Rev Regular b Fp

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Regular Derivations in BasicSuperposition-Based Calculi

Vladimir Aleksic and Anatoli Degtyarev

Department of Computer Science, King’s College, Strand, London WC2R 2LS, U.K.vladimir,anatoli @dcs.kcl.ac.uk

Abstract. We prove the completeness of the regular strategy of deriva-tions for superposition-based calculi. The regular strategy was pioneeredby Kanger in [Kan63], who proposed that all equality inferences takeplace before all other steps in the proof. We show that the strategy is

complete with the elimination of tautologies. The implication of our res-ult is the completeness of non-standard selection functions by which innon-relational clauses only equality literals (and all of them) are selected.

1 Introduction

In this work we prove the completeness of regular strategy in derivations insuperposition-based calculi. Introducing the concept of regular derivation forsequent calculus with equality, Kanger [Kan63] proposed that all equality infer-ences take place at the beginning of the derivation, so that they precede all othersteps in the proof.

In the case of clause calculi, the possibility to regularize derivations wasformulated and proved by Robinson and Wos [RW69b], as a result about thecompleteness of paramodulation:

If a functionally reexive set of clauses S is closed under paramodulationand factoring, and if S is E-unsatisable, then S is unsatisable.

This work analyzes regular derivation strategy in clause calculi. Proving com-pleteness of regular derivations turns out not to be a trivial problem, especiallywith showing compatibility with a set of redundancy criteria. The goal is toprove the following conjecture from [DV01]:

Let S be a set of Horn clauses with respect to equality literals, with thefollowing property: the arguments of every non-equality atom in S arevariables. Then there exists a refutation of S with redundancy criteria Cin which applications of superposition precede applications of all otherrules 1 (resolution, equality solution and factoring).

1 We try to prove the existence of a more constrained form of derivations, in whichboth superposition and equality solution precede all other inferences.



The statement of the conjecture addresses clauses in which the arguments of non-equality literals are variables. Even though this may sound as a restriction,

the reason behind is that this way we can eliminate tautologies from derivations.Focus on the example by Lynch [Ly97].

Example 1. Consider the following set of clauses

→ P (c,b,b)P (c,c,b ), P (c,b,c) → b ≈ c

P (x,y,y ) → P (x,y,x )P (x,y,y ) → P (x,x,y )P (c,c,c ) →

and assume an ordering such that b c. If an equality literal is selected in eachclause, it is possible to make exactly two superposition inferences, which bothderive a tautology.

However, this set of clauses can be transformed to a logically equivalent setof at clauses , which have the property that all arguments of predicate literalsare variables.

1. x ≈ c, y ≈ b → P (x,y,y )2. x ≈ c, y ≈ b, P (x,x,y ), P (x,y,x ) → b ≈ c3. P (x,y,y ) → P (x,y,x )4. P (x,y,y ) → P (x,x,y )5. x ≈ c, P (x,x,x ) →

Let selection be the same as in the original set of clauses. Inferences that pre-viously led to tautologies can not be preformed anymore since paramodualtionsinto variables are forbidden. As a result, there is a regular derivation without

tautologies.6. y ≈ b → P (c,y,y ) [es 1]7. y ≈ b, P (c,c,y ), P (c,y,c ) → b ≈ c [es 2]8. P (c,c,b ), P (c,b,c ) → b ≈ c [es 7]9. y ≈ c, P (c,c,b ), P (c,b,c ) → P (c,y,y ) [s 8, 6]

10. P (c,c,b ), P (c,b,c ) → P (c,c,c ) [es 9]11. P (c,c,c ) → [es 5]12. → P (c,b,b) [es 6]

The Horn subset consisting of the “relational” clauses 3 , 4, 10, 11 and 12 isunsatisable, i.e. is refuted by resolution without tautologies under arbitraryselection function.

Recall the conjecture from [DV01], quoted earlier in this chapter. We prove,motivated by the previous example, that the statement of the conjecture holdstaken that the set C contains only one redundancy criterion – tautology elimin-ation.

In case of superposition calculi like basic superposition (see [BGLS92], [NR92a])and strict basic superposition (see [BG97]), it turns out that the conjecture holds



only if inference rules are applied on clauses that are Horn with respect to equal-ity literals (for general clauses, it is obvious that some factoring inferences must

precede superposition). This, however, is not a restriction, since any set of gen-eral (w.r.t. equality literals) clauses can, by renaming equality literals using newpredicate symbols, be translated into a satisability equivalent set of clauseswhich are Horn with respect to equality literals.

Example 2. For the following unsatisable set of clauses, it is obvious that factor-ing has to take place before superposition.

→ a ≈ b, a ≈ b→ a ≈ c, a ≈ c

b ≈ c →

This set of clauses can be modied to a satisability equivalent set of clausesthat is Horn with respect to equality literals. For example, it can be done in thefollowing way.

→ P, a ≈ b→ Q, a ≈ c

P → a ≈ bQ → a ≈ c

b ≈ c →It is easy to check that using only equality inferences the unsatisable set of “relational” clauses P → Q, Q → P, P, Q →, → P, Q can be derivedwithout applying factorisation on equality literals.

In terms of the latest results in paramodulation-based theorem proving, ourresult can be formulated in a few different ways, as a theorem about the com-

pleteness of a superposition-based calculus which:– employs a special selection strategy by which all equality literals and only

they are selected in non relational clauses;– employs an ordinary selection strategy, but imposes the order by which equal-

ity literals are always greater then relational literals, i.e. admits t P (t),where t is a term, P is a predicate symbol, and hence, dismissing the sub-term property.

Referring to the items listed above, there have been some results in the directionof rening superposition-based systems either by weakening term and literalorderings or by allowing arbitrary selection strategies. However, none of them iswide enough to cover our result. Here we outline such attempts.

Trying to weaken term ordering constraints, Boll and Rubio (see [BR02])proved the completeness of ordered paramodulation for Horn clauses that isbased on orderings without the sub-term property. Since their result does notimplement basic strategies, and being restricted to ordered paramodulation (notsuperposition), it can not be used to prove our claim. Moreover, it is not certainif their result can be extended to employ redundancy notions like simplicationand tautology elimination.



Regarding arbitrary selection strategies, the latest result has been presen-ted by Aleksic and Degtyarev in [AD05], where they extend the results on the

completeness of arbitrary selection strategies for basic paramodulation on Hornclauses of Boll and Godoy given in [BG01]. The result presented in [AD05] isabout the completeness of arbitrary selection strategies for basic superpositionon general constrained clauses, provided there exists a refutation with no factor-ing inferences. Our consideration, in strictly equational setting, also addressesderivations from general clauses but does not require the existence of a factor-ing free refutation. For this reason, and for the reason that it does not supportstandard redundancy elimination techniques, the result of [AD05] on arbitraryselection can not be used to cover our claim either.

Overall, the regular derivation strategy denes a framework for some rene-ments of the state-of-art superposition-based inference systems, which can beconsidered non-standard because of the following reasons:

– Superposition-based systems are normally based on reduction orderings.Regular strategy addresses weakened versions, in particular the orderingswithout the sub-term property in the sense given below.

– Classical selection strategies for superposition-based proving are paramet-rized by a given ordering on ground terms. Namely, if a positive atom isselected it has to be maximal in the clause, with respect to the given order-ing. Regular strategy means a renement of this, since it allows literals tobe selected regardless of their maximality.

– Normally, the application of superposition rules is restricted to the maximalliterals of the premises. Regular strategy enables dropping the literal order-ing constraints in order to postpone the applications of the inferences onpredicate literals (i.e. resolution and factoring) till the end of the derivation.

Even though our proof is based on transformations of derivation trees, it isessentially different to the one given by Bol and Godoy in [BG01], because:

– Their transformation method assumes that the initial derivation employseager selection of negative literals, which is not a requirement in our trans-formations.

– The calculi they use are on Horn clauses. Having to deal with multiple pos-itive literals in our case produces a plenty of difficult cases in our transform-ations.

2 Preliminaries

Here we present only notions and denitions necessary for understanding thepaper. For a more thorough overview, see [NR99]. It is assumed that the readerhas a basic knowledge in substitution and unication.

All formulae are constructed over a xed signature Σ containing at least oneconstant and a binary predicate symbol ≈ . In order to distinguish equality fromidentity, we use = to denote the latter. By X we denote a set of variables. The



set of all terms over the signature Σ with variables from X is denoted by T Σ (X )and the set of ground terms T Σ (∅) by T Σ .

An equation is an expression denoted by t1 ≈ t 2 or equivalently t2 ≈ t 1 . Fordealing with non-equality predicates, atoms P (t 1 , . . . , t n ), where P is a predic-ate symbol of arity n and t1 , . . . , t n are terms, can be expressed by equationsP (t 1 , . . . , t n ) ≈ true , where true is a new symbol. In order for this encoding tobe sound, we use a two-sorted logic, in which the sort of predicate symbols andthe symbol true is different from the sort of the function symbols. A literal is apositive or a negative equation.

The expression A[s ] p indicates that an expression A contains s as a sub-expression at a position p. A[t] p is a result of replacing the occurrence of s inA at the position p by t. We assume that the position of t in A is always thesame as the position of s. Therefore we will not explicitly mention positions of sub-terms within terms, and will write A[s ] and A[t] meaning A[s ] p and A[t] p .An instance Aσ of A is the result of applying the substitution σ to A.

A clause is a disjunction of literals, denoted by a formula L1 , L 2 , . . . , L m .This denition allows for multiple occurences of identical literals, i.e. for treatingclauses as multiset of literals. Sometimes, especially in examples to improve read-ability, we use the sequent notation by which a clause ¬A1 , . . . , ¬Ak , B 1 , . . . B l

is represented as A 1 , . . . , A k → B 1 , . . . , B l .A constraint is a possibly empty conjunction of atomic equality constraints

s = t or atomic ordering constraints s t or s t. The empty constraint isdenoted by .

A constrained clause is a pair consisting of a clause C and a constraint T ,written as C | T . The part C will be referred to as the clause part and T theconstraint part of C | T . A constrained clause C | will be identied with theunconstrained clause C .

A substitution σ is said to be a solution of an atomic equality constraint s = t ,if sσ and tσ are syntactically equivalent. It is a solution of an ordering constraint s t (with respect to a reduction ordering > which is total on ground terms),if sσ > tσ , and a solution of s t if it is a solution of s t or s = t . Generally,a substitution σ is a solution of a constraint T , if it is a simultaneous solutionto all its atomic constraints. A constraint is satisable if it has a solution.

A ground instance of a constrained clause C | T is any ground clause Cσ ,such that σ is a ground substitution and σ is a solution to T .

A tautology is a constrained clause whose all ground instances are tautologies.There are two forms of tautologies:

C, l ≈ r | T where σ is a solution to T and lσ = rσ

and

C, s ≈ t, l ≈ r | T where σ is a solution to T and lσ = sσ and rσ = tσ.

A contradiction is a constrained clause | T , with an empty clause partsuch that the constraint T is satisable. A constrained clause is called void if its



constraint is unsatisable. Void clauses have no ground instances and thereforeare redundant.

A set of constrained clauses is satisable if the set of all its ground instancesis satisable.A derivation is a possibly innite ordered sequence of sets of clauses, where

each set is obtained from the previous one either by adding a clause (conclusionof an inference) or by deleting a clause by using deletion rules. Further in thepaper, we dene a set of inference and deletion rules relevant for this work. Aderivation of the empty clause is differently called a refutation .

Throughout the paper we assume that derivations are in tree-like form, withconstrained clauses as nodes. In the tree representation, premisses of inferencesare children nodes of their conclusions. In a derivation tree, a node can nothave more than one parent. Therefore, if a clause takes part in more than oneinference, the derivation tree contains as many copies of the clause (with thewhole sub-derivation it is a conclusion of). The constrained clause which is theroot of the tree will be referred to as the root of the derivation. Similarly, theinference with the root clause as its consequence will be called the root inference of the derivation.

A derivation is regular , if all applications of superposition and equality solu-tion rules in the derivation precede all other inferences. Otherwise, a derivationis irregular .

A selection strategy is a function from a set of clauses, that maps each clauseto its sub-multiset. If a clause is non-empty, then the selected sub-multiset isnon-empty too. A derivation is compatible with a selection strategy if all theinferences are performed on the selected literals, i.e. all the literals involved inthe inferences are selected.

3 Regular transformations

We prove the completeness with tautology elimination of regular derivations inbasic setting by transforming derivation trees, where a transformation step is anapplication of a permutation rule , which we dene in a later chapter. A similar ap-proach is used in [dN96], but for derivations by resolution. For paramodulation-based calculi, in [BG01] the authors use a transformation method to provetheir result on arbitrary selection on Horn clauses. However, our transformationmethod is essentially different, for two reasons. First, we address derivations ongeneral clauses, whereas they restrict themselves to the Horn case. Secondly, op-posite to our transformation method, the application of their method may causethe appereance of tautologies in the derivations.

The starting point of our transformations is a refutation by BFP (see [Ly95]).This is the calculus of choice because, appart from being complete, it is basic,does not contain a factoring rule and allows for tautology elimination. The ab-sence of a factoring rule is essential for our result. The transformations methodthat we present is based on permuting two consecutive inferences, which is notalways possible in the presence of factoring.



Assume, for a moment, that our calculus of choice contains equality factoring inference (for example, see [BGLS92], [NR92a]). Consider the following deriva-

tion sequence:c ≈ d

a ≈ b, a ≈ c, a ≈ da ≈ b, c ≈ d, a ≈ d (eq fac )

a ≈ b, d ≈ d, a ≈ d (sup )

and assume that a b c d. Here an application of equality factoring pre-cedes a superposition inference. Effectively, the application of factoring producesthe literal c ≈ d, which is made up of the smaller terms of the literals a ≈ c anda ≈ d. To regularize this fragment of derivation, it is necessary to transform thederivation in a way that superposition precedes factoring. If the two inferenceswere to swap positions, it would mean that superposition takes place into thesmallest term of the literal a ≈ c, which is never possible by the denition of superposition.

The situation is somewhat different if the calculus contains positive factoring ,like it is the case with SBS of [BG97]. Positive factoring does not produce freshliterals, it removes literals that are sintacticly equivalent. Hence, it is possible topermute every application of positive factoring with a superposition inference.As for equality solution inferences, it is also always possible to permute themwith applications of positive factoring rule. This, however, may result in theappearance of tautologies that did not exist in the original derivation. Moreprecisely, we can not prove using transformations of derivations, that the calculusSBS allows for the elimination of tautologies.

3.1 The Calculus EBFP

The calculus EBFP (extended BFP ) of constrained clauses consists of the rules:Factored (positive and negative) overlap

l1 ≈ r 1 , . . . , l n ≈ r n , Γ 1 | T 1 s[l] t, Γ 2 | T 2s [r 1 ] t , . . . , s [r n ] t, Γ 1 , Γ 2 | T 1 ∧T 2 ∧δ

where δ is a shortcut for ( l1 r 1 ∧. . . ∧ln r n ∧s t ∧l1 = l∧. . . ∧ln = l)and ∈ ≈, ≈ .

Equality solution 2

s ≈ t, Γ | T Γ | T ∧ s = t

Relational resolution

Γ 1 , P | T 1 Γ 2 , ¬Q | T 2Γ 1 , Γ 2 | T 1 ∧T 2 ∧P = Q

2 In [Ly95] that introduced the calculus BF P , this inference is called “reection”. Weuse the terminology from the papers which present the results that our work is acontinuation of, like [DV01].



Relational factoring (positive and negative)

Γ, L 1 , L 2 | T Γ, L 1 | T ∧L 1 = L 2

where– L1 and L 2 are either both positive or both negative literals;– L1 and L 2 are identical up to variable renaming.

It is assumed that the premises of the above rules have disjoint variables,which can always be achieved by their renaming.

The calculus EBFP consists of the rules of the calculus BFP (see [Ly95]),with the addition of the explicitly stated resolution inference rule and relationalfactoring (positive and negative). Note that BFP is dened on purely equationalclauses, in which case resolution is expressed by a sequence of steps in which

factored overlap is followed by reection. Reection and factored (positive andnegative) overlap inferences will be referred to as equational inferences, whilethe ones that take place with predicate literals will be called relational.

The reason for introducing negative relational factoring rule is of a technicalnature – it is only used in the proof of regular transformations and its existencedoes not affect the completeness of the calculus. In other words, the calculusEBFP without the negative factoring rule is complete.

3.2 Permutation rules

As it has already been mentioned in the introduction, we work with the clausesthat have only variables as arguments of predicate literals ( at clauses). Thisproperty of clauses prevents superposition inferences into arguments of the pre-dicate literals (into variables), which furthermore makes it possible to character-ize the superposition inferences as strictly equational inferences, which provesessential in the denition of the below permutation rules.

The permutation rules are applied to derivation trees, and their effect isinverting the order of two consecutive inferences, whenever a relational inferenceprecedes an equality inference. In the denitions below, wherever the symbol

is used, it can represent either ≈ or ≈ .

res-es rule – Resolution precedes equality solution

Γ 1 , s ≈ t, ¬Q | T 1 Γ 2 , P | T 2Γ 1 , Γ 2 , s ≈ t | T 1 ∧T 2 ∧P = Q

(res )

Γ 1 , Γ 2 | T 1 ∧T 2 ∧ s = t ∧P = Q (sup )

This sequence transforms to:

Γ 1 , ¬Q, s ≈ t | T 1Γ 1 , ¬Q | T 1 ∧ s = t

(sup )Γ 2 , P | T 2

Γ 1 , Γ 2 | T 1 ∧T 2 ∧ s = t ∧P = Q (res )



fac-es rule – Relational factoring precedes equality solution

Γ, s ≈ t, L 1 , L 2 | T Γ, s ≈ t, L 1 | T ∧L 1 = L 2 (fac )

Γ, L 1 | T ∧s = t ∧L 1 = L 2(es )

where L 1 and L 2 are either both positive or both negative predicate literals.Similarly to the previous rule, this sequence transforms to:

Γ, L 1 , L 2 , s ≈ t | T Γ, L 1 , L 2 | T ∧s = t

(es )

Γ, L 1 | T ∧ s = t ∧L 1 = L 2(fac )

This permutation, as well as the previous one, is always possible to make,since predicate inferences always take place on predicate literals, while equal-ity solutions are always preformed on equality literals.

res-sup rule – Resolution followed by superpositionΓ 11 , l1 ≈ r 1 , . . . , l n ≈ r n , ¬Q | T 1 Γ 12 , P | T 2

Γ 11 , Γ 12 , l1 ≈ r 1 , . . . , l n ≈ r n | T 1 ∧T 2 ∧P = Q (res )

Γ 2 , u [l] v | T Γ 11 , Γ 12 , Γ 2 , u [r 1 ] v, . . . , u [r n ] v | T 1 ∧T 2 ∧T ∧P = Q ∧T 4

(sup )

where T 4 stands for ( l1 r 1 ∧ . . . ∧ ln r n ∧ s t ∧ l1 = l ∧ . . . ∧ ln = l),and ∈ ≈, ≈ . In this case, the sequence transforms to:

Γ 11 , l 1 ≈ r 1 , . . . , l n ≈ r n , ¬Q | T 1 Γ 2 , u [l] v | T Γ 11 , Γ 2 , u [r 1 ] v, . . . , u [r n ] v, ¬Q | T ∧T 1 ∧T 4

(sup )Γ 12 , P | T 2

Γ 11 , Γ 12 , Γ 2 , u [r 1 ] v, . . . , u [r n ] v | T ∧T 1 ∧T 2 ∧P = Q ∧T 4(res )

fac-sup rule – Relational factoring followed by superposition

Γ 1 , l 1 ≈ r 1 , . . . , l n ≈ r n , L 1 , L 2 | T 1Γ 1 , l1 ≈ r 1 , . . . , l n ≈ r n | T 1 ∧L 1 = L 2

(fac )Γ 2 , u [l] v | T

Γ 1 , Γ 2 , u [r 1 ] v, . . . , u [r n ] v | T 1 ∧T 2 ∧T ∧L 1 = L 2 ∧T 3(sup )

where T 3 stands for ( l1 r 1 ∧. . . ∧ ln r n ∧s t ∧ l1 = l∧ . . . ∧ln = l) and∈ ≈, ≈ . L1 and L2 are either both positive or both negative literals. In

this case, the sequence transforms to:

Γ 1 , l 1 ≈ r 1 , . . . , l n ≈ r n , L 1 , L 2 | T 1 Γ 2 , u [l] v | T Γ 1 , Γ 2 , u [r 1 ] v, . . . , u [r n ] v, L 1 , L 2 | T 1 ∧T ∧T 3

(sup )

Γ 1 , Γ 2 , u [r 1 ] v, . . . , u [r n ] v | T 1 ∧T 2 ∧T ∧L 1 = L 2 ∧T 3(fac )

By analyzing the permutation rules res-sup and fac-sup , one can easilynotice that, once applied to derivation trees, they can introduce some tautologies.In the case of the rule res-sup , this is due to the fact that, different to what itis in the original derivation, the literal ¬Q , after the transformation, appears inthe same clause with Γ 2 (which may contain a literal Q ). It is important, at thispoint, to note that tautologies introduced this way can only be tautologies withrespect to predicate literals.



Lemma 1. The above permutation rules modify BFP derivations into BFP derivations.

Proof. Every permutation rule denes a way of inverting the order of two adja-cent inference rules in a derivation tree. After changing positions, the inferencesstill take place with the same literals at the same positions in terms as it wasin the original derivation. Also, all ordering constraints are kept. Therefore, theresulting derivation is a valid BFP derivation.

3.3 A proof by transformation

In order to prove the completeness of the regular strategy for basic superposition,we start with a refutation by BFP of an unsatisable set of clauses S . Assume

that the root of the refutation is

| T , where T is a satisable constraint. Sincethe calculus employs constraint inheritance, we can nd a solution to T , and ap-ply it to the whole refutation. Having that our transformations do not introduceinferences ”from” and ”to” some fresh literals, and that they they do not changethe positions at which the inferences take place, we can consider only groundinstances of the refutation. Further in this work, all the transformations will beassumed to take place on ground derivations.

A quick word on notation. The compound

Ω C

denotes a derivation (derivation tree) Ω which is rooted by a clause C . The clauseC is a part of Ω . When it is not important which clause roots a derivation, wewill use only Ω .

Lemma 2. Any derivation by BFP can be transformed to a derivation by EBFP without introducing new tautologies.

Proof. The statement of the lemma talks about the treatment of predicate lit-erals. We can chose to treat them as predicate literals or equality atoms. Thecalculus BFP treats predicates as equality atoms. On the other hand, to makeour transformations easier we need treat them as predicate literals.

Every factored overlap with a literal of the form P (t ) ≈ true (where P isa predicate symbol of arity n and t is an n-tuple of terms) and can be turnedinto a sequence of inferences that consists of a number of positive factoring stepsfollowed by an application of resolution. It is clear that this transformation doesnot introduce new literals to clauses, it may only take some duplicate positiveliterals away. It follows that the transformation does not cause appereance of new tautologies. Therefore, if there were no tautologies in the original derivation,there will be no tautologies after the transformation has taken place.



Lemma 3. Any EBFP derivation Ω of the form:

Π 1¬P, C 1Π 2P, C 2

C 1 , C 2(res ) Π 3

DE (sup ).... (eq infs )

F

where the inferences that follow res are all equality inferences, can be split intotwo derivations Ω 1 and Ω 2 with conclusions F 1 and F 2 for which:

– The clause F 1 contains the literal P (can be written as F ∗1 , P ) and F 2 con-tains the literal ¬P (can be written as F ∗2 , ¬P ).

– The union of the literals from F ∗1 and F ∗2 contains all the literals that appear in F and only those literals, with possible duplicates.

Proof. The induction is on the number of (equality) inferences in Ω that takeplace after the inference res . Let Ω with a conclusion F be a derivation thatis obtained from Ω by cutting off its last inference. By the induction hypothesis,Ω can be split into Ω 1 and Ω 2 , rooted by F 1 and F 2 respectively.

Focus to the nal inference of Ω . It involves one or more literals from theclause F . Let the nal inference of Ω , without a loss of generality, be a positivesuperposition inference with F as the ”from” clause. Note that the conclusionof this inference is in fact the clause F .

Γ 1 , l ≈ r 1 , . . . , l ≈ r m Γ 2 , u [l] ≈ v

Γ 1 , Γ 2 , u [r 1 ] ≈ v, . . . , u [r m ] ≈ v

In case all the literals l ≈ r 1 , . . . , l ≈ rm belong to (w.l.o.g.) F 1 , we add thefollowing derivation to Ω 1 , thus dening the nal form of Ω 1 . The added inferencehas F 1 as the ”from” premise:

F ∗

1 , l ≈ r 1 , . . . , l ≈ r m Γ 2 , u [l] ≈ vΓ 2 , F ∗

1 , u [r 1 ] ≈ v, . . . , u [r m ] ≈ v

There are no added inferences to Ω 2 , which is then the same as Ω 2 . By theinduction hypothesis, the clauses F 1 and F 2 contain all the literals form Γ 1 .Besides, F 1 contains the literal P and F 2 the literal ¬P . Therefore, the conclusionF 1 of Ω 1 inherits the literal P from the F 1 , and similarly F 2 inherits ¬P fromF 2 , and the union of the literals from F 1 and F 2 contains only (and all of them)the literals from Γ 1 , Γ 2 .

Otherwise, assume that the literals l ≈ r 1 , . . . , l ≈ r k appear in F 1 , while theliterals l ≈ r k +1 , . . . l ≈ r m appear in F 2 . It is easy to see that, in order to obtainall the literals that appear in Ω , both F 1 and F 2 should paramodulate into thenegative premise of the last inference of Ω . We therefore produce Ω 1 and Ω 2 by



adding an inference to both Ω 1 and Ω 2 . These inferences have the clauses F 1and F 2 as positive premises.

F ∗

1 , l ≈ r 1 , . . . , l ≈ r k Γ 2 , u [l] ≈ vΓ 2 , F ∗

1 , u [r 1 ] ≈ v, . . . , u [r k ] ≈ v

andF ∗

2 , l ≈ r k +1 , . . . , l ≈ r m Γ 2 , u [l] ≈ vΓ 2 , F ∗

2 , u [r k +1 ] ≈ v, . . . , u [r m ] ≈ v

Similarly to the previous case, the statement of the lemma holds. It is worthpointing out that this case produces duplicate literals in the union of the literalsfrom the clauses F 1 and F 2 . It due to the fact that the ”to” clause of the nalinference of Ω appears as the ”to” clause of the nal inferences of both Ω 1 andΩ 2 , and therefore the literals from Γ 2 are inherited to both F 1 and F 2 .

Note that the same reasoning applies when the last inference of Ω is equality

solution. The consideration then forks in two sub-cases, determined by whetherthe literal inferenced upon in Ω appears in both F 1 and F 2 or just in one of them.

It can be seen, from the proof of the previous lemma, that every clause inthe two newly obtained derivations is a clause that contains no other literalsthan some clause of the original derivations. Thus, if there are no tautologies inthe original derivation, there will be no tautologies after the transformation hastaken place.

Denition 1. A clause is e-empty if it contains no equality literals (and zeroor more predicate literals). A derivation of an e-empty clause from a set of clauses which contain both predicate and equality literals is called e-refutation .An e-refutation that ends with equality inferences is called s-e-refutation (from short e-refutation). Note that the empty clause is also e-empty. Similarly, every refutation is also an e-refutation.

Lemma 4. An e-refutation by EBFP can be transformed into a regular EBFP e-refutation with the same conclusion.

Proof. In a derivation tree, a predicate inference for which there is an equalityinference following it is called a non-terminating predicate inference. Let Ω bean e-refutation by EBFP with a conclusion R . Without a loss of generality, weassume that the nal inference of Ω is an equality inference. Otherwise, we canalways neglect the predicate inferences at the end of the derivation tree, andapply the lemma on the sub-derivation obtained this way. Let n be the numberof non-terminating predicate inferences in Ω . Among all the predicate inferences

in the derivation that are not followed by other predicate inferences, pick the onethat is followed by the least number of inferences and call it inf . If the numberof the inferences that follow inf is m , the induction is on the regularity pair(n, m ), where:

(n 1 , m 1 ) > (n 2 , m 2 ) if n 1 > n 2 or

n 1 = n 2 and m 1 > m 2



A regular derivation is assigned the pair (0 , 0).Assume that inf is a resolution inference. In case of factoring, the discussion

is similar (the difference is in the permutation rules applied) and less complex.The inference that follows inf can be equality solution. In this case, the ruleres–es applies, which modies the Ω to a derivation Ω , which at least hasthe second member of the regularity pair lesser than m . This transformationdoes not change the conclusion of Ω . The induction hypothesis applies to thesub-derivation of Ω without the trailing resolution inferences.

Alternatively, the derivation Ω is of the form:

Π 1C 1

Π 2C 2

Γ 1 , l ≈ r 1 , . . . l ≈ r k(inf ) Π 3

Γ 3 , u [l] ≈ vΓ 1 , Γ 2 , u [r 1 ] ≈ v, u [r 2 ] ≈ v, . . . u [r k ] ≈ v

If all the literals l ≈ r 1 , . . . , l ≈ rk belong to either C 1 or C 2 , then similarly tothe previous case, the permutation res–sup can be applied, which also resultsin obtaining a derivation with a smaller regularity pair.

If neither of the previous two scenarios apply, then some of the literals l ≈r 1 , . . . , l ≈ rk appear in C 1 , while the others are inherited from C 2 . In otherwords,

C 1 = P, Γ 1 , l ≈ r 1 , . . . , l ≈ r l and C 2 = ¬P, Γ 2 , l ≈ r l+1 , . . . l , ≈ r m .

By the previous lemma, the derivation can be split into two e-regular derivationsΩ 1 and Ω 2 . They can be transformed, by the induction hypothesis, to regulare-refutations Ω 1 and Ω 2 with the conclusions F 1 and F 2 . The previous lemmastates that the clauses F 1 and F 2 contain the literals P and ¬P . This means that,

by performing a resolution inference on F 1 (= F 1∗, P ) and F 2 (= F ∗

2 , ¬P ), thederivations Ω 1 and Ω 2 can be joined to a derivation with the conclusion F ∗

1 , F ∗

2

that contains all the literals that appear in the conclusion of Ω , with possibleduplicates. However, the duplicates problem can be solved by applying positiveand negative factoring inference rules.

The base of the induction is a derivation with the regularity pair (1 , k ) wherek ≥ 1. More precisely, in case the previous lemma applies to a derivation withonly one non-terminating predicate inference, k is allowed to be greater than1. This is because the previous lemma makes it possible to push all predicateinferences down, below all equality inferences that follow. Otherwise, the base of the induction is any derivation which can be assigned the pair (1 , 1). By applyinga suitable permutation rule, such derivation can be made regular.

Lemma 5. Any set of unsatisable clauses has a regular refutation in which tautologies are redundant.

Proof. Because of its completeness property and compatibility with tautologyelimination, there is always a tautology-free BF P refutation from a set of unsat-isable clauses. Every such refutation is also an e-refutation, and by the previ-ous lemma, it can be transformed to a regular EBFP refutation. As it has been



already stated, the preformed transformation does not cause the appearance of tautologies w.r.t. equality literals. Having a regular derivation means that there

can be derived a set of purely predicate clauses from which the empty clausecan be derived by resolution. Each of those purely predicate clauses is actuallythe root of a regular derivation. If there are tautologies w.r.t. predicate clausesin such regular derivations, the corresponding root will be a tautology, too. Assuch, it is not needed in the further refutation by resolution, and can be dis-carded. By discarding this clause, we discard the whole sub-derivation wheretautologies appeared. This proves that even tautologies w.r.t. predicate literalscan be eliminated.

The following is an instance of the conjecture from [DV01], and is a straightforward consequence of the previous lemma.

Theorem 1. Let S be a set of Horn with respect to equality literals with the

following property: the arguments of every non-equality atom in S are variables.Then there exists a refutation of S with tautology elimination in which applica-tions of superposition precede applications of all other rules (resolution, equality solution and factoring).

4 Future work

A topic for further research is whether regular derivations are compatible withother redundancy elimination techniques, such as simplication. It would beinteresting (and challenging) to implement a theorem prover based on equal-ity elimination [DV01] (which is based on regular derivations), which would becompetitive with resolution-based provers.

5 Aknowledgements

We thank the anonymous referees for helpful comments and suggestions. Ourwork is supported by EPSRC research grants GR/S61973/01 and GR/S63175/01.



References

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