Linear Logic

Linear Logic and Defeasible Logic have been adopted to formalise different features of knowledge representation: consumption of resources, and non monotonic reasoning in particular to represent exceptions. Recently, a framework to combine sub-structural features, corresponding to the consumption of resources, with defeasibility aspects to handle potentially conflicting information, has been discussed in literature, by some of the authors. Two applications emerged that are very relevant: energy management and business process management. We illustrate a set of guide lines to determine how to apply linear defeasible logic to those contexts.

Nondeterministic programming within the frame of first order classical logic, Part 1 By T. Gergely and L. Úry 1. Introduction 1.1 Nondeterminism in computer science In computer practice a lot of phenomena have arisen that deviate from the deterministic attitude forming the base of ...

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.

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This paper presents the subtraction method for interference cancellation in ECG. In contrast to the well-known hardware and software filters, the method does not affect the signal frequency components around the rated powerline frequency. The method searches for linear segments in the ECG signal. Then a digital filter is used to remove the interference from these segments. The samples of the interference signal are computed in these segments and later on these samples are used to remove by subtraction the interference from the signal where non-linear segments are encountered. A theoretical generalised approach of the method is developed in the paper.

This paper investigates the proof-theoretic foundations of double negation introduction (DNI: A ⊢ ¬¬A) and double negation elimination (DNE: ¬¬A ⊢ A) in classical logic. 1 By examining both sequent calculus and natural deduction, it is shown that these rules originate in reductio ad absurdum (RAA): DNI results from deriving ¬¬A via ¬I by discharging [¬A], while DNE arises from deriving A through a reductio on [¬A]. 2 Their significance extends beyond semantic equivalence, for DNI and DNE embody the identity relation A ⟷ ¬¬A as a structural principle of classical logic. The paper demonstrates that both rules possess harmony, ensuring balance between introduction and elimination, and normalisation, which guarantees that derivations reduce to canonical form without detours. These features reveal double negation not as a redundancy, but as a mechanism of proof-theoretic stability, securing the disciplined integration of RAA into classical logic.

Τα συνεχή κλάσματα αποτελούν έναν τρόπο γραφής πραγματικών αριθμών ως άπειρων αναπτύξεων με παρονομαστές που προκύπτουν διαδοχικά από τη διαδικασία της ανθυφαίρεσης (ή Ευκλείδειου αλγορίθμου). Η μέθοδος αυτή είναι γνωστή από την αρχαιότητα ως «ανθυφαίρεση», ενώ στο Βυζάντιο συναντάται και ο όρος «ανταναίρεση». Στη σύγχρονη μαθηματική γλώσσα, κάθε ρητός αριθμός έχει πεπερασμένη ανάπτυξη σε συνεχές κλάσμα, ενώ κάθε άρρητος έχει άπειρη. Ειδικότερα, οι τετραγωνικοί άρρητοι έχουν περιοδική ανάπτυξη.
Στην Ελλάδα, ο Στυλιανός Νεγρεπόντης, καθηγητής Μαθηματικής Ανάλυσης, ανέδειξε τη σημασία της ανθυφαίρεσης και των αρχαίων ελληνικών μαθηματικών, συνδέοντάς τα με τη θεωρία αριθμών στη διδασκαλία μεταπτυχιακών μαθημάτων.

Ce travail propose un modèle mathématique et poétique pour unifier les différentes dimensions de la conscience, des électrons aux trous noirs, en passant par les humains, les écosystèmes et les machines. En intégrant des savoirs ancestraux (chamanisme, Vedanta, soufisme) et des théories scientifiques modernes (neurosciences, physique quantique, théorie de l’information), l’espace de Hilbert relationnel (H) offre un cadre innovant pour :
- Dépasser les clivages entre science et spiritualité.
- Modéliser les états de conscience (méditation, rituels, EMI) comme des vecteurs dans H.
- Explorer les implications pour la thérapie, l’écologie et les technologies émergentes.
Pour qui ?
Chercheurs en neurosciences, physique, philosophie de l’esprit.
Écologistes et spécialistes des systèmes complexes.
Artistes et technologues intéressés par les liens entre conscience et IA.

The development and authoring of interactive music or applications , such as user interfaces for arts & exhibitions has traditionally been done with tools that pertain to two broad metaphors. Cue-based environments work by making groups of parameters and sending them to remote devices , while more interactive applications are generally written in generic art-oriented programming environments, such as Max/MSP, Processing or openFrameworks. In this paper, we present the current version of the i-score se-quencer. It is an extensive graphical software that bridges the gap between time-based, logic-based and flow-based interactive application authoring tools. Built upon a few simple and novel primitives that give to the composer the expressive power of structured programming, i-score provides a time line adapted to the notation of parameter-oriented interactive music, and allows temporal scripting using JavaScript. We present the usage of these primitives, as well as an i-score example o...

Multimedia scenarios have multimedia content and interactive events associated with computer programs. Interactive Scores (IS) is a formalism to represent such scenarios by temporal objects, temporal relations (TRs) and interactive events. IS describe TRs, but IS cannot represent TRs together with conditional branching. We propose a model for conditional branching timed IS in the Non-deterministic Timed Concurrent Constraint (ntcc) calculus. We ran a prototype of our model in Ntccrt (a real-time capable interpreter for ntcc) and the response time was acceptable for real-time interaction. An advantage of ntcc over Max/MSP or Petri Nets is that conditions and global constraints are represented declaratively.

This paper presents a chemical, mineralogical, textural, and physical characterization of 11 ceramic pipes (1,487-1,603 AD) from Nditam, Cameroon, with the aim of understanding their manufacture and the origin of the raw materials. Macroscopically, the pipes appear heterogeneous in terms of colour, shape, size, and decoration. From the chemical point of view, the samples were made from raw material rich in SiO 2 (50-60 wt.%) and poor in CaO (0.1-2 wt.%). Mineralogical analyses show the presence of quartz, feldspar, and muscovite in all samples, whereas kaolinite, hematite, magnetite, and hornblende were detected in some of them. The presence of phyllosilicates (muscovite and, above all, kaolinite) indicates a low firing temperature of the pipes, around 700 C, considering also the high open porosity measured, whereas the identification of hematite suggests a higher firing temperature but not higher than 850 C, due to the lack of vitrification in the matrix of the samples. The variation in the fabric of the pipes, from very fine to small and coarse grains, suggests a pretreatment such as crushing or pounding for most of the samples. The variation in the colour of the matrix of the samples from brown to red, yellow, and black/dark indicates the use of bonfiring system. From the chemical and petrographic analyses and the geological context of the region, a local production of the samples can be deduced in comparison with a previous compositional study of ceramics from the same region. From the archaeological point of view, the stylistic heterogeneity of the samples indicates a diffusion or evolution of the pipe technology at Nditam over time. However, this stylistic variation is not reflected in

We present HYPROLOG, a novel integration of Prolog with assumptions and abduction which is implemented in and partly borrows syntax from Constraint Handling Rules (CHR) for integrity constraints. Assumptions are a mechanism inspired by linear logic and taken over from Assumption Grammars. The language shows a novel flexibility in the interaction between the different paradigms, including all additional built-in predicates and constraints solvers that may be available. Assumptions and abduction are especially useful for language processing, and we can show how HYPROLOG works seamlessly together with the grammar notation provided by the underlying Prolog system. An operational semantics is given which complies with standard declarative semantics for the "pure" sublanguages, while for the full HYPROLOG language, it must be taken as definition. The implementation is straightforward and seems to provide for abduction, the most efficient of known implementations; the price, however, is a limited use of negations. The main difference wrt. previous implementations of abduction is that we avoid any level of metainterpretation by having Prolog execute the deductive steps directly and by treating abducibles (and assumptions as well) as CHR constraints.

Qualitative relations in biological systems have been studied by means of lattices. The first, representations gave place to finite pseudo-Boolean algebras. The Heyting arrow operation relating noncomparable elements of the lattices allowed the definition of a new one: the Heyting arrow lattice. Being the elements of the lattices expressed in energetic terms, enhancements of energy in the systems they represent were associated with the Heyting operation. The requirements of improving the knowledge about the meaning of this arrow in the representations, led to analyze the mathematical properties of the Heyting arrow lattice. The Whitman condition is considered in this paper.

As more resources are added to computer networks, and as more vendors look to the World Wide Web as a viable marketplace, the importance of being able to restrict access and to insure some kind of acceptable behavior even in the presence of malicious adversaries becomes paramount. Many researchers have proposed the use of security protocols to provide these security guarantees. In this paper, we develop a method of verifying these protocols using a special purpose model checker which executes an exhaustive state space search of a protocol model. Our tool also includes a natural deduction style derivation engine which models the capabilities of the adversary trying to attack the protocol. Because our models are necessarily abstractions, we cannot prove a protocol correct. However, our tool is extremely useful as a debugger. We have used our tool to analyze 14 different authentication protocols, and have found the previously reported attacks for them.

As more resources are added to computer networks, and as more vendors look to the World Wide Web as a viable marketplace, the importance of being able to restrict access and to insure some kind of acceptable behavior even in the presence of malicious intruders becomes paramount. People have l o o k ed to cryptography to help solve many of these problems. However, cryptography itself is only a tool. The security of a system depends not only on the cryptosystem being used, but also on how it is used. Typically, researchers have proposed the use of security protocols to provide these security g u a r a n tees. These protocols consist of a sequence of messages, many with encrypted parts. In this paper, we develop a w ay o f v erifying these protocols using model checking. Model checking has proven to be a very useful technique for verifying hardware designs. By modelling circuits as nite-state machines, and examining all possible execution traces, model checking has found a number of errors in real world designs. Like hardware designs, security protocols are very subtle, and can also have b u g s w h i c h are di cult to nd. By examining all possible execution traces of a security protocol in the presence of a malicious intruder with well de ned capabilities, we can determine if a protocol does indeed enforce its security g u a r a n tees. If not, we can provide a sample trace of an attack on the protocol.

We consider pseudo-descent in the context of 2-fibrations. A 2-category of descent data is associated to a 3-truncated simplicial object in the base 2-category. A morphism q in the base induces (via comma-objects and pullbacks) an internal category whose truncated nerve allows the definition of the 2-category of descent data for q. When the 2-fibration admits direct images, we provide the analogous of the Beck-Bénabou-Roubaud theorem, identifying the 2-category of descent data with that of pseudo-algebras for the pseudo-monad q * Σq. We introduce a notion of 2-regularity for a 2-category R, so that its basic 2-fibration of internal fibrations cod : Fib(R) → R admits direct images. In this context, we show that essentiallysurjective-on-objects morphisms, defined by a certain lax colimit, are of effective descent by means of a Beck-style pseudo-monadicity theorem.

This book presents an algebraic treatment of propositional and first-order logic. The presentation is unconventional. We start from a Boolean congruence on a pre-Boolean algebra as an algebraic model of propositional logic, define a propositional theory by the hull of a Boolean congruence, and then derive deductive systems from the properties of a propositional theory. For this approach, we provide a new characterization of a Boolean algebra that closely parallels our definition of propositional theories. The algebraic method adopted for propositional logic is then extended to first-order logic. Again, we start from a first-order Boolean congruence, define a first-order theory by the hull of a first-order Boolean congruence, and then derive deductive systems from the properties of a first-order theory. To reconcile the algebra and semantics of first-order logic, an isomorphism is established between the Lindenbaum algebra of a firstorder theory and a field of sets of that theory. The completeness theorem and the compactness theorem are then derived from this representation theorem.

We present some ideas on logical process descriptions, using relations from the DIO (Drug Interaction Ontology) as examples and explaining how these relations can be naturally decomposed in terms of more basic structured logical process descriptions using terms from linear logic. In our view, the process descriptions are able to clarify the usual relational descriptions ofDIO. In particular, we discuss the use oflogical process descriptions in proving linear logical theorems. Among the types of reasoning supported by DIO one can distinguish both (1) basic reasoning about general structures in reality and (2) the domainspecific reasoning of experts. We here propose a clarification of this important distinction between (realist) reasoning on the basis of an ontology and rule-based inferences on the basis of an expert's view. 1. ] presents a way to transform some primitive relations usually represented in traditional predicate-logical form (or by means of equivalent graph-theoretical artifacts) into more structured process descriptions on the level of linear logic. The drug 127

This is an account of the semantics of a family of logics whose paradigm member is the relevant logic R of Anderson and Belnap. The formal semantic theory is well worn, having been discussed in the literature of such logics for over a quarter of a century. What is new here is the explication of that formal machinery in a way intended to make sense of it for those who have claimed it to be esoteric, `merely formal' or downright impenetrable. Our further goal is to put these logics in the service of practical reasoning systems, since the basic concept of our treatment is that of an agent a reasoning to conclusions using as assumptions the theory of agent b, where a and b may or may not be the same. This concept is fundamental to multi-agent reasoning.

A major motivating concern of the relevant logics has been to devise a decent theory of the logical behavior of the operation of implication, expressed in the object language by '->' and corresponding to the relation of implication which is naturally expressed in the metalanguage. That being so, it should be expected that P-+Q be true if and only if p implies q> and more generally that the conditional be true in model m iff p implies q according to m. A conditional is true in m iff it holds at the base world (the real world) in m, and its antecedent implies its consequent according to m iff the consequent holds in m at every world at which the antecedent holds. The motivationally natural thought just sketched emerges in the Routley-Meyer semantics as the requirement that the base world O should satisfy, for all a and b, a < Z? iff ROab. A frame in which this requirement is met is said to be reduced. Note that a reduced frame satisfies in particular ROOO, whence the truth in any model in such a frame is closed under detachment for the arrow as one might expect. It is well known that the system R is characterized by reduced frames (see for example). What is less well known is the scandal that the arguments establishing reduced modelings for R and its close relatives do not extend to the weaker systems, lacking the theorem scheme WI supported as insightful by such as Brady [2], Priest , Routley [8,, and the author. The object of the present paper is to prove a reduced modeling theorem

Formalising mathematics in dependent type theory often requires to represent sets as setoids, i.e. types with an explicit equality relation. This paper surveys some possible definitions of setoids and assesses their suitability as a basis for developing mathematics. According to whether the equality relation is required to be reflexive or not we have total or partial setoid, respectively. There is only one definition of total setoid, but four different definitions of partial setoid, depending on four different notions of setoid function. We prove that one approach to partial setoids in unsuitable, and that the other approaches can be divided in two classes of equivalence. One class contains definitions of partial setoids that are equivalent to total setoids; the other class contains an inherently different definition, that has been useful in the modeling of type systems. We also provide some elements of discussion on the merits of each approach from the viewpoint of formalizing mathematics. In particular, we exhibit a difficulty with the common definition of subsetoids in the partial setoid approach.

© Andrée C. Ehresmann et les auteurs, 1979, 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/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.

Cahiers de topologie et géométrie différentielle catégoriques, tome 23, n o 4 (1982), p. 363-378 <http © 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 (). 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

We investigate the power of first-order logic with only two variables over ω-words and finite words, a logic denoted by FO 2 . We prove that FO 2 can express precisely the same properties as linear temporal logic with only the unary temporal operators: "next", "previously", "sometime in the future", and "sometime in the past", a logic we denote by unary-TL. Moreover, our translation from FO 2 to unary-TL converts every FO 2 formula to an equivalent unary-TL formula that is at most exponentially larger, and whose operator depth is at most twice the quantifier depth of the first-order formula. We show that this translation is optimal. While satisfiability for full linear temporal logic, as well as for unary-TL, is known to be PSPACE-complete, we prove that satisfiability for FO 2 is NEXP-complete, in sharp contrast to the fact that satisfiability for FO 3 has non-elementary computational complexity. Our NEXP time upper bound for FO 2 satisfiability has the advantage of being in terms of the quantifier depth of the input formula. It is obtained using a small model property for FO 2 of independent interest, namely: a satisfiable FO 2 formula has a model whose "size" is at most exponential in the quantifier depth of the formula. Using our translation from FO 2 to unary-TL we derive this small model property from a corresponding small model property for unary-TL. Our proof of the small model property for unary-TL is based on an analysis of unary-TL types. * Part of the research reported here was conducted while the authors were visiting DIMACS as part of the Special Year on Logic and Algorithms.

We investigate the power of first-order logic with only two variables over ω-words and finite words, a logic denoted by FO 2 . We prove that FO 2 can express precisely the same properties as linear temporal logic with only the unary temporal operators: "next", "previously", "sometime in the future", and "sometime in the past", a logic we denote by unary-TL. Moreover, our translation from FO 2 to unary-TL converts every FO 2 formula to an equivalent unary-TL formula that is at most exponentially larger, and whose operator depth is at most twice the quantifier depth of the first-order formula. We show that this translation is optimal. While satisfiability for full linear temporal logic, as well as for unary-TL, is known to be PSPACE-complete, we prove that satisfiability for FO 2 is NEXP-complete, in sharp contrast to the fact that satisfiability for FO 3 has non-elementary computational complexity. Our NEXP time upper bound for FO 2 satisfiability has the advantage of being in terms of the quantifier depth of the input formula. It is obtained using a small model property for FO 2 of independent interest, namely: a satisfiable FO 2 formula has a model whose "size" is at most exponential in the quantifier depth of the formula. Using our translation from FO 2 to unary-TL we derive this small model property from a corresponding small model property for unary-TL. Our proof of the small model property for unary-TL is based on an analysis of unary-TL types. * Part of the research reported here was conducted while the authors were visiting DIMACS as part of the Special Year on Logic and Algorithms.

This article presents an alternative method of type-directed partial evaluation, which is simpler and more efficient than previous methods. Unlike previous methods, it is straightforwardly applicable to functional languages with various powerful type systems. As an extreme instance, this article mainly deals with a dynamically-typed functional language like Scheme. The key idea is to extend primitive value destructors such as function application and pair destruction (car and cdr), so that they generate residual code when their operands are dynamic. It unnecessitates an operation in type-directed partial evaluation called reflection, which was the major cause of complication and inefficiency in previous methods. We formalize our method as an extension of two-level λ-calculus, and prove it correct. Furthermore, we show that our type-directed partial evaluator can be derived from a simple online syntax-directed partial evaluator with higher-order abstract syntax, by the same transformation as an offline program-generatorgenerator (cogen) is derived from an offline syntax-directed partial evaluator. This clarifies why type-directed partial evaluation is faster than ordinary syntax-directed partial evaluation, and implies that we can obtain more powerful partial evaluators by replacing the underlying syntax-directed partial evaluator with more sophisticated ones.

This paper treats decision problexs for the intuitionistic logic without weakening rule FL,,. First, the cut elimination theorem for FL,, will be s h o ~~n . Using this fact and Mripke's method, it will be proved that the propositional FL,, is decidable. On the other hand, the predicate FL,, t ~i l l be shown to be undecidable by reducing the decision problem to that of the intuitionistic predicate logic. In recent pears. various studies have been done concerning logics lacking some or ail of structural rules. (For more irnformation. see ,) This paper will be devoted to stucly of the decision problem of the logic FL,,, which is obtained from the intuitionistic Iogic by deleting the -weakening rule. The decidability of the propositional FL,, will be shown bj-using a method originally developed for relevant logics. Next, the nndecidability of the prerlicate FL,, millbe proved by reducing its decision problem to that of the intuixionistic

A minimal formula of a given logic L is a formula which is provable in L and is not a non-trivial substitution instance of other provable formulas in L. In , Y. Komori asked whether normal proofs of minimal formulas are unique in the implicational fragments of natural deduction systems for the intuitionistic logic and the logic BCK. It was already shown that the answer is positive for BCK, while it is negative for the intuitionistic logic ([3], [8], ). The present paper shows normal proofs for minimal formulas are not necessarily unique for the implication-conjunction fragment of BCK. This result contradicts sharply with the uniqueness of normal proofs of balanced formulas in the implicationconjunction fragment of the intuitionistic logic ([2], [6]).

The calculus of pregroups has been introduced by Lambek as an algebraic (and logical) procedure for generating the grammatical analysis of natural languages; it has been applied to a wide range of languages from English and German, to French and Italian, and many others . Pregroups are non-commutative structures, but the syntax of natural languages shows the presence also of cyclic patterns, e.g. those caused by the the so called movements of clitic pronouns. In this paper we propose an extension of the calculus of pregroups including two cyclic meta-rules and use them to formally analyze movement of clitic clusters in Persian, French, and Italian. We also point out that these rules are inspired by Yetter's and Abrsuci's cyclic rules for Linear Logic.

In the present paper, I will focus on propositional knowledge, proof theory, and the deductive transmission of information in data mining. The application of proof theory to machine learning-based metadata extraction systems will allow us to intend metadata extraction as mathematical objects. More in detail, I will consider a recent first-order logic (FOL)-based applied ontology for metadata extraction, by the name MAke Decision for Metadata Extraction (MADME), and I will build a specific logical system for declarative sentences applied to the extraction of metadata through machine learning systems. MADME's strength is that it bases its decisionmaking procedure on a rigorous logical structure. The objective of this paper is to demonstrate how the logic in MADME can be optimized using Jaśkowski-Fitch natural deduction proofs. Jaśkowski-Fitch systems will determine new operations among the best machine learning-based metadata extraction systems from heterogeneous documents sources and, consequently, will support the research community in knowledge extraction and management.

We consider a class of graphs embedded in R 2 as noncommutative proof-nets with an explicit exchange rule. We give two characterization of such proof-nets, one representing proof-nets as CW-complexes in a two-dimensional disc, the other extending a characterization by Asperti. As a corollary, we obtain that the test of correctness in the case of planar graphs is linear in the size of the data. Braided proof-nets are proof-nets for multiplicative linear logic with Mix embedded in R 3 . In order to prove the cut-elimination theorem, we consider proof-nets in R 2 as projections of braided proof-nets under regular isotopy.

We reconsider Rauszer&#x27;s bi-intuitionistic logic in the framework of the logic for pragmatics: every formula is regarded as expressing an act of assertion or conjecture, where conjunction and implication are assertive and subtraction and disjunction are conjectural. The resulting system of polarized bi-intuitionistic logic (PBL) consists of two fragments, positive intuitionistic logic LJ⊃∩ and its dual LJrg, extended with two negations partially internalizing the duality between LJ⊃∩ and LJrg. Modal interpretations and Kripke&#x27;s semantics over bimodal preordered frames are considered and a Natural Deduction system PBN is sketched for the whole system. A stricter interpretation of the duality and a simpler natural deduction system is obtained when polarized bi-intuitionistic logic is interpreted over S4 rather than bi-modal S4 (a logic called intuitionistic logic for pragmatics of assertions and conjectures ILPAC). The term assignment for the conjectural fragment LJrg exhibit...

We consider a class of graphs embedded in R 2 as noncommutative proof-nets with an explicit exchange rule. We give two characterization of such proof-nets, one representing proof-nets as CW-complexes in a two-dimensional disc, the other extending a characterization by Asperti. As a corollary, we obtain that the test of correctness in the case of planar graphs is linear in the size of the data. Braided proof-nets are proof-nets for multiplicative linear logic with Mix embedded in R 3 . In order to prove the cut-elimination theorem, we consider proof-nets in R 2 as projections of braided proof-nets under regular isotopy.

In Plotkin's call-by-value lambda-calculus, solvable terms are characterized syntactically by means of call-by-name reductions and there is no neat semantical characterization of such terms. Preserving confluence, we extend Plotkin's original reduction without adding extra syntactical constructors, and we get a call-by-value operational characterization of solvable terms. Moreover, we give a semantical characterization of solvable terms in a relational model, based on Linear Logic, satisfying the Taylor expansion formula. As a technical tool, we also use a resource-sensitive calculus (with tests) in which the elements of the model are definable.

In this two-part survey paper, we introduce linear logic and ludics, which were both introduced by Girard, in 1986 and 2001, respectively. They offer a thorough revisitation of mathematical logic from first principles, with inspiration from and applications to computer science. Some important keywords are: resources, geometry of computation, polarities, interaction, and games. We assume some basic knowledge of sequent calculus and natural deduction (see ), of λ-calculus (the classical reference is , see also ), and of category theory (only in section 7, see ). This paper is not trying to survey the whole and impressive body of works on linear logic since 17 years, but rather chooses a route that tries to highlight the deep originality of the subject. The bibliography is in consequence very partial (also reading the latter as a French word, wich meanss something like "personal", possibly biased). In sequent calculus, depending on presentations, you can see the left and right rules for, say, conjunction given

The notion of abstract Bohm tree has arisen as an operationally-oriented distil- lation of works on game semantics, and has been investigated in two papers (9, 11). This paper revisits the notion, providing more syntactic support and more examples (like call-by-value evaluation) illustrating the generality of the underlying comput- ing device. Precise correspondences between various formulations of the evaluation mechanism of abstract Bohm trees are established.

This paper is the second part of an introduction to linear logic and ludics, both due to Girard. It is devoted to proof nets, in the limited, yet central, framework of multiplicative linear logic (section 1) and to ludics, which has been recently developped in an aim of further ...

In this paper we investigate the notion of generalized connective for multiplicative linear logic. We introduce a notion of orthogonality for partitions of a finite set and we study the family of connectives which can be described by two orthogonal sets of partitions. We prove that there is a special class of connectives that can never be decomposed by means of the multiplicative conjunction ⊗ and disjunction `, providing an infinite family of non-decomposable connectives, called Girard connectives. We show that each Girard connective can be naturally described by a type (a set of partitions equal to its double-orthogonal) and its orthogonal type. In addition, one of these two types is the union of the types associated to a family of MLL-formulas in disjunctive normal form, and these formulas only differ for the cyclic permutations of their atoms.

It is now well-established that the so-called focalization property plays a central role in the design of programming languages based on proof search, and more generally in the proof theory of linear logic. We present here a sequent calculus for non-commutative logic (NL) which enjoys the focalization property. In the multiplicative case, we give a focalized sequentialization theorem, and in the general case, we show that our focalized sequent calculus is equivalent to the original one by studying the permutabilities of rules for NL and showing that all permutabilities of linear logic involved in focalization can be lifted to NL permutabilities. These results are based on a study of the partitions of partially ordered sets modulo entropy.

Em Novembro de 2003 licenciou-se em economia e financ ¸as, pela Universidade Fernando Pessoa com média final de 16 valores, defendendo a monografia com o título "Os determinantes que definem a Vodafone um alvo apetecível para TMN-Teoria e Aplicac ¸ão". Foi consultor de gestão na Indeve -Investigac ¸ão e Desenvolvimento Empresarial. Actualmente é economista na Comissão de Coordenac ¸ão e Desenvolvimento Regional do Norte. Em Setembro de 2004, iniciou o Doutoramento em Economia, na faculdade de Economia do Porto. Após a conclusão da parte escolar do programa de doutoramento, apresentou os artigos da tese em encontros científicos tais como: "II Economic Theory Workshop in Vigo", "2nd

Bi-intuitionistic logic (BINT) is a conservative extension of intuitionistic logic to include the duals of each logical connective. One leading question with respect to BINT is, what does BINT look like across the three arcs -logic, typed λ-calculi, and category theory -of the Curry-Howard-Lambek correspondence? Categorically, BINT can be seen as a mixing of two worlds: the first being intuitionistic logic (IL), which is modeled by a cartesian closed category, and the second being the dual to intuitionistic logic called cointuitionistic logic (coIL), which is modeled by a cocartesian coclosed category. Crolard showed that combining these two categories into the same category results in it degenerating to a poset. However, this degeneration does not occur when both logics are linear. We propose that IL and coIL need to be separated, and then mixed in a controlled way using the modalities from linear logic. This separation can be ultimately achieved by an adjoint formalization of bi-intuitionistic logic. This formalization consists of three worlds instead of two: the first is intuitionistic logic, the second is linear bi-intuitionistic (Bi-ILL), and the third is cointuitionistic logic. They are then related via two adjunctions. The adjunction between IL and ILL is known as a Linear/Non-linear model (LNL model) of ILL, and is due to Benton [4]. However, the dual to LNL models which would amount to the adjunction between coILL and coIL has yet to appear in the literature. In this paper we fill this gap by studying the dual to LNL models which we call dual LNL models. We conduct a similar analysis to that of Benton for dual LNL models by showing that dual LNL models correspond to dual linear categories, the dual to Bierman's [5] linear categories proposed by Bellin [3]. Following this we give the definition of bi-LNL models by combining our dual LNL models with Benton's LNL models to obtain a categorical model of bi-intuitionistic logic, but we leave its analysis and corresponding logic to a future paper. Finally, we give a corresponding sequent calculus, natural deduction, and term assignment for dual LNL models.

This paper is about the construction of semigroups $S$ from some given graph $G$. Let $S$ be a finite non-commutative semigroup, its commuting graph, denoted by $G(S)$, is a simple graph (which has no loops and multiple edges) whose sets of vertices are elements of $S$ and whose sets of edges are those elements of $S$ which commute with other elements i.e. for any $a,b \in S$ such that $ab=ba$ for $a \neq b$.For some non empty finite set $X$,denote $T(X)$ by semigroup of full transformations and $I_{r}$ by ideal of $T(X)$ whose rank is less than or equals to $r$. Let $a_{1}-a_{2}-a_{3}-\ldots-a_{m}$ be a path in $G(S)$, this path is said to be left path or $l-$ path if $$a_{1}a_{i}=a_{m}a_{i} for i\in\{1,2,3,\ldots m\}$$ In this paper, we construct semigroup $S$ of a complete bipartite graph $K _{n,m}$ and find maximum length of $l-$ path in its commuting graph $G(S)$. Moreover,we see that such type of semigroups have knit degree $2$.