Closure of Tree Automata Languages under Innermost Rewriting

International audiencePreservation 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 {\em 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

Computer-Aided Derivation of Multi-scale Models: A Rewriting Framework

We introduce a framework for computer-aided derivation of multi-scale models. It relies on a combination of an asymptotic method used in the field of partial differential equations with term rewriting techniques coming from computer science. In our approach, a multi-scale model derivation is characterized by the features taken into account in the asymptotic analysis. Its formulation consists in a derivation of a reference model associated to an elementary nominal model, and in a set of transformations to apply to this proof until it takes into account the wanted features. In addition to the reference model proof, the framework includes first order rewriting principles designed for asymptotic model derivations, and second order rewriting principles dedicated to transformations of model derivations. We apply the method to generate a family of homogenized models for second order elliptic equations with periodic coefficients that could be posed in multi-dimensional domains, with possibly multi-domains and/or thin domains.Comment: 26 page

Too much duplicating patterns represent nonregular languages

A constrained term pattern s:phi represents the language of all instances of the term s satisfying the constraint phi. For each variable in s, this constraint specifies the language of its allowed substitutions. The goal of this project is to find efficient and computable properties ensuring non-regularity of the language represented by a set of constrained patterns.. A constrained term pattern s:phi represents the language of all instances of the term s satisfying the constraint phi. For each variable in s, this constraint specifies the language of its allowed substitutions.The goal of this project is to find efficient and computable properties ensuring non-regularity of the language represented by a set of constrained patterns

Too much duplicating patterns represent nonregular languages

A constrained term pattern s:phi represents the language of all instances of the term s satisfying the constraint phi. For each variable in s, this constraint specifies the language of its allowed substitutions. The goal of this project is to find efficient and computable properties ensuring non-regularity of the language represented by a set of constrained patterns.. A constrained term pattern s:phi represents the language of all instances of the term s satisfying the constraint phi. For each variable in s, this constraint specifies the language of its allowed substitutions.The goal of this project is to find efficient and computable properties ensuring non-regularity of the language represented by a set of constrained patterns