The Temporal Logic of two dimensional Minkowski spacetime is decidable

We consider Minkowski spacetime, the set of all point-events of spacetime under the relation of causal accessibility. That is, ${\sf x}$ can access ${\sf y}$ if an electromagnetic or (slower than light) mechanical signal could be sent from ${\sf x}$ to ${\sf y}$. We use Prior's tense language of ${\bf F}$ and ${\bf P}$ representing causal accessibility and its converse relation. We consider two versions, one where the accessibility relation is reflexive and one where it is irreflexive. In either case it has been an open problem, for decades, whether the logic is decidable or axiomatisable. We make a small step forward by proving, for the case where the accessibility relation is irreflexive, that the set of valid formulas over two-dimensional Minkowski spacetime is decidable, decidability for the reflexive case follows from this. The complexity of either problem is PSPACE-complete. A consequence is that the temporal logic of intervals with real endpoints under either the containment relation or the strict containment relation is PSPACE-complete, the same is true if the interval accessibility relation is "each endpoint is not earlier", or its irreflexive restriction. We provide a temporal formula that distinguishes between three-dimensional and two-dimensional Minkowski spacetime and another temporal formula that distinguishes the two-dimensional case where the underlying field is the real numbers from the case where instead we use the rational numbers.Comment: 30 page

Complexity of monodic guarded fragments over linear and real time

Accepted versio

An Objection to Naturalism and Atheism from Logic

I proffer a success argument for classical logical consequence. I articulate in what sense that notion of consequence should be regarded as the privileged notion for metaphysical inquiry aimed at uncovering the fundamental nature of the world. Classical logic breeds necessitism. I use necessitism to produce problems for both ontological naturalism and atheism

Ockhamist Propositional Dynamic Logic: a natural link between PDL and CTL

International audienceWe present a new logic called Ockhamist Propositional Dynamic Logic, OPDL, which provides a natural link between PDL and CTL*. We show that both PDL and CTL* can be polynomially embedded into OPDL in a rather simple and direct way. More generally, the semantics on which OPDL is based provides a unifying framework for making the dynamic logic family and the temporal logic family converge in a single logical framework. Decidability of the satisfiability problem for OPDL is studied in the paper

Citizenship and the Politics of Civic Driven Change

Nation states are premised on the legitimizing presence of a polity comprised of citizens. The politics of this relationship is central to discourse on how societies evolve. Yet in the discipline of international development studies the topic remains peripheral. Reasons can be found in conceptual confusion, in selectivity in donor thinking and policies towards civil society and in the growth-driven political economy of NGO-ism. Remedies for the political lacunae are being sought through a concerted focus on people's rights, citizenship and qualities of leadership that all show valuable progress. This chapter will examine a comprehensive complement to such efforts referred to as civic driven change (CDC). Originating in a grounded empirical approach, the constituent principles and elements of CDC offer a lens that can both sharpen and deepen insights and advance analysis of civic agency in socio-political processes. As an ontologically grounded normative proposition, CDC allows exposure and examination of 'uncivil' forces stemming from contending claims on citizenship. These factors are typically ignored or denied in an historical harmony model of societal change. A CDC narrative is illustrated by reference to contemporary examples of citizen action that play out at multiple sites of governance

Towards a Formal Verification of Process Model's Properties - SimplePDL and TOCL Case Study

International audienceMore and more, models, through Domain Specific Languages (DSL), tend to be the solution to define complex systems. Expressing properties specific to these metamodels and checking them appear as an urgent need. Until now, the only complete industrial solutions that are available consider structural properties such as the ones that could be expressed in OCL. There are although some attempts on behavioural properties for DSL. This paper addresses a method to specify and then check temporal properties over models. The case study is SimplePDL, a process metamodel. We propose a way to use a temporal extension of OCL, TOCL, to express properties. We specify a models transformation to Petri Nets and LTL formulae for both the process model and its associated temporal properties. We check these properties using a model checker and enrich the model with the analysis results. This work is a first step towards a generic framework to specify and effectively check temporal properties over arbitrary models

Labeled natural deduction for temporal logics

Nonostante la notevole rilevanza delle logiche temporali in molti campi dell'informatica, la loro analisi teorica non è certo da ritenersi conclusa. In particolare, molti sono i punti ancora aperti nell'ambito della teoria della dimostrazione, specialmente se consideriamo le logiche temporali di tipo branching. Il principale contributo di questa tesi consiste nella presentazione di un approccio modulare per la definizione di sistemi di deduzione naturale etichettata per un'ampia gamma di logiche temporali. Viene innanzitutto proposto un sistema per la logica temporale minimale di Prior; si mostra quindi come estenderlo in maniera modulare allo scopo di trattare logiche più complesse, quali ad esempio LTL. Viene infine proposta un'estensione al caso delle logiche branching, concentrando l'attenzione sulle logiche con semantica di tipo Ockhamist e bundled. Per i sistemi proposti, viene condotta una dettagliata analisi dal punto di vista della teoria della dimostrazione. In particolare, nel caso delle logiche del tempo discreto, per le quali si richiedono regole che modellino un principio di induzione, viene definita una procedura di normalizzazione ispirata da quelle dei sistemi per l'Aritmetica di Heyting. Come conseguenza, si ottiene una dimostrazione puramente sintattica della consistenza dei sistemi.Despite the great relevance of temporal logics in many applications of computer science, their theoretical analysis is far from being concluded. In particular, we still lack a satisfactory proof theory for temporal logics and this is especially true in the case of branching-time logics. The main contribution of this thesis consists in presenting a modular approach to the definition of labeled (natural) deduction systems for a large class of temporal logics. We start by proposing a system for the basic Priorean tense logic and show how to modularly enrich it in order to deal with more complex logics, like LTL. We also consider the extension to the branching case, focusing on the Ockhamist branching-time logics with a bundled semantics. A detailed proof-theoretical analysis of the systems is performed. In particular, in the case of discrete-time logics, for which rules modeling an induction principle are required, we define a procedure of normalization inspired to those of systems for Heyting Arithmetic. As a consequence of normalization, we obtain a purely syntactical proof of the consistency of the systems