Problem of Decidability for Some Constructive Theories of Equalities

Suppose T is a first order intuitionistic theory (more precisely, a theory of Heyting's predicate calculus, e.g., abelian groups, one unary function, dense linear order, etc.) presented to us by a set of axioms (denoted also by) T.Question. Is T decidable?One knows that if the classical counterpart of T (i.e., take the same axioms but with the classical predicate calculus as the underlying logic) is not decidable, then T cannot be decidable. The problem remains for theories whose classical counterpart is decidable. In [8], sufficient conditions for undecidability were given, and several intuitionistic theories such as abelian groups and unary functions (both with decidable equality) were shown to be undecidable. In this note we show decidability results (see Theorems 1 and 2 below), and compare these results with the undecidability results previously obtained. The method we use is the reduction-method, described fully in [12] and widely applied in [3], which is applied here roughly as follows:Let T be a given theory of Heyting's predicate calculus. We know that Heyting's predicate calculus is complete for the Kripke-model type of semantics. We choose a class M of Kripke models for which T is complete, i.e., all axioms of T are valid in any model of the class and whenever φ is not a theorem of T, φ is false in some model of M.

“…The decidability of the intuitionistic theory of decidable equality was proved syntactically by Lifshits [2] as well as the undecidability of the intuitionistic theory of equality.…”

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confidence: 99%

Decidability of some intuitionistic predicate theories

Gabbay

1972

“…As in the preceding examples, we may reduce the problem to that of finding a normal form for formulae of the form (14) V…”

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confidence: 99%

“…The proof of the equivalence of (15) with (16) is similar to that for (12) and (13) given above and also proves the equivalence of (14) with its corresponding formula of the form of (16). We omit the proof and the precise description of the corresponding formula for (14) on the grounds that both the description and the proof should, by now, be sufficiently clear to the reader. §D.…”

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confidence: 99%

“…With regard to more elementary theories, in [25], Scott proved the completeness of the propositional calculus (in the sense of the note, above) for his model of the theory of the order of the reals. As discussed in §111, Lifshits [15] established the completeness of the monadic predicate calculus on one predicate with respect to substitutional validity (there called deductive validity-see also [14]) within the theory of equality. And, finally, in §111, we established the completeness of M x within several further theories.…”

Section: Apartness ~\{X Xmentioning

confidence: 99%

See 1 more Smart Citation

Elementary intuitionistic theories

Smoryñski

1973

The present paper concerns itself primarily with the decision problem for formal elementary intuitionistic theories and the method is primarily model-theoretic. The chief tool is the Kripke model for which the reader may find sufficient background in Fitting's book Intuitionistic logic model theory and forcing (North-Holland, Amsterdam, 1969). Our notation is basically that of Fitting, the differences being to favor more standard notations in various places.The author owes a great debt to many people and would particularly like to thank S. Feferman, D. Gabbay, W. Howard, G. Kreisel, G. Mints, and R. Statman for their valuable assistance.The method of elimination of quantifiers, which has long since proven its use in classical logic, has also been applied to intuitionistic theories (i) to demonstrate decidability ([9], [15], [17]), (ii) to prove the coincidence of an intuitionistic theory with its classical extension ([9], [17]), and (iii), as we will see below, to establish relations between an intuitionistic theory and its classical extension. The most general of these results is to be obtained from the method of Lifshits' quantifier elimination for the intuitionistic theory of decidable equality.Since the details of Lifshits' proof have not been published, and since the proof yields a more general result than that stated in his abstract [15], we include the proof and several corollaries.

“…Orevkov [5] gave proofs of the undecidability of one letter in S5*. Other undecidability results for intuitionistic theories are given in Smorynski [6], and Lifshits [11]. See also [7] for an example of an intermediate logic where the theory of one unary predicate is decidable but the theory of two is not.…”

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confidence: 99%

Sufficient conditions for the undecidability of intuitionistic theories with applications

Gabbay¹

1972

Let Δ be a set of axioms of a theory Tc(Δ) of classical predicate calculus (CPC); Δ may also be considered as a set of axioms of a theory TH(Δ) of Heyting's predicate calculus (HPC). Our aim is to investigate the decision problem of TH(Δ) in HPC for various known theories Δ of CPC.Theorem I(a) of §1 states that if Δ is a finitely axiomatizable and undecidable theory of CPC then TH(Δ) is undecidable in HPC. Furthermore, the relations between theorems of HPC are more complicated and so two CPC-equivalent axiomatizations of the same theory may give rise to two different HPC theories, in fact, one decidable and the other not.Semantically, the Kripke models (for which HPC is complete) are partially ordered families of classical models. Thus a formula expresses a property of a family of classical models (i.e. of a Kripke model). A theory Θ expresses a set of such properties. It may happen that a class of Kripke models defined by a set of formulas Θ is also definable in CPC (in a possibly richer language) by a CPC-theory Θ′! This establishes a connection between the decision problem of Θ in HPC and that of Θ′ in CPC. In particular if Θ′ is undecidable, so is Θ. Theorems II and III of §1 give sufficient conditions on Θ to be such that the corresponding Θ′ is undecidable. Θ′ is shown undecidable by interpreting the CPC theory of a reflexive and symmetric relation in Θ′.