On the strength of temporal proofs (original) (raw)

A propositional linear time logic with time flow isomorphic to

Journal of Applied Logic, 2014

Primarily guided with the idea to express zero-time transitions by means of temporal propositional language, we have developed a temporal logic where the time flow is isomorphic to ordinal ω 2 (concatenation of ω copies of ω). If we think of ω 2 as lexicographically ordered ω×ω, then any particular zero-time transition can be represented by states whose indices are all elements of some {n} × ω. In order to express noninfinitesimal transitions, we have introduced a new unary temporal operator [ω] (ω-jump), whose effect on the time flow is the same as the effect of α → α + ω in ω 2 . In terms of lexicographically ordered ω × ω, [ω]φ is satisfied in i, j -th time instant iff φ is satisfied in i + 1, 0 -th time instant. Moreover, in order to formally capture the natural semantics of the until operator U, we have introduced a local variant u of the until operator. More precisely, φ uψ is satisfied in i, j -th time instant iff ψ is satisfied in i, j + k -th time instant for some nonnegative integer k, and φ is satisfied in i, j + l -th time instant for all 0 l < k. As in many of our previous publications, the leitmotif is the usage of infinitary inference rules in order to achieve the strong completeness.

Labeled Natural Deduction for Temporal Logics

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 minimal 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.

Simplifying Inductive Schemes in Temporal Logic

2019

In propositional temporal logic, the combination of the connectives “tomorrow” and “always in the future” require the use of induction tools. In this paper, we present a classification of inductive schemes for propositional linear temporal logic that allows the detection of loops in decision procedures. In the design of automatic theorem provers, these schemes are responsible for the searching of efficient solutions for the detection and management of loops. We study which of these schemes have a good behavior in order to give a set of reduction rules that allow us to compute these schemes efficiently and, therefore, be able to eliminate these loops. These reduction laws can be applied previously and during the execution of any automatic theorem prover. All the reductions introduced in this paper can be considered a part of the process for obtaining a normal form of a given formula. 2012 ACM Subject Classification Theory of computation → Modal and temporal logics

Induction in First-Order Logic with Temporal Metric Operators

A preliminary exploration of induction in the realm of first-order logic enriched with temporal metric operators is presented. The Interaction Modelling Language (IMOLA) is formally defined, it is shown to have a lattice structure with respect to an extended subsumption relation and locally finite and complete refinement operators are introduced.

Alternating-Time Temporal Logic in the Calculus of (Co)Inductive Constructions

Lecture Notes in Computer Science, 2012

This work presents a complete formalization of Alternatingtime Temporal Logic (ATL) and its semantic model, Concurrent Game Structures (CGS), in the Calculus of (Co)Inductive Constructions, using the logical framework Coq. Unlike standard ATL semantics, temporal operators are formalized in terms of inductive and coinductive types, employing a fixpoint characterization of these operators. The formalization is used to model a concurrent system with an unbounded number of players and states, and to verify some properties expressed as ATL formulas. Unlike automatic techniques, our formal model has no restrictions in the size of the CGS, and arbitrary state predicates can be used as atomic propositions of ATL.

Temporal logics need their clocks

Theoretical Computer Science, 1992

In this paper we solve some open problems raised in recent publications of the Computer Science Temporal Logic school represented by Manna-Pnueli [11], [12], Abadi-Manna [5], Abadi [1]-[4]. These problems concern the proof theoretic powers of the following inference systems: T 0 introduced in [11], [12] and reformulated in [1]-[4]; the resolution system R of [5]; and T 1 , T 2 of [1]-[4]. We use first-order temporal logic (FTL) with modalities , [F ], and U denoting "nexttime", "always-in-the-future", and "until" respectively. Given a first-order similarity type or language L, the usual predicate etc. symbols of L are considered to be rigid, i.e. their meanings do not change in time. Similarly, individual variables x i (i ∈ ω) are rigid. To this we add an infinity y i (i ∈ ω) of flexible constants. That is, the meaning of y i is allowed to change in time. Other authors, see e.g. Abadi [1]-[4], add flexible predicates too, but we will not need them here though we will mention them occasionally. Our theorems remain true even if we allow flexible predicateand function symbols, as it will be very easy to see. F m(F T L) denotes the set of all FTL-formulas (of some fixed similarity type L) defined above. For semantic purposes, we use classical two-sorted models M = < T, D, f 0 ,. .. , f i ,. .. > i∈ω where D is a classical first-order structure of similarity type L, T = < T, 0, suc, ≤, +, × > is a structure of the same similarity type as the standard model N = < ω, 0, suc, ≤, +, × > of arithmetic, and for i ∈ ω, f i , a function from T into D, serves to interpret the flexible constant y i. T is called the time-frame of M, and, except for its language, is arbitrary. M od denotes the class of all models M of the above kind. (The members of M od are basically the same as Kripke models known from the

A Complete Proof System for First Order Interval Temporal Logic with Projection

This paper presents an ω-complete proof system for the extension of first order Interval Temporal Logic (ITL,) by a projection operator . Alternative earlier approaches to the axiomatisation of projection in ITL are briefly presented and discussed. An extension of the proof system which is complete for the extension of Duration Calculus (DC ,) by projection is also given.