Sufficient Conditions For Completeness Research Articles

On involutive division on monoids

We consider an arbitrary monoid MM, on which an involutive division is introduced, and the set of all its finite subsets SetMM. Division is considered as a mapping d:SetMM{d:SetM \times M}, whose image d(U,m){d(U,m)} is the set of divisors of mm in UU. The properties of division and involutive division are defined axiomatically. Involutive division was introduced in accordance with the definition of involutive monomial division, introduced by V.P. Gerdt and Yu.A. Blinkov. New notation is proposed that provides brief but explicit allowance for the dependence of division on the SetMM element. The theory of involutive completion (closures) of sets is presented for arbitrary monoids, necessary and sufficient conditions for completeness (closedness) - for monoids generated by a finite set XX. The analogy between this theory and the theory of completely continuous operators is emphasized. In the last section, we discuss the possibility of solving the problem of replenishing a given set by successively expanding the original domain and its connection with the axioms used in the definition of division. All results are illustrated with examples of Thomas monomial division.

Simple motion pursuit differential game problem of many players with integral and geometric constraints on controls function.

We study a simple motion pursuit differential game of many pursuers and one evader in a Hilbert space $l_{2}$. The control functions of the pursuers and evader are subject to integral and geometric constraints respectively. Duration of the game is denoted by positive number $\theta $. Pursuit is said to be completed if there exist strategies $u_{j}$ of the pursuers $P_{j}$ such that for any admissible control $v(\cdot)$ of the evader $E$ the inequality $\|y(\tau)-x_{j}(\tau)\|\leq l_{j}$ is satisfied for some $ j\in \{1,2, \dots\}$ and some time $\tau$. In this paper, sufficient conditions for completion of pursuit were obtained. Consequently strategies of the pursuers that ensure completion of pursuit are constructed.

On Topological Properties of Metrics Defined via Generalized “Linking Construction”

We analyze topological properties of metric spaces obtained by using Száz’s construction, which we used to call generalized “linking construction.” In particular, we provide necessary and sufficient conditions for completeness of metric spaces obtained in this way. Moreover, we examine the relation between Száz’s construction and the “linking construction.” A particular attention is drawn to the “floor” metric, the analysis of which provides some interesting observations.

Proving completeness of logic programs with the cut

Abstract Completeness of a logic program means that the program produces all the answers required by its specification. The cut is an important construct of programming language Prolog. It prunes part of the search space, this may result in a loss of completeness. This paper proposes a way of proving completeness of programs with the cut. The semantics of the cut is formalized by describing how SLD-trees are pruned. A sufficient condition for completeness is presented, proved sound, and illustrated by examples.

Completeness Conditions of Systems of Moore Automata

Abstract This paper describes sufficient completeness conditions for the class of Moore automata, compositions with feedback on this class and the behavior realization. It is shown how to realize a delay by automata from the set satisfying sufficient conditions for completeness. The ideas of these constructions demonstrate how from a set of automata with long cycles of transients can be realized a delay with short cycle of transient. It is shown that there are large finite complete sets of automata, for which any proper subsets are not complete.

The completable digraphs for the totally nonnegative completion problem

In this paper, we study the totally nonnegative completion problem when the partial totally nonnegative matrix is non-combinatorially symmetric. In general, this type of partial matrix does not have a totally nonnegative completion. Here, we give necessary and sufficient conditions for completion of a partial totally nonnegative matrix to a totally nonnegative matrix in the cases where the digraph of the off-diagonal specified entries takes certain forms as path, cycles, alternate paths, block graphs, etc., distinguishing between the monotonically and non-monotonically labeled case.

The Basis Properties of Some Systems of Exponential Functions, Cosines, and Sines

We consider some systems of exponential functions, cosines, and sines with complex-valued coefficients and establish a necessary and sufficient condition for completeness and minimality of these systems in Lebesgue spaces.

On the intrinsic complexity of learning recursive functions

The intrinsic complexity of learning compares the difficulty of learning classes of objects by using some reducibility notion. For several types of learning recursive functions, both natural complete classes are exhibited and necessary and sufficient conditions for completeness are derived. Informally, a class is complete iff both its topological structure is highly complex while its algorithmic structure is easy. Some self-describing classes turn out to be complete. Furthermore, the structure of the intrinsic complexity is shown to be much richer than the structure of the mind change complexity, though in general, intrinsic complexity and mind change complexity can behave “orthogonally”.

Control Game Problems for Quasilinear Systems with Riemann-Liouville Fractional Derivatives

Conflict-controlled processes for systems with Riemann-Liouville fractional derivatives of arbitrary order are studied. The solutions to such systems are presented in the form of an analog of the Cauchy formula. Sufficient conditions for completion of the game are obtained using the method of resolving functions. These conditions are based on the modified Pontrjagin condition expressed in terms of the Mittag-Leffler matrix functions. To find them, the Lagrange-Sylvester interpolating polynomials are applied.

Termination and confluence in infinitary term rewriting

AbstractThe basic notions of the theory of term rewriting are defined for terms that may involve function letters of infinite arity. A sufficient condition for completeness is derived, and its use demonstrated by the example of abstract clones over infinitary signatures.

Structures and methods of dynamical decision-making

The problem of dynamical decision making is considered here. The system approach to the problem is developed. The notions of decision making system and its structure are introduced in accordance with this approach. Three main pans can be singled out within the structure: base, envelope and deciser. The notion of base completeness is introduced. Sufficient conditions of completeness are obtained. Different variants of complete bases depending on uncertainty conditions are considered. The corresponding methods of optimal decision making are obtained.

Completeness of arbitrarily sampled discrete time wavelet transforms

An arbitrarily sampled discrete time wavelet transform is said to be complete if it is uniquely invertible, i.e., if the underlying signal can be uniquely recovered from the available samples of the wavelet transform. We develop easy-to-compute necessary and sufficient conditions and necessary but not sufficient conditions for the completeness of an arbitrarily sampled dyadic discrete time wavelet transform of a periodic signal. In particular, we provide necessary and sufficient conditions for completeness of the sampled wavelet transform when the lowpass filter associated with the dyadic wavelet filter bank has no unit circle zeros other than those at z=1. We present necessary but not sufficient conditions for completeness when the lowpass filter associated with the dyadic wavelet filter bank has arbitrary unit circle zeros. We also provide necessary and sufficient conditions for completeness of a set of samples of both the lowpass approximation to the signal and its wavelet transform. All the conditions we derive use only information about the location of the retained samples and the analyzing wavelet filter bank. They can easily be checked without explicitly computing of the rank of a matrix. Finally, we present a simple signal reconstruction procedure that can be used once we have determined the arbitrarily sampled discrete time wavelet transform is complete.

A remark on the spectral domain of nonstationary processes

The question of completeness of the spectral domain generated by a bimeasure is discussed. The bimeasure is derived from a spectral representation of a given process of weak class (C). In particular by an example it is shown that the spectral domain of a weak class (C) process is not necessarily complete. A sufficient condition for completeness is established.

Negation-as-failure rule for general logic programs with equality

General logic programs are programs that have clauses containing inequations and negative literals in their bodies. In this paper, the success and failure of general logic programs are defined mutually recursively. The pair of their interpretations are shown to be the least fixpoint of the inference function T defined on pairs of interpretations. Consequentially, with a complete equality theory, every model of the completed program contains the interpretation of the success set and does not meet that of the failure set. As a result, the soundness of SLDNF resolution and the negation-as-failure rule are proved. Necessary and sufficient conditions for completeness are also presented.

Completion of linear differential games

The concept of completion of differential games was introduced by Pontryagin in Ref. 1. Here, necessary and sufficient conditions for completion of linear differential games are presented.

Functional completeness in one variable

It is extremely difficult to make general statements about functional completeness. (For the main reference on the subject see Post [2].) In this paper we restrict ourselves to the case of unary functions in a finite valued logic, and prove a result concerning minimal functionally complete sets, along with a necessary and sufficient condition for completeness. A basic familiarity with group theory would be helpful.

Completeness conditions for finite automata

This paper contains results obtained by the author during work in the seminar on the theory of finite automata held at the Computer Centre of the Ukrainian Academy of Sciences under V.M. Glushkov. The problem of completeness for finite automata was formulated by Glushkov in his lectures on the theory of finite automata at Kiev University. Some sufficient conditions of completeness were given there. In the present paper the problem of completeness is solved for a certain kind of composition and some methods of realization of automata.

A criterion for completeness of degrees of unsolvability

Kleene and Post [2] have shown that between each degree a and its completion a′ there are an infinity of mutually incomparable degrees of unsolvability, and that between a and its nth. completion a(n) there are an infinity of mutually incomparable degrees each incomparable with a′, …, a(n−1). They have left open the following questions: whether there exist complete degrees in similar profusion; whether indeed there exist incomparable complete degrees at all; and whether certain degrees, such as a(ω) ([2] p. 401), are complete. All these questions are answered affirmatively by the corollary of the present paper, which states that every degree greater than or equal to 0′ is complete. Since every complete degree is already known to be greater than or equal to 0′, it follows from the present paper that being greater than or equal to 0′ is both a necessary and a sufficient condition for completeness.