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Investigation on fragments of first order branching temporal logic. (English) Zbl 1002.03016
Various fragments of first-order computational tree logic (FOCTL) are investigated. The fragment of the logic with only the operator F (sometimes in the future) is not axiomatizable. This is shown by a possible embedding of arithmetic into it. The proof can be extended to first-order linear time logic. It is also proved that the logic with the past operator H (always in the past) is not axiomatizable as well. The proof is done by showing that the set of valid formulae of $$\mathbf{FOCTL}_{\mathbf H}$$ is $$\Pi^0_2$$-complete. The only axiomatizable fragment is the one with the next operator (X).

##### MSC:
 03B44 Temporal logic 03B25 Decidability of theories and sets of sentences 03B70 Logic in computer science
##### Keywords:
first-order branching temporal logic; axiomatization; FOCTL
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##### References:
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