Cost Automata, Safe Schemes, and Downward Closures

Higher-order recursion schemes are an expressive formalism used to define languages of possibly infinite ranked trees. They extend regular and context-free grammars, and are equivalent to simply typed ?Y-calculus and collapsible pushdown automata. In this work we prove, under a syntactical constraint called safety, decidability of the model-checking problem for recursion schemes against properties defined by alternating B-automata, an extension of alternating parity automata for infinite trees with a boundedness acceptance condition. We then exploit this result to show how to compute downward closures of languages of finite trees recognized by safe recursion schemes

Collapsible Pushdown Automata and Recursion Schemes

International audienceWe consider recursion schemes (not assumed to be homogeneously typed, and hence not necessarily safe) and use them as generators of (possibly infinite) ranked trees. A recursion scheme is essentially a finite typed {deterministic term} rewriting system that generates, when one applies the rewriting rules ad infinitum, an infinite tree, called its value tree. A fundamental question is to provide an equivalent description of the trees generated by recursion schemes by a class of machines. In this paper we answer this open question by introducing collapsible pushdown automata (CPDA), which are an extension of deterministic (higher-order) pushdown automata. A CPDA generates a tree as follows. One considers its transition graph, unfolds it and contracts its silent transitions, which leads to an infinite tree which is finally node labelled thanks to a map from the set of control states of the CPDA to a ranked alphabet. Our contribution is to prove that these two models, higher-order recursion schemes and collapsible pushdown automata, are equi-expressive for generating infinite ranked trees. This is achieved by giving an effective transformations in both directions

Typing weak MSOL properties

International audienceWe consider non-interpreted functional programs: the result of the execution of a program is its normal form, that can be seen as the tree of calls to built-in operations. Weak monadic second-order logic (wMSO) is well suited to express properties of such trees. This is an extension of first order logic with quantification over finite sets. Many behavioral properties of programs can be expressed in wMSO. We use the simply typed lambda calculus with the fixpoint operator, $\lambda Y$-calculus, as an abstraction of functional programs that faithfully represents the higher-order control flow. We give a type system for ensuring that the result of the execution of a $\lambda Y$-program satisfies a given wMSO property. The type system is an extension of a standard intersection type system with both: the least-fixpoint rule, and a restricted version of the greatest-fixpoint rule. In order to prove soundness and completeness of the system we construct a denotational semantics of $\lambda Y$-calculus that is capable of computing properties expressed in wMSO. The model presents many symmetries reflecting dualities in the logic and has also other applications on its own. The type system is obtained from the model following the domain in logical form approach

On Model-Checking Higher-Order Effectful Programs (Long Version)

Model-checking is one of the most powerful techniques for verifying systems and programs, which since the pioneering results by Knapik et al., Ong, and Kobayashi, is known to be applicable to functional programs with higher-order types against properties expressed by formulas of monadic second-order logic. What happens when the program in question, in addition to higher-order functions, also exhibits algebraic effects such as probabilistic choice or global store? The results in the literature range from those, mostly positive, about nondeterministic effects, to those about probabilistic effects, in the presence of which even mere reachability becomes undecidable. This work takes a fresh and general look at the problem, first of all showing that there is an elegant and natural way of viewing higher-order programs producing algebraic effects as ordinary higher-order recursion schemes. We then move on to consider effect handlers, showing that in their presence the model checking problem is bound to be undecidable in the general case, while it stays decidable when handlers have a simple syntactic form, still sufficient to capture so-called generic effects. Along the way we hint at how a general specification language could look like, this way justifying some of the results in the literature, and deriving new ones

Décidabilité et Complexité

International audienceL'informatique fondamentale est un vaste sujet, comme en témoignent les 2 283 et 3 176 pages des "Handbooks" (228; 1). Couvrir en quelques dizaines de pages, l'ensemble de l'in- formatique nous a semblé une entreprise hors de notre portée. De ce fait, nous nous sommes concentrés sur la notion de calcul, sujet qui reflète le goût et la passion des auteurs de ce chapitre. La notion de calcul est omniprésente et aussi ancienne que les mathématiques

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 22nd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2019, which took place in Prague, Czech Republic, in April 2019, held as part of the European Joint Conference on Theory and Practice of Software, ETAPS 2019. The 29 papers presented in this volume were carefully reviewed and selected from 85 submissions. They deal with foundational research with a clear significance for software science