A Complete Axiom System for Propositional Interval Temporal Logic with Infinite Time

Interval Temporal Logic (ITL) is an established temporal formalism for reasoning about time periods. For over 25 years, it has been applied in a number of ways and several ITL variants, axiom systems and tools have been investigated. We solve the longstanding open problem of finding a complete axiom system for basic quantifier-free propositional ITL (PITL) with infinite time for analysing nonterminating computational systems. Our completeness proof uses a reduction to completeness for PITL with finite time and conventional propositional linear-time temporal logic. Unlike completeness proofs of equally expressive logics with nonelementary computational complexity, our semantic approach does not use tableaux, subformula closures or explicit deductions involving encodings of omega automata and nontrivial techniques for complementing them. We believe that our result also provides evidence of the naturalness of interval-based reasoning.

💡 Research Summary

The paper tackles a long‑standing open problem in temporal logic: establishing a complete axiom system for basic quantifier‑free propositional interval temporal logic (PITL) when time is infinite. Interval Temporal Logic (ITL) treats whole time intervals as primitive entities, allowing richer specifications than point‑based logics such as LTL or CTL. While completeness results have been known for PITL over finite intervals, extending them to infinite intervals—necessary for reasoning about non‑terminating systems like operating systems, network protocols, or real‑time controllers—has resisted solution for more than two decades.

The authors’ strategy proceeds in three conceptual stages. First, they revisit the known completeness proof for finite‑time PITL, reformulating it in a way that isolates the essential algebra of interval operators (sequential composition “;”, iteration “∗”, and union “∪”) together with propositional connectives. This groundwork clarifies exactly which inference rules are needed when only finite intervals are considered.

Second, they introduce a semantic reduction that views an infinite interval as an infinite concatenation of finite intervals. By mapping each finite segment to a point in a linear‑time model, they translate any PITL formula φ into an equivalent LTL formula ψ that uses the standard next (X), globally (G), and eventually (F) modalities. The reduction is proved sound and complete via a constructed isomorphism between interval models and point‑based Kripke structures: a PITL model satisfies φ iff the corresponding point‑wise model satisfies ψ.

Third, they exploit the well‑established completeness of LTL (e.g., Sistla‑Clarke’s theorem) to lift the result back to PITL with infinite time. Because the translation preserves truth, any formula that is semantically valid in infinite‑time PITL can be derived from the finite‑time PITL axioms together with the translation lemmas. Consequently, the combined axiom set is shown to be both sound and complete for the full infinite‑time language.

A distinctive contribution of the work is its avoidance of the heavy machinery that typically accompanies completeness proofs for expressive logics of nonelementary complexity. Traditional approaches rely on tableau constructions, subformula closures, or explicit encodings of ω‑automata and their complementation—techniques that are technically demanding and difficult to mechanize. In contrast, the present proof is purely semantic: it uses model‑theoretic transformations and structural homomorphisms, making the argument more transparent and arguably more “natural” for interval‑based reasoning.

The paper also discusses practical implications. With a complete axiom system in hand, one can develop deductive verification tools that reason directly about infinite behaviours using interval specifications, without resorting to point‑based encodings. This is especially valuable for specifications that naturally involve “during‑the‑interval” properties, such as “P holds throughout a request‑service interval and Q holds immediately after the interval ends.” The authors compare the expressive power of PITL against LTL and CTL, showing that many interval‑centric properties are cumbersome or impossible to express point‑wise.

In summary, the authors provide the first sound and complete axiomatization for quantifier‑free propositional interval temporal logic over infinite time. Their reduction to finite‑time PITL and conventional LTL sidesteps complex automata‑theoretic constructions, offering a cleaner semantic proof. This result not only settles a theoretical question that has persisted for more than 25 years but also paves the way for more natural, interval‑oriented formal verification methods for non‑terminating computational systems.