Horn Fragments of the Halpern-Shoham Interval Temporal Logic

Abstract

We investigate the satisfiability problem for Horn fragments of the Halpern-Shoham interval temporal logic depending on the type (box or diamond) of the interval modal operators, the type of the underlying linear order (discrete or dense), and the type of semantics for the interval relations (reflexive or irreflexive). For example, we show that satisfiability of Horn formulas with diamonds is undecidable for any type of linear orders and semantics. On the contrary, satisfiability of Horn formulas with boxes is tractable over both discrete and dense orders under the reflexive semantics and over dense orders under the irreflexive semantics, but becomes undecidable over discrete orders under the irreflexive semantics. Satisfiability of binary Horn formulas with both boxes and diamonds is always undecidable under the irreflexive semantics.

The Temporal Logic of Two Dimensional Minkowski Spacetime Is Decidable

“…If f + is constantly equal to c on an open line segment l and defined at the left end of l then f + also equals c there. 3 Similarly for f − with right ends. All diagonal duals of statements in this paragraph (that is, statements for vertical lines) hold similarly.…”

“…In [6], Halpern and Shoham introduced a family of modal logics in which the entities under discussion are intervals. These logics have subsequently come to be highly influential, and their axiomatisability, decidability, and complexity extensively studied [19,2,9,4,11,10,3]. In Section 7, we describe how the proof of the complexity of validity for two-dimensional Minkowski spacetime can be adapted to prove PSPACE-completeness of the temporal logic of intervals where the accessibility relation is overlaps ∪ meets ∪ before, or its reflexive closure.…”

On Disallowing Punctual Intervals in Reflexive Semantics of Halpern-Shoham Logic

“…To obtain the new results we show a trick that enables us to mimic under (≤, S, Dis) proof techniques used in the irreflexive semantics [5]. We observe that under reflexive semantics the adjacent relation between intervals is not irreflexive if punctual intervals are allowed.…”

“…Since the satisfiability problem of HS-formulas is undecidable a number of syntactical and semantical modifications of the logic have been studied [8,3,4,5,6,11]. One of the ideas is to weaken semantics of Allen's relations, by the so-called softening [10].…”

“…The obtained relations are known as reflexive semantics, as most of them become reflexive. Other modifications related to this paper consist of imposing additional conditions on the order of time points (e.g., discreteness or density), disallowing punctual intervals, and syntactically restricting the set of well-formed formulas [2,5].…”