We introduce "synchronous algebras", an algebraic structure tailored to recognize automatic relations (a.k.a. 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. Since this construction is effective, this implies that the membership problem is decidable for (infinitely) many natural classes of automatic relations. 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 theory 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.

We show that the problem of whether a query is equivalent to a query of tree-width $k$ is decidable, for the class of Unions of Conjunctive Regular Path Queries with two-way navigation (UC2RPQs). A previous result by Barcel\'o, Romero, and Vardi [SIAM Journal on Computing, 2016] has shown decidability for the case $k=1$, and here we extend this result showing that decidability in fact holds for any arbitrary $k\geq 1$. The algorithm is in 2ExpSpace, but for the restricted but practically relevant case where all regular expressions of the query are of the form $a^*$ or $(a_1 + \dotsb + a_n)$ we show that the complexity of the problem drops to $\Pi^P_2$. We also investigate the related problem of approximating a UC2RPQ by queries of small tree-width. We exhibit an algorithm which, for any fixed number $k$, builds the maximal under-approximation of tree-width $k$ of a UC2RPQ. The maximal under-approximation of tree-width $k$ of a query $q$ is a query $q'$ of tree-width $k$ which is contained in $q$ in a maximal and unique way, that is, such that for every query $q''$ of tree-width $k$, if $q''$ is contained in $q$ then $q''$ is also contained in $q'$. Our approach is shown to be robust, in the sense that it allows also to test equivalence with queries of a given path-width, it also covers the previously known result for $k=1$, and it allows to test for equivalence of whether a (one-way) UCRPQ is equivalent to a UCRPQ of a given tree-width (or path-width).

We study the minimization problem for Conjunctive Regular Path Queries (CRPQs) and unions of CRPQs (UCRPQs). This is the problem of checking, given a query and a number k , whether the query is equivalent to one of size at most k . For CRPQs we consider the size to be the number of atoms, and for UCRPQs the maximum number of atoms in a CRPQ therein, motivated by the fact that the number of atoms has a leading influence on the cost of query evaluation. We show that the minimization problem is decidable, both for CRPQs and UCRPQs. We provide a 2ExpSpace upper-bound for CRPQ minimization, based on a brute-force enumeration algorithm, and an ExpSpace lower-bound. For UCRPQs, we show that the problem is ExpSpace-complete, having thus the same complexity as the classical containment problem. The upper bound is obtained by defining and computing a notion of maximal under-approximation. Moreover, we show that for UCRPQs using the so-called simple regular expressions consisting of concatenations of expressions of the form a + or a 1 + ⋅⋅⋅ + a k , the minimization problem becomes "PiP2"-complete, again matching the complexity of containment.

We show that the problem of whether a query is equivalent to a query of tree-width $k$ is decidable, for the class of Unions of Conjunctive Regular Path Queries with two-way navigation (UC2RPQs). A previous result by Barceló, Romero, and Vardi [SIAM Journal on Computing, 2016] has shown decidability for the case $k=1$, and here we extend this result showing that decidability in fact holds for any arbitrary $k\geq 1$. The algorithm is in 2ExpSpace, but for the restricted but practically relevant case where all regular expressions of the query are of the form $a^*$ or $(a_1 + \dotsb + a_n)$ we show that the complexity of the problem drops to $Π^P_2$. We also investigate the related problem of approximating a UC2RPQ by queries of small tree-width. We exhibit an algorithm which, for any fixed number $k$, builds the maximal under-approximation of tree-width $k$ of a UC2RPQ. The maximal under-approximation of tree-width $k$ of a query $q$ is a query $q'$ of tree-width $k$ which is contained in $q$ in a maximal and unique way, that is, such that for every query $q''$ of tree-width $k$, if $q''$ is contained in $q$ then $q''$ is also contained in $q'$. Our approach is shown to be robust, in the sense that it allows also to test equivalence with queries of a given path-width, it also covers the previously known result for $k=1$, and it allows to test for equivalence of whether a (one-way) UCRPQ is equivalent to a UCRPQ of a given tree-width (or path-width).

We study the separability problem for automatic relations (i.e., relations on finite words definable by synchronous automata) in terms of recognizable relations (i.e., finite unions of products of regular languages). This problem takes as input two automatic relations $R$ and $R'$, and asks if there exists a recognizable relation $S$ that contains $R$ and does not intersect $R'$. We show this problem to be undecidable when the number of products allowed in the recognizable relation is fixed. In particular, checking if there exists a recognizable relation $S$ with at most $k$ products of regular languages that separates $R$ from $R'$ is undecidable, for each fixed $k \geq 2$. Our proofs reveal tight connections, of independent interest, between the separability problem and the finite coloring problem for automatic graphs, where colors are regular languages.

Abstract We show that the existence of a first-order formula separating two monadic second order formulas over countable ordinal words is decidable. This extends the work of Henckell and Almeida on finite words, and of Place and Zeitoun on $$\omega $$ ω -words. For this, we develop the algebraic concept of monoid (resp. $$\omega $$ ω -semigroup, resp. ordinal monoid) with aperiodic merge, an extension of monoids (resp. $$\omega $$ ω -semigroup, resp. ordinal monoid) that explicitly includes a new operation capturing the loss of precision induced by first-order indistinguishability. We also show the computability of FO-pointlike sets, and the decidability of the covering problem for first-order logic on countable ordinal words.