2 T. MACKLING The answer is partial because, in this paper, we neglect the functional substitutivity axioms for equality, and provide appropriate gene

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1 PC <, EPC <, AND THE EQUALITY PREDICATE AND SUBSUMPTION IN RESOLUTION-BASED AUTOMATIC-THEOREM-PROVING: THE FIRST FOUR AXIOMS T. Mackling Department of Electrical Engineering, McGill University, 3480 University Street, Montreal, PQ, Canada, H3A 2A7 Abstract In this paper we show that the standard notions of tautology and subsumption can be naturally generalized, (so that refutation completeness is preserved with respect to the associated deletion), within the context of (a specied set of) modied deduction rules for binary clausal resolution-refutation which build-in the reexivity, symmetry, transitivity and predicate substitutivity axioms for equality. To this end, rst a resolution system, PC <, in which equality has no special status, is introduced and its refutation completeness under subsumption and tautology deletion shown. The generalized notions of subsumption and tautology, 0 -subsumption and 0 -tautology, respectively, are then formulated and analyzed. EPC <, a system which provides an adequate deductive context for refutation completeness under deletion with respect to these generalized notions, is then introduced. To prove this completeness, some technical notions (principally, to allow an induction) are introduced, and a number of technical results are obtained. Additionally, some clause \normalization" techniques including generalized forms of replacement factoring and a form of demodulation, are suitably formulated, and the preservation of refutation completeness with the further introduction of these normalization procedures is shown. Some renements for the rules are oered, as well as a possible further generalization for subsumption, and a generalized form of a special case of replacement resolution, folding. Key Words Resolution, generalizing deduction and deletion rules, subsumption, tautology, building in the equality axioms, refutation completeness Introduction. 0. Introduction and Some Formal Preliminaries In this paper we provide a positive partial answer to the following informally posed question: To what extent can the standard notions of tautology and subsumption be naturally generalized, (so that refutation completeness is preserved with respect to the associated deletion), within a context of reasonably modied deduction rules for binary clausal resolution-refutation which builds-in the equality axioms? Date: September 24, This work was supported by NSERC Grant A1329, and the NCE-IRIS-I-B5 and NCE-IRIS-II-IS5 grants. 1

2 2 T. MACKLING The answer is partial because, in this paper, we neglect the functional substitutivity axioms for equality, and provide appropriate generalizations for the notions of subsumption and tautology, together with a specication of a set of modied binary resolution-style deduction rules, (in which refutation completeness is preserved under deletion with respect to these generalized notions), for the theory consisting of only the reexivity, symmetry, transitivity and predicate substitutivity axioms for equality. Nevertheless, we feel this paper is an instructive rst step towards a system which provides the complete solution. The equality axioms have been identied as a source of ineciency or even incapacity for clause resolution/deduction oriented refutation provers (see e.g. [3, 5, 6, 9, 12, 13, 17] ). A number of approaches to \building in equality", have been oered in the literature of automated deduction. These include paramodulation, hyper-paramodulation, and demodulation [13, 17], E-Resolution [10], RUE-Resolution [5, 6], Z-Resolution [7], and the approach suggested by Sibert in [14]. More recently, restrictions of ordered paramodulation are described in, for example, [1, 2]. However, with the exception of [14], and the work described in [1, 2], nothing signicant has yet been written on how the notions of tautology and subsumption can be appropriately and feasibly generalized within the framework of such techniques. We present EPC <, (Equality Predicate Colouring), (which in the terminology of [16], is essentially an instance of binary partial narrow theory resolution), as one approach to \the rst four" equality axioms, and describe how such a generalization is possible. Of course, generalizing these notions allows a strengthened deletion strategy, which potentially results in a signicant pruning of the clause set search space. Before the introducing EPC < and our notions, 0 -subsumption and 0 -tautology, (which generalize those of subsumption and tautology), we rst introduce a resolution system PC <, (Predicate Colouring), in which equality has no special status, and show its refutation complete under subsumption and tautology deletion. We do this for the following reasons: 1) simply to introduce PC <, which we feel is, as a conceptual model at least, an important binary clausal resolution/refutation system in its own right, and 2) to provide a conceptual springboard for the system EPC <, and in particular, for the proof-theoretical arguments we needed to justify those properties we claim it possesses, (such as its refutation completeness with the generalized subsumption and tautology deletion). Though \independently invented" (I was not aware of the depelopment called Ordered Resolution during the preparation of this thesis), PC < is closely related to ordered resolution [15] 1, and Ordered Resolution may be seen as a kind of corollary to Predicate Colouring. Our formulation and treatment of PC <, (in Section 1), serves as a (partial) template for what we wish to do for EPC <, and for most results obtained for PC <, there are analogous ones later obtained for EPC <. We then, (in Section 2), formulate and analyze 0 -subsumption and 0 -tautology, our generalized notions of subsumption and tautology. In particular, we give a kind of semantic characterization for these syntactic predicates. We follow this, (in Section 3), with the introduction of EPC <, our system which provides an adequate deductive context for refutation completeness under deletion with respect to these generalized notions. Unfortunately, our proof-theoretical argument of this completeness is not brief, and we require numerous technical results. Often we require a conclusion C from hypotheses H, were we have already shown that C 0 follows from H 0, for C 0 and H 0 closely resembling C and H, respectively. However, possibly because of the very syntactic nature of our subject, we shied away from the problem of trying to establish, and prove our required results within, an appropriate and adequate abstract setting. 1 There is slick formulation presented here, even though, it seems to me, that the proof given for completeness on pp , is inadequate.

3 PC <, EPC <, AND THE EQUALITY PREDICATE AND SUBSUMPTION IN RESOLUTION-BASED ATP 3 Instead, we settled on avoiding redundant argumentation where possible through \modularization by lemma's", (using the analogy of `lemma is to proof' as `subroutine is to program'), and, where it seemed expedient, by specifying proofs for results by indicating sucient modication to the proofs given for similar results. 2 There is an obvious need for something better here, presumably, either through adequate abstraction or (the use of) some sort of polymorphic proof-theory, but neither of these were felt to be immediately available and amenable to our task. In any case, in contrast to the situation typically assumed in automated theorem proving, the kind of proof structure found in proof-theoretical arguments we present, is, in a very transparent sense, (that of lemmas depending on other lemmas, et cetera), far from at, and it is at least hoped that some of the proofs within this paper, can at some point be better (and more clearly) stratied and/or encapsulated, to the extent that, in particular, one may show that a property P, which we showed for EPC <, 0 -subsume, 0 -tautology, QR full, et cetera, must also hold for some other specic system and corresponding denitions, by similarly showing, all of the attendant \atomic lemmas" hold. Of course, we will not try to make exact the nature of such an undertaking here, but evidently, any such enterprise, if successful, should at least be suggestive as to the kind of abstraction and/or polymorphic proof-theory which would prove most useful. Most of our proofs are by nitary induction, and virtually all of them are constructive. Though none of them utilize anything exotic, some are quite large. Incidentally, with respect to the \mathematization" of the proof-theory, (obtaining the results in a more general setting), almost every time it was discovered that some \nice mathematics" could be used, (for example, general properties of relations or simple algebraic structures), it was omitted, typically because a ner, more concrete characterization was ultimately required, and it could be swept aside by simply providing that concrete characterization in the rst place. All apologies aside, once we have established the refutation completeness of EPC < (in Sections 4), a battery of technical results, (in Sections 5 and 6), and thereby, a few big ones, (notably, Corollary 6.6), more or less as icing, we examine (in Section 7), the potential role of some clause \normalization" techniques including generalized forms of replacement factoring and a form of demodulation, within the context of EPC <, and oer some renements for the rules, possible further generalization for subsumption, and a generalized form of a special case of replacement resolution, which we refer to as folding. A few words on the function substitutivity equality axioms, and some concluding remarks are given in Section 8. Each section begins with an overview, which briey outlines and surveys the contents of that section. All proofs have been placed in an Appendix. Actually the work of [1, 2] is quite related to what is done here, and deserves greater comparison. Unfortunately, comparison is not straight forward. In [1, 2], the notion of redundancy, a somewhat dierent notion of \generalized subsumption", is introduced. This notion is based of a clause ordering which is well-founded on ground clauses, and is basically an equality generalization of the notion of redundancy introduced for Ordered Resolution (see eg. [15]). The system based on strict superposition, [1, 2], Bachmair and Ganzinger present is refutation complete with \redundancy deletion", and conditional equality rewriting, (the latter essentially corresponding to the \normalization techniques" we describe). Further, the given proof of this completeness, based on the construction of a model for a given saturated set of consistent clauses via a canonical rewrite system, is certainly much briefer than the proof-theoretical argument we provide for the completeness of our system. Finally, the modied superposition based calculus presented builds in all the the equality clauses and is arguably of much greater practical signicance, and hence, theoretical interest, than EPC <. 2 We take advantage of the presentation of PC < here, for example, by sometimes referring to a proof of a result for PC <, for the proof of a similar result for EPC <.

4 4 T. MACKLING Still, the approaches taken and results correspondingly obtained simply dier. Unlike those of the modied superposition rules of [1, 2], the deduction rules for EPC < do not enjoy a straight forward lifting property, and this is a primary complication in thwarting relatively brief semantic arguements of the sort found in [1]. More importantly, 0 -subsumption our generalized notion of subsumption, and the notion of redundancy, as introducted in [1] are independent. For example, if C 1 and C 2 are the clauses fx 6= a ; a 6= dg and fx 6= a ; x 6= dg, respectively, (where x is a variable and a and d denote distinct ground terms), then, in general, neither is C 1 redundant in fc 2 g, nor is C 2 redundant in fc 1 g. However both C 1 0 -subsumes C 2 and C 2 0 -subsumes C 1. Finally, the notion of redundancy (and its associated deletion) is intended to be well founded (with respect to an introduced clause ordering which is total on ground clauses), while no such assumption is imposed on 0 -subsumption. The semantic characterization we provide for 0 -subsumption indicates that in some sense, 0 -subsumption is the correct generalization for subsumption over the built-in theory 0, and since redundancy is not a generalization of 0 -subsumption, redundancy can not, in the same sense, be the correct generalization for subsumption over the equality axioms. Finally, it is perhaps worth pointing out that it is unclear in [1, 2] to what extent the non-local condition of redundancy can be eectively established. In contrast, the 0 -subsumption condition is local, (like subsumption), and methods for eective detection are available Some Formal Preliminaries. We assume familiarity with the standard nomenclature and folklore of rst-order clause-based resolution/refutation. In particular, we assume familiarity with ordinary binary merge resolution (OBMR): where a clause is simply regarded as a (nite) set of literals, (the clause representing a disjunction of those literals), and for any clauses A and B, (such that the variables of B are disjoint from those of A ) 3, the set of (OBMR-)resolvents, OBMR(A; B), of A and B are all those clauses C such that for some non-empty subsets ; 6= A 0 A and ; 6= B 0 B, the set A 0 [ f:l j L 2 B 0 g unies, to L 0, say, with most general unier, and C A n fl 0 g [ B n f:l 0 g. By standard denition, a clause is a tautology precisely when it contains a pair of pair of oppositely signed but otherwise identical literals. A clause C subsumes a clause D precisely when there is some substitution, such that C D. With regards to OBMR, we assume familiarity with what we shall term the principle of subsumption (for OBMR) 4 : Suppose? B is an OBMR-deduction of a clause B, from a set of clauses S [ fdg, 0 in which D occurs, and C subsumes D. Then there is an OBMR-deduction? B of a clause B 0, where B 0 subsumes B, from S [ fcg, in which at least one instance of C occurs, but in which no instance of D occurs, and where, if, (for? either? B or? 0 B ), F (?) denotes the underlying graph of?, (forget the clauses which label the nodes), regarded as a poset, (so that n < n 0 if n 0 is a parent node of n ), and for any node n of?,?(n) denotes the clause which labels n in?, there is an order preserving embedding j : F (? 0 B )! F (? B ), such that for any node n of? 0 B,? 0 B (n) subsumes? B (j(n)). 3 Otherwise, A or B is replaced by some alphabetic variant, so as to make their sets of variables disjoint, with the (OBMR-)resolvents dened only up to alphabetic equivalence. 4 Though this is a fairly straight forward or even obvious result, and the formulation here is our own, a reasonable reference is [4].

5 PC <, EPC <, AND THE EQUALITY PREDICATE AND SUBSUMPTION IN RESOLUTION-BASED ATP 5 Of course, in practice, a clause is (typically) represented by a repetition-free list of literals, and by an ordered clause, we simply mean such a list. Similarly, by the left-most literal of an ordered clause C, which satises some property, P, we simply mean the rst literal of C which satises P. For any (ordered) clause C, jcj will denote the underlying set of literals of C. By a clause ordering function, o, we simply mean any function o which orders the literals of any given (unordered) clause, so that, for any clause C, o(c) is an ordered clause whose underlying set of literals, jo(c)j, is C. By a parameterized clause ordering function, o, we simply mean any clause ordering function which takes additional arguments, (so that, o has arity n + 1 for some n 0, and for any additional arguments a 1 ; : : : ; a n, of the appropriate sort, for any unordered clause C, o(~a; C) is an ordered clause whose underlying set of literals is C ). For any, possibly ordered, clauses A and B, whenever A and/or B is ordered, for economy, by the set expressions A [ B, A n B, A B, we shall simply mean the unordered clauses jaj [ jbj, jaj n jbj, and the condition that jaj jbj, respectively. Similarly, for any literal L, by L 2 A, we mean L 2 jaj. We shall also say that A = B, modulo ordering, to indicate that jaj = jbj. Of course, for any (possibly ordered) clause A, and any literal L, L occurs in A when L 2 A. For any clause A, and any predicate symbol P, (we will say that), P occurs positively in A if for some tuple, ~t t 1 ; : : : ; t n, of terms, (where n 0 is the arity of P ), the literal P (~t) occurs in A. Similarly, P occurs negatively in A if for some tuple, ~t, of terms, the literal :P (~t) occurs in A. A predicate symbol P occurs in a clause A if P occurs positively or negatively in A. We may refer to a literal P (~t) ( :P (~t) ) as a positive (negative) P instance, (respectively). For any clause A, A is pure-positive, or simply positive, if A contains no negatively signed literal. A is mixed if A contains at least one negatively signed literal, and pure-negative, or simply negative if A contains no positively signed literal. For any clause A, and any predicate symbol P, A is pure-positive with respect to (or pure-positive in) P, if P occurs at most positively in A, (that is, P does not occur negatively in A ). A is mixed with respect to (or mixed in) P, if P occurs negatively in A, and A is pure-negative with respect to (or pure-negative in) P, if P does not occur positively in A. Denition 0.1. For any sequences, and 0, let ; 0 denote the concatenation of the sequences, and 0, and let jj denote the underlying set of, (thus, for example, if is a sequence of clauses, jj denotes the set of those clauses which occur in the sequence, ). Denition 0.2. (Proof Trees) Let R be a set of (sound) deduction rules, such that each rule r 2 R is a function of a xed nite arity k, mapping each k -tuple! C of clauses lying in the domain of r to a set of clauses r(! C). Let D be a clause, and S be a set of clauses. Then by a ( R-)proof tree of D from S, we mean a nite labelled tree such that: i. each node of is labelled with some unique clause, and the root node of is labelled with D, ii. the label C associated with any leaf node of is a member of S, and iii. if n 1 ; : : : ; n k are the sequenced parent nodes of a node n 2, labelled with C 1 ; : : : ; C k and C respectively, then for some r 2 R of arity k, C 2 r(c 1 ; : : : ; C k ). We will also say such a is a proof tree of D from S in R, or simply that is an R -deduction of D (from S ).

6 6 T. MACKLING Denition 0.3. We will say a node n 2 lies (strictly) below n 0 2, if n 0 is an (strict) ancestor of n in, or, equivalently, if n lies within the branch segment which travels from the root node of to terminate at (but does not include) n 0. If the context prevents ambiguity, we will also say a clause D occurs below a clause C, if there is a node n 2 with label D which lies below the node n 0 2 which is being implicitly specied by the label C. If a clause C is associated to a leaf node n 2, we will say that (with respect to ) C is a leaf clause. If a clause C is associated to a node n 2, and at least one of the parent nodes of n is a leaf node, we will say that (with respect to ), C is a leaf resolvent. Denition 0.4. ( -unsatisable) Given a set of clauses, we will say a set S of clauses is -unsatisable if [S is an unsatisable set of clauses. Denition 0.5. (leaf-index) Given a deduction of a clause D from (or refutation of) a set S, for any C 2 S, let us say that C has leaf-index k in, if C occurs at a leaf node in exactly k 0 times. We will denote the leaf-index of a clause C in a deduction by k (C) Overview. 1. PC <, A System For Binary Clausal Refutation/Resolution In this section the binary resolution/refutation system PC < is introduced, and various results for it are presented. In Subsection 1.1 the deduction rules for PC < are presented (in Sub-subsection 1.1.1), soundness noted, and (in Sub-subsection 1.1.2), some technical notions are introduced for later use. In Subsection 1.2 some technical results for lifting are presented. These culminate in Lemma 1.4, which, essentially, states the sense in which lifting holds for PC <. Subsection 1.3 simply contains Theorem 1.1, which states the refutation completeness of PC <. In Subsection 1.4 some more technical results are obtained, (analogous to those of Subsection 1.2, but), this time for subsumption in PC <, which culminate in Lemma 1.8, (and Corollary 1.1). In Subsection 1.5 (PC < -) deduction sequences are introduced, and in this framework, the refutation completeness of PC < with subsumption (and tautology) deletion is formulated and established. Subsection 1.6 consists of a relatively informal discussion about deduction/deletion strategies, with Proposition 1.1 its main result. The subsection addresses the more traditional setting of refutation completeness with subsumption (and tautology) deletion within the context of a given deduction/deletion strategy, and illustrates (with Proposition 1.1), how the main results of Subsection 1.5 may be usefully interpreted in such a setting. Some Comments: The principal complication, in particular for lifting (addressed in Subsection 1.2, and the analogous condition for `subsumption' in place of `substitution instance', addressed in Subsection 1.4), in PC <, over the more typical, (and relatively trivial), situation as found in, say, OBMR, is the left-most (negative instance of the <-maximal predicate symbol ) restriction of PC <. Were this restriction dropped, (there would no longer be any reason to take clauses as ordered, and moreover), it is not hard to see that the deduction rules would lift directly: that is, it would be the case that

7 PC <, EPC <, AND THE EQUALITY PREDICATE AND SUBSUMPTION IN RESOLUTION-BASED ATP 7 whenever A is a substitution instance of A 0, ( A 0 subsumes A ), B is a substitution instance of B 0, ( B 0 subsumes B ), and C 2 PC < (A; B), there exists a C 0 2 PC < (A 0 ; B 0 ) such that C is a substitution instance of C 0, ( C 0 subsumes C, respectively). The restriction invalidates this property, and it is precisely because such \direct lifting" does not hold in PC <, that much of the complication arises. In particular, if this property held, then the need for the technical notions introduced in Sub-subsection of Subsection 1.1 vanishes. Nevertheless, in principle at least, the left-most restriction in PC < is desirable, (because, for example, it helps further reduce the number of potential resolvents), and in principle at least, it is worth while to accept the extra \burden of proof", in order to obtain the desired results for the unadulterated system PC < Predicate Colouring (PC < ): The Deduction Rules and Some Technical Apparatus The Deduction Rules. The basic idea behind PC < is to restrict ordinary binary merge resolution in a number of ways. We rst restrict resolution between clauses to resolution on a maximal predicate symbol. In other words, we totally order the predicate symbols, allow resolution between two clauses only if their maximal predicate symbol is the same, and then allow resolution only on literals of that predicate name. Thus, in passing to the empty clause, we eliminate literals, predicate name by predicate name. Further, given that two clauses have the same maximal predicate symbol, we only allow resolution between them if (at least) one of them is pure-positive with respect to that predicate name. (This \at least one parent must be pure-positive (with respect to the maximal predicate name)" restriction, is similar to the requirement that electrons be pure-positive in hyper-resolution, [12].) Finally, given that these restrictions on resolution are satised, we further demand that we merge resolve only on (unifying sub-clauses of the non-pure-positive clause which contain) the left-most negative literal whose predicate name is mutually maximal. (This literal occurs in the clause which is not pure-positive with respect to the \active" predicate symbol). This deduction system we tentatively entitle predicate colouring, and denote it by PC <. (The subscript, <, in the name `PC < ', is to indicate a particular (strict) total ordering on (unsigned) predicate symbols, and so we actually intend here the denition of a class of (restricted) ordinary binary merge resolution systems, where the ordering, <, is a parameter. The class is also (implicitly) parameterized by the ordering functions o, as seen below. Denition 1.1. (PC < -resolution) Let < be any strict linear order on predicate names. Then for any ordered clauses A and B, such that A and B share no variables in common, and any unordered clause C, C is an unordered PC < -resolvent of A and B if: 1) the <-maximal predicate symbol P, say, occurring in A is the <-maximal predicate symbol which occurs in B, 2) A is pure-positive with respect to P, ( B is pure-positive with respect to P ), and B is mixed with respect to P, ( A is mixed with respect to P ), 3) when L 0, say, denotes the left-most negative P instance, occurring in B, (occurring in A ), there are subclauses ; 6= A 0 A and L 0 2 B 0 B, ( L 0 2 A 0 A and ; 6= B 0 B ),

8 8 T. MACKLING and a most general substitution which unies A 0 [ f:l j L 2 B 0 g, ( f:l j L 2 A 0 g [ B 0 ), (to the singleton f:l 0 g ), and 4) C A n f:l 0 g j B n fl 0 g, ( C A n fl 0 g j B n f:l 0 g, respectively). For any ordered clauses A and B sharing no variables in common, let jpc < (A; B)j denote the set of all unordered PC < -resolvents C, of A and B. If A and B do have variables in common, let jpc < (A; B)j denote jpc < (A 0 ; B 0 )j, where A 0 and B 0 are some variable disjoint alphabetic variants of A and B respectively. 5 Let o be any xed parameterized clause ordering function, so that for any ordered clauses A and B, and any unordered clause C, o(a; B; C) is an ordered clause C 0 such that jc 0 j = C. Then for any clauses A and B, let PC < (A; B) denote the collection of ordered clauses: fo(a; B; C) j C 2 jpc < (A; B)jg. (Thus, PC < (A; B) is the set jpc < (A; B)j, where each clause in the set has been endowed with some (arbitrary) ordering.) Soundness of the deduction rules for PC < is obvious, (since, for example, for any clauses A and B, clearly, jpc < (A; B)j OBMR(A; B) ) Some Technical Apparatus for PC <. Denition 1.2. Given a clause B, with <-maximal predicate symbol P, say, ( P possibly `='), B mixed with respect to P, and a nite collection C of clauses, such that 8C 2 C, C is pure-positive with respect to P, let us say that D is PC < + 1 -deducible from fbg [ C, if D is pure-positive with respect to P, (or has <-maximal predicate symbol Q < P ), and there exist sequences (for some k 1 ), C 0 ; : : : ; C k?1, and D 0 ; : : : ; D k?1 ; D k, such that: i. D 0 B and D k D, ii. 8i: 0 i k? 1, C i has <-maximal predicate symbol P ; C i 2 C, and iii. 8i: 0 i k? 1, D i+1 2 PC < 1 (D i ; C i ). (Thus a D which is PC < + 1 -deducible from fbg[c is a deduction-ordering maximal clause obtained from B by PC < -resolutions with clauses in C, which has <-maximal predicate symbol less than or equal to P, and is pure-positive with respect to P.) Further, let us say that is a PC < + 1 -deduction of D from fbg [ C, if is a (\linear"-) PC < - deduction of depth k, (for some k 1 ), with k + 1 leaves B D 0 ; C 0 ; : : : ; C k?1, and k non-leaf nodes (labelled by) D 1 ; : : : ; D k, as above, and as illustrated in Figure 1 below. Note that since C is nite by assumption, and each PC < -resolution eliminates at least one (in particular, the leftmost) negative P literal, clearly there are at most nitely many clauses D + PC 1 + < -deducible from fbg [ C, and every associated PC 1 < -deduction (of D from fbg [ C ), can be of depth at most n, where n is the number of distinct negative P instances which occur in B. Also note that in any PC < + 1 -deduction of D from fbg [ C, (as above), B occurs exactly once, that occurrence as a leaf, and every resolvent in has exactly one parent which is a leaf in C. Denition 1.3. Given a PC < -refutation of (or a PC < -deduction of a clause D from) a set S, let us say that a node n 2, with label C n, say, is PC < -initial if: 5 Thus unordered PC < -resolvents are only dened up to alphabetic equivalence.

9 PC <, EPC <, AND THE EQUALITY PREDICATE AND SUBSUMPTION IN RESOLUTION-BASED ATP 9 C 0 D 0 B C 1 D 1 D 2 C k?1 D k?1 D k D Figure 1. A Linear Deduction i. n is a leaf, or ii. for some parent node n 0, say, of n, the clause C n 0, labelling n 0 is mixed in its <-maximal predicate symbol P, say, ( P possibly `='), while C n is pure-positive with respect to P. Note that if is a PC < + 1 -deduction, (of a clause D from fbg [ C, say) as in Denition 1.2, then the root node of, (labelled by D ) is PC < -initial. In particular, if is any PC < -refutation (of an unsatisable set S ), the root node of, (labelled by the empty clause) is PC < -initial. Observation 1.1. If n, with label C n, say, is any non-leaf node in a PC < -refutation of (or a PC < -deduction of a clause D from) a set S, then: i) n has exactly one deduction-ordering maximal (proper) ancestor n 0, with label C n 0, say, such that n 0 is PC < -initial and C n 0 is mixed with respect to its <-maximal predicate symbol, P (possibly `='), say, and ii) there is a nite sequence of nodes m 1 ; : : : ; m k, (for some k 1 ) with labels C m1 ; : : : ; C mk, say, such that for j : 1 j k, C mj has <-maximal predicate symbol P and is pure-positive with respect to P, and such that the fragment? (say,) of, rooted at n and pruned in fm 1 ; : : : ; m k g, is a depth k ("linear"-) PC < -deduction of C n from fc n 0g[fC m1 ; : : : ; C mk g, with k + 1 leaves, (labelled by) C n 0; C m1 ; : : : ; C mk, and k non-leaf nodes. In particular, if C n is pure-positive with respect to P, then the fragment?, above, is a PC < deduction of C n from fc n 0; C m1 ; : : : ; C mk g. Denition 1.4. For any ; n; C n ; n 0 ; C n 0; m 1 ; : : : ; m k ; C m1 ; : : : ; C mk as in Observation 1.1, above, let us say n (and that C n ) has PC < -initial principal ancestor n 0, (C n 0 ) and ancillary parents m 1 ; : : : ; m k, (C m1 ; : : : ; C mk ). Observation 1.2. Note that (in PC < ), every ancillary parent (being pure-positive in its <-maximal predicate symbol) is PC < -initial. Also, if is any non-trivial PC < -deduction of a clause D, where D (at the root), is PC < -initial, +1 (in ), then may be decomposed into a PC < -deduction 0, of D, from principal ancestor B, (say), and ancillary parents C 0 ; : : : ; C k?1, (say, for some k 1 ), a (possibly trivial) PC < -deduction B of B, and (possibly trivial), PC < -deductions, Ci, of C i, for i: 0 i k? 1, respectively.

10 10 T. MACKLING 1.2. Lifting in PC <. Lemma 1.1. Let B and C be as in Denition 1.2, and suppose B 0 is any re-ordering of +1 (in particular, the negative P literals in) B. Then, if is any PC < -deduction of D from +1 fbg [ C, there is a PC < -deduction 0 of D 0 from fb 0 g [ C, where D 0 is some re-ordering of D, and such that 8C 2 C, k 0(C) = k (C). Of course, for any re-ordering D 0 of D, D 0 subsumes D. Lemma 1.2. Suppose B 2 PC 1 < (A; D), for some clauses A, B, and D, such that A has <-maximal predicate symbol P and is pure-positive with respect to P, ( P possibly `='), (and D has <-maximal predicate symbol P and is mixed with respect to P ). Suppose also that for some clauses A 0 and D 0, (modulo clause ordering), A is a substitution instance of A 0, and D is a substitution instance, D 0 D, (for some substitution D, say), of D 0, such that with L 1, say, the left-most negative P instance in D 0, L 1 D occurs as the left-most negative P instance in D. Then there exists B 0 2 PC 1 < (A 0 ; D 0 ) such that (modulo ordering), B is a substitution instance of B 0. Lemma 1.3. Let B be a clause with <-maximal predicate symbol P, say, ( P possibly `='), B mixed with respect to P, and suppose that (for some k 1 ), C 0 ; : : : ; C k?1, (the C i not + necessarily distinct), and B = D 0 ; : : : ; D k?1 ; D k = D, represents a PC 1 < -deduction, (as in Denition 1.2). Suppose B is (modulo ordering), a substitution instance of some clause B 0, and that C0 0 ; : : : ; C0 k?1, (the C0 i not necessarily distinct), is a sequence of (not necessarily distinct) clauses such that 8i: 0 i k? 1, C i is (modulo ordering) a substitution instance of C 0 i. + Then there exists a PC 1 < -deduction 0, of some clause D 0, from principal ancestor B 0 and ancillary parents fc0 0 ; : : : ; C0 k?1 g, where D is (modulo ordering) a substitution instance of D0, and such that 8C 0 2 fc0 0 ; : : : ; C0 k?1 g, k 0(C0 ) = jjfi: 0 i k? 1 ; C 0 i = C 0 gjj. Lemma 1.4. Suppose is a PC < -deduction of a clause A, from a set S, and ( A, at) the root is PC < -initial. Suppose L denotes the set of leaf nodes n of, and for each n 2 L, let B n denote the clause (in ) at n. Suppose that for each n 2 L, B 0 n is a clause, of which (modulo ordering), B n is a substitution instance, and that S 0 denotes the set comprised of the clauses B 0 n, (as n ranges over the leaf nodes of ). Then there is a PC < -deduction 0 of a clause A 0, PC < -initial (in 0 ), of which (modulo ordering), A is a substitution instance, from S 0, such that 8B 0 2 S 0, k 0(B 0 ) = jjfn 2 L: B 0 n = B 0 gjj. Obviously then, the total number ( 0 ) of leaf nodes of 0, must be equal to the total number (), of leaf nodes of Refutation Completeness of PC <. Theorem 1.1. P C < is refutation complete Subsumption and PC <. Lemma 1.5. Suppose B 2 PC < 1 (A; D), for some clauses A, B, and D, such that A has <-maximal predicate symbol P and is pure-positive with respect to P, ( P possibly `='),

11 PC <, EPC <, AND THE EQUALITY PREDICATE AND SUBSUMPTION IN RESOLUTION-BASED ATP 11 (and D has <-maximal predicate symbol P and is mixed with respect to P ). Suppose also that for some clauses A 0 and D 0, A 0 subsumes A, and D 0 subsumes D, and that either: i) D 0 is pure-positive with respect to P, or ii) with D some substitution such that D 0 D D, and L 1 say, the left-most negative P instance in D 0, L 1 D occurs as the left-most negative P instance in D. Then, in case (i), D 0 subsumes B, and in case (ii), either A 0 subsumes B, or there exists B 0 2 PC < 1 (A 0 ; D 0 ) such that B 0 subsumes B. Lemma 1.6. Suppose is a PC < +1 -deduction of a clause D, from ancillary parents in C, (say), and principal ancestor B, B with <-maximal predicate symbol (and mixed with respect to) P, (say, P possibly `='). Then if B 0 subsumes B and B 0 is pure-positive with respect to P, B 0 subsumes D. Lemma 1.7. Let B be a clause with <-maximal predicate symbol P, say, ( P possibly `='), B mixed with respect to P, and suppose that (for some k 1 ), C 0 ; : : : ; C k?1, (the C i not necessarily distinct), and B = D 0 ; : : : ; D k?1 ; D k = D, represents a PC 1 < -deduction, + (as in Denition 1.2). Suppose B 0 subsumes B, and that C0; 0 : : : ; Ck?1 0, is a sequence of (not necessarily distinct) clauses such that 8i: 0 i k? 1, C 0 i subsumes C i. Then either: i. B 0 subsumes D, ii. C 0 i subsumes D, for some i: 0 i k? 1, such that (modulo ordering) C 0 i 6= C i, or iii. there exists a PC < + 1 -deduction 0, of a clause D 0, from principal ancestor B 0 and ancillary parents in fc 0 0 ; : : : ; C0 k?1 g, where D0 subsumes D, and such that 8C 0 2 fc 0 0 ; : : : ; C0 k?1 g, k 0(C 0 ) jjfi: 0 i k? 1 ; C 0 i = C 0 gjj. Lemma 1.8. Suppose is a PC < -deduction of a clause A, from a set S, and ( A, at) the root is PC < -initial. Suppose L denotes the set of leaf nodes n of, and for each n 2 L, let B n denote the clause (in ) at n. Suppose that for each n 2 L, B 0 n is a clause which subsumes B n, and that S 0 denotes the set comprised of the clauses B 0 n, (as n ranges over the leaf nodes of ). Then there is a PC < -deduction 0 of a clause A 0, PC < -initial (in 0 ), which subsumes A, from S 0, such that 8B 0 2 S 0, k 0(B 0 ) jjfn 2 L: B 0 n = B 0 gjj. Obviously then, the total number ( 0 ) of leaf nodes of 0, must be less than or equal to the total number (), of leaf nodes of. As the empty clause, 2 is necessarily PC < -initial, we also immediately have the following: Corollary 1.1. Suppose is a PC < -refutation of S, that L denotes the set of leaf nodes n of, and for each n 2 L, B n denotes the clause (in ) at n. Suppose that for each n 2 L, B 0 n is a clause which subsumes B n, and that S 0 denotes the set comprised of the clauses B 0 n, (as n ranges over the leaf nodes of ). Then there is a PC < -refutation 0 of S 0, such that 8B 0 2 S 0, k 0(B 0 ) jjfn 2 L: B 0 n = B 0 gjj. Again, obviously, the total number ( 0 ) of leaf nodes of 0, must be less than or equal to the total number (), of leaf nodes of.

12 12 T. MACKLING 1.5. Refutation Completeness of PC < Under Subsumption (and Tautology) Deletion. Denition 1.5. For any nite non-empty set S of clauses, by a PC < -deduction sequence from S, we shall mean any sequence of clauses C 0 ; C 1 ; : : :, such that either for some k 0, C k = 2, or: 1) 8C 2 S : 9i 0, such that modulo alphabetic equivalence, C i = C, and 2) 8i 0 : 8j 0 : 8C 2 PC < 1 (C i ; C j ) : 9k 0, such that modulo alphabetic equivalence, C k = C. By a partial PC < -deduction sequence from S, we mean any (possibly empty) initial (consecutive) subsequence C 1 ; C 2 ; : : : ; C k, (for some k 0 ), of some PC < -deduction sequence from S, = C 1 ; C 2 ; : : :. Denition 1.6. For any nite non-empty set S of clauses, by a subsumption deleted PC < -deduction sequence from S, we shall mean any sequence of clauses C 0 ; C 1 ; : : :, such that: 1) 8C 2 S : 9i 0, such that C i subsumes C, and 2) 8i 0 : 8j 0 : 8C 2 PC < 1 (C i ; C j ) : 9k 0, such that C k subsumes C. Lemma 1.9. Let C 0 ; C 1 ; : : :, be a subsumption deleted PC < -deduction sequence from S, (for some nite non-empty set of clauses S ), and let S 0 be any nite non-empty set of clauses such that, for every clause C 2 S 0, C i subsumes C, for some i 0. If is any PC < -deduction of (a clause) D, from S 0, in which D, (at the root) is PC < -initial, (in ), then for some n 0, C n subsumes D. Theorem 1.2. (PC < is refutation complete under subsumption deletion.) Given any nite unsatisable set S of clauses, for any subsumption deleted PC < -deduction sequence from S, C 0 ; C 1 ; : : :, it is the case that for some n 0, C n is the empty clause, 2. Lemma If D is a PC < -resolvent of a clause B with a tautology C, then either: i) D is a tautology, or ii) B subsumes D. Denition 1.7. For any nite non-empty set S of clauses, by a subsumption and tautology deleted PC < -deduction sequence from S, we shall mean any sequence of clauses C 0 ; C 1 ; : : :, such that: 1) 8C 2 S, either C is a tautology, or 9i 0, such that C i subsumes C, and 2) 8i 0 : 8j 0 : 8C 2 PC < 1 (C i ; C j ), either C is a tautology, or 9k 0, such that C k subsumes C. Lemma Let S be a nite non-empty set of clauses. Given a non-empty subsumption and tautology deleted PC < -deduction sequence from S, C 0 ; C 1 ; : : :, there exists a subsumption deleted PC < -deduction sequence from S, 0 C 0 0; C 0 1; : : :, such that 8j 0, either: a) 9i 0 such that C 0 j = C i, or b) C 0 j is a tautology. Corollary 1.2. (PC < is refutation complete under subsumption and tautology deletion.) Given any nite unsatisable set S of clauses, for any subsumption and tautology deleted PC < - deduction sequence from S, C 0 ; C 1 ; : : :, it is the case that for some n 0, C n is the empty clause, 2.

13 PC <, EPC <, AND THE EQUALITY PREDICATE AND SUBSUMPTION IN RESOLUTION-BASED ATP Generating Deduction Sequences with Deletion. Refutation completeness under subsumption deletion, (for a resolution system R ), is normally formulated in the context of (a particular deletion strategy, within) a particular deduction strategy. Our approach, via Denition 1.5, and Denition 1.7, (and their EPC < analogues, found in the next section), was to deal with the issue of refutation completeness with subsumption deletion, in fashion relatively free of the context of a specic deduction/deletion strategy. There are of course, various strategies for generating (sound) (PC < -)deduction sequences, (with or without deletion), which are guaranteed to eventually deduce the empty clause, from any initially given nite unsatisable set of clauses, S. This statement is more generally true of almost any (nitary, rst order) refutation complete (clausal, or clause-like) resolution system R, (where, for instance, for any nite set, X, of \clauses", the set of R -resolvents of the \clauses" in X, R(X), say, is nite). But rather than going through the rather lengthy 6 formulation required to dene the appropriate generalized notions and specify exactly which systems the following discussion is amenable to, in the remainder of this section, we will simply use the terms `tautology', and `subsumption', `deduction', `clause', `resolvent', `2 ', et cetera, somewhat informally, and leave the obvious generality (and, for dierent systems, the necessary modications) of our remarks for the reader to observe. Because some of the material we will present (particularly that describing the basic deduction strategies) is folklore, presented only to provide a conceptual backdrop for what we wish to convey, and because (in our opinion) the content of this section is relatively trivial, this section will be presented quite informally. The most basic deduction strategy is the \shell method", where, essentially: With S 0 denoting the given initial given nite clause set, inductively, for any i 0, one successively constructs S i+1 S i [ R(S i ), where, (for each i 0 ) R(S i ) denotes the set of all resolvents generated from (pairs of, -in those cases where R consists of only binary deduction rules), clauses in S i. Of course, it is intended that the construction of the successor shells, S i+1 ; : : :, may be stopped once 2 2 S i. Also, of course, by refutation completeness, by denition, for any (nite) unsatisable S, there is an R -deduction of the empty clause, 2, from S 0 S, and if C has a depth k deduction from S, clearly C 2 S k, so there is some nite n 0 such that 2 2 S n. In the context of the shell method strategy 7, the property of refutation completeness under subsumption (and tautology) deletion is naturally formulated in terms of claiming that in passing from S i to S i+1, we can rst \thin" S i to any S 0 i S i such that for any clause C 2 S i, (either C is a tautology, or), there is some C 0 2 S 0 i, such that C 0 subsumes C, (and then take S i+1 S 0 i [ R(S0 i ) ). Presumably better than the shell method strategy, is a \best-rst" strategy, where, for example, essentially: 6 It would be, at least relative to the intended content of this section. 7 It is in this context, that Sibert [14] shows the refutation completeness of his system under (his) subsumption deletion. However his subsumption relation is little more than the \classical" one, and, (basically because of an absence of a \left-most literal restriction"), unlike our rules, his deduction rules (directly) lift with respect to subsumption.

14 14 T. MACKLING a sequential data base of clauses is dynamically maintained, where with S 0 denoting the given initial given nite clause set, the data base initially contains S 0, a \resolution table" indicating which resolutions have (or have not yet) been performed, is dynamically maintained, there is a positive integer \tness function", f( ), which assigns to each clause, C, a tness, f(c) 0, such that for any integer n 0, any any initially given nite set S of clauses, there are at most nitely many 8 clauses C deducible from S with f(c) n, and at each stage, a clause C such that a resolution with a clause C 0, (where all resolvents between C and C 0 are generated), which precedes C in the data base, has not yet been performed, for which f(c) is minimal (amongst such clauses), is selected from the data base, together with some C 0 as above, all resolvents of C and C 0 are generated and each resolvent that is not, modulo alphabetic equivalence, already in the data base, is added to the data base, and nally the resolution table is updated. (Usually other things are going on here, as well. For example, clauses may be assigned integer names, and, as well, the names of the parent clauses from which they were resolved. In this way, at any stage, for any clause C in the data-base, it is a relatively easy matter to construct the deduction tree of C, and thereby determine, for any clause C 0 in the data base, if C 0 was an ancestor of C. Also, for proof retrieval, for instance, sometimes deleted clauses aren't actually deleted from the data base, but merely marked/tagged as such.) The procedure is, of course, expected to stop once the empty clause has been deduced. Again, by refutation completeness, by denition, for any (nite) unsatisable S, there is an R - deduction of the empty clause, 2, from S 0 S. Since clearly any clause C in the data base has a deduction from S, by our assumption on the tness function f( ), for any pair of clauses C 0 and C in the data base, eventually we will (either deduce the empty clause, or) resolve C against C 0. Thus, if C has a depth k deduction from S, we will eventually (unless at some stage before, we have the empty clause in the data base), have C in the data base, and so, (if S is unsatisable), at some stage we will have deduced the empty clause. In this context, the question of refutation completeness under subsumption (and tautology) deletion leads naturally to the notions of \forward" versus \backward" (subsumption-) deletion. The property of refutation completeness under forward subsumption (and tautology) deletion, is then (formulated as) the assertion that the above procedure will eventually deduce the empty clause, when it is modied so that, where before: \ : : : all resolvents of C and C 0 are generated and each resolvent that is not, modulo alphabetic equivalence, already in the data base, is added : : : " now: \ : : : all resolvents of C and C 0 are generated and each resolvent that is not (a tautology and is not) subsumed by some clause already in the data base, is added : : : ". A (best-rst) deduction/deletion strategy involving backward (subsumption) deletion is any such (best-rst) strategy as above, where, however, one is allowed to delete clauses from the data base, if they are subsumed by some newly deduced (or introduced) clause, and typically, providing some 8 For example, f(c) might be f 1 (C)+f 2 (C), where f 1 ( ) is positive and monotonically increasing with respect to the clause's length, and f 2 ( ) is positive and monotonically increasing with the maximum of the (syntactic) lengths of the terms occurring in the clause.

15 PC <, EPC <, AND THE EQUALITY PREDICATE AND SUBSUMPTION IN RESOLUTION-BASED ATP 15 extra criterion is satised, this criterion specied by the strategy. Depending on the particular deletion strategy, when backward subsumption is detected, one may also be allowed to delete additional clauses. For example, when a new clause C subsumes a clause C 0 already in the data base and the criterion is satised, (depending on the particular deletion strategy), one may also be allowed to delete from the data base, all clauses, D, other than C, which have C 0 as an ancestor, (according to their recorded deduction tree). The criterion stipulates that the subsuming clause is somehow better 9 than the clause subsumed, and, as we later show, may play an important role in the preservation of refutation completeness. The criterion for backward (subsumption) deletion specied in the context of a particular best-rst deduction strategy, (in application to a specic resolution system R ), together with the description of which clauses should be deleted, can be referred to as that strategy's backward deletion policy. Refutation completeness under (restricted) backward subsumption deletion of course asserts that some such strategy, (when applied to some specic system), involving backward deletion is refutation complete, (in the obvious sense), and (in this best-rst context), a system (really, the specic strategy should also be given), is said to be refutation complete under both forward and (restricted) backward subsumption (as well as tautology) deletion if any such strategy involving backward deletion can be given, such that when modied so that forward subsumption (and tautology) deletion is present, the resulting strategy remains refutation complete. Evidently without any deletion involved, both the shell method as well as any best-rst deduction strategy (as above), (when applied to PC < ), for any given initial (nite) clause set S, naturally yield a (PC < -)deduction sequence,, as in Denition 1.5, such that for any clause C, C 2 jj if and only if the strategy deduced C at some point. Specically: For the shell method, let 0 be some sequencing of S 0, for i 0, let i+1 denote some sequencing of those clauses in S i+1 n S i, and let 0 ; 1 ; : : :. For a best-rst strategy (as above), C i, the i th clause in, is the i th clause in sequence of clauses given in the data base at any stage where the data base contains i clauses. Moreover, (evidently) any implementation of the shell method (without deletion), can, without loss of eciency, be equivalently described in terms of the sequential construction of, as above, (where one still keeps track of which shell (set) each C i 2 jj belongs). Here at any stage i, the sequence 0 ; 1 ; : : : ; i, is the current data base, which is now sequentially constructed. In terms of any such sequential description of the shell method deduction strategy, the notions of forward subsumption and backward subsumption deletion are naturally formulated: At any stage, i 0, we have a nite repetition free sequence: 0 ; 1 ; : : : ; i C 1 ; C 2 ; : : : ; C m, say, (for some m 0 ), where S i = j 0 ; 1 ; : : : ; i j. In passing from S i to S 0 i S i, clauses may be sequentially deleted, so that if S 0 i has k 0 fewer clauses than S i, then there is a sequence: S i T 0 T 1 T k S 0 i, of clause sets T j, such that, ( 80 j < k ), T j contains exactly T j+1 together with one clause not in T j+1. In each transition from T j to T j+1, some clause, C l 2 S i \ T j is subsumed by some other (distinct) clause C n 2 S i \ T j, and C l is the clause deleted, (so that T j = T j+1 [ fc l g ). If l < n, then the deletion is an instance of backward subsumption deletion, and otherwise, (if l > n ), the deletion is an instance of forward subsumption deletion. 9 For example, the criterion might be that the subsuming clause is shorter than the one subsumed.