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Decidable Kripke models of intuitionistic theories

Hajime Ishihara, Bakhadyr Khoussainov, Anil Nerode
1998 Annals of Pure and Applied Logic  
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

A Semantic Approach to Conservativity

Tomasz Połacik
2016 Studia Logica: An International Journal for Symbolic Logic  
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

Computable Kripke Models and Intermediate Logics

Hajime Ishihara, Bakhadyr Khoussainov, Anil Nerode
1998 Information and Computation  
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

Comparing Constructive Arithmetical Theories Based on NP-PIND and coNP-PIND

M. Moniri
2003 Journal of Logic and Computation  
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

Some remarks on the complexity of Henkin-Kripke models

A.S. Troelstra
1978 Indagationes Mathematicae (Proceedings)  
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

An Independence Result for Intuitionistic Bounded Arithmetic

Morteza Moniri
2006 Journal of Logic and Computation  
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

Some weak fragments of HA and certain closure properties

Morteza Moniri, Mojtaba Moniri
2002 Journal of Symbolic Logic (JSL)  
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

Some remarks on the complexity of Henkin-Kripke models

A.S. Troelstra
1978 Indagationes Mathematicae (Proceedings)  
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

Intuitionistic weak arithmetic

Morteza Moniri
2003 Archive for Mathematical Logic  
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

Page 568 of Mathematical Reviews Vol. , Issue 93b [page]

1993 Mathematical Reviews  
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.  ... 

Page 4622 of Mathematical Reviews Vol. , Issue 2002G [page]

2002 Mathematical Reviews  
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.  ... 

Preservation theorems for bounded formulas

Morteza Moniri
2006 Archive for Mathematical Logic  
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

Intuitionisitic Tableau Extracted [chapter]

James Caldwell
1999 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

Indefinite Reasoning with Definite Rules

L. Thorne McCarty, Ron van der Meyden
1991 International Joint Conference on Artificial Intelligence  
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

Page 5245 of Mathematical Reviews Vol. , Issue 91J [page]

1991 Mathematical Reviews  
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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