Guillem Godoy

Confluence of shallow right-linear rewrite systems

We show that confluence of shallow and right-linear term rewriting systems is decidable. This class of rewriting system is expressive enough to include nontrivial nonground rules such as commutativity, identity, and idempotence. Our proof uses the fact that this class of rewrite systems is known to be regularity-preserving, which implies that its reachability and joinability problems are decidable. The new decidability result is obtained by building upon our prior work for the class of ground term rewriting systems and shallow linear term rewriting systems. The proof relies on the concept of extracting more general rewrite derivations from a given rewrite derivation.

The confluence of ground term rewrite systems is decidable in polynomial time

The confluence property of ground (i.e., variable-free) term rewrite systems (GTRS) is well-known to be decidable. This was proved independently by M. Dauchet et al. (1987; 1990) and by M. Oyamaguchi (1987) using tree automata techniques and ground tree transducer techniques (originated from this problem), yielding EXPTIME decision procedures (PSPACE for strings). Since then, it has been a well-known longstanding open question whether this bound is optimal. The authors give a polynomial-time algorithm for deciding the confluence of GTRS, and hence alsofor the particular case of suffix- and prefix string rewrite systems or Thue systems. We show that this bound is optimal for all these problems by proving PTIME-hardness for the string case. This result may have some impact on other areas of formal language theory, and in particular on the theory of tree automata.

The Emptiness Problem for Tree Automata with Global Constraints

We define tree automata with global constraints (TAGC), generalizing the class of tree automata with global equality and disequality constraints (TAGED). TAGC can test for equality and disequality between subterms whose positions are defined by the states reached during a computation. In particular, TAGC can check that all the subterms reaching a given state are distinct. This constraint is related to monadic key constraints for XML documents, meaning that every two distinct positions of a given type have different values. We prove decidability of the emptiness problem for TAGC. This solves, in particular, the open question of decidability of emptiness for TAGED. We further extend our result by allowing global arithmetic constraints for counting the number of occurrences of some state or the number of different subterms reaching some state during a computation. We also allow local equality and disequality tests between sibling positions and the extension to unranked ordered trees. As a consequence of our results for TAGC, we prove the decidability of a fragment of the monadic second order logic on trees extended with predicates for equality and disequality between subtrees, and cardinality.

Decidable Classes of Tree Automata Mixing Local and Global Constraints Modulo Flat Theories

We define a class of ranked tree automata TABG generalizing both the tree automata with local brother tests of Bogaert and Tison (1992) and with global equality and disequality constraints (TAGED) of Filiot et al. (2007). TABG can test for equality and disequality modulo a given flat equational theory between brother subterms and between subterms whose positions are defined by the states reached during a computation. In particular, TABG can check that all the subterms reaching a given state are distinct. This constraint is related to monadic key constraints for XML documents, meaning that every two distinct positions of a given type have different values. We prove decidability of the emptiness problem for TABG. This solves, in particular, the open question of decidability of emptiness for TAGED. We further extend our result by allowing global arithmetic constraints for counting the number of occurrences of some state or the number of different equivalence classes of subterms (modulo a given flat equational theory) reaching some state during a computation. We also adapt the model to unranked ordered terms. As a consequence of our results for TABG, we prove the decidability of a fragment of the monadic second order logic on trees extended with predicates for equality and disequality between subtrees, and cardinality.

Unification and matching on compressed terms

Term unification plays an important role in many areas of computer science, especially in those related to logic. The universal mechanism of grammar-based compression for terms, in particular the so-called singleton tree grammars (STGAs), have recently drawn considerable attention. Using STGs, terms of exponential size and height can be represented in linear space. Furthermore, the term representation by directed acyclic graphs (dags) can be efficiently simulated. The present article is the result of an investigation on term unification and matching when the terms given as input are represented using different compression mechanisms for terms such as dags and singleton tree grammars. We describe a polynomial time algorithm for context matching with dags, when the number of different context variables is fixed for the problem. For the same problem, NP-completeness is obtained when the terms are represented using the more general formalism of singleton tree grammars. For first-order unification and matching polynomial time algorithms are presented, each of them improving previous results for those problems.

Deciding Fundamental Properties of Right-(Ground or Variable) Rewrite Systems by Rewrite Closure

Right-(ground or variable) rewrite systems (RGV systems for short) are term rewrite systems where all right hand sides of rules are restricted to be either ground or a variable. We define a minimal rewrite extension \(\overline{R}\) of the rewrite relation induced by a RGV system R. This extension admits a rewrite closure presentation, which can be effectively constructed from R. The rewrite closure is used to obtain decidability of the reachability, joinability, termination, and confluence properties of the RGV system R. We also show that the word problem and the unification problem are decidable for confluent RGV systems. We analyze the time complexity of the obtained procedures; for shallow RGV systems, termination and confluence are exponential, which is the best possible result since all these problems are EXPTIME-hard for shallow RGV systems.

Closure of Tree Automata Languages under Innermost Rewriting

Preservation of regularity by a term rewriting system (TRS) states that the set of reachable terms from a tree automata (TA) language (aka regular term set) is also a TA language. It is an important and useful property, and there have been many works on identifying classes of TRS ensuring it; unfortunately, regularity is not preserved for restricted classes of TRS like shallow TRS. Nevertheless, this property has not been studied for important strategies of rewriting like the innermost strategy - which corresponds to the call by value computation of programming languages. We prove that the set of innermost-reachable terms from a TA language by a shallow TRS is not necessarily regular, but it can be recognized by a TA with equality and disequality constraints between brothers. As a consequence we conclude decidability of regularity of the reachable set of terms from a TA language by innermost rewriting and shallow TRS. This result is in contrast with plain (not necessarily innermost) rewriting for which we prove undecidability. We also show that, like for plain rewriting, innermost rewriting with linear and right-shallow TRS preserves regularity.

Context Matching for Compressed Terms

This paper is an investigation of the matching problem for term equations s = t where s contains context variables and first-order variables, and both terms s and t are given using some kind of compressed representation. The main result is a polynomial time algorithm for context matching with dags, when the number of different context variables is fixed for the problem. NP-completeness is obtained when the terms are represented using the more general formalism of singleton tree grammars. As an ingredient of this proof, we also show that the special case of first-order matching with singleton tree grammars is decidable in polynomial time.

The HOM problem is decidable

We provide an algorithm that, given a tree homomorphism H and a regular tree language L represented by a tree automaton, determines whether H(L) is regular. This settles a question that has been open for a long time. Along the way, we develop new constructions and techniques which are interesting by themselves, and provide several significant intermediate results. For example, we prove that the universality problem is decidable for languages represented by tree automata with equality constraints, and that the equivalence and inclusion problems are decidable for images of regular tree languages through tree homomorphisms. Our algorithms are based on the following constructions. We describe a simple transformation for converting a tree automaton with equality constraints into a tree automaton with inequality constraints recognizing the complementary language. We also define a new class of automata with arbitrary inequality constraints and a particular kind of equality constraints. An automaton of this new class essentially recognizes the intersection of a tree automaton with inequality constraints and the image of a regular tree language through a tree homomorphism. We prove decidability of emptiness and finiteness for this class by a pumping mechanism.

Characterizing Confluence by Rewrite Closure and Right Ground Term Rewrite Systems

Abstract.Just as saturation under an appropriate superposition calculus leads to a convergent presentation of a given equational theory, saturation under a suitable chaining calculus gives, what we call, a rewrite closure. We present a theorem that characterizes confluence of (possibly nonterminating) term rewrite systems that admit a rewrite closure presentation, in terms of local confluence of a related terminating term rewrite system and joinability inclusion between these two rewrite systems. Using constraints to avoid variable chaining, we obtain a finite and computable rewrite closure presentation for right ground systems. This gives an alternate method to decide the reachability and joinability properties for right ground systems. The characterization of confluence, combined with the rewrite closure presentation, is used to obtain a decision procedure for confluence of right ground systems (this problem has been open for quite some time [8]), and a simple decision procedure for the unification problem of confluent right ground systems (result recently obtained in [17). An EXPTIME-hardness result is also proved for reachability and confluence of right ground systems.