Incompleteness of first-order temporal logic with until

First Order Logic and Gödel Incompleteness

Incompleteness of a First-Order Gödel Logic and Some Temporal Logics of Programs

It is shown that the infinite-valued first-order Gödel logic G based on the set of truth values {1/k : k ∈ ω \ {0}} ∪ {0} is not r.e. The logic G is the same as that obtained from the Kripke semantics for first-order intuitionistic logic with constant domains and where the order structure of the model is linear. From this, the unaxiomatizability of Kröger’s temporal logic of programs (even of t...

On Incompleteness of Multi-dimensional First-order Temporal Logics

In this paper we show that first-order temporal logics form a proper expressiveness hierarchy with respect to dimensionality and quantifier depth of temporal connectives. This result resolves (negatively) the open question concerning the existence of an expressively complete first-order temporal logic, even when allowing multi-dimensional temporal connectives.

First-Order Logic with Two Variables and Unary Temporal Logic

We investigate the power of first-order logic with only two variables over ω-words and finite words, a logic denoted by FO. We prove that FO can express precisely the same properties as linear temporal logic with only the unary temporal operators: “next”, “previously”, “sometime in the future”, and “sometime in the past”, a logic we denote by unary-TL. Moreover, our translation from FO to unary...

Proof Planning for First-Order Temporal Logic

Proof planning is an automated reasoning technique which improves proof search by raising it to a meta-level. In this paper we apply proof planning to First-Order Linear Temporal Logic (FOLTL), which can be seen as a quantified version of Linear Temporal Logic, overcoming its finitary limitation. Automated reasoning in FOLTL is hard, since it is non-recursively enumerable; but its expressivenes...