The Resolution Calculus by Alexander Leitsch

By Alexander Leitsch

This is a very new presentation of answer as a logical calculus and as a foundation for computational algorithms and choice procedures.
The first half offers with the conventional themes (Herbrand's theorem, completeness of solution, refinements and deletion) yet with many new beneficial properties and ideas like normalization of clauses, solution operators, and seek complexity.
Building in this origin, the second one half supplies a scientific therapy of contemporary examine subject matters. it's proven how resolution selection procedures might be utilized to resolve the choice challenge for a few vital first-order sessions. The complexity of resolution is analyzed when it comes to Herbrand complexity, and new thoughts like flooring projection are used to categorise the complexity of refinements. ultimately, the tactic of useful extension is brought; mixed with solution it supplies a computational calculus that's more suitable than such a lot others.

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Then there is aPE PS(C) and terms tl, ... , tn such that L = P(t l , ... , t n ) or L = ,P(t l , ... , t n ). In order to prove vp (e) = F for a r' '" r mod Vee) - thus vr( {C}) = F - it is enough to show: There is a r' such that r' '" r mod V (e) and VpH (P(h, ... ,tn )) =VP(P(tl, ... ,tn )). By direct inspection of VpH (P(tl, ... , t n )) we find: VpH (P(tl, ... ,tn)) = PH(P)(UpH (h), ... ,UpH (t n )) where type: u( ) ur~ is the semantic function for terms, which - in this case - is of the : T(C) -+ H(C).

Ld V B:n} U R. for A~ = Ai \ L, Bj = B j \ L. 2a) Suppose that V' is satisfiable. Then there exists an H-interpretation r such that vr (V') = T. By definition of vr, vr ({L }) must be true. By definition of "or" we obtain vr(V') =vr{{B~, ... ,B:n}UR) =vr(V'~). We conclude that V'~ is satisfiable. 2b) Suppose that V'~ is satisfiable and vr{V'~) = T. Let r' be like r with the exception vHL) = T. Because neither L nor Ld occurs in V'~ we get vHV'~) = vr(V'), but also vH{L, Lv A~, ... , Lv A~}) = T.

3 (Herbrand's theorem). A set of clauses C is unsatisfiable iff there exists a finite unsatisfiable set of clauses C' such that C' consists of ground instances of clauses in C. Proof. 1. Suppose that there exists a finite, unsatisfiable set of ground clauses C' defined by C. 3 Term Models and Herbrand's Theorem If C' = {C~, ... ,q} then F(C') = I\~=l 39 c: (note that for ground clauses C' F({C'}) = C'). Because F(C) -+ F({C}) is valid for every C E C and F({C}) -+ C' is valid for every ground instance C' of C, F(C) -+ C~ is valid for every C~ above; but then, clearly, F(C) -+ F(C') is valid too.

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