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The Boolean satisfiability problem is NP-complete, and consequently, tautology is co-NP-complete. It is widely believed that (equivalently for all NP-complete problems) no polynomial-time algorithm can solve the satisfiability problem, although some algorithms perform well on special classes of formulas, or terminate quickly on many instances. [8]
One example of a co-NP-complete problem is tautology, the problem of determining whether a given Boolean formula is a tautology; that is, whether every possible assignment of true/false values to variables yields a true statement.
In propositional logic, tautology is either of two commonly used rules of replacement. [ 1 ] [ 2 ] [ 3 ] The rules are used to eliminate redundancy in disjunctions and conjunctions when they occur in logical proofs .
In literary criticism and rhetoric, a tautology is a statement that repeats an idea using near-synonymous morphemes, words or phrases, effectively "saying the same thing twice". [ 1 ] [ 2 ] Tautology and pleonasm are not consistently differentiated in literature. [ 3 ]
Tautology may refer to: Tautology (language), a redundant statement in literature and rhetoric; Tautology (logic), in formal logic, a statement that is true in every ...
Tautological consequence can also be defined as ∧ ∧ ... ∧ → is a substitution instance of a tautology, with the same effect. [2]It follows from the definition that if a proposition p is a contradiction then p tautologically implies every proposition, because there is no truth valuation that causes p to be true and so the definition of tautological implication is trivially satisfied.
For example, the word 睡 ('to sleep') is an intransitive verb, but may express different meaning when coupled with objects of prepositions as in "to sleep with". However, in Mandarin, 睡 is usually coupled with a pseudo-character 觉 , yet it is not entirely a cognate object, to express the act of resting.
Semantic completeness is the converse of soundness for formal systems. A formal system is complete with respect to tautologousness or "semantically complete" when all its tautologies are theorems, whereas a formal system is "sound" when all theorems are tautologies (that is, they are semantically valid formulas: formulas that are true under every interpretation of the language of the system ...