Linear Logic

In this paper we study a version of constructive linear-time temporal logic (LTL) with the "next" temporal operator. The logic is originally due to Davies, who has shown that the proof system of the logic corresponds to a type system for binding-time analysis via the Curry-Howard isomorphism. However, he did not investigate the logic itself in detail; he has proved only that the logic augmented with negation and classical reasoning is equivalent to (the "next" fragment of) the standard formulation of classical lineartime temporal logic. We give natural deduction, sequent calculus and Hilbert-style proof systems for constructive LTL with conjunction, disjunction and falsehood, and show that the sequent calculus enjoys cut elimination. Moreover, we also consider Kripke semantics and prove soundness and completeness. One distinguishing feature of this logic is that distributivity of the "next" operator over disjunction " (A ∨ B) ⊃ A ∨ B" is rejected in view of a type-theoretic interpretation.

There are currently dozens of freely available tools to help combat phishing and other web-based scams. Many of these tools come in the form of web browser extensions that warn users when they are browsing a suspected phishing site. We used verified phishing URLs and legitimate URLs to test the effectiveness of 10 popular antiphishing toolbars. Overall, we found that the anti-phishing toolbars that were examined in this study left a lot to be desired. SpoofGuard did a very good job at identifying fraudulent sites, but it also incorrectly identified a large fraction of legitimate sites as fraudulent. EarthLink, Google, Netcraft, Cloudmark, and Internet Explorer 7 identified most fraudulent sites correctly and had few, if any, false positives, but they still missed more than 15% of fraudulent sites. The TrustWatch, eBay, and Netscape 8 toolbars could correctly identify less than half the fraudulent sites, and McAfee SiteAdvisor did not correctly identify any fraudulent sites. Many of the toolbars we tested were vulnerable to some simple exploits as well. In this paper we describe the anti-phishing toolbar test bed we developed, summarize our findings, and offer observations about the usability and overall effectiveness of these toolbars. Finally, we suggest ways to improve anti-phishing toolbars.

We develop a logic for reasoning about adjacency and separation of memory blocks, as well as aliasing of pointers. We provide a memory model for our logic and present a sound set of natural deduction-style inference rules. We deploy the logic in a simple type system for a stack-based assembly language. The connectives for the logic provide a flexible yet concise mechanism for controlling allocation, deallocation and access to both heap-allocated and stack-allocated data.

Why did it take us 50 years since the birth of the quantum mechanical formalism to discover that unknown quantum states cannot be cloned? Yet, the proof of the 'no-cloning theorem' is easy, and its consequences and potential for applications are immense. Similarly, why did it take us 60 years to discover the conceptually intriguing and easily derivable physical phenomenon of 'quantum teleportation' ? We claim that the quantum mechanical formalism doesn't support our intuition, nor does it elucidate the key concepts that govern the behaviour of the entities that are subject to the laws of quantum physics. The arrays of complex numbers are kin to the arrays of 0s and 1s of the early days of computer programming practice. Using a technical term from computer science, the quantum mechanical formalism is 'low-level'. In this review we present steps towards a diagrammatic 'high-level' alternative for the Hilbert space formalism, one which appeals to our intuition. The diagrammatic language as it currently stands allows for intuitive reasoning about interacting quantum systems, and trivialises many otherwise involved and tedious computations. It clearly exposes limitations such as the no-cloning theorem, and phenomena such as quantum teleportation. As a logic, it supports 'automation': it enables a (classical) computer to reason about interacting quantum systems, prove theorems, and design protocols. It allows for a wider variety of underlying theories, and can be easily modified, having the potential to provide the required step-stone towards a deeper conceptual understanding of quantum theory, as well as its unification with other physical theories. Specific applications discussed here are purely diagrammatic proofs of several quantum computational schemes, as well as an analysis of the structural origin of quantum non-locality. The underlying mathematical foundation of this high-level diagrammatic formalism relies on so-called monoidal categories, a product of a fairly recent development in mathematics. Its logical underpinning is linear logic, an even more recent product of research in logic and computer science. These monoidal categories do not only provide a natural foundation for physical theories, but also for proof theory, logic, programming languages, biology, cooking, ... So the challenge is to discover the necessary additional pieces of structure that allow us to predict genuine quantum phenomena. These additional pieces of structure represent the capabilities nature has provided us with to manipulate entities subject to the laws of quantum theory.

Is a scientific explanation of free will possible? Biologically based on the ability to generate alternative pattern of actions and non-predictive behaviors. Emerging with articulated language (being able to say NO, OR, IF... THEN) and symbolic consciousness in the Paleolithic period (the birth of art and symbolic detachment). Coevolving with the production of ever more complex symbolic environments worked by recursive consciousness and symbolic causality. It is embodied in the great scientific figures, reflecting their free will in it methodical testing, in physics, in biology, in neuroscience, in paleoanthropology. To deny free will, like Einstein or Libet, is to reaffirm it more. The phenosystems, an extension of the concept of ecosystem in the cultural sphere (such as "quantum physics," "contemporary music" or "strategy games"), are the milieus of existence and exercise of free will in the modern sense, which is not a pure act of the mind but is embodied in the action and in the management of symbolic resources, formalized by linear logic.

Keywords : emergence, coevolution, exaptation, neuronal recycling, symbolic consciousness, recursive consciousness, parietal art, laws and constraints, epistemic cut, entanglement, conscious content, sensitive correlate, semantic externalism, spread mind, symbolic causality, circular causality, spiral causality, integral experience, illusionism, exaptation, dynamic coupling, teleodynamics, semiodynamics, dynamic depth, enaction, cogeneration, generative passage, exosomatic tool, gregorian creature, subjective ontology, syntactic recursion, semantic recursion, linear logic, neurophenomenology, heterophenomenology, theoretical freedom, experimental freedom, critical freedom, non-locality, phenosystem, pheno-symbolic world, The three worlds, chaosmos.

Une explication scientifique du libre arbitre est-elle possible ? Fondée biologiquement sur la capacité de générer des modèles d’actions alternatifs et des comportements non prédictibles. Emergeant avec le langage articulé (pouvoir dire NON, OU, SI… ALORS) et la conscience symbolique au Paléolithique (la naissance de l’art et le détachement symbolique). Coévoluant avec la production d’environnements symboliques toujours plus complexes travaillés par la conscience récursive et la causalité symbolique. Il s’incarne dans les grandes figures scientifiques, réfléchissant leur libre arbitre dans sa mise en épreuve méthodique, en physique, en biologie, dans les neurosciences, en paléoanthropologie. Nier le libre arbitre, comme Einstein ou Libet, revient à le réaffirmer encore davantage. Les phénosystèmes, extension du concept d’écosystème à la sphère culturelle, (comme « la physique quantique », « la musique contemporaine » ou « les jeux de stratégies »), sont les milieux d’existence et d’exercice du libre arbitre au sens moderne, qui n’est pas un acte pur de l’esprit mais s’incarne dans l’action et dans la gestion des ressources symboliques, formalisée par la logique linéaire. 

Mots-clés : émergence, coévolution, exaptation, recyclage neuronal, conscience symbolique, conscience récursive, art pariétal, lois et contraintes, coupure épistémique, intrication, contenu conscient, corrélat sensible, externalisme sémantique, esprit étalé, causalité symbolique, causalité circulaire, causalité spiralaire, expérience intégrale, illusionnisme, couplage dynamique, téléodynamique, sémiodynamique, profondeur dynamique, énaction, cogénération, passage génératif, outil exosomatique, créature grégorienne, ontologie subjective, récursion syntaxique, récursion sémantique, logique linéaire, neurophénoménologie, hétérophénoménologie, liberté théorique, liberté expérimentale, liberté critique, non localité, phénosystème, monde phéno-symbolique, les trois mondes, le chaosmos.

Albert Dechambre, mai 2021 – [email protected]

Welcome to the proceedings of the 14th WoLLIC meeting, which was held in Rio de Janeiro, Brazil, July 2 -5, 2007. The Workshop on Logic, Language, Information and Computation (WoLLIC) is an annual international forum on inter-disciplinary research involving formal logic, computing and programming theory, and natural language and reasoning. The WoLLIC meetings alternate between Brazil (and Latin America) and other countries, with the aim of fostering interest in applied logic among Latin American scientists and students, and facilitating their interaction with the international applied logic community.

Η μαθηματική γλωσσολογία είναι το πεδίο των μαθηματικών, όπου μελετώνται γλωσσικά φαινόμενα και οι σχέσεις μεταξύ τους ως αντικείμενα μαθηματικών θεωριών. Στην παρούσα εργασία, παρουσιάζονται οι κατηγοριακές γραμματικές ξεκινώντας από τη Θεωρία Κατηγοριών. Αρχικά, αναπτύσσουμε τις βασικές έννοιες της Θεωρίας Κατηγοριών, όπως αυτές της κατηγορίας, του συναρτητή και του φυσικού μετασχηματισμού. Έπειτα, εισερχόμαστε στον κόσμο των κατηγοριακών γραμματικών και, πιο συγκεκριμένα, παρουσιάζουμε τη γραμματική του Αϊντουκιέβιτς, τη γραμματική Αϊντουκιέβιτς-Μπαρ-Χιλλέλ (AB), τόσο το προσεταιριστικό όσο και το μη προσεταιριστικό Λογισμό Λάμπεκ (L και NL) και το Λογισμό Λάμπεκ- Φαν Μπένθεμ (LP). Αναφέρουμε επίσης και τη δύναμη των κατηγοριακών γραμματικών εξετάζοντας τη σχέση τους με τις ασυμφραστικές γραμματικές.
Στη συνέχεια, διατυπώνονται οι κατηγοριακές γραμματικές στο πλαίσιο της γραμμικής λογικής και οδηγούμαστε στα δίκτυα αποδείξεων. Τέλος, αφιερώνεται ένα κεφάλαιο αποκλειστικά στην εφαρμογή των κατηγοριακών γραμματικών στην ελληνική γλώσσα. Κύριος στόχος της εργασίας είναι η κατάδειξη της ευδοκίμησης των κατηγοριακών γραμματικών στην ελληνική.

Residuation is a fundamental concept of ordered structures and categories. In this survey we consider the consequences of adding a residuated monoid operation to lattices. The resulting residuated lattices have been studied in several branches of mathematics, including the areas of lattice-ordered groups, ideal lattices of rings, linear logic and multi-valued logic. Our exposition aims to cover basic results and current developments, concentrating on the algebraic structure, the lattice of varieties, and decidability.

Humanity is on the threshold of recognizing the fundamental error in its view of life and death. Both death as well as active life is necessary to the vital formation of a larger, more essential whole. In this paper, I apply the sociology of knowledge and change as it pertains to death and focus on ways in which we can step outside its traditional frameworks and limitations. I also discuss topics related to death such as birth, aging, sickness, and war, and examine cultural differences in attitudes toward death. I offer varying perspectives including the Buddhist view and from these draw implications and conclusions. I apply the lenses of contemporary social scientists such as Edgar Morin, Kenneth Gergen, Edward Stewart, Milton Bennett, Mary Catherine Bateson, E. Doyle McCarthy, Philip Slater, and Piotr Sztompka. To these I add other relevant passages from the writings and speeches of key thinkers on the topic of death, in particular, Buddhist philosopher, peacebuilder, and educator, Daisaku Ikeda. To construct a more humanistic and sustainable view of life, it is first of all crucial to establish a culture which perceives death in its larger living context as but one cycle in the expansive eternity of life.

Abstract
Free will exists and the proof is simple: without symbolic freedom, no symbolic resources and, consequently, no free will. The existence of symbolic resources presupposes symbolic freedom, and this was only possible by the emergence of symbolic consciousness during the cultural evolution of Sapiens that triggered such a liberation. In return, symbolic resources nourish and amplify this symbolic freedom. As symbolic resources are indubitable and the milieu of exercise and ordeal of free will, its existence is beyond doubt even if its scope is still limited.
I just said that Free Will exist, but not in the absolute or naïve forms that the history of ideas established: a gift from God to man (Saint Augustine), a poisoned gift to punish him (Nietzsche), a pure act of the mind that would be a kind of unmoved mover, an instant consciousness that Libet's experiments in neuroscience could have adopt, at the beginning, so it could figure in an experimental protocol in a laboratory.       
Free will is the biological capacity of a particular physical system, Sapiens, to make conscious choices, drawing on symbolic resources that he has elaborate since the Upper Paleolithic. Selected, along with other species, as a "choice machine", a generator of diversity and unpredictibility and a Bayesian machine, like surprising a predator by suddenly changing its trajectory or choosing a shelter, this ability has become conscious through the combination of two types of innovations: the development of phenomenal consciousness - increased neuronal plasticity, new cognitive functions such as articulated language - and cultural evolution, which we can subsume under a single concept: the emergence of symbolic consciousness. The first type of decisive innovation is extremely difficult to study experimentally but is based on studies debated in the field of evolutionary biology, paleoanthropology and linguistics. But the second equally decisive innovation is perfectly clear since we can study cultural evolution and symbolic consciousness from our daily experience and written data, from parietal writing to articulated writing. I discuss the emergence of symbolic consciousness in my essay Le concept scientifique du libre arbitre – Emergence et coévolution (The Scientific Concept of Free Will – Emergence and coevolution).
I will start by fixing the epistemological framework (1) and the definition of Free Will in the biological and symbolic sense (2). I will present "Einstein's axiom" of symbolic freedom (3) and James’s axiom of conscious will (4). I will define the concepts of “symbolic consciousness”, "symbolic freedom" and "symbolic resource" (5). With these tools, I will establish the proof that is immediate by sketching a formalization using linear logic (6).
The proof (6.2)
All skeptical arguments are invalid, either they do not talk about the subject but about outdated or disembodied concepts of free will, or they rely on outdated conceptions of science such as the model of the manifest image and scientific image of Wilfrid Sellars or Arthur Eddington, or they distort scientific thinking, or, finally, they actually presuppose the existence of their free will in an attempt to refute it.
This brief essay is both an introduction and an outcome to the four essays I have devoted to the Scientific Concept of Free Will. In addition to the one mentioned, two essays delve into the interpretation of quantum physics where the freedom of the physicist was exercised with extraordinary acuity, a Free Will in action in the face of the conceptual upheavals produced by the quantum revolution. The last delves into the little-explored field of biosemiotics, the emergence of symbolic consciousness that accompanies biosemiosis at every stage of its development and its overtaking. 
La liberté du physicien – Complément au Concept scientifique du libre arbitre (Physicist Freedom - Complement to the Scientific Concept of Free Will)
La conscience symbolique de Erwin Schrödinger – Complément au Concept scientifique du libre arbitre (en cours d’écriture) (The Symbolic Consciousness of Erwin Schrödinger - Complement to the Scientific Concept of Free Will)
De la biosémiose primaire à la liberté sémiotique - Emergence de la conscience symbolique (From Primary Biosemiosis to Semiotic Freedom – Emergence of Symbolic Consciousness)
***
Résumé
Le libre arbitre existe et la preuve est simple : sans liberté symbolique, pas de ressources symboliques et, par voie de conséquence, pas de libre arbitre. L’existence de ressources symboliques présuppose la liberté symbolique et celle-ci n’a été possible que par l’émergence de la conscience symbolique au cours de l’évolution culturelle de Sapiens qui a déclenché une telle libération. En retour, les ressources symboliques alimentent et amplifient cette liberté symbolique. Les ressources symboliques étant indubitables et le milieu d’exercice et d’épreuve du libre arbitre, son existence ne fait aucun doute même si sa portée est encore limitée.
Le libre arbitre existe, ai-je affirmé, mais pas sous les formes absolues ou naïves que l’histoire des idées a pu établir : un don de Dieu à l’homme (Saint Augustin), cadeau empoisonné pour pouvoir le punir (Nietzsche), un acte pur de l’esprit qui serait une espèce de moteur non mû, une conscience instantanée que les expériences de Libet en neurosciences ont pu adopter au début pour qu’il puisse figurer dans un protocole expérimental dans un laboratoire .       
Le libre arbitre est la capacité biologique d’un système physique particulier, Sapiens, de faire des choix conscients, en s’appuyant sur des ressources symboliques qu’il a élaborées depuis le Paléolithique supérieur. Sélectionnée, avec d’autres espèces, comme une « machine à faire des choix », un générateur de diversité et d’imprédictibilité, comme surprendre un prédateur en changeant subitement de trajectoire ou choisir un abri, cette capacité est devenue consciente par la combinaison de deux types d’innovations : le développement de la conscience phénoménale - une plasticité neuronale accrue, de nouvelles fonctions cognitives comme le langage articulé - et l’évolution culturelle, que nous pouvons subsumer sous un seul concept : l’émergence de la conscience symbolique. Le premier type d’innovation déterminante est extrêmement difficile à étudier expérimentalement mais repose sur des études débattues dans le champ de la biologie évolutionniste, de la paléoanthropologie et de la linguistique. Mais la seconde innovation tout aussi déterminante est parfaitement claire puisque nous pouvons étudier l’évolution culturelle et la conscience symbolique en partant de notre expérience quotidienne et de données écrites, de l’écriture pariétale à l’écriture articulée. Je traite l’émergence de la conscience symbolique dans mon essai  Le concept scientifique du libre arbitre – Emergence et coévolution, où tous les éléments de la preuve sont exposés, mais demandaient à être réunis ici de manière explicite.
Je commencerai par fixer le cadre épistémologique (1). Je présenterai « l’axiome d’Einstein » de la liberté symbolique  (2) et l’axiome de James de la volonté consciente (3). Je définirai les concepts de « conscience symbolique », de « liberté symbolique » et de « ressource symbolique » (5). Avec ces outils, j’établirai la preuve qui est immédiate en esquissant une formalisation au moyen de la logique linéaire (6).
En annexes des extraits de mon essai : La causalité symbolique et la conscience symbolique (résumé du chapitre 4) et Les critiques des expériences de Libet sur le libre arbitre (extrait du chapitre 13) qui illustrent l’exercice du libre arbitre en acte dans la vie quotidienne et dans les neurosciences.
La preuve (6)
Tous les arguments sceptiques sont invalides, soit qu’ils ne parlent pas du sujet mais de concepts du libre arbitre désuets ou désincarnés, soit qu’ils s’appuient sur des conceptions de la science dépassées comme le modèle de l’image manifeste et de l’image scientifique de Wilfrid Sellars ou Arthur Eddington, soit qu’ils dénaturent la pensée scientifique, soit enfin, c’est le cas le plus fréquent, qu’ils présupposent en réalité l’existence de leur libre arbitre pour tenter de le réfuter.
Ce bref essai est à la fois une introduction et un aboutissement des quatre essais que j’ai consacrés au concept scientifique du libre arbitre. Outre celui qui a été cité, deux essais plongent dans l’interprétation de la physique quantique où la liberté du physicien s’est exercée avec une acuité extraordinaire, un libre arbitre en acte face aux bouleversements conceptuels produits par la révolution quantique. Le dernier plonge dans le champ peu exploré de la biosémiotique, de l’émergence de la conscience symbolique qui accompagne la biosémiose à chacune des étapes de son développement et de son dépassement. 
La liberté du physicien - Complément au Concept scientifique du libre arbitre (Physicist Freedom - Complement to the Scientific Concept of Free Will)
La conscience symbolique de Erwin Schrödinger – Complément au Concept scientifique du libre arbitre (en cours d’écriture) (The Symbolic Consciousness of Erwin Schrödinger - Complement to the Scientific Concept of Free Will)
De la biosémiose primaire à la liberté sémiotique - Emergence de la conscience symbolique (From Primary Biosemiosis to Semiotic Freedom – Emergence of Symbolic Consciousness)

In the Tractatus it is stated that questions about logical formatting ("why just these rules?") cannot be meaningfully formulated, since it is precisely the application of logical rules which enables the formulation of a question whatsoever; analogously, Wittgenstein's celebrated infinite regress argument on rule-following seems to undermine any explanation of deduction, as relying on a logical argument.

In this paper, I discuss the analysis of logic in the pragmatic approach recently proposed by Brandom. I consider different consequence relations, formalized by classical, intuitionistic and linear logic, and I will argue that the formal theory developed by Brandom, even if provides powerful foundational insights on the relationship between logic and discursive practices, cannot account for important reasoning patterns represented by non-monotonic or resource-sensitive inferences. Then, I will present an incompatibility semantics in the framework of linear logic which allow to refine Brandom’s concept of defeasible inference and to account for those non-monotonic and relevant inferences that are expressible in linear logic. Moreover, I will suggest an interpretation of discursive practices based on an abstract notion of agreement on what counts as a reason which is deeply connected with linear logic semantics.

If we go through a list of some of the main problematiques 1 that are defining the new Century, such as water, forced migrations, poverty, environmental crises, violence, terrorism, neo-imperialism, destruction of social fabric, we must conclude that none of them can be ...

We discuss paradoxes of implication in the setting of a proof-conditional theory of meaning for logical constants. We argue that a proper logic of implication should be not only relevant, but also constructive and nonmonotonic. This leads us to select as a plausible candidate LL, a fragment of linear logic which di¤ers from R in that it rejects both contraction and distribution.

Message passing is a key ingredient of concurrent programming. The purpose of this paper is to describe the equivalence between the proof theory, the categorical semantics, and term calculus of message passing. In order to achieve this we introduce the categorical notion of a linear actegory and the related polycategorical notion of a poly-actegory. Not surprisingly the notation used for the term calculus borrows heavily from the (synchronous) pi-calculus. The cut elimination procedure for the system provides an operational semantics.

We define proof nets for cyclic multiplicative linear logic as edge bi-coloured graphs. Our characterization is purely graph theoretical and works without further complication for proof nets with cuts, which are usually harder to handle in the non-commutative case. This also provides a new characterization of the proof nets for the Lambek calculus (with the empty sequence) which simply are a restriction on the formulas to be considered (which are asked to be intuitionistic).

Why did it take us 50 years since the birth of the quantum mechanical formalism to discover that unknown quantum states cannot be cloned? Yet, the proof of the 'no-cloning theorem'is easy, and its consequences and potential for applications are immense. Similarly, why did it take us 60 years to discover the conceptually intriguing and easily derivable physical phenomenon of 'quantum teleportation'? We claim that the quantum mechanical formalism doesn't support our intuition, nor does it elucidate the key concepts that govern the ...

We present the general theory of the method of glueing and associated technique of orthogonality for constructing categorical models of all the structure of linear logic: in particular we treat the exponentials in detail. We indicate simple applications of the methods and show that they cover familiar examples.

La geometria dell'interazione (GoI) nasce alla fine degli anni '80 come riformulazione geometrica del λ-calcolo, attraverso una profonda riflessione matematica sulla nozione di linearità in logica. A più di vent'anni dalla sua comparsa, si può dire che il più ambizioso risultato di GoI sia quello di aver presentato una morfologia radicalmente "monista" per la logica, ossia ben lontana dall'abituale dualismo tra sistemi formali (la sintassi) e modelli (la semantica). Piuttosto, questa morfologia, incentrata sulla dualità tra prova e test, sembra fornire condizioni di possibilità puramente logiche per una tale, ormai paradigmatica, distinzione. In questa breve presentazione proverò a introdurre alcuni aspetti caratteristici e alcune motivazioni di una tale "svolta" geometrica.

In this paper I disuss logic in the pragmatic apporach of (Brandom, 2008). I consider differnt logical consequence relations (classical, intuitionistic and linear) and I will argue that the formal treatment proposed by Brandom, even if I believe it provides powerful intuions and an interesting framework on logic in general, doesn't allow to state properly the relationship between differnt logics. I propose an alternative account of the elaboration of logical vocabuilaries not based on incompatibility semantics, rather on a particular notion of interaction, which I claim is implicit in the practice of giving and asking for reasons, which allows to state the relationship between differnt logics in terms of differnt aspects of the

Affine Logic (EAL , LAL resp.) has been studied for two different reasons: on the one hand the evaluation of typed terms using LAL (EAL resp.) proof-nets admits a guaranteed polynomial (elementary, resp.) bound; on the other hand these terms can also be evaluated by optimal reduction using the abstract version of Lamping's algorithm. The first reduction is global while the second one is local and asynchronous. We prove that for LAL (EAL resp.) typed terms, Lamping's abstract algorithm also admits a polynomial (elementary, resp.) bound. We also show its soundness and completeness (for EAL and LAL with type fixpoints), by using a simple geometry of interaction model (context semantics).

A Chu space is a binary relation =| from a set A to an antiset X defined as a set which transforms via converse functions. Chu spaces admit a great many interpretations by virtue of realizing all small concrete categories and most large ones arising in mathematical and computational practice. Of particular interest for computer science is their interpretation as computational processes, which takes A to be a schedule of events distributed in time, X to be an automaton of states forming an information system in the sense of Scott, and the pairs (a, x) in the =| relation to be the individual transcriptions of the making of history. The traditional homogeneous binary relations of transition on X and precedence on A are recovered as respectively the right and left residuals of the heterogeneous binary relation =| with itself. The natural algebra of Chu spaces is that of linear logic, made a process algebra by the process interpretation.

A long standing problem in logic programming is how to impose directionality on programs in a safe fashion. The benefits of directionality include freedom from explicit sequential control, the ability to reason about algorithmic properties of programs (such as termination, complexity and deadlock-freedom) and controlling concurrency. By using Girard's linear logic, we are able to devise a type system that combines types and modes into a unified framework, and enables one to express directionality declaratively. The rich power of the type system allows outputs to be embedded in inputs and vice versa. Type checking guarantees that values have unique producers, but multiple consumers are still possible. From a theoretical point of view, this work provides a “logic programming interpretation” of (the proofs of) linear logic, adding to the concurrency and functional programming interpretations that are already known. It also brings logic programming into the broader world of typed languages and types-as-propositions paradigm, enriching it with static scoping and higher-order features.

The solution of games is a key decision problem in the context of verification of open systems and program synthesis. We present an automata-theoretic approach to solve timed games. Our solution gives a general framework to solve many classes of timed games via a translation to tree automata, extending to timed games a successful approach to solve discrete games. Our approach relies on translating a timed automaton into a tree automaton that accepts all the trees corresponding to a given strategy of the protagonist. This construction exploits the region automaton introduced by Alur and Dill. We use our framework to solve timed Büchi games in exponential time, timed Rabin games in exponential time, Ctl games in exponential time and Ltl games in doubly exponential time. All these results are tight in the sense that they match the known lower bounds on these decision problems.

In this paper, we propose a method for automated web service composition by applying Linear Logic (LL) theorem proving. We distinguish value-added web services and core service by assuming that the core ser- vice is already selected by the user, but its functionality does not completely match the user's requirement. Our method enables au- tomation for combining the core service

We introduce the Danos–Régnier category DR(M) of a linear inverse monoid M, as a categorical description of geometries of interaction (GOI) inspired from the weight algebra. The natural setting for GOI is that of a so-called weakly Cantorian linear inverse monoid, in which case DR(M) is a kind of symmetrized version of the classical Abramsky–Haghverdi–Scott construction of a weak linear category from a GOI situation. It is well-known that GOI is perfectly suited to describe the multiplicative fragment of linear logic, and indeed DR(M) will be a ∗-autonomous category in this case. It is also well-known that the categorical interpretation of the other linear connectives conflicts with GOI interpretations. We make this precise, and show that DR(M) has no terminal object, no Cartesian product of any two objects, and no exponential—whatever M is, unless M is trivial. However, a form of coherence completion of DR(M) à la Hu–Joyal (which for additives resembles a layered approach à la Hugh...

We introduce a number of logics to reason about collective propositional attitudes that are defined by means of the majority rule. It is well known that majoritarian aggregation is subject to irrationality, as the results in social choice theory and judgment aggregation show. The proposed logics for modelling collective attitudes are based on a substructural propositional logic that allows for circumventing inconsistent outcomes. Individual and collective propositional attitudes, such as beliefs, desires, obligations, are then modelled by means of minimal modalities to ensure a number of basic principles. In this way, a viable consistent modelling of collective attitudes is obtained.

This work contributes to the theory of judgment aggregation by discussing a number of significant non-classical logics. After adapting the standard framework of judgment aggregation to cope with non-classical logics, we discuss in particular results for the case of Intuitionistic Logic, the Lambek calculus, Linear Logic and Relevant Logics. The motivation for studying judgment aggregation in non-classical logics is that they offer a number of modelling choices to represent agents' reasoning in aggregation problems. By studying judgment aggregation in logics that are weaker than classical logic, we investigate whether some well-known impossibility results, that were tailored for classical logic, still apply to those weak systems.

A Chu space is a binary relation between two sets. In this thesis we show that Chuspaces form a non-interleaving model of concurrency which extends event structureswhile endowing them with an algebraic structure whose natural logic is linear logic.We provide several equivalent definitions of Chu spaces, including two pictorialrepresentations. Chu spaces represent processes as automata or schedules, and Chuduality gives a simple way of converting between schedules and automata. We showthat ...

We wish to thank all the authors who submitted papers and demonstrations to this workshop, the members of the programme committee, the invited speaker, and the CP-2005 Tutorial and Workshop Chairs, Alan Frisch and Ian Miguel. We would like to especially acknowledge the financial sponsorship received from

We present a game-based semantic framework into which the time complexity of any IMELL proof can be read out of its interpretation. This gives a compositional view of the geometry of interaction framework introduced by the first author. In our model the time measure is given by means of slots, as introduced by Ghica in a recent paper. The cost associated to a strategy is polynomially related to the normalization time of the interpreted proof, in the style of a complexity-theoretical full abstraction result.

Quantum logic is static, describing automata having uncertain states but no state transitions and no Heisenberg uncertainty tradeoff. We cast Girard's linear logic in the role of a dynamic quantum logic, regarded as an extension of quantum logic with time nonstandardly interpreted over a domain of linear automata and their dual linear schedules. In this extension the uncertainty tradeoff emerges via the "structure veil." When VLSI shrinks to where quantum effects are felt, their computer-aided design systems may benefit from such logics of computational behavior having a strong connection to quantum mechanics.

Although 'glue semantics' is the most extensively developed theory of semantic composition for LFG, it is not very well integrated into the LFG projection architecture, due to the absence of a simple and well-explained correspondence between glue-proofs and f-structures. In this paper I will show that we can improve this situation with two steps: (1) Replace the current quantificational formulations of glue (either Girard's system F, or first order linear logic) with strictly propositional linear logic (the quantifier, unit and exponential free version of either MILL or ILL, depending on whether or not tensors are used). (2) Reverse the direction of the standard r-projection from f-structure to meaning, giving one going from the (atomic nodes of) the glue-proof to the f-structure, rather than from the f-structure to a 'semantic projection' which is itself somehow related to the glue-proof. As a side effect, the standard semantic projection of LFG glue semantics can be dispensed with. A result is that LFG sentence structures acquire a level composed of strictly binary trees, constructed out of nodes representing function application and lambda abstraction, with a significant resemblance to external and internal merge in the Minimalist Program. This increased resemblance between frameworks might assist in making useful comparisons.

Abstract: The relationship between Lexical-Functional Grammar (LFG) functional structures (f-structures) for sentences and their semantic interpretations can be expressed directly in a fragment of linear logic in a way that explains correctly the constrained interactions between quantifier scope ambiguity and bound anaphora. The use of a deductive framework to account for the compositional properties of quantifying expressions in natural language obviates the need for additional mechanisms, such as Cooper storage, to represent the ...

Reddy introduced an extended intuitionistic linearcalculus, called LLMS (for Linear Logic Model ofState), to model some features of state manipulation. His calculus includes the connective" before" and an associated modality y. De Paiva presentsa (collection of) dialectica categorical models forClassical Linear Logic, the categories GC. Thesecategories contain an extra tensor product functorfi and a comonad structure corresponding to amodality related to this tensor. It is surprisingthat these works arising from completely ...

Concurrent computation can be given an abstract mathematical treatment very similar to that provided for sequential computation by domain theory and denotational semantics of Scott and Strachey. A simple domain theory for concurrency is presented. Based on a categorical model of linear logic and associated comonads, it highlights the role of linearity in concurrent computation. Two choices of comonad yield two expressive metalanguages for higher-order processes, both arising from canonical constructions in the model. Their denotational semantics are fully abstract with respect to contextual equivalence. One language, called HOPLA for Higher-Order Process LAnguage, derives from an exponential of linear logic. It can be viewed as an extension of the simply-typed lambda calculus with CCS-like nondeterministic sum and prefix operations, in which types express the form of computation path of which a process is capable. HOPLA can directly encode calculi like CCS, CCS with process passing, and mobile ambients with public names, and it can be given a straightforward operational semantics supporting a standard bisimulation congruence. The denotational and operational semantics are related with simple proofs of soundness and adequacy. Full abstraction implies that contextual equivalence coincides with logical equivalence for a fragment of Hennessy-Milner logic, linking up with simulation equivalence. The other language is called Affine HOPLA and is based on a weakening comonad that yields a model of affine-linear logic. This language adds to HOPLA an interesting tensor operation at the price of linearity constraints on the occurrences of variables. The tensor can be understood as a juxtaposition of independent processes, and allows Affine HOPLA to encode processes of the kind found in treatments of nondeterministic dataflow. The domain theory can be generalised to presheaf models, providing a more refined treatment of nondeterministic branching and supporting notions of bisimulation. The operational semantics for HOPLA is guided by the idea that derivations of transitions in the operational semantics should correspond to elements of the presheaf denotations. Similar guidelines lead to an operational semantics for the first-order fragment of Affine HOPLA. An extension of the operational semantics to the full language is based on a stable denotational semantics which associates to each computation the minimal input necessary for it. Such a semantics is provided, based on event structures; it agrees with the presheaf semantics at first order and exposes the tensor operation as a simple parallel composition of event structures. The categorical model obtained from presheaves is very rich in structure and points towards more expressive languages than HOPLA and Affine HOPLA-in particular concerning extensions to cover independence models. The thesis concludes with a discussion of related work towards a fully fledged domain theory for concurrency. v Thanks to my supervisor, Glynn Winskel, for four years of joint work under his expert leadership-and for introducing me to some of the good things in life besides research, like Jalfrezi and Habit Ale; my committee members Pierre-Louis Curien and Guy McCusker for their thoughtful and detailed comments, corrections and suggestions for improving the thesis; Pino Rosolini and the people at DISI, University of Genoa, for their hospitality during my stay there and for helpful comments on my work-and especially to Matías Menni for his friendship; Marcelo Fiore for his insightful suggestions at my Part A exam; Erik Meineche Schmidt and Mogens Nielsen for their early encouragement; the staff and students at Daimi/BRICS for creating a stimulating working environment; "Laesegruppen" and "Frokostklubben" for forcing me to have a life besides my studies (or, at least for trying); my family for boldly asking questions about my research even though my answers were often incomprehensible. Last, but certainly not least, I wish to thank my wife Mette for her support, her patience, and her unwavering faith in me.

For the given logical calculus we investigate the size of the proportion of the number of true formulas of a certain length n against the number of all formulas of such length. We are especially interested in asymptotic behavior of this fraction when n tends to infinity. If the limit of fractions exists it represents the real number between 0 and 1 which we may call the density of truth for the investigated logic. In this paper we apply this approach to the intuitionistic logic of one variable with implication and negation. The result is obtained by reducing it to the same problem of Dummett's intermediate linear logic of one variable ( see [?]). Actually, this paper shows the exact density of intuitionistic logic and demonstrates that it covers a substantial part (more then 93%) of classical propositional calculus. Despite using strictly mathematical means to solve all discussed problems, this paper in fact, may have a philosophical impact on understanding how much the phenomenon of truth is sporadic or frequent in random mathematics sentences.

We recently introduced an extensional model of the pure λ-calculus living in a canonical cartesian closed category of sets and relations [6]. In the present paper, we study the non-deterministic features of this model. Unlike most traditional approaches, our way of interpreting non-determinism does not require any additional powerdomain construction: we show that our model provides a straightforward semantics of non-determinism (may convergence) by means of unions of interpretations as well as of parallelism (must convergence) by means of a binary, non-idempotent, operation available on the model, which is related to the mix rule of Linear Logic. More precisely, we introduce a λ-calculus extended with non-deterministic choice and parallel composition, and we define its operational semantics (based on the may and must intuitions underlying our two additional operations). We describe the interpretation of this calculus in our model and show that this interpretation is sensible with respect to our operational semantics: a term converges if, and only if, it has a non-empty interpretation.