Horn Fragments of the Halpern-Shoham Interval Temporal Logic (Technical Report)

TL;DR

This paper analyzes the computational complexity of Horn fragments of the Halpern-Shoham interval temporal logic, revealing decidability and undecidability results depending on the operators, order types, and semantics.

Contribution

It provides a comprehensive classification of satisfiability problems for Horn fragments of the logic based on various parameters, highlighting new decidability boundaries.

Findings

Satisfiability with diamonds is undecidable for all order types and semantics.

Satisfiability with boxes is decidable in several cases, including over dense orders with reflexive semantics.

Binary Horn formulas with both boxes and diamonds are always undecidable under irreflexive semantics.

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.