We formulate some term rewriting systems in which the number of computation steps is finite for each output, but this number cannot be bounded by a provably total computable function in Peano arithmetic PA. Thus, the termination of such systems is unprovable in PA. These systems are derived from an independent combinatorial result known as the Worm principle; they can also be viewed as versions of the well-known Hercules-Hydra game introduced by Paris and Kirby. Bibliography: 16 titles.
In this paper we study the propositional fragment
of the joint logic of problems and propositions introduced by Melikhov. We provide Kripke semantics for this logic and show that
is complete with respect to those models and has the finite model property. We consider examples of the use of
-models usage. In particular, we prove that
is a conservative extension of the logic
. We also show that the logic
is complete with respect to Kripke frames with sets of audit worlds introduced by Artemov and Protopopescu (who called them audit set models).
Bibliography: 31 titles.
We consider the joint logic of problems and propositions $\mathrm{QHC}$ introduced by Melikhov. We construct Kripke models with audit worlds for this logic and prove the soundness and completeness of $\mathrm{QHC}$ with respect to this type of model. The conservativity of the logic $\mathrm{QHC}$ over the intuitionistic modal logic $\mathrm{QH4}$, which coincides with the ‘lax logic’ $\mathrm{QLL}^+$, is established. We construct Kripke models with audit worlds for the logic $\mathrm{QH4}$ and prove the corresponding soundness and completeness theorems. We also prove that the logics $\mathrm{QHC}$ and $\mathrm{QH4}$ have the disjunction and existence properties.
Bibliography: 33 titles.
Формулируются системы преобразований термов, число шагов работы которых на произвольном входе конечно, но не ограничивается никакой вычислимой функцией, доказуемо тотальной в арифметике Пеано $\mathsf{PA}$. Тем самым, утверждение о сходимости таких систем не доказуемо в $\mathsf{PA}$. Эти системы получаются из независимого комбинаторного утверждения, известного как принцип червя; их также можно рассматривать как вариант хорошо известной игры Геракла и гидры, введенной Дж. Парисом и Л. Кирби.
Библиография: 16 названий.
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