Counting finite models

1997 ◽  
Vol 62 (3) ◽  
pp. 925-949 ◽  
Author(s):  
Alan R. Woods
Keyword(s):  
Additional Condition ◽  
Fixed Number ◽  
Second Order ◽  
Single Component ◽  
Partial Function ◽  
Unary Function ◽  
Finite Models ◽  
Finite Structure ◽  
Generating Series

AbstractLetφbe a monadic second order sentence about a finite structure from a classwhich is closed under disjoint unions and has components. Compton has conjectured that if the number ofnelement structures has appropriate asymptotics, then unlabelled (labelled) asymptotic probabilitiesν(φ)(μ(φ)respectively) forφalways exist. By applying generating series methods to count finite models, and a tailor made Tauberian lemma, this conjecture is proved under a mild additional condition on the asymptotics of the number of single component-structures. Prominent among examples covered, are structures consisting of a single unary function (or partial function) and a fixed number of unary predicates.

On first-order sentences without finite models

2004 ◽  
Vol 69 (2) ◽  
pp. 329-339 ◽  
Author(s):  
Marko Djordjević
Keyword(s):  
Binary Relation ◽  
Function Symbol ◽  
Complete Theory ◽  
Order Property ◽  
Unary Function ◽  
First Order ◽  
Finite Models

We will mainly be concerned with a result which refutes a stronger variant of a conjecture of Macpherson about finitely axiomatizable ω-categorical theories. Then we prove a result which implies that the ω-categorical stable pseudoplanes of Hrushovski do not have the finite submodel property.Let's call a consistent first-order sentence without finite models an axiom of infinity. Can we somehow describe the axioms of infinity? Two standard examples are:ϕ1: A first-order sentence which expresses that a binary relation < on a nonempty universe is transitive and irreflexive and that for every x there is y such that x < y.ϕ2: A first-order sentence which expresses that there is a unique x such that, (0) for every y, s(y) ≠ x (where s is a unary function symbol),and, for every x, if x does not satisfy (0) then there is a unique y such that s(y) = x.Every complete theory T such that ϕ1 ϵ T has the strict order property (as defined in [10]), since the formula x < y will have the strict order property for T. Let's say that if Ψ is an axiom of infinity and every complete theory T with Ψ ϵ T has the strict order property, then Ψ has the strict order property.Every complete theory T such that ϕ2 ϵ T is not ω-categorical. This is the case because a complete theory T without finite models is ω-categorical if and only if, for every 0 < n < ω, there are only finitely many formulas in the variables x1,…,xn, up to equivalence, in any model of T.

Theories with finitely many models

1986 ◽  
Vol 51 (2) ◽  
pp. 374-376 ◽  
Author(s):  
Simon Thomas
Keyword(s):  
Natural Number ◽  
Complete Theory ◽  
Elementary Equivalence ◽  
First Order ◽  
Finite Models ◽  
Simple Observation ◽  
Finite Structure ◽  
Order Language ◽  
Complete Theories

If L is a first order language and n is a natural number, then Ln is the set of formulas which only make use of the variables x1,…,xn. While every finite structure is determined up to isomorphism by its theory in L, the same is no longer true in Ln. This simple observation is the source of a number of intriguing questions. For example, Poizat [2] has asked whether a complete theory in Ln which has at least two nonisomorphic finite models must necessarily also have an infinite one. The purpose of this paper is to present some counterexamples to this conjecture.Theorem. For each n ≤ 3 there are complete theories in L2n−2andL2n−1having exactly n + 1 models.In our notation and definitions, we follow Poizat [2]. To test structures for elementary equivalence in Ln, we shall use the modified Ehrenfeucht-Fraïssé games of Immerman [1]. For convenience, we repeat his definition here.Suppose that L is a purely relational language, each of the relations having arity at most n. Let and ℬ be two structures for L. Define the Ln game on and ℬ as follows. There are two players, I and II, and there are n pairs of counters a1, b1, …, an, bn. On each move, player I picks up any of the counters and places it on an element of the appropriate structure.

scholarly journals Rogue waves in the three-level defocusing coupled Maxwell–Bloch equations

2021 ◽  
Vol 477 (2256) ◽  
Author(s):  
Xin Wang ◽  
Lei Wang ◽  
Jiao Wei ◽  
Bowen Guo ◽  
Jingfeng Kang
Keyword(s):  
Rogue Wave ◽  
Laser Pulses ◽  
Rogue Waves ◽  
Fixed Number ◽  
Second Order ◽  
Ultrashort Laser Pulses ◽  
Resonant Medium ◽  
Schur Polynomials ◽  
Rogue Wave Solution ◽  
Integrable Generalization

The coupled Maxwell–Bloch (CMB) system is a fundamental model describing the propagation of ultrashort laser pulses in a resonant medium with coherent three-level atomic transitions. In this paper, we consider an integrable generalization of the CMB equations with the defocusing case. The CMB hierarchy is derived with the aid of a 3 × 3 matrix eigenvalue problem and the Lenard recursion equation, from which the defocusing CMB model is proposed as a special reduction of the general CMB equations. The n -fold Darboux transformation as well as the multiparametric n th-order rogue wave solution of the defocusing CMB equations are put forward in terms of Schur polynomials. As an application, the explicit rogue wave solutions from first to second order are presented. Apart from the traditional dark rogue wave, bright rogue wave and four-petalled rogue wave, some novel rogue wave structures such as the dark four-peaked rogue wave and the double-ridged rogue wave are found. Moreover, the second-order rogue wave triplets which contain a fixed number of these rogue waves are shown.

scholarly journals The evolution of anti-social rewarding and its countermeasures in public goods games

2015 ◽  
Vol 282 (1798) ◽  
pp. 20141994 ◽  
Author(s):  
Miguel dos Santos
Keyword(s):  
Public Goods ◽  
Public Good ◽  
Additional Condition ◽  
Social Dilemmas ◽  
Second Order ◽  
The Public ◽  
Common Pool ◽  
Public Goods Games ◽  
The Commons ◽  
Social Trap

Cooperation in joint enterprises can easily break down when self-interests are in conflict with collective benefits, causing a tragedy of the commons. In such social dilemmas, the possibility for contributors to invest in a common pool-rewards fund, which will be shared exclusively among contributors, can be powerful for averting the tragedy, as long as the second-order dilemma (i.e. withdrawing contribution to reward funds) can be overcome (e.g. with second-order sanctions). However, the present paper reveals the vulnerability of such pool-rewarding mechanisms to the presence of reward funds raised by defectors and shared among them (i.e. anti-social rewarding), as it causes a cooperation breakdown, even when second-order sanctions are possible. I demonstrate that escaping this social trap requires the additional condition that coalitions of defectors fare poorly compared with pro-socials, with either (i) better rewarding abilities for the latter or (ii) reward funds that are contingent upon the public good produced beforehand, allowing groups of contributors to invest more in reward funds than groups of defectors. These results suggest that the establishment of cooperation through a collective positive incentive mechanism is highly vulnerable to anti-social rewarding and requires additional countermeasures to act in combination with second-order sanctions.

scholarly journals A dichotomy in classifying quantifiers for finite models

2005 ◽  
Vol 70 (4) ◽  
pp. 1297-1324
Author(s):  
Saharon Shelah ◽  
Mor Doron
Keyword(s):  
Partial Order ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
Natural Partial Order ◽  
Existential Quantifier ◽  
First Order ◽  
Finite Models ◽  
One To One

AbstractWe consider a family of finite universes. The second order existential quantifier Qℜ means for each U Є quantifying over a set of n(ℜ)-place relations isomorphic to a given relation. We define a natural partial order on such quantifiers called interpretability. We show that for every Qℜ, either Qℜ is interpretable by quantifying over subsets of U and one to one functions on U both of bounded order, or the logic L(Qℜ) (first order logic plus the quantifier Qℜ) is undecidable.

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