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Set relations are particularly suitable for specifying the small- step operational semantics of s... more Set relations are particularly suitable for specifying the small- step operational semantics of synchronous languages. We present a frame- work for the denition, formal verication of properties, and execution of binary set relations. The framework consists of two parts. The rst part is a library in the Prototype Verication System (PVS) that contains deni- tions and proofs of properties, such
The Plan Execution Interchange Language (PLEXIL) is a synchronous language developed by NASA to s... more The Plan Execution Interchange Language (PLEXIL) is a synchronous language developed by NASA to support autonomous spacecraft operations. In this paper, we propose a rewriting logic semantics of PLEXIL in Maude, a high-performance logical engine. The rewriting logic semantics is by itself a formal interpreter of the language and can be used as a semantic benchmark for the implementation of
The lambda-Pi-calculus allows to express proofs of minimal predicate logic. It can be extended, i... more The lambda-Pi-calculus allows to express proofs of minimal predicate logic. It can be extended, in a very simple way, by adding computation rules. This leads to the lambda-Pi-calculus modulo. We show in this paper that this simple extension is surprisingly expressive and, in particular, that all functional Pure Type Systems, such as the system F, or the Calculus of Constructions, can be embedded in it. And, moreover, that this embedding is conservative under termination hypothesis.
We provide a computational definition of the notions of vector space and bilinear functions. We u... more We provide a computational definition of the notions of vector space and bilinear functions. We use this result to introduce a minimal language combining higherorder computation and linear algebra. This language extends the λ-calculus with the possibility to make arbitrary linear combinations of terms α.t + β.u. We describe how to "execute" this language in terms of a few rewrite rules, and justify them through the two fundamental requirements that the language be a language of linear operators, and that it be higher-order. We mention the perspectives of this work in the field of quantum computation, whose circuits we show can be easily encoded in the calculus. Finally, we prove the confluence of the entire calculus.
Set relations are particularly suitable for specifying the smallstep operational semantics of syn... more Set relations are particularly suitable for specifying the smallstep operational semantics of synchronous languages. We present a framework for the definition, formal verification of properties, and execution of binary set relations. The framework consists of two parts. The first part is a library in the Prototype Verification System (PVS) that contains definitions and proofs of properties, such as determinism and compositionality, for synchronous relations. The second part is a rewriting logic specification that animates in the Maude system synchronous binary relations on sets. We illustrate the use of this framework with the operational semantics of the Plan Execution Interchange Language (PLEXIL), a rich concurrent and reactive plan execution language developed by NASA to support autonomous spacecraft operations.
The notion of computable function can be extended from the natural numbers to other domains, usin... more The notion of computable function can be extended from the natural numbers to other domains, using an indexing. Usually, the choice of the index- ing affects the set of computable functions and there is no intrinsic notion of computability over these domains. In this paper, we show that, in contrast, when we extend the notion of computable function to algebraic structures, we obtain an intrinsic notion in many cases. We give examples of such structures having an intrinsic notion of computability and characterize them as finitely generated structures.
We give a new proof of a theorem of Mints that the positive fragment of minimal predicate logic i... more We give a new proof of a theorem of Mints that the positive fragment of minimal predicate logic is decidable. The idea of the proof is to replace the eigenvariable condition of sequent calculus by an appropriate scoping mechanism. The algorithm given by this proof seems to be more practical than that given by the original proof. A naive implementation is given at the end of the paper. Another contribution is to show that this result extends to a large class of theories, including simple type theory (higher-order logic) and second-order propositional logic. We obtain this way a new proof of the decidability of the inhabitation problem for positive types in system F.
Higher order unification is equational unification for βη-conversion. But it is not first order e... more Higher order unification is equational unification for βη-conversion. But it is not first order equational unification, as substitution has to avoid capture. Thus, the methods for equational unification (such as narrowing) built upon grafting (i.e., substitution without renaming) cannot be used for higher order unification, which needs specific algorithms. Our goal in this paper is to reduce higher order unification
We generalize the theory of Cellular Automata to arbitrary, time-varying graphs. In other words w... more We generalize the theory of Cellular Automata to arbitrary, time-varying graphs. In other words we formalize, and prove theorems about, the intuitive idea of a labelled graph which evolves in time -but under the natural constraint that information can only ever be transmitted at a bounded speed, with respect to the distance given by the graph. The notion of translation-invariance is also generalized. The definition we provide for these 'causal graph dynamics' is simple and axiomatic. The theorems we provide also show that it is robust. For instance, causal graph dynamics are stable under composition and under restriction to radius one. In the finite case some fundamental facts of Cellular Automata theory carry through: causal graph dynamics admit a characterization as continuous functions, and they are stable under inversion. The provided examples suggest a wide range of applications of this mathematical object, from complex systems science to theoretical physics.
We present a calculus, called the scheme-calculus, that permits to express natural deduction proo... more We present a calculus, called the scheme-calculus, that permits to express natural deduction proofs in various theories. Unlike λ-calculus, the syntax of this calculus sticks closely to the syntax of proofs, in particular, no names are introduced for the hypotheses. We show that despite its non-determinism, some typed scheme-calculi have the same expressivity as the corresponding typed λ-calculi.
In 6] we have proposed a general higher-order uni cation method using a theory of explicit substi... more In 6] we have proposed a general higher-order uni cation method using a theory of explicit substitutions and we have proved its completeness. In this paper, we investigate the case of higher-order patterns as introduced by Miller. We show that our general algorithm specializes in a very convenient way to patterns. We also sketch an e cient implementation of the abstract algorithm and its generalization to constraint simpli cation, which has yielded good experimental results at the core of a higher-order constraint logic programming language.
Résumé: L'informatique fondamentale est un vaste sujet, comme en témoignent les 2 283 et 3 1... more Résumé: L'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.
Page 1. inria-00070034, version 1 - 19 May 2006 Page 2. inria-00070034, version 1 - 19 May 2006 P... more Page 1. inria-00070034, version 1 - 19 May 2006 Page 2. inria-00070034, version 1 - 19 May 2006 Page 3. inria-00070034, version 1 - 19 May 2006 Page 4. inria-00070034, version 1 - 19 May 2006 Page 5. inria-00070034, version 1 - 19 May 2006 Page 6. ...
Set relations are particularly suitable for specifying the small- step operational semantics of s... more Set relations are particularly suitable for specifying the small- step operational semantics of synchronous languages. We present a frame- work for the denition, formal verication of properties, and execution of binary set relations. The framework consists of two parts. The rst part is a library in the Prototype Verication System (PVS) that contains deni- tions and proofs of properties, such
The Plan Execution Interchange Language (PLEXIL) is a synchronous language developed by NASA to s... more The Plan Execution Interchange Language (PLEXIL) is a synchronous language developed by NASA to support autonomous spacecraft operations. In this paper, we propose a rewriting logic semantics of PLEXIL in Maude, a high-performance logical engine. The rewriting logic semantics is by itself a formal interpreter of the language and can be used as a semantic benchmark for the implementation of
The lambda-Pi-calculus allows to express proofs of minimal predicate logic. It can be extended, i... more The lambda-Pi-calculus allows to express proofs of minimal predicate logic. It can be extended, in a very simple way, by adding computation rules. This leads to the lambda-Pi-calculus modulo. We show in this paper that this simple extension is surprisingly expressive and, in particular, that all functional Pure Type Systems, such as the system F, or the Calculus of Constructions, can be embedded in it. And, moreover, that this embedding is conservative under termination hypothesis.
We provide a computational definition of the notions of vector space and bilinear functions. We u... more We provide a computational definition of the notions of vector space and bilinear functions. We use this result to introduce a minimal language combining higherorder computation and linear algebra. This language extends the λ-calculus with the possibility to make arbitrary linear combinations of terms α.t + β.u. We describe how to "execute" this language in terms of a few rewrite rules, and justify them through the two fundamental requirements that the language be a language of linear operators, and that it be higher-order. We mention the perspectives of this work in the field of quantum computation, whose circuits we show can be easily encoded in the calculus. Finally, we prove the confluence of the entire calculus.
Set relations are particularly suitable for specifying the smallstep operational semantics of syn... more Set relations are particularly suitable for specifying the smallstep operational semantics of synchronous languages. We present a framework for the definition, formal verification of properties, and execution of binary set relations. The framework consists of two parts. The first part is a library in the Prototype Verification System (PVS) that contains definitions and proofs of properties, such as determinism and compositionality, for synchronous relations. The second part is a rewriting logic specification that animates in the Maude system synchronous binary relations on sets. We illustrate the use of this framework with the operational semantics of the Plan Execution Interchange Language (PLEXIL), a rich concurrent and reactive plan execution language developed by NASA to support autonomous spacecraft operations.
The notion of computable function can be extended from the natural numbers to other domains, usin... more The notion of computable function can be extended from the natural numbers to other domains, using an indexing. Usually, the choice of the index- ing affects the set of computable functions and there is no intrinsic notion of computability over these domains. In this paper, we show that, in contrast, when we extend the notion of computable function to algebraic structures, we obtain an intrinsic notion in many cases. We give examples of such structures having an intrinsic notion of computability and characterize them as finitely generated structures.
We give a new proof of a theorem of Mints that the positive fragment of minimal predicate logic i... more We give a new proof of a theorem of Mints that the positive fragment of minimal predicate logic is decidable. The idea of the proof is to replace the eigenvariable condition of sequent calculus by an appropriate scoping mechanism. The algorithm given by this proof seems to be more practical than that given by the original proof. A naive implementation is given at the end of the paper. Another contribution is to show that this result extends to a large class of theories, including simple type theory (higher-order logic) and second-order propositional logic. We obtain this way a new proof of the decidability of the inhabitation problem for positive types in system F.
Higher order unification is equational unification for βη-conversion. But it is not first order e... more Higher order unification is equational unification for βη-conversion. But it is not first order equational unification, as substitution has to avoid capture. Thus, the methods for equational unification (such as narrowing) built upon grafting (i.e., substitution without renaming) cannot be used for higher order unification, which needs specific algorithms. Our goal in this paper is to reduce higher order unification
We generalize the theory of Cellular Automata to arbitrary, time-varying graphs. In other words w... more We generalize the theory of Cellular Automata to arbitrary, time-varying graphs. In other words we formalize, and prove theorems about, the intuitive idea of a labelled graph which evolves in time -but under the natural constraint that information can only ever be transmitted at a bounded speed, with respect to the distance given by the graph. The notion of translation-invariance is also generalized. The definition we provide for these 'causal graph dynamics' is simple and axiomatic. The theorems we provide also show that it is robust. For instance, causal graph dynamics are stable under composition and under restriction to radius one. In the finite case some fundamental facts of Cellular Automata theory carry through: causal graph dynamics admit a characterization as continuous functions, and they are stable under inversion. The provided examples suggest a wide range of applications of this mathematical object, from complex systems science to theoretical physics.
We present a calculus, called the scheme-calculus, that permits to express natural deduction proo... more We present a calculus, called the scheme-calculus, that permits to express natural deduction proofs in various theories. Unlike λ-calculus, the syntax of this calculus sticks closely to the syntax of proofs, in particular, no names are introduced for the hypotheses. We show that despite its non-determinism, some typed scheme-calculi have the same expressivity as the corresponding typed λ-calculi.
In 6] we have proposed a general higher-order uni cation method using a theory of explicit substi... more In 6] we have proposed a general higher-order uni cation method using a theory of explicit substitutions and we have proved its completeness. In this paper, we investigate the case of higher-order patterns as introduced by Miller. We show that our general algorithm specializes in a very convenient way to patterns. We also sketch an e cient implementation of the abstract algorithm and its generalization to constraint simpli cation, which has yielded good experimental results at the core of a higher-order constraint logic programming language.
Résumé: L'informatique fondamentale est un vaste sujet, comme en témoignent les 2 283 et 3 1... more Résumé: L'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.
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