G\"odel-Dummett linear temporal logic
- URL: http://arxiv.org/abs/2306.15805v1
- Date: Tue, 27 Jun 2023 21:19:52 GMT
- Title: G\"odel-Dummett linear temporal logic
- Authors: Juan Pablo Aguilera and Mart\'in Di\'eguez and David Fern\'andez-Duque
and Brett McLean
- Abstract summary: We investigate a version of linear temporal logic whose propositional fragment is G"odel-Dummett logic.
We define the logic using two natural semantics first a real-valued semantics, where statements have a degree of truth in the real unit interval.
We show that these two semantics indeed define one and the same logic.
This G"odel temporal logic does not have any form of the finite model property for these two semantics.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: We investigate a version of linear temporal logic whose propositional
fragment is G\"odel-Dummett logic (which is well known both as a
superintuitionistic logic and a t-norm fuzzy logic). We define the logic using
two natural semantics: first a real-valued semantics, where statements have a
degree of truth in the real unit interval and second a `bi-relational'
semantics. We then show that these two semantics indeed define one and the same
logic: the statements that are valid for the real-valued semantics are the same
as those that are valid for the bi-relational semantics. This G\"odel temporal
logic does not have any form of the finite model property for these two
semantics: there are non-valid statements that can only be falsified on an
infinite model. However, by using the technical notion of a quasimodel, we show
that every falsifiable statement is falsifiable on a finite quasimodel,
yielding an algorithm for deciding if a statement is valid or not. Later, we
strengthen this decidability result by giving an algorithm that uses only a
polynomial amount of memory, proving that G\"odel temporal logic is
PSPACE-complete. We also provide a deductive calculus for G\"odel temporal
logic, and show this calculus to be sound and complete for the above-mentioned
semantics, so that all (and only) the valid statements can be proved with this
calculus.
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