Introduction

Regular word languages

• regular expressions: denotational

• finite automata: operational

• MSO: logic

• finite moinoids: algebraic, reason about the infixes of the language

  • $$\{\begin{array}{l}\phi :{\Sigma }^{\ast }⟶M\\ \epsilon \mapsto 1\\ uvw⟼\phi (u)\phi (v)\phi (w)\end{array}$$
  • aperiodic monoids ⇔ first order logic

regular tree expressions, finite algebra: not that easy to manipulate…

Definitions

Tree

A ranked alphabet:

is a pair $⟨𝔉,arity⟩$ where $arity:𝔉⟶\mathbb{N}$

notations:

• $f^{(n)}$ for $f\in 𝔉$ with arity $n$
• ${𝔉}_n\stackrel{\scriptscriptstyle\mathrm{def}}{=}\{f\in 𝔉\mid arity(f)=n\}$, $𝔉\stackrel{\scriptscriptstyle\mathrm{def}}{=}\bigcup _{n\in \mathbb{N}}{𝔉}_n$

Rooted, ordered, labelled, finite trees

as partial functions:

let $𝔉$ be a ranked alphabet. A partial function $t:{\mathbb{N}}_{>0}^{\ast }⟶𝔉$ is a tree iff

• $\mathrm{d}\mathrm{o}\mathrm{m}(t)$ is finite an non-empty

• $\mathrm{d}\mathrm{o}\mathrm{m}(t)$ is prefixed-closed: $\forall p,p^{\prime }\in {\mathbb{N}}_{>0}^{\ast },p{p}^{\prime }\in \mathrm{d}\mathrm{o}\mathrm{m}(t)⟹p\in \mathrm{d}\mathrm{o}\mathrm{m}(t)$

• labels are consistent with $𝔉$: $\forall p\in \mathrm{d}\mathrm{o}\mathrm{m}(t),t(p)\in {𝔉}_n$ for some $n$ implies $\{pi\in \mathrm{d}\mathrm{o}\mathrm{m}(t)\mid i\in {\mathbb{N}}_{>0}^{\ast }\}=\{p1,\dots ,pn\}$

notation: $T(𝔉)$ is the set of trees labelled by $𝔉$

the subtree of $t$ at $p\in \mathrm{d}\mathrm{o}\mathrm{m}(t)$ is:

$t_{|p}$, defined by $\{\begin{array}{l}\mathrm{d}\mathrm{o}\mathrm{m}(t_{|p})\stackrel{\scriptscriptstyle\mathrm{def}}{=}\{p^{\prime }\mid p{p}^{\prime }\in \mathrm{d}\mathrm{o}\mathrm{m}(t)\}\\ t_{|p}(p^{\prime })\stackrel{\scriptscriptstyle\mathrm{def}}{=}t(p{p}^{\prime })\end{array}$

A non-deterministic finite tree automaton (NFTA) is:

a tuple $𝒜\stackrel{\scriptscriptstyle\mathrm{def}}{=}⟨Q,𝔉,Q_f,\Delta ⟩$ where

• $Q$ is a finite set of states
• $𝔉$ is a finite ranked alphabet
• $Q_f\subseteq Q$ is a set of final states
• $\Delta \subseteq \bigcup _{n}Q\times {𝔉}_n\times Q^n$

A run of $𝒜$ on a tree $t$ is:

a tree in $T(Q\times \mathbb{N})$ where $arity(q,n)\stackrel{\scriptscriptstyle\mathrm{def}}{=}n$ such that

• $\mathrm{dom}\rho =\mathrm{dom}t$
• $$\forall p\in \mathrm{dom}\rho =\mathrm{dom}t,\text{ if }\rho (p)=(q,n)\text{ for some }q,n\text{ then }\exists (q,t(p),q_1,\dots ,q_n)\in \Delta \text{ s.t. }t(p)\in {𝔉}_n$$
• $$\forall 1\le i\le n,\rho (pi)=(q_i,n_i)\text{ for some }{n}_i$$

A run $\rho$ is accepting if $\rho (\epsilon )\in Q_f$

The language of $𝒜$ is:

$$L_{𝒜}\stackrel{\scriptscriptstyle\mathrm{def}}{=}\{t\in T(𝔉)\mid \exists \text{ accepting run on }t\}$$

Example:

• $Q\stackrel{\scriptscriptstyle\mathrm{def}}{=}\{q_f,q_g,q_a,q_b\}$

• $Q_f\stackrel{\scriptscriptstyle\mathrm{def}}{=}\{q_f\}$

• $𝔉\stackrel{\scriptscriptstyle\mathrm{def}}{=}\{f^{(2)},g^{(1)},a^{(0)},b^{(0)}\}$

• $$\begin{array}{c}\Delta \stackrel{\scriptscriptstyle\mathrm{def}}{=}\{(q_f,f^{(2)},q_g,q_g),\\ (q_g,g^{(1)},q_g),\\ (q_g,g^{(1)},q_a),\\ (q_g,g^{(1)},q_b),\\ (q_a,a^{(0)}),\\ (q_b,b^{(0)})\}\end{array}$$

Example:

$𝔉\stackrel{\scriptscriptstyle\mathrm{def}}{=}\{\vee ,\wedge ,\neg ,\mathrm{\top },\mathrm{\perp }\}$ (with obvious arities)

$𝒜$ s.t. $L(𝒜)$ is the set of Boolean formulae that evaluate to true

• $Q\stackrel{\scriptscriptstyle\mathrm{def}}{=}\{q_0,q_1\}$, $Q_f\stackrel{\scriptscriptstyle\mathrm{def}}{=}\{q_1\}$

• $$\begin{array}{c}\Delta \stackrel{\scriptscriptstyle\mathrm{def}}{=}\{(q_1,\mathrm{\top }),\\ (q_0,\mathrm{\perp }),\\ (q_1,\neg ,q_0),\\ (q_0,\neg ,q_1),\\ (q_1,\wedge ,q_1,q_1),(q_0,\wedge ,q_1,q_0),\\ (q_1,\wedge ,q_1,q_1),(q_0,\wedge ,q_0,q_0),\\ (q_1,\wedge ,q_0,q_1),(q_0,\wedge ,q_1,q_0),\\ ⋮\\ \}\end{array}$$

Bottom-up

Inductive definition:

let $𝔉$ be a ranked alphabet s.t.

• $a^{(0)}\in {𝔉}_0$ is a tree
• if $t_1,\dots ,t_n$ are trees and $f^{(n)}\in {𝔉}_n$, then $f^{(n)}(t_1,\dots ,t_n)$ is a tree

Let $𝒳$ be a countable set of variables $𝒳\cap 𝔉=\varnothing$.

We set $arity(x)=0,\forall x\in 𝒳$.

A tree in $T(𝔉\cup 𝒳)$ is called a term and we rather write $T(𝔉,𝒳)$ in that case.

A substitution:

is a function $\sigma :𝒳⟶T(𝔉,𝒳)$ with $\{x\in 𝒳\mid \sigma (x)\ne x\}$ finite

It defines a function $T(𝔉,𝒳)⟶T(𝔉,𝒳)$ by congruence:

• $x\sigma \stackrel{\scriptscriptstyle\mathrm{def}}{=}\sigma (x)$
• $f^{(n)}(t_1,\dots ,t_n)\sigma \stackrel{\scriptscriptstyle\mathrm{def}}{=}{f}^{(n)}(t_1\sigma ,\dots ,t_n\sigma )$

thus $a^{(0)}\sigma =a^{(0)}$

Example:

if $\sigma :\{\begin{array}{l}x⟼a^{(0)}\\ y⟼g(x)\end{array}$

$t$:

⇓

$t\sigma$:

A term $t\in T(𝔉,𝒳)$ is linear:

if every variable of $𝒳$ appears at most once in $t$

A context:

is a term in $T(𝔉,\{◻\})$ where $◻$ occurs exactly once.

notation :

• $t\sigma$ is an instance of $t$.
• a term in $T(𝔉,𝒳)$ with no variables is a ground term.

Example:

$t$:

⇓

$t=C[t^{\prime }]\stackrel{\scriptscriptstyle\mathrm{def}}{=}C\sigma$ where $C\sigma$ for $\sigma (◻)=g(b)$ and

$C\stackrel{\scriptscriptstyle\mathrm{def}}{=}$

and

$t^{\prime }\stackrel{\scriptscriptstyle\mathrm{def}}{=}$

notations: $C(𝔉)$ for the set of contexts over $𝔉$

A term rewriting system $R$ over $T(𝔉,𝒳)$:

is a set of pairs $l⟶r$ with $l,r\in T(𝔉,𝒳)$ and $vars(r)\subseteq vars(l)$, where $vars(t)\stackrel{\scriptscriptstyle\mathrm{def}}{=}\{x\in 𝒳\mid \exists p\in \mathrm{dom}t;t(p)=x\}$

$R$ defines a rewriting relation ${⟶}_R\subseteq T(𝔉)\times T(𝔉)$ by:

• $t{⟶}_R{t}^{\prime }$
• iff $\exists C\in C(𝔉)$ and a substitution $\sigma$ s.t. $t\stackrel{\scriptscriptstyle\mathrm{def}}{=}C[l\sigma ]$ and $t^{\prime }\stackrel{\scriptscriptstyle\mathrm{def}}{=}C[r\sigma ]$

Example:

$t$:

1) $C=f(◻,g(b))$, $\sigma =x⟼a$

$t{⟶}_R$

2) $C=f(g(a),◻)$, $\sigma =x⟼b$

$t{⟶}_R$

Example 2: It doesn’t work there:

$t$:

${⟶}_R$

where $C=◻$, $\sigma :x⟼g(a)$

Bottom-up tree automata

Given a NFTA $𝒜\stackrel{\scriptscriptstyle\mathrm{def}}{=}⟨Q,𝔉,Q_f,\Delta ⟩$, we define the top-down rewrite rules by:

• $f^{(n)}(q_1,\dots ,q_n){⟶}_{𝒜}q$ for all $(q,f^{(n)},q_1,\dots ,q_n)\in \Delta$ and using $arity(q)=0$ for all $q\in Q$

Proposition: $L(𝒜)\stackrel{\scriptscriptstyle\mathrm{def}}{=}\{t\in T(𝔉)\mid \exists q\in Q_f;t{⟶}_{𝒜}^{\ast }q\}$

Example: when it comes to the previous example related to logical formulas:

Other view:

$arity(q)\stackrel{\scriptscriptstyle\mathrm{def}}{=}1$ for all $q\in Q$

and

in that view:

a NFTA is complete:

if $\forall n,\forall f\in {𝔉}_n,\forall q_1,\dots ,q_n\in Q,\exists q\in Q$ s.t. $(q,f^{(n)},q_1,\dots ,q_n)\in \Delta$

NB: If $𝒜$ is complete, for all $t$, there exists a $q$ s.t. $t{⟶}_{𝒜}^{\ast }q$

a NFTA is deterministic:

if $\forall n,\forall f\in {𝔉}_n,\forall q_1,\dots ,q_n\in Q,\exists q\in Q$ s.t. $|\{q\in Q\mid (q,f^{(n)},q_1,\dots ,q_n)\in \Delta \}|\le 1$

NB: If $𝒜$ is deterministic, for all $t$, there exists at most one $q$ s.t. $t{⟶}_{𝒜}^{\ast }q$

NB: If $𝒜$ is complete and deterministic, for all $t$, there exists a unique $q$ s.t. $t{⟶}_{𝒜}^{\ast }q$

Closure properties

As for word automata, one can complete and determinize any NFTA.

Proposition: Given a NTFA $𝒜$, we can construct an equivalent complete NTFA ${𝒜}^{\prime }$.

$$Q^{\prime }\stackrel{\scriptscriptstyle\mathrm{def}}{=}Q\bigsqcup \{sink\}$$

We send any missing transition to the sink:

Proposition: Given a NTFA $𝒜$, we can construct an equivalent complete and deterministic TFA ${𝒜}^{\prime }$.

• $Q^{\prime }\stackrel{\scriptscriptstyle\mathrm{def}}{=}{2}^Q$
• $Q_f^{\prime }\stackrel{\scriptscriptstyle\mathrm{def}}{=}\{E\subseteq Q\mid E\cap Q_f\ne \varnothing \}$
• $$\begin{array}{c}{\Delta }^{\prime }\stackrel{\scriptscriptstyle\mathrm{def}}{=}\{(E,f^{(n)},E_1,\dots ,E_n)\mid E_1,\dots ,E_n\subseteq Q\text{ and }\\ E\stackrel{\scriptscriptstyle\mathrm{def}}{=}\{q\in Q\mid \exists (q_1,\dots ,q_n)\in E_1\times \cdots \times E_n;(q,f^{(n)},q_1,\dots ,q_n)\in \Delta \}\}\end{array}$$

Closure properties

Boolean closure quite similar to the word case:

• Union is disjoint union of tree automata

• Complementation: determinize, then complement $Q_f$

• Intersection: product automaton

Top-down tree automata

Given a NFTA $𝒜\stackrel{\scriptscriptstyle\mathrm{def}}{=}⟨Q,𝔉,Q_f,\Delta ⟩$, we define the bottom-up rewrite rules by:

• $q{⟶}_{𝒜}{f}^{(n)}(q_1,\dots ,q_n)$ for all $(q,f^{(n)},q_1,\dots ,q_n)\in \Delta$ and using $arity(q)=0$ for all $q\in Q$

Proposition: $L(𝒜)\stackrel{\scriptscriptstyle\mathrm{def}}{=}\{t\in T(𝔉)\mid \exists q\in Q_f;q{⟶}_{𝒜}^{\ast }t\}$

Example:

If the language is $\{f(a,b),f(b,a)\}$

then

(chap. 1, 3, 8)