Dallas S. Lankford. Canonical inference. Memo ATP-32, Automatic Theorem Proving Project, University of Texas, Austin, TX, December 1975.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....rules, and has as its goal the production of confluent (Church Rosser) systems of rules that can be used to decide validity. To achieve this, the larger sides of rules are overlapped on (non variable) subterms of each other, producing equations that are called critical pairs . Brown [ 1975 ] and Lankford [ 1975 ] first suggested combining completion for oriented unconditional equations, with paramodulation for unorientable ones and resolution for non equality atoms. Paul [ 1986 ] studied the application of completion to sets of Horn clauses with equality. Completion was extended to conditional equations ....

....of the maximal condition, since a drop with non empty subproofs must have a step emanating from the larger side of its largest condition. Non critical unconditional peaks t 0 t t 00 have alternative, smaller proofs t 0 t t 00 in E by the version of the Critical Pair Lemma in [ Lankford, 1975 ] Consider a non critical drop w[uoe] qoe)uoe voe w[voe] with unconditional subproof poe p 0 p 00 p 000 , where poe is no smaller than any other term in the subproof qoe. Suppose p has a variable x at position and the first step applies within the variable part pj . ....

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Dallas S. Lankford. Canonical inference. Memo ATP-32, Automatic Theorem Proving Project, University of Texas, Austin, TX, December 1975.

....completion [4] a variant of Knuth Bendix completion that avoids failure due to incomparable critical pairs presupposes such an ordering. In addition, using a reduction ordering that is total on ground terms, one can show that any finite set of ground equations has a decidable word problem [13, 20]. It is very easy to obtain such orderings. Indeed, many of the standard methods for constructing reduction orderings yield orderings that are total on ground terms: both Knuth Bendix orderings [12] and lexicographic path orderings [10] are total on ground terms if they are based on a total ....

D. S. Lankford. Canonical inference. Memo ATP-32, Automatic Theorem Proving Project, University of Texas, Austin, TX, December 1975.

....completion [4] a variant of Knuth Bendix completion that avoids failure due to incomparable critical pairs presupposes such an ordering. In addition, using a reduction ordering that is total on ground terms, one can show that any finite set of ground equations has a decidable word problem [15, 9]. It is, in fact, very easy to obtain such orderings: many of the standard methods for constructing reduction orderings yield orderings that are total on ground terms. For instance, both Knuth Bendix orderings [14] and lexicographic path orderings [11] are total on ground terms if they are based ....

D. S. Lankford. Canonical inference. Memo ATP-32, Automatic Theorem Proving Project, University of Texas, Austin, TX, December 1975.

....first clause of p q j p; p q j q; p q r j p q rewrites the others (assuming p q r) to q j p and p r j p. Then, the first becomes p p j p and is deleted, leaving the Horn clauses p ) q, q ) p, and p ) r. This algorithm is based on the completion based congruence closure method in [ Lankford, 1975 ] shown to be doable with low polynomial time complexity in [ Gallier et al. 1988 ] By the theorem in [ Lankford and Ballantyne, 1983 ] for uniqueness of canonical associative commutative rewriting systems, it results in a unique set of equivalences, determined by the ordering . The ....

.... is used to guide the inference mechanism, so that only maximal terms are used in any inference step. When is the empty ordering, the method reduces to special paramodulation, in which the functional reflexive axioms are not needed and paramodulation into variables is not performed (see [ Lankford, 1975 ] Most important, a nonempty ordering allows conditional equations that are simplifiable to be replaced without compromising (refutational) completeness. Hence, the power of the method, both in minimizing possible inferences and maximizing potential simplifications, is brought to bear by ....

Dallas S. Lankford. Canonical inference. Memo ATP-32, Automatic Theorem Proving Project, University of Texas, Austin, TX, December 1975.

....if the relation R=EQ is terminating, i.e. if there are no infinite sequences of the form: s 0 =EQ s 0 0 R s 1 =EQ s 0 1 R s 2 =EQ s 0 2 R s 3 : An equational rewrite system R=EQ is compatible with a quasi order = on T (F ; X ) if =EQ and R . We have that ([11, 16, 17]) 1 By linear context we mean a context with a single occurrence of the trivial context 2. Theorem 2.3 A rewrite system is terminating if and only if it is compatible with a reduction order. An equational rewrite system is terminating if and only if it is compatible with a reduction ....

D. S. Lankford. Canonical inference. Technical Report Memo ATP-36, Automatic Theorem Proving Project, University of Texas, Austin, 1975.

....completion is used for forward reasoning, while rewriting is a form of backward reasoning. Completion utilizes an ordering on terms to provide strong guidance during forward reasoning and to direct the simplification of equations. Besides the use of convergent systems as decision procedures, Lankford [ 1975 ] proposed that completion like methods supplant paramodulation for equational deduction within resolution based theorem provers; later, Hsiang [ 1982 ] showed how a variant of completion can be used in place of resolution for (refutational) theorem proving in first order predicate calculus. ....

....of a term, since (not(or(x; y) 2 x y 1 is greater than (and (not (x) not(y) 2 x y , for all x and y. A class of exponential interpretations were used for termination arguments in [ Iturriaga, 1967 ] The use, in particular, of polynomial interpretations was developed in [ Lankford, 1975; Lankford, 1979 ] Here a multivariate integer polynomial f (x 1 ; xn ) of degree n is associated with each n ary symbol f in Fn , for all n. The choice of coefficients must satisfy the monotonicity condition and ensure that terms are mapped into nonnegative integers only, as is the ....

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D. S. Lankford, Canonical inference, Memo ATP-32, Automatic Theorem Proving Project, University of Texas, Austin, TX (December 1975).

....while equations are used in whichever direction reduces the term it is applied to in the given ordering. An identity is proved when both sides reduce via rules and equations to the identical term. For completeness of this theorem proving methodology, we require a completable ordering. Theorem25 [75][44] 33] Suppose s = t is a theorem of E. For any fair completion strategy starting with E and a completable simplification ordering, at some point s and t will rewrite to the identical term using the generated rules and equations. The word problem in arbitrary equational theories can always be ....

Dallas S. Lankford. Canonical inference. Memo ATP-32, Automatic Theorem Proving Project, University of Texas, Austin, TX, December 1975.

....have been found (see [ Jouannaud and Kirchner, 1991 ] Combining constraints with deduction, whether equational [ Kirchner and Kirchner, 1989 ] or full first order [ Kirchner et al. 1990 ] is another potential growth area. Theorem proving and symbolic computation Since the pioneering work of Lankford [ 1975 ] research on the application of ideas from rewriting to more traditional refutational theorem provers for first order predicate calculus has proceeded in bits and spurts. Recent work has shown that using orderings on terms and formulae helps restrict deduction and increase the amount of ....

Dallas S. Lankford. Canonical inference. Memo ATP-32, Automatic Theorem Proving Project, University of Texas, Austin, TX, December 1975.

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Dallas S. Lankford. Canonical inference. Memo ATP-32, Automatic Theorem Proving Project, University of Texas, Austin, TX, December 1975.

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Lankford, D. S. 1975a. Canonical Inference, Technical Report, ATP-32, Automatic Theorem Proving Project, University of Texas at Austin.

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D. S. Lankford. Canonical Inference. MemoATP-32, Automatic Theorem Proving Project, University of Texas, Austin, TX, December 1975.

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