Combinatory Categorial Grammar is a generalization of classical Categorial Grammar that is at the least greater level of expressive power than context-free grammar that has yet been identified. The paper reviews its use for the explanatory analysis of natural language grammar, particularly with respect to bounded and unbounded constructions, multiple dependencies, relativization, and coordination.

/pdf/combinatory-categorial-grammar-2zn6uhjktz.pdf

Combinatory Categorial Grammar

Motivated by reasoning tasks in the context of XML languages, the satisfiability problem of logics on data trees is investigated. The nodes of a data tree have a label from a finite set and a data value from a possibly infinite set. It is shown that satisfiability for two-variable first-order logic is decidable if the tree structure can be accessed only through the child and the next sibling predicates and the access to data values is restricted to equality tests. From this main result decidability of satisfiability and containment for a data-aware fragment of XPath and of the implication problem for unary key and inclusion constraints is concluded.

/pdf/two-variable-logic-on-data-trees-and-xml-reasoning-53ke4jb9i0.pdf

Two-variable logic on data trees and XML reasoning

Preface 1. Introduction Part I. Semantic Acts and Intentions: 2. Speaker meaning 3. Expression 4. Alternative analyses 5. Communication 6. Reference Part II. Languages and Semantic Acts: 7. Languages 8. Basic word meaning 9. Conventions 10. Compositional word meaning 11. Living languages Part III. Thoughts and Ideas: 12. Thought 13. Sentences, propositions and thoughts 14. The constituency thesis 15. Ideas or concepts 16. The possession of concepts 17. The acquisition of concepts 18. The association of ideas 19. Objects, images and conceptions 20. The language of thought hypothesis Part IV. Ideational Theories of Meaning: 21. Objections to ideational theories 22. Priority objections 23. Incompleteness objections References Index.

/pdf/meaning-expression-and-thought-4brv9bwvjh.pdf

Meaning, Expression and Thought

In a data word or a data tree each position carries a label from a finite alphabet and a data value from some infinite domain. These models have been considered in the realm of semistructured data, timed automata and extended temporal logics.

This paper survey several know results on automata and logics manipulating data words and data trees, the focus being on their relative expressive power and decidability.

/pdf/automata-and-logics-for-words-and-trees-over-an-infinite-3hvmgnmvnk.pdf

Automata and logics for words and trees over an infinite alphabet

Collapsible pushdown automata (CPDA) are a new kind of higher-order pushdown automata in which every symbol in the stack has a link to a stack situated somewhere below it. In addition to the higher-order push and pop operations, CPDA have an important operation called collapse, whose effect is to "collapse" a stack s to the prefix as indicated by the link from the topmost symbol of s. Our first result is that CPDA are equi-expressive with recursion schemes as generators of (possibly infinite) ranked trees. In one direction, we give a simple algorithm that transforms an order-n CPDA to an order-n recursion scheme that generates the same tree, uniformly for all n Gt= 0. In the other direction, using ideas from game semantics, we give an effective transformation of order-n recursion schemes (not assumed to be homogeneously typed, and hence not necessarily safe) to order-n CPDA that compute traversals over an abstract syntax graph of the scheme, and hence paths in the tree generated by the scheme. Our equi-expressivity result is the first automata-theoretic characterization of higher-order recursion schemes. Thus CPDA are also a characterization of the simply-typed lambda calculus with recursion (generated from uninterpreted 1st-order symbols) and of (pure) innocent strategies. An important consequence of the equi-expressivity result is that it allows us to reduce decision problems on trees generated by recursion schemes to equivalent problems on CPDA and vice versa. Thus we show, as a consequence of a recent result by Ong (modal mu-calculus model-checking of trees generated by recursion schemes is n-EXPTIME complete), that the problem of solving parity games over the configuration graphs of order-n CPDA is n-EXPTIME complete, subsuming several well-known results about the solvability of games over higher-order pushdown graphs by (respectively) Walukiewicz, Cachat, and Knapik et al. Another contribution of our work is a self-contained proof of the same solvability result by generalizing standard techniques in the field. By appealing to our equi-expressivity result, we obtain a new proof of Ong's result. In contrast to higher-order pushdown graphs, we show that the monadic second-order theories of the configuration graphs of CPDA are undecidable. It follows that -- as generators of graphs -- CPDA are strictly more expressive than higher-order pushdown automata.

/pdf/collapsible-pushdown-automata-and-recursion-schemes-2m1u3yyxki.pdf

Collapsible Pushdown Automata and Recursion Schemes

We introduce a new class of automata, which we call vector addition tree automata. These automata are a natural generalization of vector addition systems with states, which are themselves equivalent to Petri-nets. Then, we prove that the decidability of provability in multiplicative exponential linear logic (which is an open problem) is equivalent to the decidability of the reachability relation for vector addition tree automata. This result generalizes the well-known connection existing between Petri nets and the !-horn fragment of multiplicative exponential linear logic.

Vector addition tree automata

We propose a new approach to analysing higher-order recursive schemes. Many results in the literature use automata models generalising pushdown automata, most notably higher-order pushdown automata with collapse (CPDA). Instead, we propose to use the Krivine machine model. Compared to CPDA, this model is closer to lambda-calculus, and incorporates nicely many invariants of computations, as for example the typing information. The usefulness of the proposed approach is demonstrated with new proofs of two central results in the field: the decidability of the local and global model checking problems for higher-order schemes with respect to the mu-calculus.

https://hal.inria.fr/inria-00589407/document

Krivine machines and higher-order schemes

A non-deterministic recursion scheme recognizes a language of finite trees. This very expressive model can simulate, among others, higher-order pushdown automata with collapse. We show decidability of the diagonal problem for schemes. This result has several interesting consequences. In particular, it gives an algorithm that computes the downward closure of languages of words recognized by schemes. In turn, this has immediate application to separability problems and reachability analysis of concurrent systems.

The Diagonal Problem for Higher-Order Recursion Schemes is Decidable

In this paper, we aim at understanding the derivations of minimalist grammars without the shortest move constraint. This leads us to study the relationship of those derivations with logic. In particular we show that the membership problem of minimalist grammars without the shortest move constraint is as difficult as provability in Multiplicative Exponential Linear Logic. As a byproduct, this result gives us a new representation of those derivations with linear λ-terms. We show how to interpret those terms in a homomorphic way so as to recover the sentence they analyse. As the homorphisms we describe are rather evolved, we turn to a proof-net representation and explain how Monadic Second Order Logic and related techniques allow us both to define those proof-nets and to retrieve the sentence they analyse.

/pdf/minimalist-grammars-in-the-light-of-logic-m59ebznktx.pdf

Minimalist grammars in the light of logic

We propose a model-based approach to the model checking problem for recursive schemes. Since simply typed lambda calculus with the fixpoint operator, λY-calculus, is equivalent to schemes, we propose the use a model of λY to discriminate the terms that satisfy a given property. If a model is finite in every type, this gives a decision procedure. We provide a construction of such a model for every property expressed by automata with trivial acceptance conditions and divergence testing. Such properties pose already interesting challenges for model construction. Moreover, we argue that having models capturing some class of properties has several other virtues in addition to providing decidability of the model-checking problem. As an illustration, we show a very simple construction transforming a scheme to a scheme reflecting a property captured by a given model.

https://hal.inria.fr/docs/00/74/12/39/PDF/m.pdf

Using models to model-check recursive schemes

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