Synthesis of multifunction and universal logic elements over residue class rings. I

Translated from Kibernetika, No. 4, pp. 89–94, July–August, 1988.     

Singular shift registers over residue class rings

The study of shift registers over the ringZ _p ^r (p a prime,r 1), in particular the singular case, is carried out, giving necessary and sufficient conditions to make a shift register singular. Its state graph is characterized as a tree, whose form depends on the characteristic polynomial of the shift register. The relationships with singular shift register overZ _p are explored and a proof of their equivalence is presented.Research sponsored by the National Science Foundation, Grant GK-1065x, and the Joint Services Electronics Program, Grant AFOSR-68-1488.Part of this work was done while the author was at Electronics Research Laboratory, University of California at Berkeley.     

Occurrence indices of elements in linear recurrence sequences over primary residue rings

We study distances to the first occurrence (occurrence indices) of a given element in a linear recurrence sequence over a primary residue ring \(\mathbb{Z}_{p^n } \). We give conditions on the characteristic polynomial F(x) of a linear recurrence sequence u which guarantee that all elements of the ring occur in u. For the case where F(x) is a reversible Galois polynomial over \(\mathbb{Z}_{p^n } \), we give upper bounds for occurrence indices of elements in a linear recurrence sequence u. A situation where the characteristic polynomial F(x) of a linear recurrence sequence u is a trinomial of a special form over ?₄ is considered separately. In this case we give tight upper bounds for occurrence indices of elements of u.     

Distribution of r-tuples in one class of uniformly distributed sequences over residue rings

We consider the distribution of r-tuples in one class of uniformly distributed linear recurring sequences over residue rings. We establish bounds for the number of occurrences of a given r-tuple and prove that these bounds are asymptotically the best possible.     

Equidistributed sequences over finite fields produced by one class of linear recurring sequences over residue rings

We consider the distribution of r-patterns in one class of uniformly distributed sequences over a finite field. We establish bounds for the number of occurrences of a given r-pattern and prove upper bounds for the cross-correlation function of these sequences.     

Tunable multifunction logic modules based on parametric representation of functions of k-valued logic over quasifields

We propose a new class of tunable logic modules that realize functions of k-valued logic. These modules are characterized by homogeneous structure and linear dependence of the number of tunable inputs on the number of input variables.Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 24–29, July–August, 1991.     

A universal logic element over the complex field

We propose a mathematical model of the universal logic element using a parametric representation of Boolean functions over the complex field.Translated from Kibernetika, No. 3, pp. 116–121, May–June, 1991.     

A class of 1-generator quasi-cyclic codes over finite chain rings

Using another method, which is different with the one given by Cao [1-generator quasi-cyclic (QC) codes over finite chain rings, Appl. Algebra Eng. Commun. Comput. 24 (2013), pp. 53–72], we investigate the structural properties of a class of 1-generator QC codes over finite chain rings. We give the structure of the annihilator of 1-generator QC codes, the conditions for 1-generator QC codes to be free and the minimum distance bounds on 1-generator QC codes. Under some conditions, we also discuss the enumeration of 1-generator QC codes and describe how to obtain the one and only one generator for each 1-generator QC code.     

Synthesis of feedback control logic for a class of controlled Petrinets

An efficient solution is developed for a class of forbidden state problems for discrete event systems (DESs). DESs are considered which can be modeled as cyclic controlled marked graphs (CMGs), a special class of controlled Petri nets (CPNs). The distributed representation of the DES state in terms of the CMG marking permits an efficient specification of the forbidden states in terms of individual place markings. More important, it is shown that the graphical representations of the state transition logic in a CMG can be used to synthesize state feedback logic which is maximally permissive while guaranteeing the forbidden states will not occur. The practical application of the theoretical results is illustrated for an example of automated guided vehicle (AGV) coordination in a flexible manufacturing facility     

A comparison of voice and multifunction controls: logic design is the key

The proper design of the human-computer interface is a key feature for the optimum use of advanced manned systems. In future aircraft cockpits, one of the pilot's principle tasks will be to interact with a very sophisticated computer system—an electronic crewmember. This paper discusses the experimental comparison of two communication mechanisms: voice control and multifunction keyboards. The paper also discusses the crucial role of the software logic used with these devices.     

泛逻辑学中UB代数系统的若干性质

文[1]提出了理想状态（h=k=0.5）下泛逻辑学对应的代数系统-UB代数,给出了它的一些性质。在[1]的基础上,进一步讨论了UB代数系统,得到一些有趣的结果,这些结果对UB代数的进一步研究是有用的。     

基于零级泛与运算的泛逻辑中广义重言式理论

对泛逻辑的广义重言式理论进行了研究,给出了基于零级泛与运算的广义重言式的一系列性质。主要结果有:当c∈[0.75,1]时,关于Ih=c而言,F(S)中只有3种不同的广义重言式,即可达0-重言式、0+-重言式和重言式;当h=0时,关于Ih=0而言,F(S)中存在0、1可达重言式,当α∈(0,1),不存在α-重言式、α+-重言式和可达α-重言式;用h=0.5时的广义重言式对h∈(0,0.75)时的广义重言式进行了刻画。     

泛逻辑学中UB代数系统的滤子与商代数

给出了理想状态下泛逻辑学的形式演绎系统,证明了此系统是可靠的。提出了在理想状态（h=k=0.5）下泛逻辑学对应的代数系统-UB代数,进一步讨论了UB代数滤子与商代数,得到一些有用的结果。     

A Method for Minimization Design of Two-Level Logic Networks Using Multiplexer Universal Logic Modules

A decomposition approach of the combinational functions is discussed.A design method,by which the minimization or near minimization of two-level combinational network can be obtained,is presented for a combinational function realized by using multiplexer universal logic modules.Using the method,the automated synthesis of the combinational functions can be accomplished on a computer.     

Universal abstract consistency class and universal refutation

Refutation methods based on the resolution principle are generally applied to a (finite)set of sentences,which must have a series of pre-transformations(prenex normalization,Skolemization and conjunction normalization)before starting the refutation.In this paper,the authors first generalize the concept of abstract consistency class to the most general form-universal abstract consistency class,and prove its universal univfying principle.Then,based on the R-refutation,a universal refutation method is proposed and its soundness and completeness are proved by means of the universal unifying principle.This method can be applied directly to any finite set of wffs sithout preprocessing the wffs at all so that the refutation procedure is more natural.     

Clausal intuitionistic logic II. tableau proof procedures

Clausal intuitionistic logic is an extension of Horn-clause logic which permits the appearance of negations and embedded implications on the right-hand side of a rule, and interprets these new rules intuitionistically in a set of partial models. In this article the second of a pair, clausal intuitionistic logic, is shown to have a tableau proof procedure that generalizes Horn-clause refutation proofs. The proof procedures are explained by means of several detailed examples. Soundness and completeness theorems are stated and proven in full detail.     

Efficient logic design of reliable control units. II

Two autogram-based formal methods of construction of control units from nontransparent flip-flops are described. One of them is a simplification of a frequently used (classical) method for design of control units. The results of synthesis under this method are easily interpreted in terms of the initial statement of a design problem. The other method is a generalization of the classical method. It allows one to optimize areas and realize control units using standard unified blocks. Both methods can be easily used within the framework of CAD tools.     

泛逻辑学中UB代数系统的fuzzy滤子

文献[2]给出了理想状态下泛逻辑学的形式演绎系统!,证明了此系统是可靠的。文献[3]提出了在理想状态(h=k=0.5)下泛逻辑学对应的代数系统-UB代数。文献[5]研究了泛逻辑学中UB代数系统的若干性质。在文献[3,5]的基础上,进一步讨论了UB代数fuzzy滤子与商代数。