Back from the future

Abstract

Until is a notoriously difficult temporal operator as it is both existential and universal at the same time: AUB 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 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 N(LTL ∇ ) for a linear-time logic LTL ∇ endowed with the new history operator. We show that LTL ∇ is equivalent to the linear temporal logic LTL with until, which follows by formalizing back and forth translations between the two logics. We also define an indirect translation from LTL ∇ into LTL via temporal logics with past operators; such a result provides an upper bound to the problem of satisfiability for LTL ∇ formulas.

A journey in modal proof theory: From minimal normal modal logic to discrete linear temporal logic

“…The purpose of this paper is to provide a uniform proof theoretic treatment of the modal standard universal (namely ✷) and existential (namely ✸) quantifiers, in various contexts, from the simplest modal logics (the minimal K system) to multimodal systems like LTL. For this reason, following [2,4,19,20,3], we study only the Until-free fragment of LTL. Indeed, Until is complex, 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 an instant w ′ in the future at which B holds and such that A holds at all instants between w and w ′ .…”

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

“…One of the most successful proof-theoretical formulations of modal logics are the labelled systems of [24,28,30], which extend ordinary natural deduction by explicitly mirroring in the deductive apparatus the accessibility relation of Kripke models (see also [3,5,6,[19][20][21][22][23]). In a sense, they may look like a formalization of Kripke semantics in a first-order deductive fashion (see Section 9.1, below, for a more complete discussion).…”

“…The strengths of these systems, on the other hand, become apparent when expressivity comes into the spotlight-both Simpson's and Viganò's proposals accommodate a large class of complex modal and temporal logics [3,4,6,[19][20][21][22][23], and have been successively formulated also as sequent calculi [24].…”