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The second pattern of potentially globally redundant proofs appearing in global redundancy definition is related to the well-known [further explanation needed] notion of regularity [further explanation needed]. Informally, a proof is irregular if there is a path from a node to the root of the proof such that a literal is used more than once as ...
In mathematical logic, a literal is an atomic formula (also known as an atom or prime formula) or its negation. [1] [2] The definition mostly appears in proof theory (of classical logic), e.g. in conjunctive normal form and the method of resolution. Literals can be divided into two types: [2] A positive literal is just an atom (e.g., ).
The resolution rule in propositional logic is a single valid inference rule that produces a new clause implied by two clauses containing complementary literals. A literal is a propositional variable or the negation of a propositional variable.
But a rule of inference's action is purely syntactic, and does not need to preserve any semantic property: any function from sets of formulae to formulae counts as a rule of inference. Usually only rules that are recursive are important; i.e. rules such that there is an effective procedure for determining whether any given formula is the ...
A set of rules can be used to infer any valid conclusion if it is complete, while never inferring an invalid conclusion, if it is sound. A sound and complete set of rules need not include every rule in the following list, as many of the rules are redundant, and can be proven with the other rules.
A definition of a function, set, or other mathematical object that is defined in terms of itself, using a base case and a rule for generating subsequent elements. recursive function A function that can be computed by a procedure that calls itself, directly or indirectly, with a base case to prevent infinite recursion. recursive function theory
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Mathematical logic is the study of formal logic within mathematics.Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory).