J. D. Horton and B. Spencer. Clause trees: a tool for understanding and implementing resolution in automated reasoning. Articial Intelligence, 92:25-89, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....this chapter will introduce the proof theory involved in automated reasoning which utilizes this logic. 2.2 Clause Trees In order to provide a means for implementation of the previously defined logic in the construction of an automated theorem prover, a data structure is needed. The clause tree [HS97] developed for the purpose of automated theorem proving by J.D. Horton and Bruce Spencer at the University of New Brunswick, will now be introduced as the primary tool for implementation. Any clause C = fp 1 ; p 2 ; p n g; 1 i n can be represented by a clause tree T = hN; E; L; Mi, ....

....atoms, is shown in Figure 2.1. a b c d Figure 2.1: Elementary Clause Tree Representing the Clause fa; b; c; dg When dealing with first order logic, a unification of literals may sometimes be performed. The next two definitions are taken from [NM95] followed by definitions from [HS97] Definition 2.15 (Substitution) A substitution is a finite set of pairs of terms fX 1 =t 1 ; X n =t n g where each t i is a term, each X i is a unique variable, and X i 6= t i . Definition 2.16 (Variable Renaming Substitution) A variable renaming substitution on an expression E is a ....

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J.D. Horton and Bruce Spencer. Clause trees: A tool for understanding and implementing resolution in automated reasoning. In Artificial Intelligence, pages 25--89. Elsevier Science B.V., 1997.

....natural measure of proof size than the size of the tree. But often if one tree is smaller than another tree then the dag underlying the rst tree is smaller than the dag underlying the other. Support ordered resolution depends on the notion of support between two resolution steps, rst de ned in [4] and de ned for proof trees in [7] When compared with the literal ordered restriction, used by many theorem provers and explored in [3] support ordered resolution is less restrictive, in that it admits a very speci c additional resolution step. On the other hand the support ordered restriction ....

J. D. Horton and B. Spencer. Clause trees: a tool for understanding and implementing resolution in automated reasoning. Articial Intelligence, 92:25-89, 1997.

....results to minimal ones if desired. In the next section we use history paths to characterize support: the condition where one node in a binary resolution tree must be a descendant of another after any sequence of rotations. Both minimality and surgery were first developed for clause trees [5]. The clause tree is a tool for developing ideas in automated reasoning, and the binary resolution tree is an efficient, compact data structure to implement clause trees. In the second last section we show the close relationship between binary resolution trees and clause trees. Thus, this paper ....

.... b a c d a b e f g b a c d a h h Figure 12. Clause trees corresponding to Figures 1, 4, and 9 and to Figure 2 8. Relation to Clause Trees The results in this paper were developed after we understood clause trees [5], and then primarily as a means to implement clause trees. We eventually found that most of our ideas from clause trees could be expressed in binary resolution trees. Binary resolution trees are simpler in some ways since they are easier to implement, but often clause trees are easier to use when ....

J. D. Horton and B. Spencer. Clause trees: a tool for understanding and implementing resolution in automated reasoning. Artificial Intelligence, 92:25-- 89, 1997.

....reconvergent fanouts is addressed by the new technique. In the paper we develop as our main contribution in a more general way a sound and complete calculus for propositional circumscriptive reasoning in the presence of minimized and varying predicates. 1 Introduction Recently clause trees [6], a data structure and calculus for automated theorem proving, introduced a general method to close branches based on so called merge paths . In this paper we bring these merge paths to tableaux for minimal model reasoning (e.g. 4, 11 13] We use the framework of hyper tableau for this, which ....

....for automated theorem proving, introduced a general method to close branches based on so called merge paths . In this paper we bring these merge paths to tableaux for minimal model reasoning (e.g. 4, 11 13] We use the framework of hyper tableau for this, which began with [2] The paper [6] is devoted to refutational theorem proving. Merge paths allow branches to close earlier than it would be possible without them or when using merge paths to simulate known instances such as folding down [8] Expressed from the viewpoint of complement splitting [9] one advantage is that the ....

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J. D. Horton and B. Spencer. Clause trees: a tool for understanding and implementing resolution in automated reasoning. Artificial Intelligence, 92:25--89, 1997.

....of whether the literals in the parent clauses are active or inactive. This exception is very important as completeness is lost if this is not done. Instead of just keeping the clauses produced, it is important to remember also how a clause is derived. This can be done using clause trees [5], which is where this material was first derived. In this paper we develop the ideas in binary resolution trees [7] which are much more closely related to the proof trees seen in many papers on resolution. One simple rank function is to assign ranks from 1 to n arbitrarily in an n literal input ....

....in the size of the binary resolution tree, this amounts to a considerable saving of work compared to a proof procedure that does not use this restriction. 4 Combining with minimality The rank activity restriction combines well with minimality, another restriction of binary resolution. Minimality [5, 7, 8] is an extension of the better known regularity restriction [9] A binary resolution tree is regular if, for every internal node N , the atom label of N is not in the clause label of any descendant of N . The tree in Figure 1 is irregular because al(N 1 ) is a and a occurs in cl(N 4 ) Irregular ....

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J. D. Horton and Bruce Spencer. Clause trees: a tool for understanding and implementing resolution in automated reasoning. Artificial Intelligence, 92:25--89, 1997.

....labourious, since the straightforward application of the definition, as is done in the proof of Theorem 8, checks every possible sequence of rotations, and there can be exponentially many. In this section we define the notion visibility for binary resolution trees, first defined for clause trees (Horton Spencer 1997). We also give a linear algorithm for deciding whether two minimal binary resolution trees can be combined to give a minimal tree. Definition 9 (History Path) A history path P for an atom a in a binary resolution tree T is a sequence (N 0 ; Nn ) of nodes such that N 0 is a leaf, each N ....

Horton, J. D., and Spencer, B. 1997. Clause trees: a tool for understanding and implementing resolution in automated reasoning. Artificial Intelligence approximately 62 pages. Forthcoming 1997, available as http://www.cs.unb.ca/profs/bspencer/htm/ clause trees/TR95-95.ps.

....results to minimal ones if desired. In the next section we use history paths to characterize support: the condition where one node in a binary resolution tree must be a descendant of another after any sequence of rotations. Both minimality and surgery were first developed for clause trees [5]. The clause tree is a tool for developing ideas in automated reasoning, and the binary resolution tree is an efficient, compact data structure to implement clause trees. In the second last section we show the close relation between binary resolution trees and clause trees. Thus, this paper ....

....R i 1 for i = 1; k Gamma 1 such that S is on R 1 and R k closes at D. Thus D holds M via the sequences P 1 ; Pm ; R 1 ; R k and Q 1 ; Q n ; R 1 ; R k . 2 8. Relation to Clause Trees The results in this paper were developed after we understood clause trees [5], and then primarily as a means to implement clause trees. We eventually found that most of our ideas from clause trees could be expressed in binary resolution trees. Binary resolution trees are simpler in some ways since they are easier to implement, but more often clause trees are easier to use ....

J. D. Horton and B. Spencer. Clause trees: a tool for understanding and implementing resolution in automated reasoning. Artificial Intelligence, 92:25-- 89, 1997.

....can be applied to problems in programming language semantics, knowledge representation and diagnosis. The second problem is equivalent to automated theorem proving, where one refutes the complement of theorem to be proved. In this paper we concentrate on the first problem. Recently clause trees [Horton and Spencer, 1997a] a data structure and calculus for automated theorem proving, introduced a general merge operation to eliminate open goals in the tree, based on so called merge paths (cf. Section 1.1) In essence the clause tree allows one to build just one tree, but it implicitly represents a (usually large) ....

....Besides hooks there is another kind called deep merge paths which correspond to folding up. They have to be applied with care, as they can introduce some computational overhead. This holds in particular when computing models is demanded (as opposed to refutational theorem proving) The paper [Horton and Spencer, 1997a] is devoted to refutational theorem proving, and the results from there do not directly carry over to our task of computing minimal models (nethertheless, many general results about cyclic dependencies of paths in [Horton and Spencer, 1997a] can readily be taken for our case) The same can be ....

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J. D. Horton and Bruce Spencer. Clause trees: a tool for understanding and implementing resolution in automated reasoning. Artificial Intelligence, 92:25--89, 1997.

....choose merge paths that end on internal nodes. A set M of paths in a clause tree is legal if the OE relation on M can be extended to a partial order. A path P is legal in T = N;E; L; M if M [fPg is legal. If the path joining t to h is legal in T , we say that h is visible from t. Theorem 2. [1] S j= C iff there is a clause tree T on S such that cl(T ) C: If cl(T ) OE, T represents a refutation proof, indeed possibly many proofs. A tautology path joins two complementary literals. Since tautologies generally do not help in building contradictions, we want to avoid legal tautology ....

....proofs. A tautology path joins two complementary literals. Since tautologies generally do not help in building contradictions, we want to avoid legal tautology paths. Definition3. A clause tree is minimal if it does not contain an unchosen legal merge path or a legal tautology path. Theorem 4. [1] If a clause T on a set S of clauses is not minimal, there is a minimal clause tree T 0 on S such that cl(T 0 ) cl(T ) Thus all non minimal clause tree proofs are redundant, and should be avoided. We introduce MinALPOC [1] an algorithm that produces only minimal clause trees. We start with ....

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J. D. Horton and B. Spencer, Clause trees: a tool for understanding and implementing resolution in automated reasoning, submitted for publication, available as TR95-095, Faculty of Computer Science, University of New Brunswick, and via http://www.cs.unb.ca/research areas/ar/argroup.html.

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