By Vadim Vasyukevich

This ebook is devoted to new mathematical tools assigned for logical modeling of the reminiscence of electronic units. The working example is logic-dynamical operation named venjunction and venjunctive functionality in addition to sequention and sequentional functionality. Venjunction and sequention function in the framework of sequential common sense. In a kind of the corresponding equations, they organically healthy analytical expressions of Boolean algebra. therefore, a type of symbiosis is shaped utilizing components of asynchronous sequential common sense at the one hand and combinational good judgment however. So, asynchronous good judgment is represented within the kind of greater Boolean common sense. The publication comprises preliminary thoughts, basic definitions, statements, ideas and principles wanted for theoretical justification of the mathematical equipment and its validity for asynchronous common sense. Asynchronous operators named venjunctor and sequentor are designed for functional implementation. those uncomplicated components are assigned for figuring out of reminiscence features in sequential circuits. current learn paintings is the ultimate level of generalization and systematization of all these rules and investigations, author’s curiosity to which alternately flashed up and light over decades and for varied purposes till shaped “critical mass”, and all findings have been prepared definitively as a mathematical foundation of a thought properly linked lower than a typical subject – asynchronous sequential common sense, basically labeled as switching common sense, which falls into classification of algebraic logics.

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**Extra info for Asynchronous Operators of Sequential Logic: Venjunction & Sequention: Digital Circuit Analysis and Design (1st Edition)**

**Sample text**

2. 2) where X ∈ {1, 0/1, 1/0, 0}, Y ∈ {1, 0/1, 1/0, 0}, Z ∈ {1, 0}. 2 Bistable cell (Fig. 1) functioning: a) output ZX; b) output ZY. 1 Bistable Cell 27 Transforming tabulated switchings into venjunctions, and applying these venjunctions to the given above dependences, the following logical expressions are formed: Z X = ( X ∠ Y ) ∨ ( X ∠ Y ) ∨ (Y ∠ X ) ∨ (Y Z Y = (Y ∠ X ) ∨ (Y ∠ ∠ X ) ∨ (X ∠ Y ) ∨ (X X )∨ (X ∠ Y ) , ∠Y) ∨ (Y ∠ X) . 2) These expressions are output functions presented in their venjunctive complete form.

8, where each eight venjunctions becomes unity exclusively at one set of the binary sequence. It is notable that under this set all other venjunctions are reduced to zero. Therefore, it is not difficult to construct any of possible 28 functions by means of the Boolean disjunctive operation. 1 Venjunctive Form Completed by Disjunction Common expression for the venjunctive complete form is given by the following: ∨ (k ∧ V ), i∈{1, 2 ... 8} . 24) i Coefficient k regulates the sampling of such venjunctions that are chosen for representation of one or another function in its complete form.

On the circuit outputs, two inventors (NOT-elements) are placed. According to the logical circuit, dependence of output signal ZX from input signals X and Y is represented by the following formula: Z X = ( X ∧ ( X ∨ X ∠ Y )) ∨ (( X ∧ ( X ∨ X ∠ Y )) ∠ (Y ∧ (Y ∨ Y ∠ X ))) . 23) 38 2 Venjunctive Expressions Taking into account equalities – ( X ∧ ( X ∨ X ∠ Y )) = X ∠ Y and (Y ∧ (Y ∨ Y ∠ X )) = Y ∠ X , the given expression is minimized as follows: Z X = ( X ∠ Y ) ∨ (( X ∠ Y ) ∠ (Y ∠ X )) . 24) Further logical actions, including zeroing ( X ∠ Y ) ∠ (Y ∠ X ) = 0 , produce a venjunction as a result of the following operation: ZX = ( X ∠ Y ) = X ∠ Y .