Multiplicative-Additive Proof Equivalence is Logspace-complete, via Binary Decision Trees

Given a logic presented in a sequent calculus, a natural question is that of equivalence of proofs: to determine whether two given proofs are equated by any denotational semantics, ie any categorical interpretation of the logic compatible with its cut-elimination procedure. This notion can usually be captured syntactically by a set of rule permutations. Very generally, proofnets can be defined as combinatorial objects which provide canonical representatives of equivalence classes of proofs. In particular, the existence of proof nets for a logic provides a solution to the equivalence problem of this logic. In certain fragments of linear logic, it is possible to give a notion of proofnet with good computational properties, making it a suitable representation of proofs for studying the cut-elimination procedure, among other things. It has recently been proved that there cannot be such a notion of proofnets for the multiplicative (with units) fragment of linear logic, due to the equivalence problem for this logic being Pspace-complete. We investigate the multiplicative-additive (without unit) fragment of linear logic and show it is closely related to binary decision trees: we build a representation of proofs based on binary decision trees, reducing proof equivalence to decision tree equivalence, and give a converse encoding of binary decision trees as proofs. We get as our main result that the complexity of the proof equivalence problem of the studied fragment is Logspace-complete.Comment: arXiv admin note: text overlap with arXiv:1502.0199

MALL proof equivalence is Logspace-complete, via binary decision diagrams

Proof equivalence in a logic is the problem of deciding whether two proofs are equivalent modulo a set of permutation of rules that reflects the commutative conversions of its cut-elimination procedure. As such, it is related to the question of proofnets: finding canonical representatives of equivalence classes of proofs that have good computational properties. It can also be seen as the word problem for the notion of free category corresponding to the logic. It has been recently shown that proof equivalence in MLL (the multiplicative with units fragment of linear logic) is PSPACE-complete, which rules out any low-complexity notion of proofnet for this particular logic. Since it is another fragment of linear logic for which attempts to define a fully satisfactory low-complexity notion of proofnet have not been successful so far, we study proof equivalence in MALL- (multiplicative-additive without units fragment of linear logic) and discover a situation that is totally different from the MLL case. Indeed, we show that proof equivalence in MALL- corresponds (under AC0 reductions) to equivalence of binary decision diagrams, a data structure widely used to represent and analyze Boolean functions efficiently. We show these two equivalent problems to be LOGSPACE-complete. If this technically leaves open the possibility for a complete solution to the question of proofnets for MALL-, the established relation with binary decision diagrams actually suggests a negative solution to this problem.Comment: in TLCA 201

Gender, self, multiple identities, violence and magical interpretations in lovolo practices in Southern Mozambique

Includes bibliographical references (p. 237-273)

On the Resolution Semiring

In this thesis, we study a semiring structure with a product based on theresolution rule of logic programming. This mathematical object was introducedinitially in the setting of the geometry of interaction program in order to modelthe cut-elimination procedure of linear logic. It provides us with an algebraicand abstract setting, while being presented in a syntactic and concrete way, inwhich a theoretical study of computation can be carried on.We will review first the interactive interpretation of proof theory withinthis semiring via the categorical axiomatization of the geometry of interactionapproach. This interpretation establishes a way to translate functional programsinto a very simple form of logic programs.Secondly, complexity theory problematics will be considered: while thenilpotency problem in the semiring we study is undecidable in general, it willappear that certain restrictions allow for characterizations of (deterministicand non-deterministic) logarithmic space and (deterministic) polynomial timecomputation

Memoization for Unary Logic Programming: Characterizing PTIME

We give a characterization of deterministic polynomial time computation based on an algebraic structure called the resolution semiring, whose elements can be understood as logic programs or sets of rewriting rules over first-order terms. More precisely, we study the restriction of this framework to terms (and logic programs, rewriting rules) using only unary symbols. We prove it is complete for polynomial time computation, using an encoding of pushdown automata. We then introduce an algebraic counterpart of the memoization technique in order to show its PTIME soundness. We finally relate our approach and complexity results to complexity of logic programming. As an application of our techniques, we show a PTIME-completeness result for a class of logic programming queries which use only unary function symbols.Comment: Soumis {\`a} LICS 201

Unification and Logarithmic Space: Journal Version

Soumis au numéro spécial de LMCS pour RTA/TLCA 2014 ( http://www.lmcs-online.org/ojs/specialIssues.php?id=67 )We present an algebraic characterization of the complexity classes Logspace and NLogspace, using an algebra with a composition law based on unification. This new bridge between unification and complexity classes is rooted in proof theory and more specifically linear logic and geometry of interaction. We show how to build a model of computation in the unification algebra and then, by means of a syntactic representation of finite permutations in the algebra, we prove that whether an observation (the algebraic counterpart of a program) accepts a word can be decided within logarithmic space. Finally, we show that the construction naturally corresponds to pointer machines, an convenient way of understanding logarithmic space computation

Unification and Logarithmic Space

We present an algebraic characterization of the complexity classes Logspace and Nlogspace, using an algebra with a composition law based on unification. This new bridge between unification and complexity classes is rooted in proof theory and more specifically linear logic and geometry of interaction. We show how to build a model of computation in the unification algebra and then, by means of a syntactic representation of finite permutations in the algebra, we prove that whether an observation (the algebraic counterpart of a program) accepts a word can be decided within logarithmic space. Finally, we show that the construction naturally corresponds to pointer machines, a convenient way of understanding logarithmic space computation.Comment: arXiv admin note: text overlap with arXiv:1402.432

Barbaridades e violências

The author outlines her trajectory of intimate coexistence with the violence and the consequences inflicted on the mind and body of all the people who are involved in them. With examples from Mozambique, South Africa and Rwanda and violence linked to war, hunger, HIV / AIDS and gender, she articulates, through fragments of her experiences, the different layers and forms of violence that afflict the most unprotected. The approach of physical violence, structural violence, symbolic violence and its various combinations destroy human beings and, above all, those who have to live with the multiplicity of their forms. The proposal is to find ways to develop a new ethic of citizenship, empathy and solidarity that seeks to disarm these violences.A autora delineia a sua trajetória de convivência íntima com as violências e com as consequências infligidas na mente e no corpo de todas as pessoas que estão envolvidas nelas. Com exemplo de Moçambique, África do Sul e Ruanda e violências ligadas a guerra, a fome, ao HIV/AIDS e ao gênero ela articula, percorrendo fragmentos de suas experiências, as diferentes camadas e formas de violência que afligem os/as mais desemparados/as. A aproximação da violência física, da violência estrutural, da violência simbólica e das suas várias combinações destroem os seres humanas e, sobretudo, os/as que têm de conviver com a multiplicidade das suas formas. A proposta é de encontrar formas de desenvolver uma nova ética de cidadania, de empatia e solidariedade que procura desarmar estas violências

Les députés héraultais et la viticulture dans l’entre-deux-guerres : organes de décision, relais de pouvoir, législation

L’approche du sujet et les sources À l’origine de ma recherche, il y avait une volonté de m’intéresser à l’histoire rurale de l’Hérault et plus particulièrement à un secteur qui a marqué profondément les destinées de ce territoire : la viticulture, plusieurs fois séculaire. J’ai donc entrepris d’analyser l’évolution de ce secteur en le mettant en interaction avec une passion personnelle pour l’analyse politique. J’ai choisi la Troisième République qui avait été, jusqu’à une date récente, peu ..

Les députés héraultais et la viticulture dans l’entre-deux-guerres : organes de décision, relais de pouvoir, législation

L’approche du sujet et les sources À l’origine de ma recherche, il y avait une volonté de m’intéresser à l’histoire rurale de l’Hérault et plus particulièrement à un secteur qui a marqué profondément les destinées de ce territoire : la viticulture, plusieurs fois séculaire. J’ai donc entrepris d’analyser l’évolution de ce secteur en le mettant en interaction avec une passion personnelle pour l’analyse politique. J’ai choisi la Troisième République qui avait été, jusqu’à une date récente, peu ..