Esta tese de doutoramento trata da integração da teoria piagetiana com a Ciência da Computação, mais especificamente, a análise de ferramentas computacionais do ponto de vista da lógica operatória. Para isso, foi preciso investigar, em primeiro lugar, a teoria do sujeito individual, no que se refere a função simbólica e reinterpretar estes conceitos no objeto. Neste caso, o objeto e a ferramenta computacional de uso individual. Portanto, somente a partir deste estudo foi possível construir o modelo geral de interação de um sujeito qualquer com uma ferramenta computacional, para depois analisá-la operatoriamente. Em um segundo momento, introduziu-se a teoria do sujeito coletivo que, para caracterizá-lo, foi necessária a compreensão de alguns conceitos relevantes da teoria em questão envolvidos nas noções de interação interindividual e cooperação. A partir de então, foram construídos os modelos interativos mais simples de um sujeito coletivo com três tipos de ferramentas computacionais de uso cooperativo. Para isso, foram abordados alguns aspectos relativos a Computação Cooperativa ou CSCW - Computer Supported Cooperative Work. Da mesma forma que ocorre no nível individual, foram analisadas operatoriamente ferramentas computacionais de uso cooperativo.; This thesis integrates the piagetian theory with Computer Science...

The complexity of a computational problem is traditionally quantified based on the hardness of its worst case. This approach has many advantages and has led to a deep and beautiful theory. However, from the practical perspective, this leaves much to be desired. In application areas, practically interesting instances very often occupy just a tiny part of an algorithm's space of instances, and the vast majority of instances are simply irrelevant. Addressing these issues is a major challenge for theoretical computer science which may make theory more relevant to the practice of computer science. Following Bilu and Linial, we apply this perspective to MAXCUT, viewed as a clustering problem. Using a variety of techniques, we investigate practically interesting instances of this problem. Specifically, we show how to solve in polynomial time distinguished, metric, expanding and dense instances of MAXCUT under mild stability assumptions. In particular, $(1+\epsilon)$-stability (which is optimal) suffices for metric and dense MAXCUT. We also show how to solve in polynomial time $\Omega(\sqrt{n})$-stable instances of MAXCUT, substantially improving the best previously known result.

We introduce two-sorted theories in the style of [CN10] for the complexity classes \oplusL and DET, whose complete problems include determinants over Z2 and Z, respectively. We then describe interpretations of Soltys' linear algebra theory LAp over arbitrary integral domains, into each of our new theories. The result shows equivalences of standard theorems of linear algebra over Z2 and Z can be proved in the corresponding theory, but leaves open the interesting question of whether the theorems themselves can be proved.; Comment: This is a revised journal version of the paper "Formal Theories for Linear Algebra" (Computer Science Logic) for the journal Logical Methods in Computer Science

We present a general method for introducing finitely axiomatizable "minimal" two-sorted theories for various subclasses of P (problems solvable in polynomial time). The two sorts are natural numbers and finite sets of natural numbers. The latter are essentially the finite binary strings, which provide a natural domain for defining the functions and sets in small complexity classes. We concentrate on the complexity class TC^0, whose problems are defined by uniform polynomial-size families of bounded-depth Boolean circuits with majority gates. We present an elegant theory VTC^0 in which the provably-total functions are those associated with TC^0, and then prove that VTC^0 is "isomorphic" to a different-looking single-sorted theory introduced by Johannsen and Pollet. The most technical part of the isomorphism proof is defining binary number multiplication in terms a bit-counting function, and showing how to formalize the proofs of its algebraic properties.; Comment: 40 pages, Logical Methods in Computer Science

We propose a simple global computing framework, whose main concern is code migration. Systems are structured in sites, and each site is divided into two parts: a computing body, and a membrane, which regulates the interactions between the computing body and the external environment. More precisely, membranes are filters which control access to the associated site, and they also rely on the well-established notion of trust between sites. We develop a basic theory to express and enforce security policies via membranes. Initially, these only control the actions incoming agents intend to perform locally. We then adapt the basic theory to encompass more sophisticated policies, where the number of actions an agent wants to perform, and also their order, are considered.; Comment: 23 pages; to appear in Logical Methods in Computer Science

Physical systems can fail. For this reason the problem of identifying and reacting to faults has received a large attention in the control and computer science communities. In this paper we study the fault diagnosis problem for hybrid systems from a game-theoretical point of view. A hybrid system is a system mixing continuous and discrete behaviours that cannot be faithfully modeled neither by using a formalism with continuous dynamics only nor by a formalism including only discrete dynamics. We use the well known framework of hybrid automata for modeling hybrid systems, and we define a Fault Diagnosis Game on them, using two players: the environment and the diagnoser. The environment controls the evolution of the system and chooses whether and when a fault occurs. The diagnoser observes the external behaviour of the system and announces whether a fault has occurred or not. Existence of a winning strategy for the diagnoser implies that faults can be detected correctly, while computing such a winning strategy corresponds to implement a diagnoser for the system. We will show how to determine the existence of a winning strategy, and how to compute it, for some decidable classes of hybrid automata like o-minimal hybrid automata.; Comment: In Proceedings GandALF 2011...

First, we extend Leifer-Milner RPO theory, by giving general conditions to obtain IPO labelled transition systems (and bisimilarities) with a reduced set of transitions, and possibly finitely branching. Moreover, we study the weak variant of Leifer-Milner theory, by giving general conditions under which the weak bisimilarity is a congruence. Then, we apply such extended RPO technique to the lambda-calculus, endowed with lazy and call by value reduction strategies. We show that, contrary to process calculi, one can deal directly with the lambda-calculus syntax and apply Leifer-Milner technique to a category of contexts, provided that we work in the framework of weak bisimilarities. However, even in the case of the transition system with minimal contexts, the resulting bisimilarity is infinitely branching, due to the fact that, in standard context categories, parametric rules such as the beta-rule can be represented only by infinitely many ground rules. To overcome this problem, we introduce the general notion of second-order context category. We show that, by carrying out the RPO construction in this setting, the lazy observational equivalence can be captured as a weak bisimilarity equivalence on a finitely branching transition system. This result is achieved by considering an encoding of lambda-calculus in Combinatory Logic.; Comment: 35 pages...

Fair allocation has been studied intensively in both economics and computer science, and fair sharing of resources has aroused renewed interest with the advent of virtualization and cloud computing. Prior work has typically focused on mechanisms for fair sharing of a single resource. We provide a new definition for the simultaneous fair allocation of multiple continuously-divisible resources. Roughly speaking, we define fairness as the situation where every user either gets all the resources he wishes for, or else gets at least his entitlement on some bottleneck resource, and therefore cannot complain about not getting more. This definition has the same desirable properties as the recently suggested dominant resource fairness, and also handles the case of multiple bottlenecks. We then prove that a fair allocation according to this definition is guaranteed to exist for any combination of user requests and entitlements (where a user's relative use of the different resources is fixed). The proof, which uses tools from the theory of ordinary differential equations, is constructive and provides a method to compute the allocations numerically.

We consider the one-round Voronoi game, where player one (``White'', called ``Wilma'') places a set of n points in a rectangular area of aspect ratio r <=1, followed by the second player (``Black'', called ``Barney''), who places the same number of points. Each player wins the fraction of the board closest to one of his points, and the goal is to win more than half of the total area. This problem has been studied by Cheong et al., who showed that for large enough $n$ and r=1, Barney has a strategy that guarantees a fraction of 1/2+a, for some small fixed a. We resolve a number of open problems raised by that paper. In particular, we give a precise characterization of the outcome of the game for optimal play: We show that Barney has a winning strategy for n>2 and r>sqrt{2}/n, and for n=2 and r>sqrt{3}/2. Wilma wins in all remaining cases, i.e., for n>=3 and r<=sqrt{2}/n, for n=2 and r<=sqrt{3}/2, and for n=1. We also discuss complexity aspects of the game on more general boards, by proving that for a polygon with holes, it is NP-hard to maximize the area Barney can win against a given set of points by Wilma.; Comment: 14 pages, 6 figures, Latex; revised for journal version, to appear in Computational Geometry: Theory and Applications. Extended abstract version appeared in Workshop on Algorithms and Data Structures...

This thesis presents a series of theoretical results and practical realisations about the theory of computation in distributive categories. Distributive categories have been proposed as a foundational tool for Computer Science in the last years, starting from the papers of R.F.C. Walters. We shall focus on two major topics: distributive computability, i.e., a generalized theory of computability based on distributive categories, and the Imp(G) language, which is a language based on the syntax of distributive categories. The link between the former and the latter is that the functions computed by Imp(G) programs are exactly the distributively computable functions.

This paper presents a novel theoretical framework for the state space reduction of Kripke structures. We define two equivalence relations, Kripke minimization equivalence (KME) and weak Kripke minimization equivalence (WKME). We define the quotient system under these relations and show that these relations are strictly coarser than strong (bi)simulation and divergence-sensitive stutter (bi)simulation, respectively. We prove that the quotient system obtained under KME and WKME preserves linear-time and stutter-insensitive linear-time properties. Finally, we show that KME is compositional w.r.t. synchronous parallel composition.; Comment: Accepted for Student Research Forum, 39th International Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM 2013)

This paper aims to provide insight into stability of collaboration choices in P2P networks. We study networks where exchanges between nodes are driven by the desire to receive the best service available. This is the case for most existing P2P networks. We explore an evolution model derived from stable roommates theory that accounts for heterogeneity between nodes. We show that most P2P applications can be modeled using stable matching theory. This is the case whenever preference lists can be deduced from the exchange policy. In many cases, the preferences lists are characterized by an interesting acyclic property. We show that P2P networks with acyclic preferences possess a unique stable state with good convergence properties.

Basic results in combinatorial mathematics provide the foundation for a theory and calculus for reasoning about sequential behavior. A key concept of the theory is a generalization of Boolean implicant which deals with statements of the form: A sequence of Boolean expressions alpha is an implicant of a set of sequences of Boolean expressions A This notion of a generalized implicant takes on special significance when each of the sequences in the set A describes a disallowed pattern of behavior. That is because a disallowed sequence of Boolean expressions represents a logical/temporal dependency, and because the implicants of a set of disallowed Boolean sequences A are themselves disallowed and represent precisely those dependencies that follow as a logical consequence from the dependencies represented by A. The main result of the theory is a necessary and sufficient condition for a sequence of Boolean expressions to be an implicant of a regular set of sequences of Boolean expressions. This result is the foundation for two new proof methods. Sequential resolution is a generalization of Boolean resolution which allows new logical/temporal dependencies to be inferred from existing dependencies. Normalization starts with a model (system) and a set of logical/temporal dependencies and determines which of those dependencies are satisfied by the model.; Comment: 102 double-spaced pages...

We investigate the possibility of modelling the syntax and semantics of natural language by constraints, or rules, imposed by the multi-dimensional type theory Nabla. The only multiplicity we explicitly consider is two, namely one dimension for the syntax and one dimension for the semantics, but the general perspective is important. For example, issues of pragmatics could be handled as additional dimensions. One of the main problems addressed is the rather complicated repertoire of operations that exists besides the notion of categories in traditional Montague grammar. For the syntax we use a categorial grammar along the lines of Lambek. For the semantics we use so-called lexical and logical combinators inspired by work in natural logic. Nabla provides a concise interpretation and a sequent calculus as the basis for implementations.; Comment: 20 pages

The standard approach to logic in the literature in philosophy and mathematics, which has also been adopted in computer science, is to define a language (the syntax), an appropriate class of models together with an interpretation of formulas in the language (the semantics), a collection of axioms and rules of inference characterizing reasoning (the proof theory), and then relate the proof theory to the semantics via soundness and completeness results. Here we consider an approach that is more common in the economics literature, which works purely at the semantic, set-theoretic level. We provide set-theoretic completeness results for a number of epistemic and conditional logics, and contrast the expressive power of the syntactic and set-theoretic approaches; Comment: This is an expanded version of a paper that appeared in AI and Mathematics, 1998

The notion of optimality naturally arises in many areas of applied mathematics and computer science concerned with decision making. Here we consider this notion in the context of three formalisms used for different purposes in reasoning about multi-agent systems: strategic games, CP-nets, and soft constraints. To relate the notions of optimality in these formalisms we introduce a natural qualitative modification of the notion of a strategic game. We show then that the optimal outcomes of a CP-net are exactly the Nash equilibria of such games. This allows us to use the techniques of game theory to search for optimal outcomes of CP-nets and vice-versa, to use techniques developed for CP-nets to search for Nash equilibria of the considered games. Then, we relate the notion of optimality used in the area of soft constraints to that used in a generalization of strategic games, called graphical games. In particular we prove that for a natural class of soft constraints that includes weighted constraints every optimal solution is both a Nash equilibrium and Pareto efficient joint strategy. For a natural mapping in the other direction we show that Pareto efficient joint strategies coincide with the optimal solutions of soft constraints.; Comment: 39 pages. To appear in Annals of Mathematics and Artificial Intelligence

Refinement is a powerful mechanism for mastering the complexities that arise when formally modelling systems. Refinement also brings with it additional proof obligations -- requiring a developer to discover properties relating to their design decisions. With the goal of reducing this burden, we have investigated how a general purpose theory formation tool, HR, can be used to automate the discovery of such properties within the context of Event-B. Here we develop a heuristic approach to the automatic discovery of invariants and report upon a series of experiments that we undertook in order to evaluate our approach. The set of heuristics developed provides systematic guidance in tailoring HR for a given Event-B development. These heuristics are based upon proof-failure analysis, and have given rise to some promising results.; Comment: In Proceedings Refine 2011, arXiv:1106.3488

We introduce a set of eight universal Rules of Inference by which computer programs with known properties (axioms) are transformed into new programs with known properties (theorems). Axioms are presented to formalize a segment of Number Theory, DataBase retrieval and Computability Theory. The resulting Program Calculus is used to generate programs to (1) Determine if one number is a factor of another. (2) List all employees who earn more than their manager. (3) List the set of programs that halt no on themselves, thus proving that it is recursively enumerable. The well-known fact that the set of programs that do not halt yes on themselves is not recursively enumerable is formalized as a program requirement that has no solution, an Incompleteness Axiom. Thus, any axioms (programs) which could be used to generate this program are themselves unattainable. Such proofs are presented to formally generate several additional theorems, including (4) The halting problem is unsolvable. Open problems and future research is discussed, including the use of temporary sort files, programs that calculate statistics (such as counts and sums), the synthesis of programs to solve other well-known problems from Number Theory, Logic, DataBase retrieval and Computability Theory...

Consider a game where Alice generates an integer and Bob wins if he can factor that integer. Traditional game theory tells us that Bob will always win this game even though in practice Alice will win given our usual assumptions about the hardness of factoring. We define a new notion of bounded rationality, where the payoffs of players are discounted by the computation time they take to produce their actions. We use this notion to give a direct correspondence between the existence of equilibria where Alice has a winning strategy and the hardness of factoring. Namely, under a natural assumption on the discount rates, there is an equilibriumwhere Alice has a winning strategy iff there is a linear-time samplable distribution with respect to which Factoring is hard on average. We also give general results for discounted games over countable action spaces, including showing that any game with bounded and computable payoffs has an equilibrium in our model, even if each player is allowed a countable number of actions. It follows, for example, that the Largest Integer game has an equilibrium in our model though it has no Nash equilibria or epsilon-Nash equilibria.; Comment: To appear in Proceedings of The First Symposium on Innovations in Computer Science

A type theory is presented that combines (intuitionistic) linear types with type dependency, thus properly generalising both intuitionistic dependent type theory and full linear logic. A syntax and complete categorical semantics are developed, the latter in terms of (strict) indexed symmetric monoidal categories with comprehension. Various optional type formers are treated in a modular way. In particular, we will see that the historically much-debated multiplicative quantifiers and identity types arise naturally from categorical considerations. These new multiplicative connectives are further characterised by several identities relating them to the usual connectives from dependent type theory and linear logic. Finally, one important class of models, given by families with values in some symmetric monoidal category, is investigated in detail.; Comment: Based on the technical report arXiv:1405.0033 . To appear in the proceedings of FoSSaCS 2015, in the Advanced Research in Computing and Software Science (ARCoSS) subline of Springer's Lecture Notes in Computer Science series