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A literal is either a variable (in which case it is called a positive literal) or the negation of a variable (called a negative literal). A clause is a disjunction of literals (or a single literal). A clause is called a Horn clause if it contains at most one positive literal.
In propositional calculus a literal is simply a propositional variable or its negation.. In predicate calculus a literal is an atomic formula or its negation, where an atomic formula is a predicate symbol applied to some terms, (, …,) with the terms recursively defined starting from constant symbols, variable symbols, and function symbols.
In computer science, a literal is a textual representation (notation) of a value as it is written in source code. [1] [2] Almost all programming languages have notations for atomic values such as integers, floating-point numbers, and strings, and usually for Booleans and characters; some also have notations for elements of enumerated types and compound values such as arrays, records, and objects.
In the context of functions, the term variable refers commonly to the arguments of the functions. This is typically the case in sentences like "function of a real variable", "x is the variable of the function f: x ↦ f(x)", "f is a function of the variable x" (meaning that the argument of the function is referred to by the variable x).
The pseudocode DPLL function only returns whether the final assignment satisfies the formula or not. In a real implementation, the partial satisfying assignment typically is also returned on success; this can be derived by keeping track of branching literals and of the literal assignments made during unit propagation and pure literal elimination.
Formally a Zhegalkin monomial is the product of a finite set of distinct variables (hence square-free), including the empty set whose product is denoted 1.There are 2 n possible Zhegalkin monomials in n variables, since each monomial is fully specified by the presence or absence of each variable.
The resulting assignment is the minimal model of the Horn formula, that is, the assignment having a minimal set of variables assigned to true, where comparison is made using set containment. Using a linear algorithm for unit propagation, the algorithm is linear in the size of the formula.
Boole's expansion theorem, often referred to as the Shannon expansion or decomposition, is the identity: = + ′ ′, where is any Boolean function, is a variable, ′ is the complement of , and and ′ are with the argument set equal to and to respectively.