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Gödel logics and the fully boxed fragment of LTL
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In this paper we show that a very basic fragment of FO-LTL, the monadic fully boxed fragment (all connectives and quantifiers are guarded by P) is not recursively enumerable wrt validity and 1-satisfiability if three predicates are present. This result is obtained by reduction of the fully boxed fragment of FO-LTL to the Gödel logic G↓, the infinitely valued Gödel logic with truth values in [0,1] such that all but 0 are isolated. The result on 1-satisfiability is in no way symmetric to the result on validity as in classical logic: this is demonstrated by the analysis of G↑, the related infinitely-valued Gödel logic with truth values in [0, 1] such that all but 1 are isolated. Validity of the monadic fragment with at least two predicates is not recursively enumerable, 1-satisfiability of the monadic fragment is decidable.
Title: Gödel logics and the fully boxed fragment of LTL
Description:
In this paper we show that a very basic fragment of FO-LTL, the monadic fully boxed fragment (all connectives and quantifiers are guarded by P) is not recursively enumerable wrt validity and 1-satisfiability if three predicates are present.
This result is obtained by reduction of the fully boxed fragment of FO-LTL to the Gödel logic G↓, the infinitely valued Gödel logic with truth values in [0,1] such that all but 0 are isolated.
The result on 1-satisfiability is in no way symmetric to the result on validity as in classical logic: this is demonstrated by the analysis of G↑, the related infinitely-valued Gödel logic with truth values in [0, 1] such that all but 1 are isolated.
Validity of the monadic fragment with at least two predicates is not recursively enumerable, 1-satisfiability of the monadic fragment is decidable.
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