Between quantum logic and concurrency

We start from two closure operators defined on the elements of a special kind of partially ordered sets, called causal nets. Causal nets are used to model histories of concurrent processes, recording occurrences of local states and of events. If every maximal chain (line) of such a partially ordered set meets every maximal antichain (cut), then the two closure operators coincide, and generate a complete orthomodular lattice. In this paper we recall that, for any closed set in this lattice, every line meets either it or its orthocomplement in the lattice, and show that to any line, a two-valued state on the lattice can be associated. Starting from this result, we delineate a logical language whose formulas are interpreted over closed sets of a causal net, where every line induces an assignment of truth values to formulas. The resulting logic is non-classical; we show that maximal antichains in a causal net are associated to Boolean (hence "classical") substructures of the overall quantum logic.Comment: In Proceedings QPL 2012, arXiv:1407.842

Orthomodular Lattices Induced by the Concurrency Relation

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Cyclic Ordering through Partial Orders *

International audienceThe orientation problem for ternary cyclic order relations has been attacked in the literature from combinatorial perspectives, through rotations , and by connection with Petri nets. We propose here a twofold characterization of orientable cyclic orders in terms of symmetries of partial orders as well as in terms of separating sets (cuts). The results are inspired by properties of non-sequential discrete processeses, but also apply to dense structures of any cardinality

Labelled Tableaux for Distributed Temporal Logic

The distributed temporal logic DTL is a logic for reasoning about temporal properties of discrete distributed systems from the local point of view of the system's agents, which are assumed to execute sequentially and to interact by means of synchronous event sharing. We present a sound and complete labelled tableaux system for full DTL. To achieve this, we first formalize a labelled tableaux system for reasoning locally at each agent and afterwards we combine the local systems into a global one by adding rules that capture the distributed nature of DTL. We also provide examples illustrating the use of DTL and our tableaux syste

The Limit of Splitn-Language Equivalence

AbstractSplitting is a simple form of action refinement that may be used to express the duration of actions. In particular,splitnsubdivides each action intonphases. Petri netsNandN′ aresplitn-language equivalent ifsplitn(N) andsplitn(N′) are language equivalent. It is known that these equivalences get finer and finer with increasingn. This paper characterizes the limit of this sequence by a newly defined partial order semantics. This semantics is obtained from the interval-semiword semantics, which is fully abstract for action refinement and language equivalence, by closing it under a special swap operation. The new swap equivalence lies strictly between interval-semiword and step-sequence equivalence

Elements of Petri nets and processes

We present a formalism for Petri nets based on polynomial-style finite-set configurations and etale maps. The formalism supports both a geometric semantics in the style of Goltz and Reisig (processes are etale maps from graphs) and an algebraic semantics in terms of free coloured props: the Segal space of P-processes is shown to be the free coloured prop-in-groupoids on P. There is also an unfolding semantics \`a la Winskel, which bypasses the classical symmetry problems. Since everything is encoded with explicit sets, Petri nets and their processes have elements. In particular, individual-token semantics is native, and the benefits of pre-nets in this respect can be obtained without the need of numberings. (Collective-token semantics emerges from rather drastic quotient constructions \`a la Best--Devillers, involving taking $\pi_0$ of the groupoids of states.)Comment: 44 pages. The math is intended to be in reasonably final form, but the paper may well contain some misconceptions regarding the place of this material in the theory of Petri nets. All feedback and help will be greatly appreciated. v2: fixed a mistake in Section

Foundations of program refinement by calculation

Tese de doutoramento em Informática (ramo de conhecimento em Fundamentos da Computação)Embora não seja prática generalizada, aceita-se hoje o valor da especificação formal de aplicações como ingrediente essencial ao desenvolvimento de software fiável. Isso pressupõe uma noção adicional — a de refinamento — capaz de sistematizar a derivação de implementações correctas a partir de modelos abstractos (ie. especificações). No chamado estilo construtivo de desenvolvimento, faz-se refinamento passo-a-passo, provando que cada passo decorre do anterior por regras que garantem a correcção. Estas provas, que são vulgarmente feitas na lógica de predicados e teoria de conjuntos, têm, porém, problemas de escalabilidade: por um lado, não é prático provar factos envolvendo muitas variáveis e quantificações. Por outro, o nível relativamente pouco ágil em que decorrem as provas impede a sua progressão e pede ferramentas automáticas de prova. Esta tese desenvolve uma técnica alternativa de refinamento baseada na chamada transformada-pointfree. A ideia é desenvolver um cálculo ágil capaz de calcular implementações a partir das suas especificações por transformações algébricas simples. A transformada actua sempre que pretendemos raciocinar, mapeando expressões da lógica de predicados em expressões do cálculo relacional com implosão das quantificações e outras construções baseadas em variáveis. Nesse sentido, esta tese aborda os fundamentos do refinamento de programas por cálculo, através de raciocínios ao nível do cálculo de relações binárias dito pointfree, nos seus dois níveis essenciais: dados e algoritmos. Para esse efeito, desenvolvem-se e generalizam-se algumas construções do cálculo relacional, nomeadamente a transposição funcional, uma técnica que tem por objectivo converter relações em funções, de modo a exprimir a álgebra de relações através da álgebra de funções. É utilizada nesta dissertação como leit-motiv. No sentido de potenciar ao máximo a pretendida algebrização do processo de cálculo de programas, a abordagem proposta capitaliza no conceito de conexão de Galois. Em particular, mostra-se como as principais leis de refinamento de dados podem ser vistas como esse tipo de conexão. No plano do refinamento algorítmico, estuda-se a ordem padrão de refinamento ao nível pointfree e calcula-se a sua factorização em duas subordens com comportamentos opostos: redução de não-determinismo e aumento da definição. Essa factorização torna a ordem original mais tratável matematicamente. Apresenta-se a sua teoria em estilo pointfree, que inclui uma prova simples do refinamento estrutural, para tipos paramétricos arbitrários. Finalmente, mostramos que só precisamos de uma regra completa de refinamento relacional—para provar o refinamento coalgébrico—e utilizámo-la para testemunhar o refinamento por cálculo de relações de transição correspondentes a coalgebras.Design of trustworthy software calls for technologies which discuss software reliability formally, ie. by writing and reasoning about mathematical models of real-life objects and activities (vulg. specifications). Such technologies involve the additional notion of refinement (or reification), which means the systematic process of ensuring correct implementations for formal specifications. In the well-known constructive style for software development, design is factored in several steps, each intermediate step being first proposed and then proved to follow from its antecedent. However, such an ”invent-and-verify” style is often impractical due to the complexity of the mathematical reasoning involved in real-size software problems. Moreover, program reasoning is normally carried out in predicate/ temporal logic and na¨ıve set theory — notations which don’t scale up to fully detailed models of complex problems. This thesis is concerned with the foundations of an alternative technique for program refinement based on so-called pointfree calculation. The idea is to develop a calculus allowing for programs to be actually calculated from their specifications. Instead of doing proofs from first principles, this strategy leads to implementations which are “correct by construction”. Conventional refinement rules are transformed into simple, elegant equations dispensing with points and involving only binary relation combinators. The pointfree binary relational calculus is therefore at the heart of the proposed refinement theory. This thesis adds to such a mathematical framework in two ways: on the one hand it shows how to apply it to data and algorithimc refinement problems. On the other hand, some constructions are proposed which prove useful not only in refinement but also in general. This includes generic functional transposition, a technique for converting relations into functions aimed at developing relational algebra via the algebra of functions. It is employed in this dissertation as a leit motiv. Our proposed theory of data refinement draws heavily on the Galois connection approach to mathematical reasoning. This includes a simple way to calculate refinement invariants induced by the Galois connected laws. Algorithmic refinement is addressed in the same way. The standard operation refinement ordering is given a pointfree treatmentwhich includes a simple calculation of Groves’ factorization and its direct application in structural refinement involving arbitrary parametric types. Finally, coalgebraic refinement is done using an equivalent single complete rule for data refinement which is used to witness refinement by calculation of transition relations corresponding to coalgebras