IO vs OI in Higher-Order Recursion Schemes

We propose a study of the modes of derivation of higher-order recursion schemes, proving that value trees obtained from schemes using innermost-outermost derivations (IO) are the same as those obtained using unrestricted derivations. Given that higher-order recursion schemes can be used as a model of functional programs, innermost-outermost derivations policy represents a theoretical view point of call by value evaluation strategy.Comment: In Proceedings FICS 2012, arXiv:1202.317

Specifications with Observable Formulae and Observational Satisfaction Relation

We consider algebraic specifications with observational features. Axioms as well as observations are formulae of full (ManySorted) First Order Logic with Equality. The associated semantics is based on a non standard interpretation of equality called observational equality. We study the adequacy of this semantics for software specification and the relationship with behavioural equivalence of algebras. We show that this framework defines an institution. Keywords: algebraic specification, observability, semantics. 1 Introduction Within an observational approach the loose semantics of a specification may either be defined as a class of algebras observationally equivalent to models satisfying the specification in the usual sense or as a class of algebras observationally satisfying the specification. The former way has already been deeply explored in [13] while in the latter one, the following problems remains open: 1. How to define an observational satisfaction relation w.r.t. more sophi..

Higher order indexed monadic systems

A word rewriting system is called monadic if each of its right hand sides is either a single letter or the empty word. We study the images of higher order indexed languages (defined by Maslov) under inverse derivations of infinite monadic systems. We show that the inverse derivations of deterministic level n indexed languages by confluent regular monadic systems are deterministic level n+1 languages, and that the inverse derivations of level n indexed monadic systems preserve level n indexed languages. Both results are established using a fine structural study of classes of infinite automata accepting level n indexed languages. Our work generalizes formerly known results about regular and context-free languages which form the first two levels of the indexed language hierarchy

The evaluation of first-order substitution is monadic second-order compatible

AbstractWe denote first-order substitutions of finite and infinite terms by function symbols indexed by the sequences of first-order variables to which substitutions are made. We consider the evaluation mapping from infinite terms to infinite terms that evaluates these substitution operations. This mapping may perform infinitely many nested substitutions, so that a term which has the structure of an infinite string can be transformed into one isomorphic to an infinite binary tree. We prove that this mapping is monadic second-order compatible which means that a monadic second-order formula expressing a property of the output term produced by the evaluation mapping can be translated into a monadic second-order formula expressing this property over the input term. This implies that, deciding the monadic second-order theory of the output term reduces to deciding that of the input term. As an application, we obtain another proof that the monadic second-order properties of the algebraic trees, which represent the behaviours of recursive applicative program schemes, are decidable. This proof extends to hyperalgebraic trees. These infinite trees correspond to certain recursive program schemes with functional parameters of arbitrary high type

Shelah-Stupp's iteration and Muchnik's iteration

International audienceIn the early seventies, Shelah proposed a model-theoretic construction, nowadays called "iteration". This construction is an infinite replication in a tree-like manner where every vertex possesses its own copy of the original structure. Stupp proved that the decidabil-ity of the monadic second-order (MSO) theory is transferred from the original structure onto the iterated one. In its extended version discovered by Muchnik and introduced by Semenov, the iteration became popular in computer science logic thanks to a paper by Walukiewicz. Compared to the basic iteration, Muchnik's iteration has an additional unary predicate which, in every copy, marks the vertex that is the clone of the possessor of the copy. A widely spread belief that this extension is crucial is formally confirmed in the paper. Two hierarchies of relational structures generated from finite structures by MSO interpretations and either Shelah-Stupp's iteration or Muchnik's iteration are compared. It turns out that the two hierarchies coincide at level 1. Every level of the latter hierarchy is closed under Shelah-Stupp's interation. In particular, the former hierarchy collapses at level 1. It is also established that the relations from level 1 may be encoded as sets of tuples of words defined by certain relational expressions built on top of regular expressions

On a Chomsky like Hierarchy of Infinite Graphs

We consider a strict fourlevel hierarchy of graphs closely related to the Chomsky hierarchy of formal languages. We provide a uniform presentation of the four families by means of string rewriting