Linear Logic

This paper introduces Chronolog(Z), a logic programming language based on a discrete linear-time temporal logic with unbounded past and future. Chronolog(Z) is suitable for applications involving the notion of dynamic change, such as modeling non-terminating computations, the simulation of sequential circuits, and temporal databases. The execution of the programs of the language is based on a resolution-type proof procedure called TiSLDresolution. A modular extension of Chronolog(Z) is proposed which can be used to model objects with internal memory.

An XSAT 𝜙 is satisfiable iff there exists a satisfying assignment 𝛼 to 𝜙. This paper shows that 𝜙 is satisfiable iff 𝜙 is reducible to 𝛼, viz., 𝜙 → 𝛼, thus 𝜙 ∧ 𝛼 is true, where 𝛼 = 𝑟 1 ∧ 𝑟 2 ∧ • • • ∧ 𝑟 𝑛 for some 𝑟 𝑙 ∈ {𝑥 𝑙 , 𝑥 𝑙 }. That is, the formula 𝜙 is unsatisfiable if it reduces to a simple formula 𝜓 inconsistent, otherwise 𝜙 → 𝜓 ∧𝜙 ′ such that 𝜙 ′ → 𝛼 ′ , and that 𝛼 = 𝜓 ∧ 𝛼 ′. Also, 𝛼 ′ = 𝜓 0 ∧ • • • ∧ 𝜓 𝑛 ′ and 𝜓 0 = (𝑟 ℓ 01 ∧ • • • ∧ 𝑟 ℓ 0 ñ),. .. ,𝜓 𝑛 ′ = (𝑟 ℓ 𝑛 ′ 1 ∧ • • • ∧ 𝑟 ℓ 𝑛 ′ ñ) such that {ℓ 01 , .

We present a novel approach to the automatic verification and falsification of LTL requirements of non-linear discrete-time hybrid systems. The verification tool uses an interval-based constraint solver for non-linear robust constraints to compute incrementally refined abstractions. Although the problem is in general undecidable, we prove termination of abstraction refinement based verification and falsification of such properties for the class of non-linear robust discrete-time hybrid systems. We argue, that—in industrial practice—safety critical control applications give rise to hybrid systems that are robust. We give first results on the application of this approach to a variant of an aircraft collision avoidance protocol.

We present a general solution to the problem of optimized ezecution of logic programs containing linear recursive rules. Our solution is based on eztensions of the classical counting method, which is known to be efficient but of limited applicability. In fact, the range of applicability of the counting method, and its variants proposed by previous researchers, suffer from one or more of the following limitations: the method can be applied only when (1) the adorned program contains one recursive rule, (~) the 'left part' and the 'right part' of the recursive rule do not have any common variable and (3) the relation associated with the left part of the recursive rule is 'acyclic'. In this paper, a simple and unified framework is presented, where those limitations are removed, and the counting method thus become applicable to all programs with linear rules. This framework also allows a simple treatment of programs factorizable into segments, which can be computed separately yielding a much faster ezecution. A simple factorization technique based on argument reduction is presented that produces optimizations similar to those defined in the literature for RLC-linear programs (i.e., programs with right-linear rules, leftlinear rules and a combination of the two).

Given any symmetric monoidal (closed) category C and any suitable collections W of objects of C, it is shown how to construct C[W], a polynomial such category, the result of freely adjoining to C a system of monoidal indeterminates for every element of W. It is then shown that all of the categories of "possible worlds" used to treat languages that allow for dynamic creation of "new" variables, locations, or names are instances of this construction and hence have appropriate universality properties.

Bellin, G. and J. Ketonen, A decision procedure revisited: Notes on direct logic, linear logic and its implementation, Theoretical Computer Science 95 (1992) 115-142. This paper studies decidable fragments of predicate calculus. We will focus on the structure of direct predicate calculus as defined in Ketonen and Weyhrauch (1984) in the light of the recent work of Girard (Girard 1987) on linear logic. Several graph-theoretic results are used to prove correspondences between systems of natural deduction, direct predicate logic, and linear logic. In addition, the implementation of a decision procedure for direct predicate logic is sketched.

We provide a context semantics for Multiplicative-Additive Linear Logic (MALL), together with proofnets whose reduction preserves semantics, where proofnet reduction is equated with cut-elimination on MALL sequents. The results extend the program of Gonthier, Abadi, and Lévy, who provided a "geometry of optimal λ-reduction" (context semantics) for λ-calculus and Multiplicative-Exponential Linear Logic (MELL). We integrate three features: a semantics that uses buses to implement slicing; a proofnet technology that allows multidimensional boxes and generalized garbage, preserving the linearity of additive reduction; and finally, a read-back procedure that computes a cut-free proof from the semantics, which is closely related to full abstraction theorems. Peut-être que la logique se trompe. -Yves Lafont Linear Logic appeared in 1987 and turned out quickly to be an interesting tool to model programming languages, specifically reasoning that is sensitive to the notion of consumable resources. Indeed the multiplicative fragment of Linear Logic (⊗, ) allows linear products (pairing and unpairing), implementing functions: a context pairs a continuation and an argument, a function unpairs and connects the two. The additive fragment (⊕, &) allows linear sums (injection and case dispatch), implementing features of processes in the style of CSP or CCS . The crucial difference between these two components is the way they take care with consumption of resources. The exponential fragment implements sharing of resources: arguments, control contexts. We can then implement, for example, graph reduction technology for λ-calculus with control operators (call/cc, abort, jumps), and related mechanical proof systems for classical logic-taking care of the sharing and copying, implicit in these calculi . Linear Logic was initially a sequent calculus, but this sequentialized structure was too strong and the design of a nice cut-elimination procedure was complicated. Therefore, proofnets were introduced as a more flexible representation of proofs . Further, Geometry of Interaction (GoI) developed the idea that the reduction of proofs can be seen as a local interaction process . Its intensional features provide a mediating Purgatory between the Heaven of denotational semantics, and the Hell of operational semantics. GoI was simplified in the "geometry of optimal λ-reduction" by Gonthier, Abadi and Lévy in the context of the MELL fragment. They reduced Hilbert spaces to simple data-structures, known as context semantics, and developed a proofnet technology which implemented the context semantics locally. Reduction on proofnets preserves the semantics, and Lamping's algorithm for optimal reduction of λ-terms [14] is a method of graph reduction. They further indicated how to read back any part of the Böhm tree (normal form) of a λ-term from its context semantics. Can this program be carried out for full Linear Logic? In this paper we extend these result to the MALL fragment (multiplicatives and additives): this may be a step towards a satisfactory proofnet syntax for full Linear Logic with a good characterization of proofs. The MALL fragment is quite problematic since it does not have a nice cutelimination procedure-unlike MLL, which enjoys a straightforward one. Part of the work will be to understand and improve the reduction procedure for MALL. The main contributions of this paper are to provide an integrated development of (1) a context semantics for the MALL fragment; (2) a proofnet technology allowing normalization of MALL proofs, using the ideas of multidimensional boxes and generalized garbage; and (3) a read-back procedure that inputs a valid context semantics and outputs a normalized proofnet. In Section 1, we recall some basic definitions on MALL, proofnets and Linear Logic. We introduce in Section ?? a form of context semantics for MALL proofs, and we derive from it a bus-notation based proofnet syntax in Section 3. Then, we envisage the main problems that come from the reduction on the additives in Section 4. The first difficulty is that the additive cut-elimination is not really linear, since an additive reduction step erases a whole part of the proof (this is also a problem for the locality of the reduction we would like to achieve). The second problem also stems from the additives: the way one should reduce a cut involving auxiliary formulas of two &-links is unclear. The solution to this problem will come from an extension of the MALL syntax. After this adaptation, we will be able to get a much more satisfactory cut-elimination procedure. In Section 5, we show how a normalized proof can be read-back from the context semantics of a proof, and we see how this result can be related to a form of full completeness. In Section 6, we compare our approach with other works.

We provide a context semantics for Multiplicative-Additive Linear Logic (MALL), together with proofnets whose reduction preserves semantics, where proofnet reduction is equated with cut-elimination on MALL sequents. The results extend the program of Gonthier, Abadi, and Lévy, who provided a "geometry of optimal λ-reduction" (context semantics) for λ-calculus and Multiplicative-Exponential Linear Logic (MELL). We integrate three features: a semantics that uses buses to implement slicing; a proofnet technology that allows multidimensional boxes and generalized garbage, preserving the linearity of additive reduction; and finally, a read-back procedure that computes a cut-free proof from the semantics, which is closely related to full abstraction theorems. Peut-être que la logique se trompe. -Yves Lafont Linear Logic appeared in 1987 and turned out quickly to be an interesting tool to model programming languages, specifically reasoning that is sensitive to the notion of consumable resources. Indeed the multiplicative fragment of Linear Logic (⊗, ) allows linear products (pairing and unpairing), implementing functions: a context pairs a continuation and an argument, a function unpairs and connects the two. The additive fragment (⊕, &) allows linear sums (injection and case dispatch), implementing features of processes in the style of CSP or CCS . The crucial difference between these two components is the way they take care with consumption of resources. The exponential fragment implements sharing of resources: arguments, control contexts. We can then implement, for example, graph reduction technology for λ-calculus with control operators (call/cc, abort, jumps), and related mechanical proof systems for classical logic-taking care of the sharing and copying, implicit in these calculi . Linear Logic was initially a sequent calculus, but this sequentialized structure was too strong and the design of a nice cut-elimination procedure was complicated. Therefore, proofnets were introduced as a more flexible representation of proofs . Further, Geometry of Interaction (GoI) developed the idea that the reduction of proofs can be seen as a local interaction process . Its intensional features provide a mediating Purgatory between the Heaven of denotational semantics, and the Hell of operational semantics. GoI was simplified in the "geometry of optimal λ-reduction" by Gonthier, Abadi and Lévy in the context of the MELL fragment. They reduced Hilbert spaces to simple data-structures, known as context semantics, and developed a proofnet technology which implemented the context semantics locally. Reduction on proofnets preserves the semantics, and Lamping's algorithm for optimal reduction of λ-terms [14] is a method of graph reduction. They further indicated how to read back any part of the Böhm tree (normal form) of a λ-term from its context semantics. Can this program be carried out for full Linear Logic? In this paper we extend these result to the MALL fragment (multiplicatives and additives): this may be a step towards a satisfactory proofnet syntax for full Linear Logic with a good characterization of proofs. The MALL fragment is quite problematic since it does not have a nice cutelimination procedure-unlike MLL, which enjoys a straightforward one. Part of the work will be to understand and improve the reduction procedure for MALL. The main contributions of this paper are to provide an integrated development of (1) a context semantics for the MALL fragment; (2) a proofnet technology allowing normalization of MALL proofs, using the ideas of multidimensional boxes and generalized garbage; and (3) a read-back procedure that inputs a valid context semantics and outputs a normalized proofnet. In Section 1, we recall some basic definitions on MALL, proofnets and Linear Logic. We introduce in Section ?? a form of context semantics for MALL proofs, and we derive from it a bus-notation based proofnet syntax in Section 3. Then, we envisage the main problems that come from the reduction on the additives in Section 4. The first difficulty is that the additive cut-elimination is not really linear, since an additive reduction step erases a whole part of the proof (this is also a problem for the locality of the reduction we would like to achieve). The second problem also stems from the additives: the way one should reduce a cut involving auxiliary formulas of two &-links is unclear. The solution to this problem will come from an extension of the MALL syntax. After this adaptation, we will be able to get a much more satisfactory cut-elimination procedure. In Section 5, we show how a normalized proof can be read-back from the context semantics of a proof, and we see how this result can be related to a form of full completeness. In Section 6, we compare our approach with other works.

String representations of events are applied to Robin Cooper's proposal that propositions in natural language semantics are types of situations. Links with the higher types of prooftheoretic semantics are forged, deepening type-theoretic interpretations of Discourse Representation Structures to encompass event structures.

We show that the physical principle "the adjoint associates to each state a 'test' for that state" fully characterises the Hermitian adjoint for pure quantum theory, therefore providing the adjoint with operational meaning beyond its standard mathematical definition. Also, we show that for general process theories, which all admit a diagrammatic representation, this physical principle induces a reflection operation.

This paper presents a method for model checking dense complex real-time systems. This approach is implemented at the meta level of the Rewriting Logic system Maude. The dense complex real-time system is specified using a syntax which has the semantics of timed automata and the property is specified with the temporal logic TLTL (Timed LTL). The well known timed automata model checkers Kronos and Uppaal only support TCTL model checking (a very limited fragment in the case of Uppaal). Specification of the TLTL property is reduced to LTL and its temporal constraints are captured in a new timed automaton. This timed automaton will be composed with the original timed automaton representing the semantics of the complex real-time system under analysis. Then, the product-timed automaton will be abstracted using partition refinement of state space based on strong bi-simulation. The result is an untimed automaton modulo the TLTL property which represents an equivalent finite state system to be model-checked using Maude LTL model checking. This approach is successfully tested on industrial designs.

We show that the karoubian envelope of the category of correspondances is equivalent to the category of sup-compatible mappings between complete completly distributive lattices. On sait que l’enveloppe karoubienne per met de ramener les notions semi-fonctor ielles à des notions fonctorielles, et en particulier produit des catégories car tésiennes fer mées à partir de catégories faiblement cartésiennes fermées, ce qui est utilisé en théorie des dolnaines pour obtenir des modèles de la logique linéaire. Voir [2], [3]. On sait aussi que la modélisation et la manipulation algébrique de systèmes informatiques fait appel à des correspondances de Galois analysables et génér alisables en termes de transversalité de correspondances difonctionnelles. Voir [9]. Enfin on sait que Raney a établi un lien étroit entre les "tight" correspondances de Galois et les treillis co1plètement distributifs, qui sont des treillis continus particuliers. Voir [5, 6, 7] et [1]. Le lien entre ces diver...

With sound unification, Definite Clause Grammars and compact expression of combinatorial generation algorithms, logic programming is shown to conveniently host a declarative playground where interesting properties and behaviors emerge from the interaction of heterogenous but deeply connected computational objects. Compact combinatorial generation algorithms are given for several families of lambda terms, including open, closed, simply typed and linear terms as well as type inference and normal order reduction algorithms. We describe a Prolog-based combined lambda term generator and type-inferrer for closed well-typed terms of a given size, in de Bruijn notation. We introduce a compressed de Bruijn representation of lambda terms and define its bijections to standard representations. Our compressed terms facilitate derivation of size-proportionate ranking and unranking algorithms of lambda terms and their inferred simple types. By taking advantage of Prolog's unique bidirectional execution model and sound unification algorithm, our generator can build "customized" closed terms of a given type. This relational view of terms and their types enables the discovery of interesting patterns about frequently used type expressions occurring in well-typed functional programs. Our study uncovers the most "popular" types that govern function applications among a about a million small-sized lambda terms and hints toward practical uses to combinatorial software testing. The S and K combinator expressions form a well-known Turing-complete subset of the lambda calculus. We specify evaluation, type inference and combinatorial generation algorithms for SKcombinator trees. In the process, we unravel properties shedding new light on interesting aspects of their structure and distribution. We study the proportion of well-typed terms among size-limited SK-expressions as well as the type-directed generation of terms of sizes smaller then the size of their simple types. We also introduce the well-typed frontier of an untypable term and we use it to design a simplification algorithm for untypable terms taking advantage of the fact that well-typed terms are normalizable. A uniform representation, as binary trees with empty leaves, is given to expressions built with Rosser's X-combinator, natural numbers, lambda terms and simple types. Using this shared representation, ranking/unranking algorithm of lambda terms to tree-based natural numbers are described. Our algorithms, expressed as an incrementally developed literate Prolog program, implement a declarative playground for exploration of representations, encodings and computations with uniformly represented lambda terms, types, combinators and tree-based arithmetic.

We consider the Curry-Howard-Lambek correspondence for effectful computation and resource management, specifically proposing polarised calculi together with presheaf-enriched adjunction models as the starting point for a comprehensive semantic theory relating logical systems, typed calculi, and categorical models in this context. Our thesis is that the combination of effects and resources should be considered orthogonally. Model theoretically, this leads to an understanding of our categorical models from two complementary perspectives: (i) as a linearisation of CBPV (Call-by-Push-Value) adjunction models, and (ii) as an extension of linear/non-linear adjunction models with an adjoint resolution of computational effects. When the linear structure is cartesian and the resource structure is trivial we recover Levy's notion of CBPV adjunction model, while when the effect structure is trivial we have Benton's linear/nonlinear adjunction models. Further instances of our model theory include the dialogue categories with a resource modality of Melliès and Tabareau, and the [E]EC ([Enriched] Effect Calculus) models of Egger, Møgelberg and Simpson. Our development substantiates the approach by providing a lifting theorem of linear models into cartesian ones. To each of our categorical models we systematically associate a typed term calculus, each of which corresponds to a variant of the sequent calculi LJ (Intuitionistic Logic) or ILL (Intuitionistic Linear Logic). The adjoint resolution of effects corresponds to polarisation whereby, syntactically, types locally determine a strict or lazy evaluation order and, semantically, the associativity of cuts is relaxed. In particular, our results show that polarisation provides a computational interpretation of CBPV in direct style. Further, we characterise depolarised models: those where the cut is associative, and where the evaluation order is unimportant. We explain possible advantages of this style of calculi for the operational semantics of effects.

Blass, A., A game semantics for linear logic, Annals of Pure and Applied Logic 56 (1992) 183-220. We present a game (or dialogue) semantics in the style of Lorenzen (1959) for Girard's linear logic (1987). Lorenzen suggested that the (constructive) meaning of a proposition 91 should be specified by telling how to conduct a debate between a proponent P who asserts p and an opponent 0 who denies q. Thus propositions are interpreted as games, connectives (almost) as operations on games, and validity as existence of a winning strategy for P. (The qualifier 'almost' will be discussed later when more details have been presented.) We propose that the connectives of linear logic can be naturally interpreted as the operations on games introduced for entirely different purposes by Blass (1972). We show that affine logic, i.e., linear logic plus the rule of weakening, is sound for this interpretation. We also obtain a completeness theorem for the additive fragment of affine logic, but we show that completeness fails for the multiplicative fragment. On the other hand, for the multiplicative fragment, we obtain a simple characterization of game-semantical validity in terms of classical tautologies. An analysis of the failure of completeness for the multiplicative fragment leads to the conclusion that the game interpretation of the connective @ is weaker than the interpretation implicit in Girard's proof rules; we discuss the differences between the two interpretations and their relative advantages and disadvantages. Finally, we discuss how Godel's Dialectica interpretation (1958), which was connected to linear logic by de Paiva (1989) fits with game semantics.

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We propose measurement-based quantum computation (MBQC) as a quantum mechanical toy model for spacetime. Within this framework, we discuss the constraints on possible temporal orders enforced by certain symmetries present in every MBQC. We provide a classification for all MBQC temporal relations compatible with a given initial quantum state and measurement setting, in terms of a matroid. Further, we find a symmetry transformation related to local complementation that leaves the temporal relations invariant. After light cones and closed time-like curves have previously been found to have MBQC counterparts, we identify event horizons as a third piece of the phenomenology of General Relativity that has an analogue in MBQC.

Bidders in online multi-unit combinatorial auctions face the acute problem of estimating the valuations of an immense number of packages. Can the seller guide the bidders to avoid placing bids that are too high or too low? In the single unit case, fast methods are now available for incrementally computing, for each package at each time instant, the recommended lower bound (Deadness Level) and upper bound (Winning Level) on the next bid. But when there are multiple units of items, it becomes difficult to compute the Deadness Level of a package accurately. An upper bound on this quantity can be derived however, and a bid that stays within this bound and the Winning Level is "safe", in the sense that it is not wasted and has the potential to become a winning bid. What is now needed is an incremental procedure for speeding up the computation of this bound.

We rene Gentzen's classical sequent calculus into a classical tableau calculus within the framework of Martin-Lof's type theory. This cal-culus has essentially the same inference gures as Gentzen's classical sequent calculus, but there are two important departures from the lat-ter: there are no structural inference gures except cut, and categories in the sense of Martin-Lof's type theory replace sequents. The classical tableau calculus admits a normalization algorithm similar to the algorithm implicit in Gentzen's Hauptsatz, however con-uent and strongly normalizing. The proof of the latter conforms with Gandy's technique, which in turn relies on the algorithm being weakly normalizing, here proved along the same line of argument as Gentzen's Hauptsatz. 1

Prucnal proved, see [8] and [9], that Medvedev's logic ML of finite problems is structurally complete. So far, ML is the only known structurally complete intermediate logic with the disjunction property. Another proof of Prucnal's theorem was given by D. Skvortsov [10]. We refresh Prucnal's original argument and prove that all structurally complete intermediate logics with the disjunction property coincide on monadic formulas. The logics are also provided with appropriate Kripke frames.

The paper presents an approach to reasoning about Web services in a temporal action theory. Web services are described by specifying their interaction protocols in an action theory based on a dynamic, linear-time, temporal logic. The proposed framework is based on a social approach to agent communication, where the effects of communicative actions allow changes in the social state, and interaction protocols are defined in terms of the creation and fulfillment of commitments and permissions among the agents. We show how to introduce epistemic operators in the action theory to deal with incomplete information, and we address the problem of verifying properties of Web services, as well as the problem of reasoning about the composition of Web services.

For a symmetric monoidal-closed category X and any object K, the category of K-Chu spaces is small-topological over X and small cotopological over X op. Its full subcategory of M-extensive K-Chu spaces is topological over X when X is M-complete, for any morphism class M. Often they may be presented as a full coreflective subcategory of Diers' category of affine K-spaces. Hence, in addition to their roots in the theory of pairs of topological vector spaces (Barr) and in the study of event structures for modeling concurrent processes (Pratt), Chu spaces seem to have a less explored link with algebraic geometry. We use the Zariski closure operator to characterize the self-dual category of M-extensive and M-coextensive K-Chu spaces.

For a symmetric monoidal-closed category X and any object K, the category of K-Chu spaces is small-topological over X and small cotopological over X op. Its full subcategory of M-extensive K-Chu spaces is topological over X when X is M-complete, for any morphism class M. Often they may be presented as a full coreflective subcategory of Diers' category of affine K-spaces. Hence, in addition to their roots in the theory of pairs of topological vector spaces (Barr) and in the study of event structures for modeling concurrent processes (Pratt), Chu spaces seem to have a less explored link with algebraic geometry. We use the Zariski closure operator to characterize the self-dual category of M-extensive and M-coextensive K-Chu spaces.

Steltzner (1996) explains how the major employer in Frege's Jena was Zeiss Optics, with its visionary leader Carl Zeiss and Ernst Abbe, and how Frege was supported all of his life through anonymous donations from this source. Modern logic owes a lot to enlightened industrialists who wanted to give back to society.

Linear Logic, we concisely write LL, has been introduced recently by Jean Yves Girard in Theoretical Computer Science ~0 (1987). Born from the semantics of second order lambda calculus, LL is more expressive than traditional logic (both classical and intuitionistic). Due to the absence of structural rules and to a partict:!ar treatment of negation, which is denoted by ~, proofs in LL do not have a "directional character". The constructive meaning of a proof of A-, B is a function mapping all proofs of A into proofs of B; in LL A-~B has a similar meaning, but B±-oA ± represents the same formula and has the same proofs: so one of such proofs can map a proof of A into one of B as well as a proof of B x into one of.4±~ In this paper we are interested in the multiplicative second order subsystem L,:~* of linear logic; we introduce a calculus (called z-calculus) whose terms are canonical represent: ~ions of proofs. The aim of the calculus is to give a be~ter comprehension of the computational aspects of the process of cut-elimination. We prove that the z-calculus obeys strong normalization and the Church-Rosser properties.

We generalize the notion of symmetries of propositional formulas in conjunctive normal form to modal formulas. Our framework uses the coinductive models introduced in [4] and, hence, the results apply to a wide class of modal logics including, for example, hybrid logics. Our main result shows that the symmetries of a modal formula preserve entailment: if σ is a symmetry of ϕ then ϕ |= ψ if and only if ϕ |= σ (ψ).

I also would like to express my warm thanks to the jury members: Radu MATEESCU and Stefan SCHWOON for accepting to review my thesis, Béatrice BÉRARD and Didier BUCHS for their interest in my work. I am especially grateful to Joël COURTOIS (director of EPITA) and Olivier RICOU (director of LRDE) for giving me the opportunity to pursue my thesis work in the EPITA Research and Development Laboratory (LRDE). I am also grateful to all my colleagues at the LRDE for the excellent and productive working environment, especially, Akim DEMAILLE, Clément DÉMOULINS,

Cahiers de topologie et géométrie différentielle catégoriques, tome 23, n o 4 (1982), p. 363-378 <http://www.numdam.org/item?id=CTGDC_1982__23_4_363_0> © Andrée C. Ehresmann et les auteurs, 1982, tous droits réservés. L'accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). 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/ 363 LIFTING TENSORPRODUCTSA LONG NON-ADJOINT FUNCTORS by Georg GREVE, Jenö SZIGETI and Walter THOLEN CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIF FÉRENTIELLE Vol. XXIII-4 (1982) This note contains an attempt to axiomatize structure properties of non-adjoint functors which still allow canonical liftings of tensorproducts. The type of functors which is adequate for this purpose is described 1. PRESEMITOPOLOGICAL FUNCTORS.

Quantum cryptographic systems have been commercially available, with a striking advantage over classical systems that their security and ability to detect the presence of eavesdropping are provable based on the principles of quantum mechanics. On the other hand, quantum protocol designers may commit more faults than classical protocol designers since human intuition is poorly adapted to the quantum world. To offer formal techniques for modeling and verification of quantum protocols, several quantum extensions of process algebra have been proposed. An important issue in quantum process algebra is to discover a quantum generalization of bisimulation preserved by various process constructs, in particular, parallel composition, where one of the major differences between classical and quantum systems, namely quantum entanglement, is present. Quite a few versions of bisimulation have been defined for quantum processes in the literature, but in the best case they are only proved to be pres...

Let [n] = {1, 2,. .. , n} be a finite chain and let Pn be the semigroup of partial transformations on [n]. Let CPn = {α ∈ Pn : (f or all x, y ∈ Dom α) |xα − yα| ≤ |x − y|}, then CPn is a subsemigroup of Pn. In this paper, we give a necessary and sufficient condition for an element in Pn to be regular and characterize all the Green's equivalences on the semigroup CPn.

We show that all extensions of the (non-associative) Gentzen system for distributive full Lambek calculus by simple structural rules have the cut elimination property. Also, extensions by such rules that do not increase complexity have the finite model property, hence many subvarieties of the variety of distributive residuated lattices have decidable equational theories. For some other extensions we prove the finite embeddability property, which implies the decidability of the universal theory, and we show that our results also apply to generalized bunched implication algebras. Our analysis is conducted in the general setting of residuated frames.

Context Edge graphs are graphs whose edges are labelled with identifiers, and nodes can have multiple edges between them. They are used to model a wide range of systems, including networks with distances or degrees of connection and complex relational data. Inquiry Unfortunately, the homogeneity of this graph structure prevents an effective representation in (functional) programs. Either their interface is riddled with partial functions, or the representations are computationally inefficient to process. Approach We present a novel data type for edge graphs, based on total and recursive definitions, that prevents usage errors from partial APIs and promotes structurally recursive computations. We follow an algebraic approach and provide a set of primitive constructors and combinators, along with equational laws that identify semantically equivalent constructions. Knowledge This algebra translates directly into an implementation using algebraic data types, and its homomorphisms give rise to functions for manipulating and transforming these edge graphs. Grounding We exploit the fact that many common graph algorithms are such homomorphisms to implement them in our framework. Importance In giving a theoretical grounding for the edge graph data type, we can formalise properties such as soundness and completeness of the representation while also minimising usage errors and maximising re-usability.

In the present paper, extensions of the intuitionistic linear logic with knotted structural rules are discussed. Each knotted structural rule is a rule of inference in sequent calculi of the form: from T,A,... ,A (n times)-• C infer Γ,A,... 9 A (k times)-+ C, which is called the (n-• k)-rule. It is a restricted form of the weakening rule when n< k, and of the contraction rule when n> k. Our aim is to explore how they behave like (or unlike) the weakening and contraction rules, from both syntactic and semantic point of view. It is shown that when either n = 1 or k = 1, strong similarities hold between logics with the (n ~* k) rule and logics with the weakening or the contraction rule, as for the cut elimination theorems, decidability and undecidability results and the finite model property.

Adopting the categorial approach to aggregating, interconnecting and constraining probabilistic automata, we start by looking for the notion of morphism that will lead to the characterization of aggregation as product and of interconnection and constraining as cartesian lifting. Having established the suitable notion of morphism between (presentations of countable) probability spaces and shown that such morphisms do not always compose, we are conducted to the new notion of precategory where composition is partial. We sketch the development of the theory of precategories, showing that some results are carried over from the theory of categories and obtaining other useful results for the objectives at hand. Finally, we establish the precategory of (nite) probabilistic automata and show that aggregation of such machines does correspond to a form of constrained product with a clear probabilistic interpretation and that interconnection and constraing do correspond to a cartesian lifting, paving the way for a future development of a (pre)categorial theory of compositional speci cation and veri cation of probabilistic systems.

We propose a measure and a metric on the sets of infinite traces generated by a set of atomic propositions. To compute these quantities, we first map properties to subsets of the real numbers and then take the Lebesgue measure of the resulting sets. We analyze how this measure is computed for Linear Temporal Logic (LTL) formulas. An implementation for computing the measure of bounded LTL properties is provided and explained. This implementation leverages SAT model counting and effects independence checks on subexpressions to compute the measure and metric compositionally.

Brief notes (in Italian) on iterated clique graphs. Given a graph G=(V,E), let K(G) be the intersection graph of the cliques of G. In general K^n(G) = K(K(...K(G))). It is interesting to study the behavior of the cardinality of the set of vertices of K^n(G) as n increases.

Linear logic is a substructural logic proposed as a refinement of classical and intuitionistic logics, with applications in programming languages, game semantics, and quantum physics. We present a template for Gentzen-style linear logic sequents that supports verification of logic inference rules using automatic theorem proving. Specifically, we use the Z3 Theorem Prover [8] to check targeted inference rules based on a set of inference rules that are presumed to be valid. To demonstrate the approach, we apply it to validate several derived inference rules for two different fragments of linear logic: MLL+Mix (Multiplicative Linear Logic extended with a Mix rule) and MILL (Multiplicative Intuitionistic Linear Logic).

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Interactive Scores (IS) is a formalism for composing and performing interactive multimedia scores with several applications in video games, live performance installations, and virtual museums. The composer defines the temporal organization of the score by asserting temporal relations (TRs) between temporal objects (TOs). At execution time, the performer may modify the start/stop times of the TOs by triggering interaction points and the system guarantees that all the TRs are satisfied. Implementations of IS and formal models of their behavior have already been proposed, but these do not provide usable means to reason about their properties. In this paper we introduce ReactiveIS, a programming language that fully captures the temporal structure of IS during both composition and execution. For that, we propose a semantics based on tree-like structures representing the execution state of the score at each point in time. The semantics captures the hierarchical aspects of IS and provides an intuitive representation of their execution. We also endow ReactiveIS with a logical semantics based on linear logic, thus widening the reasoning techniques available for IS. We show that ReactiveIS is general enough to capture the full behavior of IS and it also provides declarative ways to increase the expressivity of IS with, for instance, conditional statements and loops.

We offer a simple graphical representation for proofs of intuitionistic logic, which is inspired by proof nets and interaction nets (two formalisms originating in linear logic). This graphical calculus of proofs inherits good features from each, but is not constrained by them. By the Curry-Howard isomorphism, the representation applies equally to the lambda calculus, offering an alternative diagrammatic representation of functional computations.

This paper presents a formalism for defining higher-order systems based on the notion of graph transformation (by rewriting or interaction). The syntax is inspired by the Combinatory Reduction Systems of Klop. The rewrite rules can be used to define first-order systems, such as graph or term-graph rewriting systems, Lafont's interaction nets, the interaction systems of Asperti and Laneve, the non-deterministic nets of Alexiev, or a process calculus. They can also be used to specify higher-order systems such as hierarchical graphs and proof nets of Linear Logic, or to specify the operational semantics of graph-based languages.

Welcome to FG-2006, the 11th conference on Formal Grammar. This year's conference includes 12 contributed papers covering, as usual, a wide range of areas of formal grammar. In addition to the papers included in this volume, the conference features also two invited talks by .

We present a novel approach to the automatic verification and falsification of LTL requirements of non-linear discrete-time hybrid systems. The verification tool uses an interval-based constraint solver for non-linear robust constraints to compute incrementally refined abstractions. Although the problem is in general undecidable, we prove termination of abstraction refinement based verification and falsification of such properties for the class of non-linear robust discrete-time hybrid systems. We argue, that-in industrial practice-safety critical control applications give rise to hybrid systems that are robust. We give first results on the application of this approach to a variant of an aircraft collision avoidance protocol.