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As a special case, a propositional formula or Boolean expression is a Boolean circuit with a single output node in which every other node has fan-out of 1. Thus, a Boolean circuit can be regarded as a generalization that allows shared subformulas and multiple outputs.
Example Boolean circuit. The nodes are AND gates, the nodes are OR gates, and the nodes are NOT gates. In theoretical computer science, circuit complexity is a branch of computational complexity theory in which Boolean functions are classified according to the size or depth of the Boolean circuits that compute them.
A propositional logic formula, also called Boolean expression, is built from variables, operators AND (conjunction, also denoted by ∧), OR (disjunction, ∨), NOT (negation, ¬), and parentheses. A formula is said to be satisfiable if it can be made TRUE by assigning appropriate logical values (i.e. TRUE, FALSE) to
A law of Boolean algebra is an identity such as x ∨ (y ∨ z) = (x ∨ y) ∨ z between two Boolean terms, where a Boolean term is defined as an expression built up from variables and the constants 0 and 1 using the operations ∧, ∨, and ¬. The concept can be extended to terms involving other Boolean operations such as ⊕, →, and ≡ ...
The circuit on the left is satisfiable but the circuit on the right is not. In theoretical computer science, the circuit satisfiability problem (also known as CIRCUIT-SAT, CircuitSAT, CSAT, etc.) is the decision problem of determining whether a given Boolean circuit has an assignment of its inputs that makes the output true. [1]
The naive approach is to write the circuit as a Boolean expression, and use De Morgan's law and the distributive property to convert it to CNF. However, this can result in an exponential increase in equation size. The Tseytin transformation outputs a formula whose size grows linearly relative to the input circuit's.
A Boolean value is either true or false. A Boolean expression may be composed of a combination of the Boolean constants True/False or Yes/No, Boolean-typed variables, Boolean-valued operators, and Boolean-valued functions. [1] Boolean expressions correspond to propositional formulas in logic and are a special case of Boolean circuits. [2]
Boolean formulas can also be displayed as a graph: Propositional directed acyclic graph. Digital circuit diagram of logic gates, a Boolean circuit; And-inverter graph, using only AND and NOT; In order to optimize electronic circuits, Boolean formulas can be minimized using the Quine–McCluskey algorithm or Karnaugh map.