The monadic second-order logic of graphs I. Recognizable sets of Finite Graphs

The notion of a recognizable sef offinite graphs is introduced. Every set of finite graphs, that is definable in monadic second-order logic is recognizable, but not vice versa. The monadic second-order theory of a context-free set of graphs is decidable. 0 19W Academic Press. Inc. This paper begins an investigation of the monadic second-order logic of graphs and of sets of graphs, using techniques from universal algebra, and the theory of formal languages. (By a graph, we mean a finite directed hyperedge-labelled hypergraph, equipped with a sequence of distinguished vertices.) A survey of this research can be found in Courcelle [ 111. An algebraic structure on the set of graphs (in the above sense) has been proposed by Bauderon and Courcelle [2,7]. The notion of a recognizable set of finite graphs follows, as an instance of the general notion of recognizability introduced by Mezei and Wright in [25]. A graph can also be considered as a logical structure of a certain type. Hence, properties of graphs can be written in first-order logic or in secondorder logic. It turns out that monadic second-order logic, where quantifications over sets of vertices and sets of edges are used, is a reasonably powerful logical language (in which many usual graph properties can be written), for which one can obtain decidability results. These decidability results do not hold for second-order logic, where quantifications over binary relations can also be used. Our main theorem states that every definable set of finite graphs (i.e., every set that is the set of finite graphs satisfying a graph property expressible in monadic second-order logic) is recognizable. * This work has been supported by the “Programme de Recherches Coordonntes: Mathematiques et Informatique.

E-Generalization Using Grammars

We extend the notion of anti-unification to cover equational theories and present a method based on regular tree grammars to compute a finite representation of E-generalization sets. We present a framework to combine Inductive Logic Programming and E-generalization that includes an extension of Plotkin's lgg theorem to the equational case. We demonstrate the potential power of E-generalization by three example applications: computation of suggestions for auxiliary lemmas in equational inductive proofs, computation of construction laws for given term sequences, and learning of screen editor command sequences.Comment: 49 pages, 16 figures, author address given in header is meanwhile outdated, full version of an article in the "Artificial Intelligence Journal", appeared as technical report in 2003. An open-source C implementation and some examples are found at the Ancillary file

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 tests between brothers 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 the 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.Comment: 39 pages, to appear in LMCS journa

Rigid Tree Automata and Applications

International audienceWe introduce the class of Rigid Tree Automata (RTA), an extension of standard bottom-up automata on ranked trees with distinguished states called rigid. Rigid states define a restriction on the computation of RTA on trees: RTA can test for equality in subtrees reaching the same rigid state. RTA are able to perform local and global tests of equality between subtrees, non-linear tree pattern matching, and some inequality and disequality tests as well. Properties like determinism, pumping lemma, Boolean closure, and several decision problems are studied in detail. In particular, the emptiness problem is shown decidable in linear time for RTA whereas membership of a given tree to the language of a given RTA is NP-complete. Our main result is the decidability of whether a given tree belongs to the rewrite closure of an RTA language under a restricted family of term rewriting systems, whereas this closure is not an RTA language. This result, one of the first on rewrite closure of languages of tree automata with constraints, is enabling the extension of model checking procedures based on finite tree automata techniques, in particular for the verification of communicating processes with several local non rewritable memories, like security protocols. Finally, a comparison of RTA with several classes of tree automata with local and global equality tests, with dag automata and Horn clause formalisms is also provided

Identities and bases in plactic, hypoplactic, sylvester, and related monoids

The ubiquitous plactic monoid, also known as the monoid of Young tableaux, has deep connections to several areas of mathematics, in particular, to the theory of symmetric functions. An active research topic is the identities satisfied by the plactic monoids of finite rank. It is known that there is no “global" identity satisfied by the plactic monoid of every rank. In contrast, monoids related to the plactic monoid, such as the hypoplactic monoid (the monoid of quasi-ribbon tableaux), sylvester monoid (the monoid of binary search trees) and Baxter monoid (the monoid of pairs of twin binary search trees), satisfy global identities, and the shortest identities have been characterized. In this thesis, we present new results on the identities satisfied by the hypoplactic, sylvester, #-sylvester and Baxter monoids. We show how to embed these monoids, of any rank strictly greater than 2, into a direct product of copies of the corresponding monoid of rank 2. This confirms that all monoids of the same family, of rank greater than or equal to 2, satisfy exactly the same identities. We then give a complete characterization of those identities, thus showing that the identity checking problems of these monoids are in the complexity class P, and prove that the varieties generated by these monoids have finite axiomatic rank, by giving a finite basis for them. We also give a subdirect representation ofmultihomogeneous monoids by finite subdirectly irreducible Rees factor monoids, thus showing that they are residually finite.O ubíquo monóide plático, também conhecido como o monóide dos diagramas de Young, tem ligações profundas a várias áreas de Matemática, em particular à teoria das funções simétricas. Um tópico de pesquisa ativo é o das identidades satisfeitas pelos monóides pláticos de característica finita. Sabe-se que não existe nenhuma identidade “global” satisfeita pelos monóides pláticos de cada característica. Em contraste, sabe-se que monóides ligados ao monóide plático, como o monóide hipoplático (o monóide dos diagramas quasifita), o monóide silvestre (o monóide de árvores de busca binárias) e o monóide de Baxter (o monóide de pares de árvores de busca binária gémeas), satisfazem identidades globais, e as identidades mais curtas já foram caracterizadas. Nesta tese, apresentamos novos resultados acerca das identidades satisfeitas pelos monóides hipopláticos, silvestres, silvestres-# e de Baxter. Mostramos como mergulhar estes monóides, de característica estritamente maior que 2, num produto direto de cópias do monóide correspondente de característica 2. Confirmamos assim que todos os monóides da mesma família, de característica maior ou igual a 2, satisfazem exatamente as mesmas identidades. A seguir, damos uma caracterização completa dessas identidades, mostrando assim que os problemas de verificação de identidades destes monóides estão na classe de complexidade P, e provamos que as variedades geradas por estes monóides têm característica axiomática finita, ao apresentar uma base finita para elas. Também damos uma representação subdireta de monóides multihomogéneos por monóides fatores de Rees finitos e subdiretamente irredutíveis, mostrando assim que são residualmente finitos