Enumerating proofs of positive formulae

We provide a semi-grammatical description of the set of normal proofs of positive formulae in minimal predicate logic, i.e. a grammar that generates a set of schemes, from each of which we can produce a finite number of normal proofs. This method is complete in the sense that each normal proof-term of the formula is produced by some scheme generated by the grammar. As a corollary, we get a similar description of the set of normal proofs of positive formulae for a large class of theories including simple type theory and System F

Regular Matching and Inclusion on Compressed Tree Patterns with Context Variables

International audienceWe study the complexity of regular matching and inclusion for compressed tree patterns extended by context variables. The addition of context variables to tree patterns permits us to properly capture compressed string patterns but also compressed patterns for unranked trees with tree and hedge variables. Regular inclusion for the latter is relevant to certain query answering on Xml streams with references

Probabilistic approach to the lambda definability for fourth order types

AbstractIt has been proved by Loader [Ralph Loader, The Undecidability of λ - Definability, In Logic, Meaning and Computation: Essays in Memory of Alonzo Church, 331-342, C.A. Anderson and Zeleny editors, Kluwer Academic Publishers, 2001] that Statman-Plotkin conjecture (see [Richard. Statman. Equality of functionals revisited, in L.A. Harrington et al. (Eds.), Harvey Friedman's Research on the Foundations of Mathematics, North-Holland, Amsterdam, 1985, 331-338] and [Gordon D. Plotkin. λ definability and logical relations, Memorandum SAI-RM-4, School of Artificial Intelligence, University of Edinburgh, Ocober 1973]) fails. The Loader proof was done by encoding the word problem in the full type hierarchy based on the domain with 7 elements. The aim of this paper is to show that the lambda definability problem limited to regular fourth order types is decidable in any finite domain. Obviously λ definability is decidable for 1, 2 and 3 order types. As an additional effect of the result described we may observe that for certain types there is no finite context free grammar generating all closed terms. We prove also that probability that randomly chosen fourth order type (or type of the order not grater then 4 ) admits decidable lambda definability problem is zero