The Algebras for Automatic Relations

TL;DR

The interest of the proposed definition is that it allows to lift, in an effective way, pseudovarieties of regular languages to that of synchronous relations, and it is shown how algebraic characterizations of pseudovarieties of regular languages can be lifted to the pseudovarieties of synchronous relations that they induce.

Abstract

We introduce "synchronous algebras", an algebraic structure tailored to recognize automatic relations (aka. synchronous relations, or regular relations). They are the equivalent of monoids for regular languages, however they conceptually differ in two points: first, they are typed and second, they are equipped with a dependency relation expressing constraints between elements of different types. The interest of the proposed definition is that it allows to lift, in an effective way, pseudovarieties of regular languages to that of synchronous relations, and we show how algebraic characterizations of pseudovarieties of regular languages can be lifted to the pseudovarieties of synchronous relations that they induce. A typical example of such a pseudovariety is the class of "group relations", defined as the relations recognized by finite-state synchronous permutation automata. In order to prove this result, we adapt two pillars of algebraic language to synchronous algebras: (a) any relation admits a syntactic synchronous algebra recognizing it, and moreover, the relation is synchronous if, and only if, its syntactic algebra is finite and (b) classes of synchronous relations with desirable closure properties (i.e. pseudovarieties) correspond to pseudovarieties of synchronous algebras.

Figures (6)

• Figure 1: Encoding a pair of words of $\Sigma^* \times \Sigma^*$ into an element of $(\SigmaPair)^*$ where $\SigmaPair \defeq (\Sigma \times \Sigma) \,\cup\, (\Sigma \times \{\pad\}) \,\cup\, (\{\pad\} \times \Sigma)$ (left) and a deterministic complete "synchronous automaton" (right) over $\Sigma = \{a,b\}$ accepting the binary relation of pairs $(u,v)$ such that the number of $a$'s in $u_1\hdots u_k$ and in $v_1\hdots v_k$ are the same mod $2$, where $k = \min(|u|, |v|)$. $\textrm{Pad}$ denotes the set of transitions $\{\pair{a}{\pad}, \pair{b}{\pad}, \pair{\pad}{a}, \pair{\pad}{b}\}$.
• Figure 2: Drawing in $(\SigmaPair)^*$ of a "$\+V$-relation" $\+R$ and $\negrel\+R \defeq \{(u,v) \in \Sigma^*\times \Sigma^* \mid (u,v) \not\in \+R\}$, where $\+R$ is defined as $L \cap \WellFormed$ with $L \in \+V$.
• Figure 3: The landscape of rationality for binary relations. Dashed regions are empty: the intersection of functional relations and two-way rational relations collapses to regular functions by EH2001transduction.
• Figure 4: Minimal (deterministic complete) "classical" automaton for the binary relation of pairs $(u,v)$ such that the number of $a$'s in $u_1\hdots u_k$ and in $v_1\hdots v_k$ are the same mod $2$, where $k = \min(|u|, |v|)$, seen as a language over $\SigmaPair$. Said otherwise, this is automaton rejects exactly all words in $(\SigmaPair)^*$ which (1) are not the valid encoding of a pair of words and (2) are the encoding of a pair which does not satisfy the property above. Each label $*$ is defined so that the automaton is deterministic and complete.
• Figure 5: Representation of the "dependent set" $\Sync\Sigma$ of "synchronous words". Coloured edges represent the "dependency relation", and self-loops are not drawn.

Theorems & Definitions (34)

• Remark 1
• Theorem 2: Lifting theorem: Elementary Formulation
• Remark 3
• Example 6: Group relations
• Definition 8
• Example 9
• Definition 11
• Remark 13
• Proposition 16
• Example 17: Group relations: \ref{['ex:group-languages']}, cont'd.