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Propositional Proof System
In propositional calculus and proof complexity a propositional proof system (pps), also called a Cook–Reckhow propositional proof system, is a system for proving classical propositional tautologies. Mathematical definition Formally a pps is a polynomial-time function ''P'' whose range is the set of all propositional tautologies (denoted TAUT). If ''A'' is a formula, then any ''x'' such that ''P''(''x'') = ''A'' is called a ''P''-proof of ''A''. The condition defining pps can be broken up as follows: * Completeness: every propositional tautology has a ''P''-proof, * Soundness: if a propositional formula has a ''P''-proof then it is a tautology, * Efficiency: ''P'' runs in polynomial time. In general, a proof system for a language ''L'' is a polynomial-time function whose range is ''L''. Thus, a propositional proof system is a proof system for TAUT. Sometimes the following alternative definition is considered: a pps is given as a proof-verification algorithm ''P''(''A'',''x'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Propositional Calculus
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives. Propositions that contain no logical connectives are called atomic propositions. Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic. Explanation Logical connectives are found in natural languages. In English for example, some examples are "and" ( conjunction), "or" ( disjunction), "not" (negation) and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |