scholarly journals Random models and the Maslov class

1989 ◽  
Vol 54 (2) ◽  
pp. 460-466
Author(s):  
Warren Goldfarb
Keyword(s):  
Efficient Method ◽  
Probability Space ◽  
Random Model ◽  
Finite Model ◽  
Quantification Theory ◽  
Finite Models ◽  
Model Method ◽  
Maslov Class ◽  
Random Models ◽  
Novel Method

In [GS] Gurevich and Shelah introduce a novel method for proving that every satisfiable formula in the Gödel class has a finite model (the Gödel class is the class of prenex formulas of pure quantification theory with prefixes ∀∀∃ … ∃). They dub their method “random models”: it proceeds by delineating, given any F in the Gödel class and any integer p, a set of structures for F with universe {1, …, p} that can be treated as a finite probability space S. They then show how to calculate an upper bound on the probability that a structure chosen at random from S makes F false; from this bound they are able to infer that if p is sufficiently large, that probability will be less than one, so that there will exist a structure in S that is a model for F. The Gurevich-Shelah proof is somewhat simpler than those known heretofore. In particular, there is no need for the combinatorial partitionings of finite universes that play a central role in the earlier proofs (see [G] and [DG, p. 86]). To be sure, Gurevich and Shelah obtain a larger bound on the size of the finite models, but this is relatively unimportant, since searching for finite models is not the most efficient method to decide satisfiability.Gurevich and Shelah note that the random model method can be used to treat the Gödel class extended by initial existential quantifiers, that is, the prefix-class ∀…∀∃…∃; but they do not investigate further its range of applicability to syntactically specified classes.

Random models and solvable Skolem classes

1993 ◽  
Vol 58 (3) ◽  
pp. 908-914
Author(s):  
Warren Goldfarb
Keyword(s):  
Normal Form ◽  
Systematic Approach ◽  
Finite Model ◽  
Theoretic Property ◽  
Quantification Theory ◽  
Maslov Class ◽  
Random Models ◽  
Universal Quantifiers ◽  
Necessary And Sufficient ◽  
Made In

A Skolem class is a class of formulas of pure quantification theory in Skolem normal form: closed, prenex formulas with prefixes ∀…∀∃…∃. (Pure quantification theory contains quantifiers, truth-functions, and predicate letters, but neither the identity sign nor function letters.) The Gödel Class, in which the number of universal quantifiers is limited to two, was shown effectively solvable (for satisfiability) sixty years ago [G1]. Various solvable Skolem classes that extend the Gödel Class can be obtained by allowing more universal quantifiers but restricting the combinations of variables that may occur together in atomic subformulas [DG, Chapter 2]. The Gödel Class and these extensions are also finitely controllable, that is, every satisfiable formula in them has a finite model. In this paper we isolate a model-theoretic property that connects the usual solvability proofs for these classes and their finite controllability. For formulas in the solvable Skolem classes, the property is necessary and sufficient for satisfiability. The solvability proofs implicitly relied on this fact. Moreover, for any formula in Skolem normal form, the property implies the existence of a finite model.The proof of the latter implication uses the random models technique introduced in [GS] for the Gödel Class and modified and applied in [Go] to the Maslov Class. The proof thus substantiates the claim made in [Go] that random models can be adapted to the Skolem classes of [DG, Chapter 2]. As a whole, the results of this paper provide a more general, systematic approach to finite controllability than previous methods.

The finite controllability of the Maslov case

1974 ◽  
Vol 39 (3) ◽  
pp. 509-518 ◽  
Author(s):  
Stål Aanderaa ◽  
Warren D. Goldfarb
Keyword(s):  
Atomic Formula ◽  
Finite Model ◽  
Quantification Theory ◽  
Maslov Class

In this paper we show the finite controllability of the Maslov class of formulas of pure quantification theory (specified immediately below). That is, we show that every formula in the class has a finite model if it has a model at all. A signed atomic formula is an atomic formula or the negation of one; a binary disjunction is a disjunction of the form A1 ⋁ A2, where A1 and A2 are signed atomic formulas; and a formula is Krom if it is a conjunction of binary disjunctions. Finally, a prenex formula is Maslov if its prefix is ∃···∃∀···∀∃···∃ and its matrix is Krom.A number of decidability results have been obtained for formulas classified along these lines. It is a consequence of Theorems 1.7 and 2.5 of [4] that the following are reduction classes (for satisfiability): the class of Skolem formulas, that is, prenex formulas with prefixes ∀···∀∃···∃, whose matrices are conjunctions one conjunct of which is a ternary disjunction and the rest of which are binary disjunctions; and the class of Skolem formulas containing identity whose matrices are Krom. Moreover, the following results (for pure quantification theory, that is, without identity) are derived in [1] and [2]: the classes of prenex formulas with Krom matrices and prefixes ∃∀∃∀, or prefixes ∀∃∃∀, or prefixes ∀∃∀∀ are all reduction classes, while formulas with Krom matrices and prefixes ∀∃∀ comprise a decidable class. The latter class, however, is not finitely controllable, for it contains formulas satisfiable only over infinite universes. The Maslov class was shown decidable by Maslov in [11].

On the existence of finite models and decision procedures for propositional calculi

1958 ◽  
Vol 54 (1) ◽  
pp. 1-13 ◽  
Author(s):  
R. Harrop
Keyword(s):  
Decision Procedures ◽  
Finite Model ◽  
Finite Models ◽  
Model Method ◽  
General Properties ◽  
Counter Example ◽  
Converse Result ◽  
Definition Of ◽  
Propositional Calculi

1. Introduction. In this paper we consider certain general properties of propositional calculi. Two forms of the definition of a finite model of such a calculus are discussed, these forms differing in the manner prescribed for the satisfaction of the rules of the calculus. The methods of definition are shown to be equivalent for the application of the finite model method for the demonstration of the unprovability of formulae in the calculus. It is further proved that, although the decidability of a calculus follows from the existence of a finite model counter-example for each unprovable formula of the calculus, the converse result is not true.

Novel Method of Ranking Line Outage Contingencies from Network Loadability Limit Point of View

2005 ◽  
Vol 3 (1) ◽  
Author(s):  
P. R. Bijwe ◽  
M. Hanmandlu ◽  
V. N. Pande ◽  
S. M. Kelapure
Keyword(s):  
Efficient Method ◽  
Limit Point ◽  
Power Flow ◽  
Salient Feature ◽  
Point Of View ◽  
Continuation Power Flow ◽  
Sample Test ◽  
Novel Method ◽  
Newton Raphson ◽  
Optimal Multiplier

This paper presents novel, simple and efficient method for ranking line outage contingencies from network loadability limit considerations. The method follows conventional optimal multiplier based Newton Raphson power flow. The simulation of line outage contingencies is carried out near pre-contingency critical loading using the above power flow. A salient feature of the method is the use of the pre-contingency power flow Jacobian factors for simulation of all contingencies. Results for two sample test systems have been obtained with the new method and continuation power flow method in order to verify the potential of the former method for practical use.

Fast and Efficient Synthesis of Mono-(6-p-toluenesulfonyl)-β-cyclodextrin via Ultrasound Assisted Method

2015 ◽  
Vol 1083 ◽  
pp. 51-54 ◽  
Author(s):  
Wen Qian Zheng ◽  
Mei Xia Du ◽  
Feng Feng ◽  
Guo Lin Chen ◽  
Min Liao ◽  
...  
Keyword(s):  
Reaction Time ◽  
Efficient Method ◽  
Water Solution ◽  
Synthetic Methods ◽  
Efficient Synthesis ◽  
Alkaline Water ◽  
Novel Method ◽  
Ultrasound Assisted

The mono-(6-p-toluenesulfonyl)-β-cyclodextrin was firstly synthesized fast and efficiently by adopting ultrasound assisted method in alkaline water solution. The reaction time was only 40 min but with the yield of 31.1% under ultrasound condition. Compared with the conventional synthetic methods, the proposed novel method could shorten the reaction time and improve the yield. It is a simple, rapid and efficient method.

scholarly journals A finite model-theoretical proof of a property of bounded query classes within PH

2004 ◽  
Vol 69 (4) ◽  
pp. 1105-1116 ◽  
Author(s):  
Leszek Aleksander Kołodziejczyk
Keyword(s):  
Normal Form ◽  
Model Theory ◽  
Finite Model ◽  
Sufficient Condition ◽  
Finite Model Theory ◽  
Finite Models ◽  
Constructible Function ◽  
Simple Sufficient Condition

Abstract.We use finite model theory (in particular, the method of FM-truth definitions, introduced in [MM01] and developed in [K04], and a normal form result akin to those of [Ste93] and [G97]) to prove:Let m ≥ 2. Then:(A) If there exists k such that NP⊆ Σm TIME(nk)∩ Πm TIME(nk), then for every r there exists kr such that :(B) If there exists a superpolynomial time-constructible function f such that NTIME(f), then additionally .This strengthens a result by Mocas [M96] that for any r, .In addition, we use FM-truth definitions to give a simple sufficient condition for the arity hierarchy to be strict over finite models.

Note on a paper in tense logic

1969 ◽  
Vol 34 (2) ◽  
pp. 215-218 ◽  
Author(s):  
R. A. Bull
Keyword(s):  
Tense Logic ◽  
Finite Model ◽  
Model Property ◽  
Ordered Sets ◽  
Finite Model Property ◽  
Finite Models ◽  
Correct Proof

In [1, §4], my ‘proof’ that GH1 has the finite model property is incorrect; there are considerable obscurities towards the end of §1, particularly on p. 33; and I should have exhibited the finite models for GH1. In §1 of this paper I expand the analysis of the sub-directly irreducible models for GH1 which I give in §1 of [1]. In §2 I give a correct proof that GH1 has the finite model property. In §3 I exhibit these finite models as models on certain ordered sets.

scholarly journals Where Natural Protein Sequences Stand out From Randomness

2019 ◽  
Author(s):  
Laura Weidmann ◽  
Tjeerd Dijkstra ◽  
Oliver Kohlbacher ◽  
Andrei Lupas
Keyword(s):  
Natural Selection ◽  
Protein Structure ◽  
Sequence Space ◽  
Protein Sequences ◽  
Random Model ◽  
Biological Sequences ◽  
Natural Protein ◽  
Distance Distributions ◽  
Random Models ◽  
Residue Composition

AbstractBiological sequences are the product of natural selection, raising the expectation that they differ substantially from random sequences. We test this expectation by analyzing all fragments of a given length derived from either a natural dataset or different random models. For this, we compile all distances in sequence space between fragments within each dataset and compare the resulting distance distributions between sets. Even for 100mers, 95.4% of all distances between natural fragments are in accordance with those of a random model incorporating the natural residue composition. Hence, natural sequences are distributed almost randomly in global sequence space. When further accounting for the specific residue composition of domain-sized fragments, 99.2% of all distances between natural fragments can be modeled. Local residue composition, which might reflect biophysical constraints on protein structure, is thus the predominant feature characterizing distances between natural sequences globally, whereas homologous effects are only barely detectable.

On Equilibrium Existence in a Finite-Agent, Multi-Asset Noisy Rational Expectations Economy

2019 ◽  
Vol 20 (1) ◽  
Author(s):  
Ronaldo Carpio ◽  
Meixin Guo
Keyword(s):  
Rational Expectations ◽  
Positive Semidefinite ◽  
Finite Model ◽  
Equilibrium Existence ◽  
Equilibrium Behavior ◽  
Asset Pricing Model ◽  
Novel Method ◽  
The Difference ◽  
And Behavior ◽  
Semidefinite Matrix

AbstractWe introduce a novel method of proving existence of rational expectations equilibria (REE) in multi-dimensional CARA-Gaussian environments. Our approach is to construct a mapping from agents’ initial beliefs (which are characterized by a positive semidefinite matrix), to their updated beliefs, after reaching and observing equilibrium; we then show Brouwer’s fixed point theorem applies. We apply our approach to a finite-market version of Admati (1985), which is a multi-asset noisy REE asset pricing model with dispersed information. We present an algorithm to numerically solve for equilibrium of the finite model, as well as several examples illustrating the difference in equilibrium behavior between the finite and infinite models. Our method can be applied to any multi-dimensional REE model with Gaussian uncertainty and behavior that is linear in agents’ information.

On the Gödel class with identity

1981 ◽  
Vol 46 (2) ◽  
pp. 354-364 ◽  
Author(s):  
Warren D. Goldfarb
Keyword(s):  
Recursive Function ◽  
Decision Problem ◽  
Decision Procedure ◽  
Finite Model ◽  
Quantification Theory ◽  
Free Variables ◽  
Primitive Recursive

The Gödel Class is the class of prenex formulas of pure quantification theory whose prefixes have the form ∀y1∀y2∃x1 … ∃xn. The Gödel Class with Identity, or GCI, is the corresponding class of formulas of quantification theory extended by inclusion of the identity-sign “ = ”. Although the Gödel Class has long been kndwn to be solvable, the decision problem for the Gödel Class with Identity is open. In this paper we prove that there is no primitive recursive decision procedure for the GCI, or, indeed, for the subclass of the GCI containing just those formulas with prefixes ∀y1∀y2∃x.Throughout this paper we take quantification theory to include, aside from logical signs, infinitely many k-place predicate letters for each k > 0, but no function signs or constants. Moreover, by “prenex formula” we include only those without free variables. A decision procedure for a class of formulas is a recursive function that carries a formula in the class to 0 if the formula is satisfiable and to 1 if not. A class is solvable iff there exists a decision procedure for it. A class is finitely controllable iff every satisfiable formula in the class has a finite model. Since we speak only of effectively specified classes, finite controllability implies solvability (but not conversely).The GCI has a curious history. Gödel showed the Gödel Class (without identity) solvable in 1932 [4] and finitely controllable in 1933 [5].

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