Asynchronous logic (algebra)
Encyclopedia
Asynchronous logic is a sort of symbiosis (mixture) of combinational logic
and sequential logic
. In digital circuit
theory asynchronous logic differs from synchronous one because its propositional variables act asynchronously without common clocked regulation and control. It means that every input of digital circuit follows its own time, own clock.
, and it is also considered an applied discipline of mathematical logic
. Mathematical apparatus of asynchronous logic is served by Boolean algebra as well as by algebraic instruments of sequential logic – venjunction and sequention. The corresponding mathematical apparatus in the form of logical relationships is assigned for digital circuit representation, analysis and synthesis.
) with two propositional variables.
2. Sequention is a sequence (inside angle brackets ) of propositional variables, which being binary function takes logical unity value at the following order of switchings:
In all other cases sequential function is equal to zero.
2. Combination of venjunction and conjunction:
Combinational logic
In digital circuit theory, combinational logic is a type of digital logic which is implemented by boolean circuits, where the output is a pure function of the present input only. This is in contrast to sequential logic, in which the output depends not only on the present input but also on the...
and sequential logic
Sequential logic
In digital circuit theory, sequential logic is a type of logic circuit whose output depends not only on the present input but also on the history of the input. This is in contrast to combinational logic, whose output is a function of, and only of, the present input...
. In digital circuit
Digital circuit
Digital electronics represent signals by discrete bands of analog levels, rather than by a continuous range. All levels within a band represent the same signal state...
theory asynchronous logic differs from synchronous one because its propositional variables act asynchronously without common clocked regulation and control. It means that every input of digital circuit follows its own time, own clock.
Characteristics
Asynchronous logic is part of discrete mathematicsDiscrete mathematics
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not...
, and it is also considered an applied discipline of mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
. Mathematical apparatus of asynchronous logic is served by Boolean algebra as well as by algebraic instruments of sequential logic – venjunction and sequention. The corresponding mathematical apparatus in the form of logical relationships is assigned for digital circuit representation, analysis and synthesis.
Mathematical instruments
1. Venjunction is a logic-dynamic operation (sign) with two propositional variables.-
if 
on the background 
-
if 
or 
or 
2. Sequention is a sequence (inside angle brackets ) of propositional variables, which being binary function takes logical unity value at the following order of switchings:
In all other cases sequential function is equal to zero.
Asynchronous logic formulas
Formulas in asynchronous logic are represented as analytical expressions, whose propositional variables are connected by means of Boolean operations – conjunction, disjunction and negation, which are combined with additional logic-dynamic operation – venjunction. Alongside with binary variables, sequentions are also forming components of the formulas. Transformations of asynchronous logic formulas obey certain rules.Venjunction laws
1. Negation of venjunction:
2. Combination of venjunction and conjunction:
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3. Combination of venjunction and disjunction:
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4. Connection of venjunction and sequention:
Transformation of sequentions
1. Associativity:
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2. Zeroing:
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3. Absorption:
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4. Splitting:
5. Splicing (under condition
):
6. Decomposition:
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7. Association with conjunction (conjunctive decomposition of sequention):
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8. Association with venjunction (venjunctive decomposition of sequention):
Trigger function
Trigger function is an asynchronous logic function which serves as a template for constructing trigger (flip-flop) type circuits with paraphase outputs. It is a sequential function of two variables, represented by two equations:
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Input arguments X and Y must satisfy the following relations:
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External links
- http://asynlog.balticom.lv/Content/EN/about.html (Asynchronous logic and new algebra for digital circuits).
- An introduction to asynchronous circuit design by A. Davis, S. Nowick .
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