Probabilities of first-order sentences on sparse random relational structures: an application to definability on random CNF formulas

Journal of Logic and Computation ◽

10.1093/logcom/exaa061 ◽

2020 ◽

Author(s):

Lázaro Alberto Larrauri

Keyword(s):

Random Graphs ◽

Order Theory ◽

Boolean Satisfiability ◽

Random Model ◽

Order Property ◽

Expected Number ◽

First Order ◽

Relational Structures ◽

First Order Theory ◽

Cnf Formulas

Abstract We extend the convergence law for sparse random graphs proven by Lynch to arbitrary relational languages. We consider a finite relational vocabulary $\sigma $ and a first-order theory $T$ for $\sigma $ composed of symmetry and anti-reflexivity axioms. We define a binomial random model of finite $\sigma $-structures that satisfy $T$ and show that first-order properties have well defined asymptotic probabilities when the expected number of tuples satisfying each relation in $\sigma $ is linear. It is also shown that these limit probabilities are well behaved with respect to several parameters that represent the density of tuples in each relation $R$ in the vocabulary $\sigma $. An application of these results to the problem of random Boolean satisfiability is presented. We show that in a random $k$-CNF formula on $n$ variables, where each possible clause occurs with probability $\sim c/n^{k-1}$, independently any first-order property of $k$-CNF formulas that implies unsatisfiability does almost surely not hold as $n$ tends to infinity.

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Countable homogeneous relational structures and ℵ0-categorical theories

Journal of Symbolic Logic ◽

10.2307/2272734 ◽

1972 ◽

Vol 37 (3) ◽

pp. 494-500 ◽

Cited By ~ 41

Author(s):

C. Ward Henson

Keyword(s):

Order Theory ◽

Linear Ordering ◽

Categorical Theory ◽

Symmetric Relation ◽

First Order ◽

Relational Structures ◽

First Order Theory ◽

Partial Orderings ◽

Homogeneous Structures ◽

Binary Relation Symbol

A relational structure of cardinality ℵ0 is called homogeneous by Fraissé [1] if each isomorphism between finite substructures of can be extended to an automorphism of . In §1 of this paper it is shown that there are isomorphism types of such structures for the first order language L0 with a single (binary) relation symbol, answering a question raised by Fraissé. In fact, as is shown in §2, a family of nonisomorphic homogeneous structures for L0 can be constructed, each member of which satisfies the following conditions (where U is the homogeneous, ℵ0-universal graph, the structure of which is considered in [4]):(i) The relation R of is asymmetric (R ∩ R−1 = ∅);(ii) If A is the domain of and S is the symmetric relation R ∪ R−1, then (A, S) is isomorphic to U. That is, each may be regarded as the result of assigning a unique direction to each edge of the graph U.Let T0 be the first order theory of all homogeneous structures for L0 which have cardinality ℵ0. In §3 (which can be read independently of §2) it is shown that T0 has complete extensions (in L0), each of which is ℵ0-categorical. Moreover, among the complete extensions of T0 are theories of arbitrary (preassigned) degree of unsolvability. In particular, there exists an undecidable, ℵ0-categorieal theory in L0, which answers a question raised by Grzegorczyk [2], [3].It follows from Theorem 6 of [3] that there are ℵ0-categorical theories of partial orderings which have arbitrarily high degrees of unsolvability. This is in sharp contrast to the situation for linear orderings, which were the motivation for Fraissé's early work. Indeed, as is shown in [10], every ℵ0-categorical theory of a linear ordering is finitely axiomatizable. (W. Glassmire [12] has independently shown the existence of theories in L0 which are all ℵ0-categorical, and C. Ash [13] has independently shown that such theories exist with arbitrary degree of unsolvability.)

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Nonexistence of universal orders in many cardinals

Journal of Symbolic Logic ◽

10.2307/2275437 ◽

1992 ◽

Vol 57 (3) ◽

pp. 875-891 ◽

Cited By ~ 22

Author(s):

Menachem Kojman ◽

Saharon Shelah

Keyword(s):

Linear Order ◽

Order Theory ◽

Order Property ◽

Universal Model ◽

Covering Theorem ◽

First Order ◽

First Order Theory ◽

Ordered Fields ◽

Singular Cardinals ◽

Universal Order

AbstractOur theme is that not every interesting question in set theory is independent of ZFC. We give an example of a first order theory T with countable D(T) which cannot have a universal model at ℵ1; without CH; we prove in ZFC a covering theorem from the hypothesis of the existence of a universal model for some theory; and we prove—again in ZFC—that for a large class of cardinals there is no universal linear order (e.g. in every regular ). In fact, what we show is that if there is a universal linear order at a regular λ and its existence is not a result of a trivial cardinal arithmetical reason, then λ “resembles” ℵ1—a cardinal for which the consistency of having a universal order is known. As for singular cardinals, we show that for many singular cardinals, if they are not strong limits then they have no universal linear order. As a result of the nonexistence of a universal linear order, we show the nonexistence of universal models for all theories possessing the strict order property (for example, ordered fields and groups, Boolean algebras, p-adic rings and fields, partial orders, models of PA and so on).

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Forcing isomorphism

Journal of Symbolic Logic ◽

10.2307/2275144 ◽

1993 ◽

Vol 58 (4) ◽

pp. 1291-1301 ◽

Cited By ~ 1

Author(s):

J. T. Baldwin ◽

M. C. Laskowski ◽

S. Shelah

Keyword(s):

Partially Ordered Set ◽

Ordered Set ◽

Order Theory ◽

Dependence Structure ◽

Order Property ◽

First Order ◽

First Order Theory ◽

Omitting Types ◽

Universe Of Sets ◽

The Universe

If two models of a first-order theory are isomorphic, then they remain isomorphic in any forcing extension of the universe of sets. In general however, such a forcing extension may create new isomorphisms. For example, any forcing that collapses cardinals may easily make formerly nonisomorphic models isomorphic. However, if we place restrictions on the partially-ordered set to ensure that the forcing extension preserves certain invariants, then the ability to force nonisomorphic models of some theory T to be isomorphic implies that the invariants are not sufficient to characterize the models of T.A countable first-order theory is said to be classifiable if it is superstable and does not have either the dimensional order property (DOP) or the omitting types order property (OTOP). If T is not classifiable, Shelah has shown in [5] that sentences in L∞,λ do not characterize models of T of power λ. By contrast, in [8] Shelah showed that if a theory T is classifiable, then each model of cardinality λ is described by a sentence of L∞,λ. In fact, this sentence can be chosen in the . ( is the result of enriching the language by adding for each μ < λ a quantifier saying the dimension of a dependence structure is greater than μ) Further work ([3], [2]) shows that ⊐+ can be replaced by ℵ1.

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