Jean-Claude Raoult - Academia.edu (original) (raw)

Papers by Jean-Claude Raoult

Springer eBooks, 1992

A constructive valuation interpretation for classical logic and its use in witness extraction.- V... more A constructive valuation interpretation for classical logic and its use in witness extraction.- Varieties of increasing trees.- Origin functions in ?-calculus and term rewriting systems.- An algebraic approach to the interpretation of recursive types.- Decidability of reachability and disjoint union of term rewriting systems.- A complete type inference algorithm for simple intersection types.- Monadic second-order definable graph transductions.- CTL* and ECTL* as fragments of the modal ?-calculus.- Power domains supporting recursion and failure.- Parallel contraction of fibonacci trees and prefix computations on a family of interconnection topologies.- Must preorder in non-deterministic untyped ?-calculus.- A programming language for deriving hypergraphs.- Graph grammars as context-dependent rewriting systems: A partial ordering semantics.- Empty stack Pushdown ?-tree automata.- Modulo counting quantifiers over finite trees.- Finite tree automata with cost functions.- Partial type assignment in left linear applicative term rewriting systems.- A linear algorithm for solving fixed-point equations on transition systems.- Beyond conditional equations.

CAAP, 1984

The purpose of the present paper is twofold: Firstly, show that it is possib!e lo rewrite graphs ... more The purpose of the present paper is twofold: Firstly, show that it is possib!e lo rewrite graphs in a way equivalent to, and in fact slightly more powerful than that of Ehrig, Pfcnder and Schneider (1973), which has, since then, been developed mainly by the Berlin school. Our method consists in using a single push-out of partial morphisms and is described in Section 3. Section 1 is devoted to the elemenrary definitions concerning graphs and related terms. Section 2 contains the set-theoretic prerequisites for the sequel but the proofs have been moved into an appendix, for easier reading. Secondly, we indicate in Section 4 why this method is not really fit for rewriting graphs that represent collapsed terms (i.e., sharing common subterms) and we introduce pushouts of total functions, which are not morphisms everywhere on their domain. This method is connected to the clas: ic;! rc;.riting of the corresponding tern:s. The adequacy of these new rewriting rules is then ter, c J to prove a local confluence critcrian iI la Knuth-Bendix (1970) in Section 5, the proof of which turns out to be very short.

Programme 2 : calcul symbolique, programmation et genie logicielSIGLEAvailable at INIST (FR), Doc... more Programme 2 : calcul symbolique, programmation et genie logicielSIGLEAvailable at INIST (FR), Document Supply Service, under shelf-number : 14802 E, issue : a.1991 n.1410 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc

Bulletin of The European Association for Theoretical Computer Science, 1983

The purpose of the present paper is twofold: Firstly, show that it is possib!e lo rewrite graphs ... more The purpose of the present paper is twofold: Firstly, show that it is possib!e lo rewrite graphs in a way equivalent to, and in fact slightly more powerful than that of Ehrig, Pfcnder and Schneider (1973), which has, since then, been developed mainly by the Berlin school. Our method consists in using a single push-out of partial morphisms and is described in Section 3. Section 1 is devoted to the elemenrary definitions concerning graphs and related terms. Section 2 contains the set-theoretic prerequisites for the sequel but the proofs have been moved into an appendix, for easier reading. Secondly, we indicate in Section 4 why this method is not really fit for rewriting graphs that represent collapsed terms (i.e., sharing common subterms) and we introduce pushouts of total functions, which are not morphisms everywhere on their domain. This method is connected to the clas: ic;! rc;.riting of the corresponding tern:s. The adequacy of these new rewriting rules is then ter, c J to prove a local confluence critcrian iI la Knuth-Bendix (1970) in Section 5, the proof of which turns out to be very short.

We give a completion theorem for ordered magmas (i.e. ordered algebras with monotone operations) ... more We give a completion theorem for ordered magmas (i.e. ordered algebras with monotone operations) in a general form. Particular instances of this theorem are already known, and new results follow. The semantics of programming languages is the motivation of such investigations.

The possibly non distributive event domains which arise from Winskel's event structures with... more The possibly non distributive event domains which arise from Winskel's event structures with binary conflict are known to coincide with the domains of configurations of Stark's trace automata. We prove that whenever the transitive reduction of the order on finite elements in an event domain is a context-free graph in the sense of Muller and Schupp, that event domain may also be generated from a finite trace automaton, where both the set of states and the concurrent alphabet are finite. We show that the set of graph grammars which generate event domains is a recursive set. We obtain altogether an effective procedure which decides from an unlabelled graph grammar whether it generates an event domain and which constructs in that case a finite trace automaton recognizing that event domain. The advantage of trace automata over unlabelled graph grammars is to provide for a more concrete and therefore more tractable representation of event domains, well suited to an automated ver...

The well-known snake lemma is proved entirely within category theory, without the help of "p... more The well-known snake lemma is proved entirely within category theory, without the help of "points with value in..." \`a la Grothendieck, nor pseudo-elements as in Guglielmetti & Zaganidis. Instead, we define and use consistently semi-cartesian squares, which were promoted by C. Chevalley.

Proceedings of the 7th Colloquium on Automata Languages and Programming, 1980

We present here strategies for searching the (unique) zero of a real function, or its n-th deriva... more We present here strategies for searching the (unique) zero of a real function, or its n-th derivative; we assume no a priori bound on the value x of this zero. The proposed strategy performs logry + llogry+ ... +1 + log*ry evaluations of f to determine x = ɛy with error less than ɛ (here r depends only on n). An argument of slowly converning integrals shows that these strategies are essentially optimal.

We give a completion theorem for ordered magmas (i.e. ordered algebras with monotone operations) ... more We give a completion theorem for ordered magmas (i.e. ordered algebras with monotone operations) in a general form. Particular instances of this theorem are already known, and new results follow. The semantics of programming languages is the motivation of such investigations. Key wordscomplete partial orders, semantics of programming languages. Introduction. When defining recursive functions by systems of equations (Kleene [5]), one introduces an order relation which means that a partial result approximates another one. This partial order is complete (i.e. every ascending chain admits a least upper bound), thus allowing a minimal solution to be defined for the system. This matter has been rebuilt by Scott, and many authors after him, within the framework of complete lattices ; that last theory has been developed for its own sake by several authors , among which Birkhoff [1]. Frequently, the lattice structure does not seem necessary and creates instead additional troubles (Plotkin [9], Milner [8] for instance). The notion of complete partial order is good enough, and fits better to the most common instances. This algebraic framework is suitable for studying program schemes([2], [3]). We then need distinguish between the base functions and the program-defined functions, with the help of base functions and various control structures (recursive call, iteration, etc…). Thus, our domains will be ordered magmas, i.e. partial orders equipped with monotone operators (no information is lost during a computation). And we shall be concerned with completeness (the operators being supposed continuous). More precisely, we shall study the possible embeddings of an ordered magma into a complete ordered magma. Some of the ascending chains may keep their l.u.b., or may be added a new one ; this gives different completions, each characterized by a universal property. We shall thus define the Γ-completion as the completion which preserves the l.u.b. which already exist in a set Γ of subsetes of the magma. From this general theorem is defived the "ideal completion" of [9], [1], [2], [4], the "chain-completion" of [8], and the existence of factor objects in the category of ordered magmas. The above mentionned authors woule use neither operators not magmas, but only partial orders (except [4]). But eht "chain-completion" in the category of partial orders need not be a complete ordered magma (cf. Corollary 2). Definitions and the main theorem. Let F be a set of operators with arity. An F-magma M is a domain D M together with a function f M : D k M → D M for each f ∈ F with arity k. The homomorphisms of F-magmas, or F-morphisms, shall be compatible with the operators : ϕ : M → M′ is a F-morphism when ϕ(f M (a 1 , … , a k)) = f M′ (ϕ(a 1), … , ϕ(a k)) for all f ∈ F with arity k, and all a 1 , … , a k ∈ D M. In this paper, we shall only consider ordered magmas (therefore "magma" will mean "ordered magma"), with a partial order denoted by ≤ M , a least element Ω M (associated with the symbol Ω of arity 0 whichi is always supposed to be an element of F), and monotone operators f M. An F-morphism between ordered magmas must be monotone.

Theoretical Computer Science, 1979

We study the number of registers required for evaluating arithmetic expressions. This parameter o... more We study the number of registers required for evaluating arithmetic expressions. This parameter of binary trees appears in various computer science problems as well as in numerous natural sciences applications where it is known as the Strahler number. We give several enumeration results describing the distribution of the number of registers for trees of size n. The average number of registers has the asymptotic expansion log4 n f D(log., n) + o(1); here, function r;l is periodic of period one, and its Fourier expansion can be explicitly determined in terms of Riemann's zeta function and Euler's gamma function.

The possibly non distributive event domains which arise from Winskel's event structures with ... more The possibly non distributive event domains which arise from Winskel's event structures with binary connict are known to coincide with the domains of conngurations of Stark's trace automata. We prove that whenever the transitive reduction of the order on nite elements in an event domain is a context-free graph in the sense of M uller and Schupp, that event domain may also be generated from a nite trace automaton, where both the set of states and the concurrent alphabet are nite. We show that the set of graph grammars which generate event domains is a recursive set. We obtain altogether an eeective procedure which decides from an unlabelled graph grammar whether it generates an event domain and which constructs in that case a nite trace automaton recognizing that event domain. The advantage of trace automata over unlabelled graph grammars is to provide for a more concrete and therefore more tractable representation of event domains, well suited to an automated veriication of ...

Ita, 1981

Finiteness results on rewriting systems RAIRO-Informatique théorique, tome 15, n o 4 (1981), p. 3... more Finiteness results on rewriting systems RAIRO-Informatique théorique, tome 15, n o 4 (1981), p. 373-391. <http://www.numdam.org/item?id=ITA_1981__15_4_373_0> © AFCET, 1981, tous droits réservés. L'accès aux archives de la revue « RAIRO-Informatique théorique » implique l'accord avec les conditions générales d'utilisation (http://www.numdam. org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ R.A.I.R.O. Informatique théorique/Theoretical Informaties (vol. 15, n° 4, 1981, p. 373 à 391) FINITENESS RESULTS ON REWRITING SYSTEMS (*) by Jean-Claude RAOULT (*) Comm unica ted by M. Ni VAT Résumé.-Étant donné un système de récriture de termes du premier ordre noetherien et confluent, on considère la relation d'équivalence engendrée, et on prouve que le problème de lajinitude d'une classe (ou de toutes les classes) est indécidable, sauf si l'on se restreint aux termes sans variables. En revanche, lajinitude du nombre de classes est decidable.

Information Processing Letters, 1988

Proceedings of the 11th ACM SIGACT-SIGPLAN symposium on Principles of programming languages - POPL '84, 1984

A defining characteristic of “functional” specifications is the absence of assignments: updates o... more A defining characteristic of “functional” specifications is the absence of assignments: updates of tables and data structures are expressed by giving the relationship between the new and old values. An obvious implementation allocates separate space for new and old values and may consume a lot of storage. However, even when updates of attributes like symbol tables are expressed functionally, we

Proceedings of the tenth annual ACM symposium on Theory of computing - STOC '78, 1978

Lecture Notes in Computer Science, 1993

... case. A nice generalization by Dauchet & Tison [1992] of these ground tree transducers, i... more ... case. A nice generalization by Dauchet & Tison [1992] of these ground tree transducers, in which a finite automaton runs on a superposition of two trees, does extend the word transductions to the case of trees. ... bulv'), [! I]xu', [I o I]yv' ...

Springer eBooks, 1992

A constructive valuation interpretation for classical logic and its use in witness extraction.- V... more A constructive valuation interpretation for classical logic and its use in witness extraction.- Varieties of increasing trees.- Origin functions in ?-calculus and term rewriting systems.- An algebraic approach to the interpretation of recursive types.- Decidability of reachability and disjoint union of term rewriting systems.- A complete type inference algorithm for simple intersection types.- Monadic second-order definable graph transductions.- CTL* and ECTL* as fragments of the modal ?-calculus.- Power domains supporting recursion and failure.- Parallel contraction of fibonacci trees and prefix computations on a family of interconnection topologies.- Must preorder in non-deterministic untyped ?-calculus.- A programming language for deriving hypergraphs.- Graph grammars as context-dependent rewriting systems: A partial ordering semantics.- Empty stack Pushdown ?-tree automata.- Modulo counting quantifiers over finite trees.- Finite tree automata with cost functions.- Partial type assignment in left linear applicative term rewriting systems.- A linear algorithm for solving fixed-point equations on transition systems.- Beyond conditional equations.

CAAP, 1984

The purpose of the present paper is twofold: Firstly, show that it is possib!e lo rewrite graphs ... more The purpose of the present paper is twofold: Firstly, show that it is possib!e lo rewrite graphs in a way equivalent to, and in fact slightly more powerful than that of Ehrig, Pfcnder and Schneider (1973), which has, since then, been developed mainly by the Berlin school. Our method consists in using a single push-out of partial morphisms and is described in Section 3. Section 1 is devoted to the elemenrary definitions concerning graphs and related terms. Section 2 contains the set-theoretic prerequisites for the sequel but the proofs have been moved into an appendix, for easier reading. Secondly, we indicate in Section 4 why this method is not really fit for rewriting graphs that represent collapsed terms (i.e., sharing common subterms) and we introduce pushouts of total functions, which are not morphisms everywhere on their domain. This method is connected to the clas: ic;! rc;.riting of the corresponding tern:s. The adequacy of these new rewriting rules is then ter, c J to prove a local confluence critcrian iI la Knuth-Bendix (1970) in Section 5, the proof of which turns out to be very short.

Programme 2 : calcul symbolique, programmation et genie logicielSIGLEAvailable at INIST (FR), Doc... more Programme 2 : calcul symbolique, programmation et genie logicielSIGLEAvailable at INIST (FR), Document Supply Service, under shelf-number : 14802 E, issue : a.1991 n.1410 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc

Bulletin of The European Association for Theoretical Computer Science, 1983

The purpose of the present paper is twofold: Firstly, show that it is possib!e lo rewrite graphs ... more The purpose of the present paper is twofold: Firstly, show that it is possib!e lo rewrite graphs in a way equivalent to, and in fact slightly more powerful than that of Ehrig, Pfcnder and Schneider (1973), which has, since then, been developed mainly by the Berlin school. Our method consists in using a single push-out of partial morphisms and is described in Section 3. Section 1 is devoted to the elemenrary definitions concerning graphs and related terms. Section 2 contains the set-theoretic prerequisites for the sequel but the proofs have been moved into an appendix, for easier reading. Secondly, we indicate in Section 4 why this method is not really fit for rewriting graphs that represent collapsed terms (i.e., sharing common subterms) and we introduce pushouts of total functions, which are not morphisms everywhere on their domain. This method is connected to the clas: ic;! rc;.riting of the corresponding tern:s. The adequacy of these new rewriting rules is then ter, c J to prove a local confluence critcrian iI la Knuth-Bendix (1970) in Section 5, the proof of which turns out to be very short.

We give a completion theorem for ordered magmas (i.e. ordered algebras with monotone operations) ... more We give a completion theorem for ordered magmas (i.e. ordered algebras with monotone operations) in a general form. Particular instances of this theorem are already known, and new results follow. The semantics of programming languages is the motivation of such investigations.

The possibly non distributive event domains which arise from Winskel's event structures with... more The possibly non distributive event domains which arise from Winskel's event structures with binary conflict are known to coincide with the domains of configurations of Stark's trace automata. We prove that whenever the transitive reduction of the order on finite elements in an event domain is a context-free graph in the sense of Muller and Schupp, that event domain may also be generated from a finite trace automaton, where both the set of states and the concurrent alphabet are finite. We show that the set of graph grammars which generate event domains is a recursive set. We obtain altogether an effective procedure which decides from an unlabelled graph grammar whether it generates an event domain and which constructs in that case a finite trace automaton recognizing that event domain. The advantage of trace automata over unlabelled graph grammars is to provide for a more concrete and therefore more tractable representation of event domains, well suited to an automated ver...

The well-known snake lemma is proved entirely within category theory, without the help of "p... more The well-known snake lemma is proved entirely within category theory, without the help of "points with value in..." \`a la Grothendieck, nor pseudo-elements as in Guglielmetti & Zaganidis. Instead, we define and use consistently semi-cartesian squares, which were promoted by C. Chevalley.

Proceedings of the 7th Colloquium on Automata Languages and Programming, 1980

We present here strategies for searching the (unique) zero of a real function, or its n-th deriva... more We present here strategies for searching the (unique) zero of a real function, or its n-th derivative; we assume no a priori bound on the value x of this zero. The proposed strategy performs logry + llogry+ ... +1 + log*ry evaluations of f to determine x = ɛy with error less than ɛ (here r depends only on n). An argument of slowly converning integrals shows that these strategies are essentially optimal.

We give a completion theorem for ordered magmas (i.e. ordered algebras with monotone operations) ... more We give a completion theorem for ordered magmas (i.e. ordered algebras with monotone operations) in a general form. Particular instances of this theorem are already known, and new results follow. The semantics of programming languages is the motivation of such investigations. Key wordscomplete partial orders, semantics of programming languages. Introduction. When defining recursive functions by systems of equations (Kleene [5]), one introduces an order relation which means that a partial result approximates another one. This partial order is complete (i.e. every ascending chain admits a least upper bound), thus allowing a minimal solution to be defined for the system. This matter has been rebuilt by Scott, and many authors after him, within the framework of complete lattices ; that last theory has been developed for its own sake by several authors , among which Birkhoff [1]. Frequently, the lattice structure does not seem necessary and creates instead additional troubles (Plotkin [9], Milner [8] for instance). The notion of complete partial order is good enough, and fits better to the most common instances. This algebraic framework is suitable for studying program schemes([2], [3]). We then need distinguish between the base functions and the program-defined functions, with the help of base functions and various control structures (recursive call, iteration, etc…). Thus, our domains will be ordered magmas, i.e. partial orders equipped with monotone operators (no information is lost during a computation). And we shall be concerned with completeness (the operators being supposed continuous). More precisely, we shall study the possible embeddings of an ordered magma into a complete ordered magma. Some of the ascending chains may keep their l.u.b., or may be added a new one ; this gives different completions, each characterized by a universal property. We shall thus define the Γ-completion as the completion which preserves the l.u.b. which already exist in a set Γ of subsetes of the magma. From this general theorem is defived the "ideal completion" of [9], [1], [2], [4], the "chain-completion" of [8], and the existence of factor objects in the category of ordered magmas. The above mentionned authors woule use neither operators not magmas, but only partial orders (except [4]). But eht "chain-completion" in the category of partial orders need not be a complete ordered magma (cf. Corollary 2). Definitions and the main theorem. Let F be a set of operators with arity. An F-magma M is a domain D M together with a function f M : D k M → D M for each f ∈ F with arity k. The homomorphisms of F-magmas, or F-morphisms, shall be compatible with the operators : ϕ : M → M′ is a F-morphism when ϕ(f M (a 1 , … , a k)) = f M′ (ϕ(a 1), … , ϕ(a k)) for all f ∈ F with arity k, and all a 1 , … , a k ∈ D M. In this paper, we shall only consider ordered magmas (therefore "magma" will mean "ordered magma"), with a partial order denoted by ≤ M , a least element Ω M (associated with the symbol Ω of arity 0 whichi is always supposed to be an element of F), and monotone operators f M. An F-morphism between ordered magmas must be monotone.

Theoretical Computer Science, 1979

We study the number of registers required for evaluating arithmetic expressions. This parameter o... more We study the number of registers required for evaluating arithmetic expressions. This parameter of binary trees appears in various computer science problems as well as in numerous natural sciences applications where it is known as the Strahler number. We give several enumeration results describing the distribution of the number of registers for trees of size n. The average number of registers has the asymptotic expansion log4 n f D(log., n) + o(1); here, function r;l is periodic of period one, and its Fourier expansion can be explicitly determined in terms of Riemann's zeta function and Euler's gamma function.

The possibly non distributive event domains which arise from Winskel's event structures with ... more The possibly non distributive event domains which arise from Winskel's event structures with binary connict are known to coincide with the domains of conngurations of Stark's trace automata. We prove that whenever the transitive reduction of the order on nite elements in an event domain is a context-free graph in the sense of M uller and Schupp, that event domain may also be generated from a nite trace automaton, where both the set of states and the concurrent alphabet are nite. We show that the set of graph grammars which generate event domains is a recursive set. We obtain altogether an eeective procedure which decides from an unlabelled graph grammar whether it generates an event domain and which constructs in that case a nite trace automaton recognizing that event domain. The advantage of trace automata over unlabelled graph grammars is to provide for a more concrete and therefore more tractable representation of event domains, well suited to an automated veriication of ...

Ita, 1981

Finiteness results on rewriting systems RAIRO-Informatique théorique, tome 15, n o 4 (1981), p. 3... more Finiteness results on rewriting systems RAIRO-Informatique théorique, tome 15, n o 4 (1981), p. 373-391. <http://www.numdam.org/item?id=ITA_1981__15_4_373_0> © AFCET, 1981, tous droits réservés. L'accès aux archives de la revue « RAIRO-Informatique théorique » implique l'accord avec les conditions générales d'utilisation (http://www.numdam. org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ R.A.I.R.O. Informatique théorique/Theoretical Informaties (vol. 15, n° 4, 1981, p. 373 à 391) FINITENESS RESULTS ON REWRITING SYSTEMS (*) by Jean-Claude RAOULT (*) Comm unica ted by M. Ni VAT Résumé.-Étant donné un système de récriture de termes du premier ordre noetherien et confluent, on considère la relation d'équivalence engendrée, et on prouve que le problème de lajinitude d'une classe (ou de toutes les classes) est indécidable, sauf si l'on se restreint aux termes sans variables. En revanche, lajinitude du nombre de classes est decidable.

Information Processing Letters, 1988

Proceedings of the 11th ACM SIGACT-SIGPLAN symposium on Principles of programming languages - POPL '84, 1984

A defining characteristic of “functional” specifications is the absence of assignments: updates o... more A defining characteristic of “functional” specifications is the absence of assignments: updates of tables and data structures are expressed by giving the relationship between the new and old values. An obvious implementation allocates separate space for new and old values and may consume a lot of storage. However, even when updates of attributes like symbol tables are expressed functionally, we

Proceedings of the tenth annual ACM symposium on Theory of computing - STOC '78, 1978

Lecture Notes in Computer Science, 1993

... case. A nice generalization by Dauchet & Tison [1992] of these ground tree transducers, i... more ... case. A nice generalization by Dauchet & Tison [1992] of these ground tree transducers, in which a finite automaton runs on a superposition of two trees, does extend the word transductions to the case of trees. ... bulv'), [! I]xu', [I o I]yv' ...