On spectra, and the negative solution of the decision problem for identities having a finite nontrivial model

1975 ◽  
Vol 40 (2) ◽  
pp. 186-196 ◽  
Author(s):  
Ralph Mckenzie
Keyword(s):  
Model Problem ◽  
Recursive Algorithm ◽  
Finite Model ◽  
Positive Direction ◽  
First Order ◽  
Sharp Form ◽  
Universal Sentence ◽  
Order Language ◽  
Finite Set ◽  
The Universe

An algorithm has been described by S. Burris [3] which decides if a finite set of identities, whose function symbols are of rank at most 1, has a finite, nontrivial model. (By “nontrivial” it is meant that the universe of the model has at least two elements.) As a consequence of some results announced in the abstracts [2] and [8], it is clear that if the restriction on the ranks of function symbols is relaxed somewhat, then this finite model problem is no longer solvable by an algorithm, or at least not by a “recursive algorithm” as the term is used today.In this paper we prove a sharp form of this negative result; showing, by the way, that Burris' result is in a sense the best possible result in the positive direction. Our main result is that in a first order language whose only function or relation symbol is a 2-place function symbol (the language of groupoids), the set of identities that have no nontrivial model, is recursively inseparable from the set of identities such that the sentence has a finite model. As a corollary, we have that each of the following problems, restricted to sentences defined in the language of groupoids, is algorithmically unsolvable: (1) to decide if an identity has a finite nontrivial model; (2) to decide if an identity has a nontrivial model; (3) to decide if a universal sentence has a finite model; (4) to decide if a universal sentence has a model. We note that the undecidability of (2) was proved earlier by McNulty [13, Theorem 3.6(i)], improving results obtained by Murskiǐ [14] and by Perkins [17]. The other parts of the corollary seem to be new.

Almost strongly minimal theories. I

1972 ◽  
Vol 37 (3) ◽  
pp. 487-493 ◽  
Author(s):  
John T. Baldwin
Keyword(s):  
Algebraic Closure ◽  
Natural Extension ◽  
Stone Space ◽  
First Order ◽  
Order Language ◽  
Simple Class ◽  
The Universe ◽  
Natural Expansion ◽  
Minimal Formula

In [1] the notions of strongly minimal formula and algebraic closure were applied to the study of ℵ1-categorical theories. In this paper we study a particularly simple class of ℵ1-categorical theories. We characterize this class in terms of the analysis of the Stone space of models of T given by Morley [3].We assume familiarity with [1] and [3], but for convenience we list the principal results and definitions from those papers which are used here. Our notation is the same as in [1] with the following exceptions.We deal with a countable first order language L. We may extend the language L in several ways. If is an L-structure, there is a natural extension of L obtained by adjoining to L a constant a for each (the universe of ). For each sentence A(a1, …, an) ∈ L(A) we say satisfies A(a1, …, an) and write if in Shoenfield's notation If is an L-structure and X is a subset of , then L(X) is the language obtained by adjoining to L a name x for each is the natural expansion of to an L(X)-structure. A structure is an inessential expansion [4, p. 141] of an L-structure if for some .

An axiomatic system for the first order language with an equi-cardinality quantifier

1966 ◽  
Vol 31 (4) ◽  
pp. 633-640 ◽  
Author(s):  
Mitsuru Yasuhara
Keyword(s):  
Rational Numbers ◽  
Axiomatic System ◽  
Universal Quantifier ◽  
Real Numbers ◽  
Relational System ◽  
First Order ◽  
Order Language ◽  
Relational Systems ◽  
Finite Domains ◽  
The Universe

The equi-cardinality quantifier1 to be used in this article, written as Qx, is characterised by the following semantical rule: A formula QxA(x) is true in a relational system exactly when the cardinality of the set consisting of these elements which make A(x) true is the same as that of the universe. For instance, QxN(x) is true in 〈Rt, N〉 but false in 〈Rl, N〉 where Rt, Rl, and N are the sets of rational numbers, real numbers, and natural numbers, respectively. We notice that in finite domains the equi-cardinality quantifier is the same as the universal quantifier. For this reason, all relational systems considered in the following are assumed infinite.

Decidable properties of finite sets of equations in trivial languages

1984 ◽  
Vol 49 (4) ◽  
pp. 1333-1338
Author(s):  
Cornelia Kalfa
Keyword(s):  
Equational Theory ◽  
Order Theory ◽  
Operation Symbol ◽  
First Order ◽  
First Order Theory ◽  
Order Language ◽  
Finite Set ◽  
Joint Embedding ◽  
Algebraic Language ◽  
Joint Embedding Property

In [4] I proved that in any nontrivial algebraic language there are no algorithms which enable us to decide whether a given finite set of equations Σ has each of the following properties except P2 (for which the problem is open):P0(Σ) = the equational theory of Σ is equationally complete.P1(Σ) = the first-order theory of Σ is complete.P2(Σ) = the first-order theory of Σ is model-complete.P3(Σ) = the first-order theory of the infinite models of Σ is complete.P4(Σ) = the first-order theory of the infinite models of Σ is model-complete.P5(Σ) = Σ has the joint embedding property.In this paper I prove that, in any finite trivial algebraic language, such algorithms exist for all the above Pi's. I make use of Ehrenfeucht's result [2]: The first-order theory generated by the logical axioms of any trivial algebraic language is decidable. The results proved here are part of my Ph.D. thesis [3]. I thank Wilfrid Hodges, who supervised it.Throughout the paper is a finite trivial algebraic language, i.e. a first-order language with equality, with one operation symbol f of rank 1 and at most finitely many constant symbols.

Noninitial segments of the α-degrees

1973 ◽  
Vol 38 (3) ◽  
pp. 368-388 ◽  
Author(s):  
John M. Macintyre
Keyword(s):  
Distributive Lattice ◽  
Initial Segment ◽  
Elementary Theory ◽  
Recursion Theory ◽  
Partial Ordering ◽  
First Order ◽  
Order Language ◽  
Suitable Notion ◽  
The Universe ◽  
Degrees Of Unsolvability

Let α be an admissible ordinal and let L be the first order language with equality and a single binary relation ≤. The elementary theory of the α-degrees is the set of all sentences of L which are true in the universe of the α-degrees when ≤ is interpreted as the partial ordering of the α-degrees. Lachlan [6] showed that the elementary theory of the ω-degrees is nonaxiomatizable by proving that any countable distributive lattice with greatest and least members can be imbedded as an initial segment of the degrees of unsolvability. This paper deals with the extension of these results to α-recursion theory for an arbitrary countable admissible α > ω. Given α, we construct a set A with α-degree a such that every countable distributive lattice with greatest and least member is order isomorphic to a segment of α-degrees {d ∣ a ≤αd≤αb} for some α-degree b. As in [6] this implies that the elementary theory of the α-degrees is nonaxiomatizable and hence undecidable.A is constructed in §2. A is a set of integers which is generic with respect to a suitable notion of forcing. Additional applications of such sets are summarized at the end of the section. In §3 we define the notion of a tree and construct a particular tree T0 which is weakly α-recursive in A. Using T0 we can apply the techniques of [6] and [2] to α-recursion theory. In §4 we reduce our main results to three technical lemmas concerning systems of trees. These lemmas are proved in §5.

An interpretation of “finite” modal first-order languages in classical second-order languages

1976 ◽  
Vol 41 (2) ◽  
pp. 337-340
Author(s):  
Scott K. Lehmann
Keyword(s):  
Possible Worlds ◽  
Second Order ◽  
Universal Quantifier ◽  
Possible World ◽  
Simple Interpretation ◽  
First Order ◽  
Order Language ◽  
Finite Set ◽  
Modal Semantics ◽  
Individual Variables

This note describes a simple interpretation * of modal first-order languages K with but finitely many predicates in derived classical second-order languages L(K) such that if Γ is a set of K-formulae, Γ is satisfiable (according to Kripke's 55 semantics) iff Γ* is satisfiable (according to standard (or nonstandard) second-order semantics).The motivation for the interpretation is roughly as follows. Consider the “true” modal semantics, in which the relative possibility relation is universal. Here the necessity operator can be considered a universal quantifier over possible worlds. A possible world itself can be identified with an assignment of extensions to the predicates and of a range to the quantifiers; if the quantifiers are first relativized to an existence predicate, a possible world becomes simply an assignment of extensions to the predicates. Thus the necessity operator can be taken to be a universal quantifier over a class of assignments of extensions to the predicates. So if these predicates are regarded as naming functions from extensions to extensions, the necessity operator can be taken as a string of universal quantifiers over extensions.The alphabet of a “finite” modal first-order language K shall consist of a non-empty countable set Var of individual variables, a nonempty finite set Pred of predicates, the logical symbols ‘¬’ ‘∧’, and ‘∧’, and the operator ‘◊’. The formation rules of K generate the usual Polish notations as K-formulae. ‘ν’, ‘ν1’, … range over Var, ‘P’ over Pred, ‘A’ over K-formulae, and ‘Γ’ over sets of K-formulae.

Probabilities on finite models

1976 ◽  
Vol 41 (1) ◽  
pp. 50-58 ◽  
Author(s):  
Ronald Fagin
Keyword(s):  
Rate Of Convergence ◽  
Finite Subset ◽  
Asymptotic Estimate ◽  
Predicate Symbol ◽  
Relational Structure ◽  
Finite Model ◽  
First Order ◽  
Finite Models ◽  
Model Following ◽  
Finite Set

Let be a finite set of (nonlogical) predicate symbols. By an -structure, we mean a relational structure appropriate for . Let be the set of all -structures with universe {1, …, n}. For each first-order -sentence σ (with equality), let μn(σ) be the fraction of members of for which σ is true. We show that μn(σ) always converges to 0 or 1 as n → ∞, and that the rate of convergence is geometrically fast. In fact, if T is a certain complete, consistent set of first-order -sentences introduced by H. Gaifman [6], then we show that, for each first-order -sentence σ, μn(σ) →n 1 iff T ⊩ ω. A surprising corollary is that each finite subset of T has a finite model. Following H. Scholz [8], we define the spectrum of a sentence σ to be the set of cardinalities of finite models of σ. Another corollary is that for each first-order -sentence a, either σ or ˜σ has a cofinite spectrum (in fact, either σ or ˜σ is “nearly always“ true).Let be a subset of which contains for each in exactly one structure isomorphic to . For each first-order -sentence σ, let νn(σ) be the fraction of members of which a is true. By making use of an asymptotic estimate [3] of the cardinality of and by our previously mentioned results, we show that vn(σ) converges as n → ∞, and that limn νn(σ) = limn μn(σ).If contains at least one predicate symbol which is not unary, then the rate of convergence is geometrically fast.

A Note on the Realization of Types

1980 ◽  
Vol 23 (1) ◽  
pp. 95-98
Author(s):  
Alan Adamson
Keyword(s):  
Complete Theory ◽  
First Order ◽  
Order Language ◽  
The Universe

Let L be a countable first-order language and T a fixed complete theory in L. If is a model of T, is an n-sequence of variables, and ā=〈a1,…, an〉 is an n-sequence of elements of M, the universe of , we let where ranges over formulas of L containing freely at most the variables υ1,…υn. ā is said to realize in We let be where is the sequence of the first n variables of L.

Number of variables is equivalent to space

2001 ◽  
Vol 66 (3) ◽  
pp. 1217-1230 ◽  
Author(s):  
Neil Immerman ◽  
Jonathan F. Buss ◽  
David A. Mix Barrington
Keyword(s):  
Turing Machine ◽  
Descriptive Complexity ◽  
First Order ◽  
Trade Offs ◽  
Finite Set ◽  
The Universe

AbstractWe prove that the set of properties describable by a uniform sequence of first-order sentences using at most k + 1 distinct variables is exactly equal to the set of properties checkable by a Turing machine in DSPACE[nk] (where n is the size of the universe). This set is also equal to the set of properties describable using an iterative definition for a finite set of relations of arity k. This is a refinement of the theorem PSPACE = VAR[O[1]] [8]. We suggest some directions for exploiting this result to derive trade-offs between the number of variables and the quantifier depth in descriptive complexity.

A note on finitely generated models

1983 ◽  
Vol 48 (1) ◽  
pp. 163-166
Author(s):  
Anand Pillay
Keyword(s):  
Countable Model ◽  
Stable Theory ◽  
Prime Model ◽  
Finitely Generated ◽  
First Order ◽  
Elementary Chain ◽  
Skolem Functions ◽  
Order Language ◽  
Finite Set

A model M (of a countable first order language) is said to be finitely generated if it is prime over a finite set, namely if there is a finite tuple ā in M such that (M, ā) is a prime model of its own theory. Similarly, if A ⊂ M, then M is said to be finitely generated overA if there is finite ā in M such that M is prime over A ⋃ ā. (Note that if Th(M) has Skolem functions, then M being prime over A is equivalent to M being generated by A in the usual sense, that is, M is the closure of A under functions of the language.) We show here that if N is ā model of an ω-stable theory, M ≺ N, M is finitely generated, and N is finitely generated over M, then N is finitely generated. A corollary is that any countable model of an ω-stable theory is the union of an elementary chain of finitely generated models. Note again that all this is trivial if the theory has Skolem functions.The result here strengthens the results in [3], where we show the same thing but assuming in addition that the theory is either nonmultidimensional or with finite αT. However the proof in [3] for the case αT finite actually shows the following which does not assume ω-stability): Let A be atomic over a finite set, tp(ā / A) have finite Cantor-Bendixson rank, and B be atomic over A ⋃ ā. Then B is atomic over a finite set.

A form of feasible interpolation for constant depth Frege systems

2010 ◽  
Vol 75 (2) ◽  
pp. 774-784 ◽  
Author(s):  
Jan Krajíček
Keyword(s):  
Constant Depth ◽  
First Order ◽  
Order Language ◽  
Frege Systems ◽  
Image Position ◽  
The Universe ◽  
Dnf Formula

AbstractLet L be a first-order language and Φ and Ψ two L-sentences that cannot be satisfied simultaneously in any finite L-structure. Then obviously the following principle ChainL,Φ,Ψ(n, m) holds: For any chain of finite L-structures C1, …, Cm with the universe [n] one of the following conditions must fail:For each fixed L and parameters n, m the principle ChainL,Φ,Ψ(n,m) can be encoded into a propositional DNF formula of size polynomial in n, m.For any language L containing only constants and unary predicates we show that there is a constant CL such that the following holds: If a constant depth Frege system in DeMorgan language proves ChainL,Φ,Ψ(n, cL . n) by a size s proof then the class of finite L-structures with universe [n] satisfying Φ can be separated from the class of those L-structures on [n] satisfying ψ by a depth 3 formula of size 2log(S)O(1) and with bottom fan-in log(S)O(1).

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