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

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

On non-local propositional and weak monodic quantified CTL

Accepted versio

A History of Until

Until is a notoriously difficult temporal operator as it is both existential and universal at the same time: A until B holds at the current time instant w iff either B holds at w or there exists a time instant w' in the future at which B holds and such that A holds in all the time instants between the current one and w'. This "ambivalent" nature poses a significant challenge when attempting to give deduction rules for until. In this paper, in contrast, we make explicit this duality of until to provide well-behaved natural deduction rules for linear-time logics by introducing a new temporal operator that allows us to formalize the "history" of until, i.e., the "internal" universal quantification over the time instants between the current one and w'. This approach provides the basis for formalizing deduction systems for temporal logics endowed with the until operator. For concreteness, we give here a labeled natural deduction system for a linear-time logic endowed with the new operator and show that, via a proper translation, such a system is also sound and complete with respect to the linear temporal logic LTL with until.Comment: 24 pages, full version of paper at Methods for Modalities 2009 (M4M-6

From 2-sequents and Linear Nested Sequents to Natural Deduction for Normal Modal Logics

We extend to natural deduction the approach of Linear Nested Sequents and 2-sequents. Formulas are decorated with a spatial coordinate, which allows a formulation of formal systems in the original spirit of natural deduction---only one introduction and one elimination rule per connective, no additional (structural) rule, no explicit reference to the accessibility relation of the intended Kripke models. We give systems for the normal modal logics from K to S4. For the intuitionistic versions of the systems, we define proof reduction, and prove proof normalisation, thus obtaining a syntactical proof of consistency. For logics K and K4 we use existence predicates (following Scott) for formulating sound deduction rules

Advanced Proof Viewing in ProofTool

Sequent calculus is widely used for formalizing proofs. However, due to the proliferation of data, understanding the proofs of even simple mathematical arguments soon becomes impossible. Graphical user interfaces help in this matter, but since they normally utilize Gentzen's original notation, some of the problems persist. In this paper, we introduce a number of criteria for proof visualization which we have found out to be crucial for analyzing proofs. We then evaluate recent developments in tree visualization with regard to these criteria and propose the Sunburst Tree layout as a complement to the traditional tree structure. This layout constructs inferences as concentric circle arcs around the root inference, allowing the user to focus on the proof's structural content. Finally, we describe its integration into ProofTool and explain how it interacts with the Gentzen layout.Comment: In Proceedings UITP 2014, arXiv:1410.785

Sublogics of a Branching Time Logic of Robustness

In this paper we study sublogics of RoCTL*, a recently proposed logic for specifying robustness. RoCTL* allows specifying robustness in terms of properties that are robust to a certain number of failures. RoCTL* is an extension of the branching time logic CTL* which in turn extends CTL by removing the requirement that temporal operators be paired with path quantifiers. In this paper we consider three sublogics of RoCTL*. We present a tableau for RoBCTL*, a bundled variant of RoCTL* that allows fairness constraints to be placed on allowable paths. We then examine two CTL-like restrictions of CTL*. Pair-RoCTL* requires a temporal operator to be paired with a path quantifier; we show that Pair-RoCTL* is as hard to reason about as the full CTL*. State-RoCTL* is restricted to State formulas, and we show that there is a linear truth preserving translation of State-RoCTL into CTL, allowing State-RoCTL to be reasoned about as efficiently as CTL

Certified Computation

This paper introduces the notion of certified computation. A certified computation does not only produce a result r, but also a correctness certificate, which is a formal proof that r is correct. This can greatly enhance the credibility of the result: if we trust the axioms and inference rules that are used in the certificate,then we can be assured that r is correct. In effect,we obtain a trust reduction: we no longer have to trust the entire computation; we only have to trust the certificate. Typically, the reasoning used in the certificate is much simpler and easier to trust than the entire computation. Certified computation has two main applications: as a software engineering discipline, it can be used to increase the reliability of our code; and as a framework for cooperative computation, it can be used whenever a code consumer executes an algorithm obtained from an untrusted agent and needs to be convinced that the generated results are correct. We propose DPLs (Denotational Proof Languages)as a uniform platform for certified computation. DPLs enforce a sharp separation between logic and control and over versatile mechanicms for constructing certificates. We use Athena as a concrete DPL to illustrate our ideas, and we present two examples of certified computation, giving full working code in both cases