Decision problem for separated distributive lattices

1983 ◽  
Vol 48 (1) ◽  
pp. 193-196 ◽  
Author(s):  
Yuri Gurevich
Keyword(s):  
Decision Problem ◽  
Order Theory ◽  
Distributive Lattices ◽  
First Order ◽  
First Order Theory ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets

AbstractIt is well known that for all recursively enumerable sets X1, X2 there are disjoint recursively enumerable sets Y1 ⊆ Y2 such that Y ⊆ X1, Y2 ⊆ X2 and Y1, ⋃ Y2 = X1 ⋃ X2. Alistair Lachlan called distributive lattices satisfying this property separated. He proved that the first-order theory of finite separated distributive lattices is decidable. We prove here that the first-order theory of all separated distributive lattices is undecidable.

scholarly journals Distributive p-algebras and double p-algebras having n-permutable congruences

1992 ◽  
Vol 35 (2) ◽  
pp. 301-307 ◽  
Author(s):  
R. Beazer
Keyword(s):  
Order Theory ◽  
Distributive Lattices ◽  
Prime Ideals ◽  
First Order ◽  
First Order Theory ◽  
De Morgan Algebras ◽  
Element Chain

Recent research on aspects of distributive lattices, p-algebras, double p-algebras and de-Morgan algebras (see [2] and the references therein) has led to the consideration of the classes (n≧1) of distributive lattices having no n + 1-element chain in their poset of prime ideals. In [1] we were obliged to characterize the members of by a sentence in the first-order theory of distributive lattices. Subsequently (see [2]), it was realised that coincides with the class of distributive lattices having n+1-permutable congruences. This result is hereby employed to describe those distributive p-algebras and double p-algebras having n-permutable congruences. As an application, new characterizations of those distributive p-algebras and double p-algebras having the property that their compact congruences are principal are obtained. In addition, those varieties of distributive p-algebras and double p-algebras having n-permutable congruences are announced.

The decision problem for standard classes

1976 ◽  
Vol 41 (2) ◽  
pp. 460-464 ◽  
Author(s):  
Yuri Gurevich
Keyword(s):  
Decision Problem ◽  
Predicate Logic ◽  
Order Theory ◽  
First Order ◽  
First Order Theory

The standard classes of a first-order theory T are certain classes of prenex T-sentences defined by restrictions on prefix, number of monadic, dyadic, etc. predicate variables, and number of monadic, dyadic, etc. operation variables. In [3] it is shown that, for any theory T, (1) the decision problem for any class of prenex T-sentences specified by such restrictions reduces to that for the standard classes, and (2) there are finitely many standard classes K1, …, Kn such that any undecidable standard class contains one of K1, …, Kn. These results give direction to the study of the decision problem.Below T is predicate logic with identity and operation variables. The Main Theorem solves the decision problem for the standard classes admitting at least one operation variable.

scholarly journals Vaught's Theorem on Axiomatizability by a Scheme

2012 ◽  
Vol 18 (3) ◽  
pp. 382-402 ◽  
Author(s):  
Albert Visser
Keyword(s):  
Set Theory ◽  
Order Theory ◽  
Finite Axiomatizability ◽  
Original Result ◽  
First Order ◽  
First Order Theory ◽  
Recursively Enumerable

AbstractIn his 1967 paper Vaught used an ingenious argument to show that every recursively enumerable first order theory that directly interprets the weak system VS of set theory is axiomatizable by a scheme. In this paper we establish a strengthening of Vaught's theorem by weakening the hypothesis of direct interpretability of VS to direct interpretability of the finitely axiomatized fragment VS2 of VS. This improvement significantly increases the scope of the original result, since VS is essentially undecidable, but VS2 has decidable extensions. We also explore the ramifications of our work on finite axiomatizability of schemes in the presence of suitable comprehension principles.

Decidable subspaces and recursively enumerable subspaces

1984 ◽  
Vol 49 (4) ◽  
pp. 1137-1145 ◽  
Author(s):  
C. J. Ash ◽  
R. G. Downey
Keyword(s):  
Vector Space ◽  
Order Theory ◽  
Infinite Dimensional ◽  
First Order ◽  
Direct Sum ◽  
First Order Theory ◽  
Finite Dimensional ◽  
Recursively Enumerable ◽  
Natural Analogues

AbstractA subspace V of an infinite dimensional fully effective vector space V∞ is called decidable if V is r.e. and there exists an r.e. W such that V ⊕ W = V∞. These subspaces of V∞ are natural analogues of recursive subsets of ω. The set of r.e. subspaces forms a lattice L(V∞) and the set of decidable subspaces forms a lower semilattice S(V∞). We analyse S(V∞) and its relationship with L(V∞). We show:Proposition. Let U, V, W ∈ L(V∞) where U is infinite dimensional andU ⊕ V = W. Then there exists a decidable subspace D such that U ⊕ D = W.Corollary. Any r.e. subspace can be expressed as the direct sum of two decidable subspaces.These results allow us to show:Proposition. The first order theory of the lower semilattice of decidable subspaces, Th(S(V∞), is undecidable.This contrasts sharply with the result for recursive sets.Finally we examine various generalizations of our results. In particular we analyse S*(V∞), that is, S(V∞) modulo finite dimensional subspaces. We show S*(V∞) is not a lattice.

The decision problem for branching time logic

1985 ◽  
Vol 50 (3) ◽  
pp. 668-681 ◽  
Author(s):  
Yuri Gurevich ◽  
Saharon Shelah
Keyword(s):  
Decision Problem ◽  
Companion Paper ◽  
Order Theory ◽  
Branching Time ◽  
First Order ◽  
First Order Theory ◽  
Time Logic

AbstractThe theory of trees with additional unary predicates and quantification over nodes and branches embraces a rich branching time logic. This theory was reduced in the companion paper to the first-order theory of binary, bounded, well-founded trees with additional unary predicates. Here we prove the decidability of the latter theory.

Sign in / Sign up

Export Citation Format

Share Document