Tree Automata Geetam Chawla Stanly Samuel ATC Seminar 2018 Intro - - PowerPoint PPT Presentation

Tree Automata

Geetam Chawla Stanly Samuel

ATC Seminar 2018

Overview

• Intro
• DTA
• Pumping Lemma

Tree-regular languages

• Some interesting

results.

• NTA
• ↓NTA vs ↓DTA

Introduction: Automata on words vs automata on trees.

Automata over finite words: (Finite Automata)FA->DFA->NFA Generalizing Finite automata: Automata over infinite words: (Buchi Automata)BA->NBA->DBA Automata over (finite) trees: (Tree Automata)TA->DTA->NTA->↓NTA-> ↓DTA

Introduction: Automata on words vs automata on trees.

Definition A ranked alphabet is a non-empty finite set Σ of symbols, where every symbol a is assigned a finite set rk(a) ⊆ N (the set of all ranks or arities of a). Σi :={ a ∈ Σ | i ∈ rk(a)} Σ = Σ0∪···∪Σm (where m= max{ rk(a) | a ∈ Σ}) Note that the Σi need not be disjoint.

Ranked Alphabet Σ

TΣ : The equivalent of Σ* in the tree world.

Let Σ = Σ0 ∪ . . . ∪ Σn be a ranked alphabet. The set TΣ of trees (or terms) over Σ is defined inductively by:

• Each symbol a ∈ Σ is a tree a ∈ TΣ.
• For f ∈ Σk and t1 . . . tk ∈ TΣ , f ( t1 . . . tk ) is a tree in TΣ depicted by

Deterministic Tree Automaton

A deterministic tree automaton (DTA) is a tuple A = ( Q, Σ , δ, F ) where

• Q is a finite set of states,
• Σ is a ranked alphabet,
• F ⊆ Q is a set of final states, and
• δ : ∀i≥0 ⋃( Qi

× Σi )→Q is a transition function.

Pumping Lemma

• Let T be a regular tree language. Then, for an appropriate n (namely the

number of states of a tree automaton recognizing T ) the following holds: If the height of t ∈ T is ≥ n , then there are two nodes on a path of t that can split t₀ in three pieces t₀, s1 , s2 , such that an iteration (arbitrarily often) of s1 ⁠≠ c results in trees that also belong to T . In short notation: t = t₀ ◦ s1 ◦ s2 with t₀ ◦ (s1)ⁱ ◦ s2 ∈ T for every i ≥ 0.

The circle operator: o

T1 ◦ T2. A requirement: T2 must be “special”. I.e. T2 has a special node “c” whose rank = {0} C acts as a marker where we can “insert” T1. It implies second operand must contain exactly one c otherwise the operation is undefined. Corollary: Not commutative

NTA

A Nondeterministic Tree Automaton (NTA) is a tuple A = ( Q, Σ, ∆, F ) where

• Q is a finite state set,
• Σ = Σ1 ∪ . . . ∪ Σm is a ranked alphabet,
• F ⊆ Q is a set of final states, and
• ∆ ⊆ ∀i≥0 ⋃( Qi × Σi × Q) is a transition relation.

NTA run

By convention Q0 × Σ0 × Q = Σ0 × Q. A run of A on a tree t ∈ TΣ is a mapping ρ : domt → Q such that

• For leaf nodes u ∈ domt we have (valt(u) , ρ(u)) ∈ ∆, and
• For nodes u with successors u1 , . . . , ui we have

(ρ(u1), . . . , ρ (ui), valt(u), ρ(u)) ∈ ∆. An NTA A accepts a tree t ∈ TΣ if for some run ρ of A on t, ρ(ϵ) ∈ F . The tree-language recognized by A is T(A) = {t ∈ TΣ |A accepts t}.

Some interesting results

1) If G is a CFG, then T(G) viz. the set of all derivation trees is tree regular. 2) Not every regular tree language can be represented as a language T(G) for some CFG G.

↓NTA

A Nondeterministic Tree Automaton (NTA) is a tuple A = ( Q, Σ , ∆ , F ) where

• Q is a finite state set,
• Σ = Σ1 ∪ . . . ∪ Σm is a ranked alphabet,
• Q0 ⊆ Q is a set of initial states, and
• ∆ ⊆ ∀i≥0 ⋃( Q × Σi × Qi) is a transition relation.

↓NTA run

A run of A on a tree t ∈ TΣ is a mapping ρ : domt → Q such that

• p(ϵ) ∈ Q0
• For leaf nodes u ∈ domt we have (ρ(u), valt(u)) ∈ ∆, and
• For nodes u with successors u1 , . . . , ui we have

(ρ(u), valt(u), ρ(u1), . . . , ρ (ui)) ∈ ∆. An NTA A accepts a tree t ∈ TΣ iff there exists a run on t.

↓DTA

A Top down Deterministic Tree Automaton (↓DTA) is a tuple A = ( Q, Σ , ∆ , F ) is a special ↓NTA that has:

• a singleton set {q0} as an initial state set Q0
• δ : ∀i≥0 ⋃( Q × Σi )→Qi

↓DTA

A Top down Deterministic Tree Automaton (↓DTA) is a tuple A = ( Q, Σ , ∆ , F ) is a special ↓NTA that has:

• a singleton set {q0} as an initial state set Q0
• δ : ∀i≥0 ⋃( Q × Σi )→Qi

↓DTA is strictly less powerful than ↓NTA. E.g.

Thank you