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

!-Graphs with Trivial Overlap are Context-Free

String diagrams are a powerful tool for reasoning about composite structures in symmetric monoidal categories. By representing string diagrams as graphs, equational reasoning can be done automatically by double-pushout rewriting. !-graphs give us the means of expressing and proving properties about whole families of these graphs simultaneously. While !-graphs provide elegant proofs of surprisingly powerful theorems, little is known about the formal properties of the graph languages they define. This paper takes the first step in characterising these languages by showing that an important subclass of !-graphs--those whose repeated structures only overlap trivially--can be encoded using a (context-free) vertex replacement grammar.Comment: In Proceedings GaM 2015, arXiv:1504.0244

A Pumping Lemma Scheme for the Control Language Hierarchy

In [9] Weir introduced control grammars as a model for describing the syntactic structure of natural languages. Informally, a control grammar is a pair {G, C} where G is a context-free grammar whose productions are assigned labels from a finite set of labels II, and C (called the control set) is a set of strings over II. A derivation in a control grammar is similar to that in an ordinary context-free grammar except that the control set C is used to further constrain the set of valid derivations. In particular, if one views a derivation as a tree, then (in a manner to be described later) each edge in such a tree is given a label from II according to the production of G associated with the edge. The derivation tree is considered valid if certain paths in the tree correspond to strings which are in the control set C. The language generated by the control grammar is then the set of strings having at least one derivation tree in the sense just described

Descriptional Succinctness of Some Grammatical Formalisms for Natrual Language

We investigate the problem of describing languages compactly in different grammatical formalisms for natural languages. In particular, the problem is studied from the point of view of some newly developed natural language formalisms like linear control grammars (LCGs) and tree adjoining grammars (TAGs); these formalisms not only generate non-context-free languages that capture a wide variety of syntactic phenomena found in natural language, but also have computationally efficient polynomial time recognition algorithms. We prove that the formalisms enjoy the property of unbounded succinctness over the family of context-grammars, i.e. they are, in general, able to provide more compact representations of natural languages as compared to standard context-free grammars

Pattern overlap implies runaway growth in hierarchical tile systems

We show that in the hierarchical tile assembly model, if there is a producible assembly that overlaps a nontrivial translation of itself consistently (i.e., the pattern of tile types in the overlap region is identical in both translations), then arbitrarily large assemblies are producible. The significance of this result is that tile systems intended to controllably produce finite structures must avoid pattern repetition in their producible assemblies that would lead to such overlap. This answers an open question of Chen and Doty (SODA 2012), who showed that so-called "partial-order" systems producing a unique finite assembly *and" avoiding such overlaps must require time linear in the assembly diameter. An application of our main result is that any system producing a unique finite assembly is automatically guaranteed to avoid such overlaps, simplifying the hypothesis of Chen and Doty's main theorem

Decidability Results for the Boundedness Problem

We prove decidability of the boundedness problem for monadic least fixed-point recursion based on positive monadic second-order (MSO) formulae over trees. Given an MSO-formula phi(X,x) that is positive in X, it is decidable whether the fixed-point recursion based on phi is spurious over the class of all trees in the sense that there is some uniform finite bound for the number of iterations phi takes to reach its least fixed point, uniformly across all trees. We also identify the exact complexity of this problem. The proof uses automata-theoretic techniques. This key result extends, by means of model-theoretic interpretations, to show decidability of the boundedness problem for MSO and guarded second-order logic (GSO) over the classes of structures of fixed finite tree-width. Further model-theoretic transfer arguments allow us to derive major known decidability results for boundedness for fragments of first-order logic as well as new ones