i is a pending return. else i is a matched↝ return, and h is the call-predecessor of i. −∞ ↝ an internal position, for all remaining cases. Motivation Definitions NWA Restrictions Closure Equivalent Definitions Applications Matching Relation A position 1 i ` is referred to as a call position, if i j for some j, if i ≤ ≤ , then i is a pending call. else i is a matched↝ call, and j is the return-successor of i. ↝ +∞ an internal position, for all remaining cases. Motivation Definitions NWA Restrictions Closure Equivalent Definitions Applications Matching Relation A position 1 i ` is referred to as a call position, if i j for some j, if i ≤ ≤ , then i is a pending call. else i is a matched↝ call, and j is the return-successor of i. a return↝ position+∞ , if h i, for some h, if i, then i is a pending return. else i is a matched↝ return, and h is the call-predecessor of i. −∞ ↝ Motivation Definitions NWA Restrictions Closure Equivalent Definitions Applications Matching Relation A position 1 i ` is referred to as a call position, if i j for some j, if i ≤ ≤ , then i is a pending call. else i is a matched↝ call, and j is the return-successor of i. a return↝ position+∞ , if h i, for some h, if i, then i is a pending return. else i is a matched↝ return, and h is the call-predecessor of i. −∞ ↝ an internal position, for all remaining cases. Motivation Definitions NWA Restrictions Closure Equivalent Definitions Applications Matching Relation A position 1 i ` is referred to as a call position, if i j for some j, if i ≤ ≤ , then i is a pending call. else i is a matched↝ call, and j is the return-successor of i. a return↝ position+∞ , if h i, for some h, if i, then i is a pending return. else i is a matched↝ return, and h is the call-predecessor of i. −∞ ↝ an internal position, for all remaining cases. For 1 i `, there is a linear edge, between i and i 1. The initial node has an incoming linear edge which has no source,≤ and≤ the last node has an outgoing linear edge+ with no destination. For each matched call position i, there is a outgoing nesting edge, leading to the returning successor position. For a pending call i, there is a nesting edge that has no destination, while a pending return has a nested edge that has no source. Motivation Definitions NWA Restrictions Closure Equivalent Definitions Applications Matching Relation as a Graph A matching relation can be thought of as a directed acyclic graph over ` vertices: The initial node has an incoming linear edge which has no source, and the last node has an outgoing linear edge with no destination. For each matched call position i, there is a outgoing nesting edge, leading to the returning successor position. For a pending call i, there is a nesting edge that has no destination, while a pending return has a nested edge that has no source. Motivation Definitions NWA Restrictions Closure Equivalent Definitions Applications Matching Relation as a Graph A matching relation can be thought of as a directed acyclic graph over ` vertices: For 1 i `, there is a linear edge, between i and i 1. ≤ ≤ + For each matched call position i, there is a outgoing nesting edge, leading to the returning successor position. For a pending call i, there is a nesting edge that has no destination, while a pending return has a nested edge that has no source. Motivation Definitions NWA Restrictions Closure Equivalent Definitions Applications Matching Relation as a Graph A matching relation can be thought of as a directed acyclic graph over ` vertices: For 1 i `, there is a linear edge, between i and i 1. The initial node has an incoming linear edge which has no source,≤ and≤ the last node has an outgoing linear edge+ with no destination. For a pending call i, there is a nesting edge that has no destination, while a pending return has a nested edge that has no source. Motivation Definitions NWA Restrictions Closure Equivalent Definitions Applications Matching Relation as a Graph A matching relation can be thought of as a directed acyclic graph over ` vertices: For 1 i `, there is a linear edge, between i and i 1. The initial node has an incoming linear edge which has no source,≤ and≤ the last node has an outgoing linear edge+ with no destination. For each matched call position i, there is a outgoing nesting edge, leading to the returning successor position. Motivation Definitions NWA Restrictions Closure Equivalent Definitions Applications Matching Relation as a Graph A matching relation can be thought of as a directed acyclic graph over ` vertices: For 1 i `, there is a linear edge, between i and i 1. The initial node has an incoming linear edge which has no source,≤ and≤ the last node has an outgoing linear edge+ with no destination.