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In this paper we introduce effectiveness into model theory of intuitionistic logic. ... The main result shows that any computable theory T of intuitionistic predicate logic has a Kripke model with decidabIe forcing such that for any sentence 4, 4 is forced in the model if and only if C$ is ... Acknowledgements T is complete for the class of We would like to thank the referee for valuable comments on improvements of the paper. ...doi:10.1016/s0168-0072(97)00057-2 fatcat:7febp3f57bdyjecvq5sz6bnzta
In particular, we describe a class of formulae for which such conservativity results can be proven in case of any intuitionistic theory T which is complete with respect to a class of T-normal Kripke models ... We also prove conservativity results for intuitionistic theories which are closed under the Friedman translation and complete with respect to a class of conversely well-founded Kripke models. ... Let us note that not every T c -normal Kripke model is a model of the intuitionistic theory T i . Also, a Kripke model of a theory T i need not be T c -normal. ...doi:10.1007/s11225-015-9639-7 fatcat:ac3wgrynqzfv7p2enoj5x2lwia
We introduce effectiveness considerations into model theory of intuitionistic logic. ... We proceed by looking in detail at how one went from model theory of classical first-order logic to model theory of intuitionistic logic. ... Every computable intuitionistic theory T is complete for the class of decidable Kripke models. ...doi:10.1006/inco.1998.2704 fatcat:kjwfg7am4rcmnjlxamvxe3btde
Our method is mostly semantical, we use Kripke models of the theories. 2000 Mathematics Subject Classification: 03F30, 03F55, 03F50, 68Q15. ... In this note we show that the intuitionistic theory of polynomial induction on Π b+ 1 -formulas does not imply the intuitionistic theory IS 1 2 of polynomial induction on Σ b+ 1 -formulas. ... Acknowledgements Final version of this paper to appear in the Journal of Logic and Computation. ...doi:10.1093/logcom/13.6.881 fatcat:six76ujd45ajhprhoq7t4trxfm
An interesting corollary to be drawn immediately from this observation is the following: a decidable intuitionistic theory has a recursive Kripke model; and for decidable intuitionistic theories completeness ... (i) of the preceding theorem is the exact intuitionistic analogue of the well-known classical result that consistent decidable theories have recursive models, and improves on theorem 1 on page 84 of [ ...doi:10.1016/1385-7258(78)90047-1 fatcat:2kdvh4pqqndrzcisj2qpxrnqx4
the chain as a Kripke model. ... It is shown that the intuitionistic theory of polynomial induction on positive Π b 1 (coNP) formulas does not prove the sentence ¬¬∀x, y∃z ≤ y(x ≤ |y| → x = |z|). ... Corollary 2.3 Suppose K is a weak end extension Kripke model whose accessibility relation is ω and decides atomic formulas. ...doi:10.1093/logcom/exi085 fatcat:gxdlsvxk6fgjbnegws5ua7ap74
We include some remarks on the classical worlds in Kripke models of iop. ... This is established by constructing a Kripke model of iop + ¬L y (2y > x), where L y (2y > x) is universally quantified on x. ... The intuitionistic theory of the class of T -normal Kripke structures is denoted H(T ). It was shown in [AM, 1.2(ii, iii), 1.4, 2.3(ii) ] that Kripke models of lop (resp. ...doi:10.2178/jsl/1190150031 fatcat:fjk7xr7v7jf7lcl5mrlcfkwtre
An int erest ing corollary to be drawn immediately from this observation is the following : a decidable intuitionistic theory has a recursive Kripke model; and for decidable intuitionistic theories completeness ... If we apply the method of proof of theorem A to an r.e. theory r (instead of a decidable theory) we find the following PROPOSITION. ...doi:10.1016/s1385-7258(78)80020-1 fatcat:7rs6qiskzbaxhj3qnyra4zmgka
We construct ω-framed Kripke models of i∀ 1 and iΠ 1 non of whose worlds satisfies ∀x∃y(x = 2y ∨ x = 2y + 1) and ∀x, y∃zExp(x, y, z) respectively. ... A formula ϕ(x) is decidable in a Kripke model K whenever K ∀x(ϕ(x) ∨ ¬ϕ(x)). For a set T of sentences, T i and T c denote its intuitionistical and classical deductive closures. ... By [K, P. 69] , each node of this Kripke model is a ∆ o -elementary substructure of M (therefore models Π 1 -theory I∆ 0 ) and non of them satisfies exp. ...doi:10.1007/s00153-003-0189-8 fatcat:nmvgu6p24zefhko3vkqruoiizu
He calls the models ancestral Kripke models, inductive ancestral Kripke models, and nonhereditary Kripke models, respectively, and proves that with respect to each of these classes the Heyting propositional ... (NL-AMST-PR) The expressive force of some fragments of intuitionistic propositional logic with regard to Kripke frames. Z. Math. Logik Grundlag. Math. 37 (1991), no. 4, 357-362. ...
Define % to be a submodel of the Kripke model # if % can be obtained from # by removing nodes. Main Theorem: Let T C U be first-order theories. ... Then the class of Kripke models of U is closed under all submodels that are models of T if and only if U is axiomatizable by T plus a set of semipositive sentences. ...
We show that, if the intuitionistic theory of polynomial induction on strict Π b 2 formulas proves decidability of Σ b 1 formulas, then P = NP. ... These results provide simple proofs for some known results in the model theory of the bounded arithmetic theories like CPV and PV 1 . ... For a semantical investigation of this theory, see [M1] , where Kripke models of intuitionistic bounded arithmetic is studied. ...doi:10.1007/s00153-006-0016-0 fatcat:jqbplx6rgncjpbdi2ibi4hw53m
Lecture Notes in Computer Science
This paper presents a formalization of a sequent presentation of intuitionisitic propositional logic and proof of decidability. ... If the argument to the resulting decision procedure is a valid sequent, a formal proof of that fact is returned, otherwise a counter-example in the form of a Kripke Countermodel is returned. ... With the proof of decidability as our focus, we describe the formal development of a sequent proof theory, the tableau construction, and a formal theory of Kripke counter-examples which are used here as ...doi:10.1007/3-540-48754-9_11 fatcat:wnznurcr4rf5pbneanbhpqhaau
As a remedy, we propose the use of circumscribed definite rules, and we then investigate the proof theory of this more expressive framework. ... circumscribed theories that are not equivalent to any first order theory. ... R has a final Kripke model, which we denote by K*. ...dblp:conf/ijcai/McCartyM91 fatcat:2ecufuexqncd5bc2crpgjbrjyy
V. (2-KRAS) Admissibility of inference rules with parameters in intuitionistic logic, and intuitionistic Kripke models. (Russian) Dokl. Akad. ... Since the algorithm uses exclusively intuitionistic means because it depends only on the properties of n-characteristic Kripke models, the result can be considered as providing a solution to Problem 40 ...
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