Hennessy-Milner Logic with Greatest Fixed Points as a Complete Behavioural Specification Theory

There are two fundamentally different approaches to specifying and verifying properties of systems. The logical approach makes use of specifications given as formulae of temporal or modal logics and relies on efficient model checking algorithms; the behavioural approach exploits various equivalence or refinement checking methods, provided the specifications are given in the same formalism as implementations. In this paper we provide translations between the logical formalism of Hennessy-Milner logic with greatest fixed points and the behavioural formalism of disjunctive modal transition systems. We also introduce a new operation of quotient for the above equivalent formalisms, which is adjoint to structural composition and allows synthesis of missing specifications from partial implementations. This is a substantial generalisation of the quotient for deterministic modal transition systems defined in earlier papers

Structural Refinement for the Modal nu-Calculus

We introduce a new notion of structural refinement, a sound abstraction of logical implication, for the modal nu-calculus. Using new translations between the modal nu-calculus and disjunctive modal transition systems, we show that these two specification formalisms are structurally equivalent. Using our translations, we also transfer the structural operations of composition and quotient from disjunctive modal transition systems to the modal nu-calculus. This shows that the modal nu-calculus supports composition and decomposition of specifications.Comment: Accepted at ICTAC 201

Modal paraconsistent logic

Dissertação de mestrado integrado em Engenharia FísicaSuperconducting quantum circuits are a promising model for quantum computation, al though their physical implementation faces some adversities due to the hardly unavoidable decoherence of superconducting quantum bits. This problem may be approached from a formal perspective, using logical reasoning to perform software correctness of programs executed in the non-ideal available hardware. This is the motivation for the work devel oped in this dissertation, which is ultimately an attempt to use the formalism of transition systems to design logical tools for the engineering of quantum software. A transition system to capture the possibly unexpected behaviors of quantum circuits needs to consider the phenomena of decoherence as a possible error factor. In this way, we propose a new family of transition systems, the Paraconsistent Labelled Transition Systems (PLTS), to describe processes that may behave differently from what is expected when facing specific contexts. System states are connected through transitions which simultaneously characterize the possibility and impossibility of that being the system’s evolution. This kind of formalism may be used to represent processes whose evolution is impossible to be sharply described and, thus, should be able to cope with inconsistencies, as well as with vagueness or missing information. Besides giving the formal definition of PLTS, we establish how they are related under the notions of morphism, simulation, bisimulation and trace equivalence. It is a common practice to combine transition systems through universal constructions, in a suitable category, which forms a basis for a process description language. In this dis sertation, we define a category of PLTS and propose a number of constructions to combine them, providing a basis for such a language. Transition systems are usually associated with modal logics which provide a formal set ting to express and prove their properties. We also propose a modal logic, more specifically, a modal intuitionistic paraconsistent logic (MIPL), to talk about PLTS and express their properties, studying how the equivalence relations defined for PLTS extend to relations on MIPL models and how the satisfaction of formulas is preserved along related models. Finally, we illustrate how superconducting quantum circuits may be represented by a PLTS and propose the use of PLTS equivalence relations, namely that of trace equivalence, to compare circuit effectiveness.Os circuitos quânticos que operam qubits supercondutores são um modelo promissor para a arquitetura de computadores quânticos. No entanto, a sua implementação física pode tornar-se ineficaz, devido a fenómenos de decoerência a que os qubits em questão estão altamente sujeitos. Uma possível abordagem a este problema consiste em empregar a lógica e as suas ferramentas para a correção de programas a executar nestes dispositivos. A proposta desta dissertação é que se utilize o formalismo dos sistemas de transição para modelar e descrever o comportamento dos circuitos quânticos, que, por vezes, pode ser imprevisível. Para tal, considera-se a decoerência de qubits como um possível fator de erro nas computações. Assim surge uma nova família de sistemas de transição, os Paraconsistent Labelled Transition systems (PLTS), como um modelo para descrever processos que, em determinados contextos, se comportam de forma diferente do que é esperado. Os estados de um PLTS estão conectados por transições que caracterizam, simultaneamente, a possibilidade e a impossibilidade de o sistema evoluir transitando de um estado para o outro. Este é um modelo em que a informação acerca das transições pode ser incompleta ou mesmo contraditória. Além da definição formal dos PLTS, são também sugeridas, como relações entre PLTS, as noções de morfismo, simulação, bissimulação e equivalência por traços. Muitas vezes, os sistemas de transição são combinados através de construções universais numa categoria adequada, de forma a definir uma álgebra de processos. Também neste trabalho é definida uma categoria de PLTS e são propostas algumas construções, típicas nas álgebras de processos, para os combinar. Os sistemas de transição são geralmente associados a lógicas modais, que permitem expressar e provar as suas propriedades. A definição dos PLTS conduziu à definição de uma lógica modal, MIPL, que permitiu determinar de que forma as relações de equivalência definidas para PLTS, e estendidas para modelos da logica MIPL, se refletem na preservação da satisfação de fórmulas sobre os modelos relacionados. Por fim, propõe-se utilizar PLTS para a representação de circuitos quânticos e comparar a eficácia dos circuitos através da relação de equivalência por traços

Completeness for Flat Modal Fixpoint Logics

This paper exhibits a general and uniform method to prove completeness for certain modal fixpoint logics. Given a set \Gamma of modal formulas of the form \gamma(x, p1, . . ., pn), where x occurs only positively in \gamma, the language L\sharp (\Gamma) is obtained by adding to the language of polymodal logic a connective \sharp\_\gamma for each \gamma \epsilon. The term \sharp\_\gamma (\varphi1, . . ., \varphin) is meant to be interpreted as the least fixed point of the functional interpretation of the term \gamma(x, \varphi 1, . . ., \varphi n). We consider the following problem: given \Gamma, construct an axiom system which is sound and complete with respect to the concrete interpretation of the language L\sharp (\Gamma) on Kripke frames. We prove two results that solve this problem. First, let K\sharp (\Gamma) be the logic obtained from the basic polymodal K by adding a Kozen-Park style fixpoint axiom and a least fixpoint rule, for each fixpoint connective \sharp\_\gamma. Provided that each indexing formula \gamma satisfies the syntactic criterion of being untied in x, we prove this axiom system to be complete. Second, addressing the general case, we prove the soundness and completeness of an extension K+ (\Gamma) of K\_\sharp (\Gamma). This extension is obtained via an effective procedure that, given an indexing formula \gamma as input, returns a finite set of axioms and derivation rules for \sharp\_\gamma, of size bounded by the length of \gamma. Thus the axiom system K+ (\Gamma) is finite whenever \Gamma is finite

Order algebraizable logics

AbstractThis paper develops an order-theoretic generalization of Blok and Pigozziʼs notion of an algebraizable logic. Unavoidably, the ordered model class of a logic, when it exists, is not unique. For uniqueness, the definition must be relativized, either syntactically or semantically. In sentential systems, for instance, the order algebraization process may be required to respect a given but arbitrary polarity on the signature. With every deductive filter of an algebra of the pertinent type, the polarity associates a reflexive and transitive relation called a Leibniz order, analogous to the Leibniz congruence of abstract algebraic logic (AAL). Some core results of AAL are extended here to sentential systems with a polarity. In particular, such a system is order algebraizable if the Leibniz order operator has the following four independent properties: (i) it is injective, (ii) it is isotonic, (iii) it commutes with the inverse image operator of any algebraic homomorphism, and (iv) it produces anti-symmetric orders when applied to filters that define reduced matrix models. Conversely, if a sentential system is order algebraizable in some way, then the order algebraization process naturally induces a polarity for which the Leibniz order operator has properties (i)–(iv)

Deciding some displayable modal logics

In this paper we use display calculus to show the decidability for normal modal logic K and some of its extensions.Comment: Draf

Logics of Formal Inconsistency enriched with replacement: an algebraic and modal account

One of the most expected properties of a logical system is that it can be algebraizable, in the sense that an algebraic counterpart of the deductive machinery could be found. Since the inception of da Costa's paraconsistent calculi, an algebraic equivalent for such systems have been searched. It is known that these systems are non self-extensional (i.e., they do not satisfy the replacement property). More than this, they are not algebraizable in the sense of Blok-Pigozzi. The same negative results hold for several systems of the hierarchy of paraconsistent logics known as Logics of Formal Inconsistency (LFIs). Because of this, these logics are uniquely characterized by semantics of non-deterministic kind. This paper offers a solution for two open problems in the domain of paraconsistency, in particular connected to algebraization of LFIs, by obtaining several LFIs weaker than C1, each of one is algebraizable in the standard Lindenbaum-Tarski's sense by a suitable variety of Boolean algebras extended with operators. This means that such LFIs satisfy the replacement property. The weakest LFI satisfying replacement presented here is called RmbC, which is obtained from the basic LFI called mbC. Some axiomatic extensions of RmbC are also studied, and in addition a neighborhood semantics is defined for such systems. It is shown that RmbC can be defined within the minimal bimodal non-normal logic E+E defined by the fusion of the non-normal modal logic E with itself. Finally, the framework is extended to first-order languages. RQmbC, the quantified extension of RmbC, is shown to be sound and complete w.r.t. BALFI semantics