A Reduction Class Containing Formulas with one Monadic Predicate and one Binary Function Symbol

1976 ◽  
Vol 41 (1) ◽  
pp. 45
Author(s):  
Charles E. Hughes
Keyword(s):  
Function Symbol ◽  
Binary Function ◽  
Reduction Class

A reduction class containing formulas with one monadic predicate and one binary function symbol

1976 ◽  
Vol 41 (1) ◽  
pp. 45-49
Author(s):  
Charles E. Hughes
Keyword(s):  
Normal Form ◽  
Predicate Symbol ◽  
Conjunctive Normal Form ◽  
Function Symbol ◽  
Predicate Calculus ◽  
Binary Function ◽  
First Order ◽  
The Matrix ◽  
Reduction Class ◽  
Order Predicate Calculus

AbstractA new reduction class is presented for the satisfiability problem for well-formed formulas of the first-order predicate calculus. The members of this class are closed prenex formulas of the form ∀x∀yC. The matrix C is in conjunctive normal form and has no disjuncts with more than three literals, in fact all but one conjunct is unary. Furthermore C contains but one predicate symbol, that being unary, and one function symbol which symbol is binary.

scholarly journals Completeness of two theories on ordered abelian groups and embedding relations

1980 ◽  
Vol 77 ◽  
pp. 33-39 ◽  
Author(s):  
Yuichi Komori
Keyword(s):  
Binary Relation ◽  
Abelian Groups ◽  
Function Symbol ◽  
Binary Function ◽  
Relation Symbol ◽  
Unary Function ◽  
First Order ◽  
Order Language ◽  
Unary Relation ◽  
Binary Relation Symbol

The first order language ℒ that we consider has two nullary function symbols 0, 1, a unary function symbol –, a binary function symbol +, a unary relation symbol 0 <, and the binary relation symbol = (equality). Let ℒ′ be the language obtained from ℒ, by adding, for each integer n > 0, the unary relation symbol n| (read “n divides”).

scholarly journals Extensions of unification modulo ACUI

2019 ◽  
Vol 30 (6) ◽  
pp. 597-626 ◽  
Author(s):  
Franz Baader ◽  
Pavlos Marantidis ◽  
Antoine Mottet ◽  
Alexander Okhotin
Keyword(s):  
Function Symbol ◽  
The Other ◽  
Binary Function ◽  
Worst Case ◽  
Case Complexity ◽  
Equational Theories ◽  
Other Hand ◽  
Worst Case Complexity ◽  
Finite Set ◽  
Unification Problem

AbstractThe theory ACUI of an associative, commutative, and idempotent binary function symbol + with unit 0 was one of the first equational theories for which the complexity of testing solvability of unification problems was investigated in detail. In this paper, we investigate two extensions of ACUI. On one hand, we consider approximate ACUI-unification, where we use appropriate measures to express how close a substitution is to being a unifier. On the other hand, we extend ACUI-unification to ACUIG-unification, that is, unification in equational theories that are obtained from ACUI by adding a finite set G of ground identities. Finally, we combine the two extensions, that is, consider approximate ACUI-unification. For all cases we are able to determine the exact worst-case complexity of the unification problem.

Krom formulas with one dyadic predicate letter

1976 ◽  
Vol 41 (2) ◽  
pp. 341-362 ◽  
Author(s):  
Harry R. Lewis
Keyword(s):  
Normal Forms ◽  
Decision Problems ◽  
The Matrix ◽  
Reduction Class ◽  
Complex Construction ◽  
Full Class

Let Kr be the class of all those quantificational formulas whose matrices are conjunctions of binary disjunctions of signed atomic formulas. Decision problems for subclasses of Kr do not invariably coincide with those for the corresponding classes of quantificational formulas with unrestricted matrices; for example, the ∀∃∀ prefix subclass of Kr is solvable, but the full ∀∃∀ class is not ([AaLe],- [KMW]). Moreover, the straightforward encodings of automata which have been used to show the unsolvability of various subclasses of Kr ([Aa], [Bö], [AaLe]) yield but little information about signature subclasses, i.e. subclasses determined by the number and degrees of the predicate letters occurring in a formula. By a new and highly complex construction Theorem 1 establishes the best possible result on classification by signature.Theorem 1. Let C be the class of all formulas in Kr with a single predicate letter, which is dyadic; then C is a reduction class for satisfiability.Thus a signature subclass of Kr is solvable just in case the corresponding class of unrestricted quantificational formulas is solvable, to wit, just in case no predicate letter of degree exceeding one may occur. To obtain a richer classification by signature we consider further restrictions on the matrix. Let Or be the class of formulas in Kr having disjunctive normal forms with only two disjuncts. Theorem 2 sharpens Orevkov's proof of the unsolvability of Or ([Or]; see also [LeGo]).Theorem 2. Let D be the class of formulas in Or with just two predicate letters, both pentadic; then D is a reduction class for satisfiability.

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.

Optimization of Assembly Production Based on Imitation of Petri Nets

2021 ◽  
pp. 133-146
Author(s):  
A.N. Sochnev
Keyword(s):  
Petri Nets ◽  
Petri Net ◽  
Production Systems ◽  
Significant Degree ◽  
Practical Significance ◽  
Optimal Planning ◽  
Binary Function ◽  
Typical Situation ◽  
Branched Structure ◽  
Assembly Production

The paper describes the approach to solving the problem of optimal planning of the production process. A discrete production system represented by the operations of machining, welding and painting was chosen as the object of research. The study states the problem of optimization of assembly production, which contains a typical criterion of optimality. A mechanism for meeting the criterion using a simulation model based on a Petri net is determined. The rules for developing feedback on the state of the network model and a method for controlling the simulation of the Petri net based on the analysis of its states are given. A binary function is used to analyze the states of the model. The developed approach to process optimization develops the theory of Petri nets, makes it more suitable for modeling complex systems with a branched structure and a large number of interconnections, which is a typical situation for production systems. The most universal approaches of control theory, e.g. feedback principle, are used, which implies a significant degree of universality and replicability of the approach. On the basis of the developed theoretical provisions, a test example is presented that characterizes the effect of their application. The presence of assembly production at most mechanical-engineering enterprises determines the high practical significance of the developed approach

scholarly journals An Effective Method to Calculate Frequency Response of Distribution Networks for PLC Applications

Electronics ◽  
2019 ◽  
Vol 8 (6) ◽  
pp. 649 ◽  
Author(s):  
Hongshan Zhao ◽  
Weitao Zhang ◽  
Yan Wang
Keyword(s):  
Frequency Response ◽  
Distribution Networks ◽  
Dynamic Changes ◽  
Binary Function ◽  
Underground Cables ◽  
Overhead Lines ◽  
Medium Voltage ◽  
Response Characteristics ◽  
Network Topologies ◽  
Propagation Matrix

Modelling and estimating power-line communication (PLC) channels are complicated issues due to the complex network topologies, various junctions, and changeable loads. This paper focuses on the frequency response characteristics (FRCs) of medium-voltage (MV) PLC networks with special consideration of two scenarios that are often neglected but generally exist. In the first scenario, the MV distribution network is of the ring topology. In the second scenario, the MV overhead lines and underground cables join at junctions, and the shields of underground cables are grounded with nonzero grounding impedances at the junctions. These conditions lead to the failure of currently popular methods to different degrees. For this reason, we developed an effective method to calculate the FRCs of distribution networks for PLC applications. With this method, the frequency responses of nodes are simply expressed as the binary function of the overall tube propagation matrix and overall node scattering matrix, which is convenient for calculations and analyses. The proposed method was validated by the agreement between the calculated and measured FRCs. The results of two test examples showed that the proposed method performed better in comparison with the traditional approximate method when nonideal grounding conditions were taken into account. The proposed method is also independent of the network topology, so it can adapt to the dynamic changes of the network structure.

Skolem reduction classes

1975 ◽  
Vol 40 (1) ◽  
pp. 62-68 ◽  
Author(s):  
Warren D. Goldfarb ◽  
Harry R. Lewis
Keyword(s):  
Decision Problem ◽  
Atomic Formula ◽  
Complex Situation ◽  
Quantification Theory ◽  
Reduction Class

Among the earliest and best-known theorems on the decision problem is Skolem's result [7] that the class of all closed formulas with prefixes of the form ∀···∀∃···∃ is a reduction class for satisfiability for the whole of quantification theory. This result can be refined in various ways. If the Skolem prefix alone is considered, the best result [8] is that the ∀∀∀∃ class is a reduction class, for Gödel [3], Kalmár [4], and Schütte [6] showed the ∀∀∃···∃ class to be solvable. The purpose of this paper is to describe the more complex situation that arises when (Skolem) formulas are restricted with respect to the arguments of their atomic subformulas. Before stating our theorem, we must introduce some notation.Let x, y1, y2, be distinct variables (we shall use v1, v2, ··· and w1, w2, ··· as metavariables ranging over these variables), and for each i ≥ 1 let Y(i) be the set {y1, ···, yi}. An atomic formula Pv1 ··· vk will be said to be {v1, ···, vk}-based. For any n ≥ 1, p ≥ 1, and any subsets Y1, ··· Yp of Y(n), let C(n, Y1, ···, Yp) be the class of all those closed formulas with prefix ∀y1 ··· ∀yn∃x such that each atomic subformula not containing the variable x is Yi-based for some i, 1 ≤ i ≤ p.

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