Incremental Version of the Modern Syllogistic Method

Rushdi, Ali A.; Zarouan, Mohamed; Alshehri, Taleb; Rushdi, Muhammad A.

doi:10.4197/eng.26-1.2

Cited by 9 publications

(14 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The current paper is one of several new papers by the authors which are intended to extend the utility and sharpen the mathematics of the MSM. One of these papers [ 62 ] presents an incremental version of the MSM, in which the core work of the MSM is not completely repeated but is slightly incremented when additional premises are added. Another paper [ 63 ] utilizes the MSM in the exploration of hidden aspects in engineering ethical dilemmas by investigating different scenarios describing the situation from various perspectives.…”

Section: Discussionmentioning

confidence: 99%

A Modern Syllogistic Method in Intuitionistic Fuzzy Logic with Realistic Tautology

Rushdi

Zarouan

Alshehri

et al. 2015

The Scientific World Journal

View full text Add to dashboard Cite

The Modern Syllogistic Method (MSM) of propositional logic ferrets out from a set of premises all that can be concluded from it in the most compact form. The MSM combines the premises into a single function equated to 1 and then produces the complete product of this function. Two fuzzy versions of MSM are developed in Ordinary Fuzzy Logic (OFL) and in Intuitionistic Fuzzy Logic (IFL) with these logics augmented by the concept of Realistic Fuzzy Tautology (RFT) which is a variable whose truth exceeds 0.5. The paper formally proves each of the steps needed in the conversion of the ordinary MSM into a fuzzy one. The proofs rely mainly on the successful replacement of logic 1 (or ordinary tautology) by an RFT. An improved version of Blake-Tison algorithm for generating the complete product of a logical function is also presented and shown to be applicable to both crisp and fuzzy versions of the MSM. The fuzzy MSM methodology is illustrated by three specific examples, which delineate differences with the crisp MSM, address the question of validity values of consequences, tackle the problem of inconsistency when it arises, and demonstrate the utility of the concept of Realistic Fuzzy Tautology.

show abstract

Section: Discussionmentioning

confidence: 99%

A Modern Syllogistic Method in Intuitionistic Fuzzy Logic with Realistic Tautology

Rushdi

Zarouan

Alshehri

et al. 2015

The Scientific World Journal

View full text Add to dashboard Cite

show abstract

“…Then, we replace in (5) by its complete-sum form using the Blake-Tison Method [33][34][35][36][37][38]. We note that (5) involves four variables, which are all biform.…”

Section: Equational and Implicational Descriptionsmentioning

confidence: 99%

Novel Characterizations of the JK Bistables (Flip Flops)

Rushdi

Ghaleb

2019

JERR

View full text Add to dashboard Cite

The JK flip flop is a flexible type of bistable elements that has extensive uses in digital electronics and control circuits. It is usually described by its characteristic equation or next-state table (used for analysis purposes) and its excitation table (used for synthesis purposes). This paper explores a variety of novel characterizations of JK flip flops. First, equational and implicational descriptions are presented, and methods of logic deduction are utilized to produce complete statements of all propositions that are true for a general JK flip flop. Next, methods of Boolean-equation solving are employed to find all possible ways to express the excitations in terms of the present state and next state. The concept of Boolean quotient is used to impose certain requirements so as to find particularly useful expressions of the excitations. Relations of JK flip flops to other types of flip flops are also explored. This paper is expected to provide an immediate pedagogical benefit, and to help facilitate the analysis and synthesis of sequential digital circuits.

show abstract

“…The STP was also successfully utilized in solving deduction problems over propositional logic. We anticipate a better utility for the STP in solving these deduction problems, if it is used in conjunction with the Modern Syllogistic Method (MSM) [85][86][87][88][89] , which is a method that ferrets out (from any set of premises) all that can be concluded in the most compact form. We devote the next few sections to STP applications to Boolean and pseudo-Boolean functions.…”

Section: Stp Applications To the Matrixmentioning

confidence: 99%

A Tutorial Exposition of Semi-Tensor Products of Matrices with a Stress on Their Representation of Boolean Functions

Rushdi¹,

Ghaleb²

2016

JKAU: Comp. IT. Sci.

View full text Add to dashboard Cite

This paper is a detailed tutorial exposition of a novel and highly-useful product of matrices called the semi-tensor product (STP). This new product allows the multiplication of two matrices and that do not satisfy the conformity condition required in a conventional matrix product (CMP), namely, that the column dimension n of the first matrix be equal to the row dimension p of the second matrix B. The STP inherits almost all of the properties of the CMP, and reduces to it when n = p. The basic concepts, properties, manipulations of the STP are presented, and demonstrated via tutorial illustrative examples. Applications of the STP include multi-linear and non-linear control problems, theoretical analysis of mathematical problems, and derivation of mathematical proofs. We emphasize the STP use in the matrix expression of Boolean (Switching or logical) functions. We achieve a better understanding of this use by translating the STP terminology to conventional or more familiar representations of Boolean functions, and by explicitly identifying the bases on which the matrix expressions of the Boolean function are constructed. We compare the STP representation of Boolean functions with other well known representations such as the Variable-Entered Karnaugh Maps (VEKMs) and the Reduced Ordered Binary Decision Diagrams (ROBDDs). We show that while these latter representations are complete multi-level ones, the STP representation is a single leaf-level representation. In essence, the STP representation is a truth-table expression of both the function and its complement, and hence it involves unnecessary duplication. The readability of the STP representation of a Boolean function is hindered by its implicit reverse ordering starting at 1 rather than at 0. The STP representation is suitable for the larger class of pseudo-Boolean functions, and hence it is somewhat loose for the restricted class of Boolean functions. For example, it uses the very general and costly operation of real addition to implement the OR operation of Boolean algebra. Such an inefficient implementation cannot be tolerated when the intended applications are among the most intractable problems. Despite the aforementioned negative aspects, the utility of the STP in a variety of applications and even in handling Boolean functions is undeniable. In fact, the STP is computationally successful in handling medium-sized problems in areas ranging from Boolean synchronous networks to logic deduction. The STP is an effective tool for mathematical proofs and derivations over the Boolean domain. We demonstrate this point by using mathematical induction in a novel STP derivation of the Preparata Transformation which relates the truth-table representation of a two-valued Boolean function to its linear (Reed-Müller) one.

show abstract

Incremental Version of the Modern Syllogistic Method

Cited by 9 publications

References 27 publications

A Modern Syllogistic Method in Intuitionistic Fuzzy Logic with Realistic Tautology

A Modern Syllogistic Method in Intuitionistic Fuzzy Logic with Realistic Tautology

Novel Characterizations of the JK Bistables (Flip Flops)

A Tutorial Exposition of Semi-Tensor Products of Matrices with a Stress on Their Representation of Boolean Functions

Contact Info

Product

Resources

About