Combining typing and size constraints for checking the termination of higher-order conditional rewrite systems

In a previous work, the first author extended to higher-order rewriting and dependent types the use of size annotations in types, a termination proof technique called type or size based termination and initially developed for ML-like programs. Here, we go one step further by considering conditional rewriting and explicit quantifications and constraints on size annotations. This allows to describe more precisely how the size of the output of a function depends on the size of its inputs. Hence, we can check the termination of more functions. We first give a general type-checking algorithm based on constraint solving. Then, we give a termination criterion with constraints in Presburger arithmetic. To our knowledge, this is the first termination criterion for higher-order conditional rewriting taking into account the conditions in termination

On the Values of Reducibility Candidates

The straightforward elimination of union types is known to break subject reduction, and for some extensions of the lambda-calculus, to break strong normalization as well. Similarly, the straightforward elim- ination of implicit existential types breaks subject reduction. We propose elimination rules for union types and implicit existential quantification which use a form call-by-value issued from Girard's re- ducibility candidates. We show that these rules remedy the above men- tioned difficulties, for strong normalization and, for the existential quan- tification, for subject reduction as well. Moreover, for extensions of the lambda-calculus based on intuitionistic logic, we show that the obtained existential quantification is equivalent to its usual impredicative encoding w.r.t. provability in realizability models built from reducibility candidates and biorthogonals

Fibrations of Tree Automata

We propose a notion of morphisms between tree automata based on game semantics. Morphisms are winning strategies on a synchronous restriction of the linear implication between acceptance games. This leads to split indexed categories, with substitution based on a suitable notion of synchronous tree function. By restricting to tree functions issued from maps on alphabets, this gives a fibration of tree automata. We then discuss the (fibrewise) monoidal structure issued from the synchronous product of automata. We also discuss how a variant of the usual projection operation on automata leads to an existential quantification in the fibered sense. Our notion of morphism is correct (it respects language inclusion), and in a weaker sense also complete

A Model Theoretic Proof of Completeness of an Axiomatization of Monadic Second-Order Logic on Streams

International audienceWe discuss the completeness of an axiomatization of Monadic Second- Order Logic (MSO) on infinite words (or streams). By using model-theoretic tools, we give an alternative proof of D. Siefkes' result that a fragment with full comprehension and induction of second-order Peano's arithmetic is com- plete w.r.t. the validity of MSO-formulas on streams. We rely on Feferman- Vaught Theorems and the Ehrenfeucht-Fra ̈ıss ́e method for Henkin models of second-order arithmetic. Our main technical contribution is an infinitary Feferman-Vaught Fusion of such models. We show it using Ramseyan fac- torizations similar to those for standard infinite words. We also discuss a Ramsey's theorem for MSO-definable colorings, and show that in linearly ordered Henkin models, Ramsey's theorem for additive MSO-definable col- orings implies Ramsey's theorem for all MSO-definable colorings

On the Representation of Stream Functions in Denotational Domains

We investigate the representation of functions on streams in some denota- tional domains. As expected, a total continuous stream function can always be represented by a Scott-continuous function, and moreover by a strongly stable map in the corresponding Hypercoherence. It seems however difficult to represent an arbitrary stream function by a monotone map on Scott domains such that the stream function is continuous if and only if its representant is Scott-continuous. The difficulty is that the set of Scott-approximants of an open subset of a not (topologically) compact set of streams may not be Scott-open. We show that this problem does not occur in the compact case

Toward a General Rewriting-Based Framework for Reducibility

Reducibility is a powerful proof method which applies to various properties of typed terms in different type systems. For strong normalization, different vari- ants are known, such as Girard's reducibility candidates, Tait's saturated sets and biorthogonals. They differ by the closure conditions imposed to types interpreta- tions, called here reducibility families. This paper is about the computational and observational properties underlying untyped reducibility. Our starting point is the comparison of reducibility families w.r.t. their ability to handle rewriting, for which their possible stability by union plays an important role. Indeed, usual saturated sets are generally stable by union, but with rewriting it can be difficult to define a uniform notion of saturated sets. On the other hand, rewriting is more naturally taken into account by reducibility candidates, but they are not always stable by union. It seems that for a given rewrite relation, the stability by union of reducibility candidates should imply the ability to naturally define corresponding saturated sets. In this paper, we seek to devise a general framework in which the above claim can be substantiated. In particular, this framework should be as simple as possible, while allowing the formulation of general notions of reducibility candidates and saturated sets. We present a notion of non-interaction which allows to define neutral terms and reducibility candidates in a generic way. This notion can be formulated in a very simple and general framework, based only on a rewrite relation and a set of contexts, called elimination contexts, required to satisfy some simple properties. This provides a convenient level of abstraction to prove fundamental properties of reducibility candidates, to compare them with biorthogonals, and to study their stability by union. Moreover, we propose a general form of saturated sets, issued from the stability by union of reducibility candidates

Union of Reducibility Candidates for Orthogonal Constructor Rewriting

We revisit Girard's reducibility candidates by proposing a general of the notion of neutral terms. They are the terms which do not interact with some contexts called elimination contexts. We apply this framework to constructor rewriting, and show that for orthogonal constructor rewriting, Girard's reducibility candidates are stable by union

On the Representation of Stream Functions in Denotational Domains

We investigate the representation of functions on streams in some denota- tional domains. As expected, a total continuous stream function can always be represented by a Scott-continuous function, and moreover by a strongly stable map in the corresponding Hypercoherence. It seems however difficult to represent an arbitrary stream function by a monotone map on Scott domains such that the stream function is continuous if and only if its representant is Scott-continuous. The difficulty is that the set of Scott-approximants of an open subset of a not (topologically) compact set of streams may not be Scott-open. We show that this problem does not occur in the compact case

Monoidal-Closed Categories of Tree Automata

We propose a realizability semantics for automata on infinite trees, based on categories of games built on usual simple games, and generalizing usual acceptance games of tree automata. Our approach can be summarized with the slogan " automata as objects, strategies as morphisms ". We show that the operations on tree automata used in the translations of MSO-formulae to automata (underlying Rabin's Theorem, that is the decidability of MSO on infinite trees) can be organized in a deduction system based on the multiplica-tive fragment of intuitionistic linear logic (ILL). Namely, we equip a variant of usual alternating tree automata (that we call uniform tree automata) with a fi-bred monoidal closed structure which in particular, via game determinacy handles a linear complementation of alternating automata, as well as deduction rules for exis-tential and universal quantifications. This monoidal structure is actually Cartesian on non-deterministic automata. Moreover, an adaptation of a usual construction for the simulation of alternating automata by non-deterministic ones satisfies the deduction rules of the !(−) ILL-exponential modality. Our realizability semantics satisfies an expected property of witness extraction from proofs of existential statements. Moreover, it allows to combine realizers produced as interpretations of proofs with strategies witnessing (non-)emptiness of tree automata, possibly obtained using external algorithms

On the Stability by Union of Reducibility Candidates

International audienceWe investigate some aspects of proof methods for the termination of (extensions of) the second-order lambda-calculus in presence of union and existential types. We prove that Girard's reducibility candidates are stable by union iff they are exactly the non-empty sets of terminating terms which are downward-closed wrt a weak observational preorder. We show that this is the case for the Curry-style second-order lambda-calculus.As a corollary, we obtain that reducibility candidates are exactly the Tait's saturated sets that are stable by reduction. We then extend the proof to a system with product, co-product and positive iso-recursive types