what you always wanted to know about rigid e-unification

Journal of Automated Reasoning 20: 47–80, 1998. 47c© 1998 Kluwer Academic Publishers. Printed in the Netherlands.

What You Always Wanted to Know about RigidE-Unification

ANATOLI DEGTYAREV? and ANDREI VORONKOV??

Computing Science Department, Uppsala University

(Received: September 1996)

Abstract. This paper solves an open problem posed by a number of researchers: the constructionof a complete calculus for matrix-based methods with rigid E-unification. The use of rigid E-unification and simultaneous rigid E-unification for such methods was proposed by Gallier et al., in1987. After our proof of the undecidability of simultaneous rigid E-unification in 1995. (Degtyarevand Voronkov, 1996d), it became clear that one should look for more refined techniques to deal withequality in matrix-based methods. In this article, we define a complete proof procedure for first-order logic with equality based on an incomplete but terminating procedure for rigid E-unification.Our approach is applicable to the connection method and the tableau method and is illustrated onthe tableau method.

Key words: tableau method, equality reasoning, rigid E-unification.

1. Introduction

Algorithms for theorem-proving based on matings or tableaux in first-order logicwithout equality comprise two kinds of rules. Rules of the first kind constructmatrices or tableaux from a given formula using a suitable amplification. Rules ofthe second kind try to close paths or branches using substitutions making the pathsor branches inconsistent. These substitutions are unifiers of some atoms layingon a path or branch. Until recently, most approaches to introducing equalityin such matrix-based methods tried to generalize such algorithms by a suitablemodification of the notion of a unifier.

Such a modification using simultaneous rigid E-unification was introduced in(Gallier et al., 1987) for the method of matings (Andrews, 1981) or the connec-tion method (Bibel, 1981). It can be easily represented in the tableau formalism.The method of matings interleaves two steps: amplification by quantifier duplica-tion and search for mating for a given amplification. For formulas in disjunctivenormal form this method was formulated earlier in (Prawitz, 1983). In this caseamplification is represented by a matrix, and mating is represented by a set ofsimultaneously satisfiable substitution conditions (mated pairs). Prawitz proposeda procedure for constructing substitution conditions one by one, closing the cor-

? Supported by grants from INTAS, TFR and the Swedish Royal Academy of Sciences.?? Partially supported by a TFR grant.

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48 ANATOLI DEGTYAREV AND ANDREI VORONKOV

responding paths in the matrix through search with backtracking. Procedures ofthis kind were used in later formalizations and implementations of the tableaumethod, the method of matings, or the connection method for formulas with-out equality. For example, in (Fitting, 1990) the search for mating consists ofrepeated applications of MGU atomic closure rule.

Gallier et al. (1987) tried to describe a similar procedure for logic with equal-ity. For example, in (Gallier et al., 1992) they describe such a procedure inwhich a substitution condition is formalized via rigid E-unification and the setof substitution conditions are formalized via simultaneous rigid E-unification.

Simultaneous rigid E-unification can be formulated as follows. Given equa-tions si ' ti and finite sets of equations Ei, i ∈ {1, . . . , n}, find a substitution σsuch that ` (

∧e∈Ei eσ) ⊃ siσ ' tiσ, for all i (here ` means provability in first-

order logic with equality). The corresponding instance of the simultaneous rigidE-unification problem is denoted by the system of rigid equations Ei `∀ si ' ti.

EXAMPLE 1.1. In this and further examples we shall often omit parenthesesin terms with unary function symbols; for example, we write ffb instead off(f(b)). Assume that we want to prove the formula ϕ = ∃xyzu((a ' b ⊃g(x, u, v) ' g(y, fc, fd)) ∧(c ' d ⊃ g(u, x, y) ' g(v, fa, fb))). After severalapplications of tableau expansion rules to the negation normal form of ¬ϕ (α-,β- and γ-rules in the terminology of (Smullyan, 1968; Fitting, 1990) we obtainthe following tableau (we consider only the part of the tableau containing literals,omitting nonliteral formulas).

��

PPPPPPPa ' b c ' d

g(x, u, v) 6' g(y, fc, fd) g(u, x, y) 6' g(v, fa, fb)

Collecting formulas lying on the two branches in this tableau, we obtain thefollowing two rigid equations expressing inconsistency of this tableau.

a ' b `∀ g(x, u, v) ' g(y, fc, fd),

c ' d `∀ g(u, x, y) ' g(v, fa, fb).

This system of rigid equations has one solution {fa/x, fb/y, fc/u, fd/v}. Thissubstitution can be found by applying the functional reflexivity rule and MGUreplacement rule of (Fitting, 1990) (in fact, the reformulation of the paramodula-

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WHAT YOU ALWAYS WANTED TO KNOW ABOUT RIGID E-UNIFICATION 49

tion rule for tableaux). Obviously, this tableau cannot be closed without the useof the functional reflexivity.?

Since the invention of simultaneous rigid E-unification by Gallier et al. (1987),there have been a number of publications on simultaneous rigid E-unificationitself and its use in theorem proving, for example, (Gallier et al., 1988, 1989,1990, 1992; Baumgartner, 1992; Beckert and Hahnle, 1992; Becher and Peter-mann, 1994; Beckert, 1994; Goubault, 1994; Petermann, 1994). Some of thesearticles were based on the conjecture that simultaneous rigid E-unification isdecidable. There were several faulty proofs of the decidability of this problem(Gallier et al. , 1988, 1990; Goubault, 1994).

The refutation procedure for first-order logic with equality using simultane-ous rigid E-unification (Gallier et al., 1992) was based on a faulty assumptionthat solutions to simultaneous rigid E-unification can be found by consecutivecombination of finite complete sets of solutions for (nonsimultaneous) rigid E-unification (Gallier et al., 1990; 1988). Later, it was proved (Degtyarev andVoronkov, 1996d; 1996e) that simultaneous rigid E-unification is undecidable,which implied that Gallier et al.’s procedure cannot, in general, find solutions tosimultaneous rigid E-unification. However, it is not clear whether this impliesincompleteness of their procedure for first-order logic with equality: there areexamples when the procedure cannot find a solution for a given amplifica-tion in spite of the fact that such a solution exists, but can find a solutionfor a bigger amplification. Completeness of Gallier et al.’s procedure or exis-tence of a procedure complete for first-order logic with equality based on someset of solutions to rigid E-unification was an open problem (Petermann, 1994;Beckert, 1995). Our paper gives a positive solution to this problem.

An advantage of Gallier et al.’s procedure is that it allows one to extend theproof-search technology developed for tableaux without equality to the case withequality, using solutions to rigid E-unification instead of most general unifiers. Inparticular, for a given amplification Gallier et al.’s procedure always terminates.A procedure of this kind is used in the theorem prover 3T

AP (Hahnle et al.,1994) (R. Hahnle, private communication).

In this paper we define a procedure extending the tableau method to logic withequality based on an incomplete procedure for rigid E-unification. Nevertheless,? The system of clauses corresponding to this example improves a result proved in (Plaisted,

1995) by using a more complicated example. In this system of clauses

a ' b ∨ c ' d,a ' b ∨ g(u, x, y) 6' g(v, fa, fb),

c ' d ∨ g(x, u, v) 6' g(y, fc, fd),

g(x, u, v) 6' g(y, fc, fd) ∨ g(u, x, y) 6' g(v, fa, fb)

there is no refutation even by unrestricted rigid paramodulation (i.e., using nonordered rigidparamodulation and paramodulation into variables), while (Plaisted, 1995) gives an example show-ing incompleteness of ordered rigid paramodulation only.

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our procedure is complete for first-order logic with equality. Hence, we rehabili-tate Gallier et al.’s program for adding equality to semantic tableaux. Moreover,our procedure solves rigid equations lying on different tableau branches indepen-dently. This strongly improves Gallier et al.’s procedure, which uses solutions ofsome rigid equations to solve rigid equations on other branches.

A similar approach has already been defined in (Kanger, 1983) based ona more straightforward way of variable instantiation. As a method for finding aclosing substitution, Kanger proposed an algorithm that can now be characterizedas an incomplete (but terminating) algorithm for simultaneous rigid E-unifiability.Variables in a matrix (or a tableau) could be consecutively substituted by groundterms already occurring in the matrix. This procedure does not solve simultaneousrigid E-unifiability, but it gives a procedure complete for first-order logic withequality. In the terminology of (Fitting, 1990) it means that a closing substitutioncan be found after a sufficiently high (but not necessarily minimal) number ofapplications of the γ-rule. The approach to substitutions based on this idea hasbeen characterized as minus-normalization in (Matulis, 1963; Maslov, 1967).

However, for a language with function symbols, minus-normalization is inter-esting mostly theoretically. Even in simplest cases, minus-normalization requiresa huge number of instantiations. For example, in the tableau of Example 1.1,we have to consider 84 possible instantiations of variables x, y, u, v by terms inthe set {a, b, c, d, fa, fb, fc, fd}. Moreover, it has been proved that the use ofminus-normalization can lead to considerable growth of derivations. Some resultson minus-normalization are proved by (Norgela, 1974).

In this paper we describe a logical calculus BSE for rigid E-unification basedon the rigid basic superposition rule that is an adaptation of basic superpositionof (Bachmair et al., 1992; Nieuwenhuis and Rubio, 1992), for ‘rigid’ variables.For a given rigid E-unification problem (called rigid equation in this paper), thereare only a finite number of BSE-derivations for this problem. Thus, BSE givesus an algorithm returning a finite set of solutions to this rigid equation. We usethese solutions to close a tableau branch in the same way as most general unifiersare used to close a branch in the MGU atomic closure rule of (Fitting, 1990).

2. Preliminaries

We present here a brief overview of notions and preliminary definitions necessaryfor understanding the paper. We assume basic knowledge of substitutions andunification.

Let Σ be a signature and X be a set of variables. T (Σ,X) denotes the set ofall terms in the signature Σ with variables from X. The set of all ground termsin the signature Σ is denoted by T (Σ).

A literal is either an atomic formula or a negation of an atomic formula.An equation is a literal s ' t, where s, t ∈ T (Σ,X). We do not distinguishequations s ' t and t ' s. Literals of the form ¬(s ' t) are denoted by s 6' t

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and called disequations. For simplicity, we assume that ' is the only predicatesymbol of our first-order language. As usual, arbitrary first-order languages canbe represented in such language by introducing a sort bool and replacing anynonequational atom A by A ' true (for details see, e.g., (Bachmair et al.,1995)).

By a ground expression (i.e., term or literal) we mean an expression containingno variables. For any expression E, var(E) denotes the set of all variablesoccurring in E. For a sequence of variables x, we shall sometimes denote x alsothe corresponding set of variables. We write A[s] to indicate that an expressionA contains s as a subexpression and denote by A[t] the result of replacing thisoccurrence of s in A by t. By Aσ we denote the result of applying the substitutionσ to A. If A is a formula, we can as usual rename bound variables in A beforeapplying σ. We shall denote ϕ(x) a formula ϕ with zero or more free occurrencesof a variable x and write ϕ(t) to denote the formula ϕ{t/x}.

A substitution θ whose domain is a subset of {x1, . . . , xn} is denoted

{x1θ/x1, . . . , xnθ/xn}.

A substitution σ is called grounding for a set of variables V if for every variablev ∈ V the term vσ is ground.

Let θ1 and θ2 be two substitutions with disjoint domains. The union of θ1 andθ2, denoted θ1 ∪ θ2 is the substitution σ defined as follows. For every variable vwe have

σ(v)

θ1(v) if v ∈ dom(θ1)

θ2(v) if v ∈ dom(θ2)

v if v 6∈ dom(θ1) ∪ dom(θ2).

Note that we use the union notation θ1 ∪ θ2 only for substitutions with disjointdomains.

The inference systems used in this paper are defined with respect to a reductionordering, denoted by �, which is total on ground terms. Our results are valid forany such ordering.

A formula is in the Skolem negation normal form if it is constructed fromliterals using the connectives∧ and ∨ and the quantifier ∀. There is a satisfiability-preserving structure-preserving translation of formulas without equivalences intoformulas in Skolem negation normal form consisting of the standard Skolemiza-tion and a translation into negation normal form used, for example, in (Andrews,1981). To prove an arbitrary formula ϕ, we translate ¬ϕ in Skolem negationnormal form obtaining a formula ψ and try to establish unsatisfiability of ψ. Forthis reason, theorems in this paper are formulated in terms of unsatisfiability.

For an equation s ' t and a multiset of equations E, we write E ` s ' t todenote that the formula (

∧e∈E e) ⊃ s ' t is provable in first-order logic with

equality. For such formulas provability can be tested by the congruence closure

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algorithm (Shostak, 1978). For the inclusion of multisets we shall use notationS1 v S2.

3. Rigid Basic Superposition

The term ‘rigid paramodulation’ has already been used in (Becher and Petermann,1994; Plaisted, 1995) for systems of inference rules in which all variables aretreated as ‘rigid’. For example, rigid clause paramodulation of (Plaisted, 1995)is essentially paramodulation and resolution over a set of clauses, where all sub-stitutions are applied to the whole set of clauses. A similar system for resolutionwas proposed earlier in (Chang, 1972) as V-resolution and in (Lee and Chang,1973) for resolution with paramodulation as V-resolution and V-paramodulation.We shall use the term ‘rigid basic superposition’ to denote a ‘rigid’ version ofbasic superposition. We formalize rigid basic superposition using constraints ina way that is close to the presentation of (Nieuwenhuis and Rubio, 1995).

DEFINITION 3.1 (Constraints). By an (ordering) constraint we mean a set ofexpressions that can be of two kinds: an equality constraint s ' t, or an inequalityconstraint s � t, where s, t are terms. A substitution θ is a solution to a constraints ' t (respectively, a constraint s � t) if θ is grounding for var(s) ∪ var(t) andsθ coincides with tθ (respectively, sθ � tθ).

A substitution θ is a solution to a constraint C if θ is a solution to everyequality or inequality constraint in C. A constraint C is satisfiable if it has asolution. Constraints C1 and C2 are called equivalent if they have the same setsof solutions.

We assume that there is an effective procedure for checking constraint satisfia-bility. For example, there are efficient methods for solving ordering constraintsfor lexicographic path orderings given by (Nieuwenhuis, 1993; Nieuwenhuis andRubio, 1995).

DEFINITION 3.2. A rigid equation is an expression of the form E `∀ s ' t,where E is a finite multiset of equations and s, t are terms. Its solution is anysubstitution θ such that Eθ ` sθ ' tθ?.

Below we shall introduce a system BSE for solving rigid equations. Thederivable objects of BSE are constraint rigid equations.

DEFINITION 3.3 (Constraint rigid equation). A constraint rigid equation is apair consisting of a rigid equation R and a constraint C. Such a constraint rigidequation will be denoted R · C.

? The term ‘rigid equation’ could be more adequately expressed as ‘instance of a (nonsimulta-neous) rigid E-unification problem’, but this would be too lengthy.

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DEFINITION 3.4 (Calculus BSE). The calculus BSE of constraint rigid equa-tions consists of the following inference rules:

Left rigid basic superposition:

E ∪ {l ' r, s[p] ' t} `∀ e · CE ∪ {l ' r, s[r] ' t} `∀ e · C ∪ {l � r, s[p] � t, l ' p}

(lrbs),

Right rigid basic superposition:

E ∪ {l ' r} `∀ s[p] ' t · CE ∪ {l ' r} `∀ s[r] ' t · C ∪ {l � r, s[p] � t, l ' p}

(rrbs),

Equality resolution:

E `∀ s ' t · C`∀ s ' s · C ∪ {s ' t}

(er).

Application of the rules is restricted to the following conditions:

(1) The constraint at the conclusion of the rule is satisfiable.(2) The right-hand side of the rigid equation at the premise of the rule doesnot have the form q ' q.

(3) In the basic superposition rules, the term p is not a variable.(4) In the left basic superposition rule, s[r] 6= t.

The basic restriction in BSE is formalized by representing most general unifiersthrough equality constraints. Condition 1 has two purposes. The satisfiability ofequations in constraints is needed to preserve correctness of the method. The sat-isfiability of inequality constraints is needed to ensure termination (Theorem 3.9below). Conditions 3–4 are not essential; they are added as standard optimiza-tions used in paramodulation-based methods. Condition 2 prohibits applying anyrules to rigid equations of the form E `∀ q ' q.

We denote by R · C ; R′ · C′ the fact that R′ · C′ is obtained from R · C by anapplication of one of the inference rules of BSE . The symbol ;∗ denotes thereflexive and transitive closure of ;.

EXAMPLE 3.5. Consider the rigid equation ha ' a, hx ' a, hb ' fy `∀ y 'gfy. The ordering � is the Knuth-Bendix ordering (see (Martin, 1987)) in whichall weights of symbols are equal to 1 and which uses the precedence relationf > h > b > a. Under this ordering we have ht � a and ft � hb for everyground term t. The following is a BSE-derivation for this rigid equation:

ha ' a, hx ' a, hb ' fy `∀ y ' gfy · ∅ha ' a, hx ' a, hb ' fy `∀ y ' ghb · {fy � hb, gfy � y, fy ' fy}

(rrbs)(rrbs)

ha ' a, hx ' a, hb ' fy `∀ y ' ga·{fy � hb, gfy � y, fy ' fy, hx � a, ghb � y, hx ' hb}

`∀ y ' y · {fy � hb, gfy � y, fy ' fy, hx � a, ghb � y, hx ' hb, y ' ga}(er)

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By using constraint simplification, that is, replacement of constraints by equiva-lent ‘more simple’ constraints, we can rewrite this derivation as

ha ' a, hx ' a, hb ' fy `∀ y ' gfy · ∅ha ' a, hx ' a, hb ' fy `∀ y ' ghb · ∅

(rrbs)(rrbs)

ha ' a, hx ' a, hb ' fy `∀ y ' ga · {ghb � y, x ' b} (er)`∀ y ' y · {x ' b, y ' ga}

THEOREM 3.6 (Soundness of BSE). Let R · ∅ ;∗ E `∀ t ' t · C. Then anysubstitution satisfying C is a solution to R. In particular, R is solvable.

Proof. For any constraint C, denote by C' the constraint obtained from Cby removing all inequality constraints. First we note that for every applicationof an inference rule of BSE of the form E1 `∀ e1 · C1 ; E2 `∀ e2 · C2 wehave E1,¬e1, C'2 ` E2,¬e2. By induction on the number of inference steps andusing the fact Ci ⊆ Ci+1, we prove the same statement for multistep derivationsE1 `∀ e1 · C1 ;

∗ E2 `∀ e2 · C2.Let R have the form E0 `∀ r ' s. Applying the obtained statement to the

multistep derivation R · ∅ ;∗ E `∀ t ' t · C, we get E0, r 6' s, C' ` E, t 6' t.Hence,E0, C' ` r ' s. Let θ be any solution to C. We have E0θ, C'θ ` rθ ' sθ.Any constraint in C'θ has the form u ' u. Hence, E0θ ` rθ ' sθ, i.e., θ is asolution to E0 `∀ r ' s.

This theorem leads to the following definition.

DEFINITION 3.7 (Answer constraint). A constraint C is called an answer con-straint for a rigid equation R if for some rigid equation E `∀ t ' t we haveR · ∅ ;∗ E `∀ t ' t · C.

We note that BSE is an incomplete calculus for solving rigid equations. Itmeans that there are solvable rigid equations R that have no answer constraint.For instance, consider the rigid equation? x ' a `∀ gx ' x. It has one solution{ga/x}. However, no rule of BSE is applicable to x ' a `∀ gx ' x · ∅.

This means that BSE can yield fewer solutions to a rigid equation than anyother known procedure, for example, that in (Gallier et al., 1992), because allthese procedures are existentially complete. At the same time, BSE can yieldmore solutions than the procedure in (Gallier et al., 1992), as the followingexample shows. For the rigid equation a ' fa `∀ x ' fa the procedure in(Gallier et al., 1992) will find one solution {a/x}, but there are two answerconstraints whose solutions are the substitutions {a/x} and {fa/x}, respective-ly.

To show that there is only a finite number of derivations in BSE from a givenconstraint rigid equation, we prove an auxiliary statement.? Suggested by G. Becher (private communication).

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LEMMA 3.8. Let t0, t1, . . . be an infinite sequence of terms in a finite signatureall of whose variables belong to a finite set. Then there are numbers i, j suchthat i < j and the constraint ti � tj is unsatisfiable.

Proof. Following (Kruskal, 1960), we introduce a partial ordering > on termsas the smallest reflexive and transitive relation satisfying

(1) f(s1, . . . , sn) > si for all i ∈ {1, . . . , n};(2) if s > t then r[s] > r[t].

By Kruskal’s tree theorem (Kruskal, 1960) there exist i, j such that i < j andtj > ti. It is easy to see that the constraint ti � tj is unsatisfiable.

Similar statements have been proven in (Dershowitz, 1982; Plaisted, 1986).

THEOREM 3.9 (Termination of BSE). For any constraint rigid equation R · C,there exist a finite number of derivations from R · C.

Proof. Suppose that there exist an infinite number of derivations from R · C.Then, by Konig’s lemma there exists an infinite derivation R ·C = R0 ·C0 ; R1 ·C1 ; . . . consisting of superpositions. Consider any application of superpositionRi · Ci ; Ri+1 · Ci+1. Let it have the form

E ∪ {l ' r, s[p] ' t} `∀ e · CiE ∪ {l ' r, s[r] ' t} `∀ e · Ci+1

(lrbs)

(the case of right rigid basic superposition is similar). We prove that for everyn > i + 1 the constraint Cn is equivalent to Cn ∪ {s[p] � s[r]}. Indeed, theconstraint Ci+1 (and hence the constraint Cn) contains {l � r, l ' p}. By thedefinition of reduction orderings, if p � r, then s[p] � s[r]. This implies that Cnis equivalent to Cn ∪ {s[p] � s[r]}.

Since every application of rigid basic superposition replaces a literal s[p] ' t(or s[p] 6' t) in Ri by a literal s[r] ' t (respectively, s[r] 6' t), there is an infinitesequence of terms t0, t1, . . . , and an increasing sequence of natural numbersn1, n2, . . . , with the following property. For every positive natural number k theconstraint Cnk ∪ {tk−1 � tk} is equivalent to Cnk . Since all terms tk are in thesame finite signature and have variables in the same finite set, by Lemma 3.8,there are i, j such that i < j and the constraint ti � tj is unsatisfiable. SinceCnk ⊆ Cnj for all k 6 j, the constraint Cnj∪{tk1 � tk} is equivalent to Cnj , for allk 6 j. Hence, the constraint C = Cnj ∪{ti � ti+1, . . . , tj−1 � tj} is equivalent toCnj . Thus, C is satisfiable. But satisfiability of C implies satisfiability of ti � tj .Contradiction.

Note. In (Degtyarev and Voronkov, 1996f) the left rigid basic superpositionhas been formulated incorrectly in the following way:

E ∪ {l ' r, s[p] ' t} `∀ e · CE ∪ {l ' r, s[p] ' t, s[r] ' t} `∀ e · C ∪ {l � r, s[p] � t, l ' p}

(lrbs)

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With this formulation, termination is not guaranteed, as the following exampleshows. Consider the rigid equation fgx ' gx, gx ' a `∀ a ' b and arbitraryreduction ordering � total on ground terms. The following is an infinite sequenceof applications of (lrbs):

fgx ' gx, gx ' a `∀ a ' b · ∅fgx ' gx, gx ' a, gx ' fa `∀ a ' b · {gx ' gx, fgx � gx, gx � a}

(lrbs)(lrbs)

fgx ' gx, gx ' a, gx ' fa, gx ' ffa `∀ a ' b·{gx ' gx, fgx � gx, gx � a, gx � fa}

...fgx ' gx, gx ' a, gx ' fa, gx ' ffa, . . . , gx ' fna`∀ a ' b · {gx ' gx, fgx � gx, gx � a, gx � fa, . . . , gx � fn−1a}

...

It is easy to see that the constraint

{gx ' gx, fgx � gx, gx � a, gx � fa, . . . , gx � fn−1a}

is satisfied by the substitution {fn−1a/x}.Inequality constraints are not needed for soundness or completeness of our

method. The pragmatics behind inequality constraints is to ensure that the searchfor solutions of a rigid equation is finite. In addition, the use of ordering con-straints prunes the search space.

To illustrate this theorem, we consider Example 3.5. The rigid equation ofthis example has an infinite number of solutions including {b/x, ghna/y}, forevery natural number n. However, all possible BSE-derivations starting withha ' a, hx ' a, hb ' fy `∀ y ' gfy · ∅ give only two answer constraints. Oneis

{fy � hb, gfy � y, fy ' fy, hx � a, ghb � y, hx ' hb, y ' ga},

shown in Example 3.5. Another is {fy � hb, gfy � y, fy ' fy, y ' ghb}, obtainedfrom the following derivation:

ha ' a, hx ' a, hb ' fy `∀ y ' gfy · ∅ha ' a, hx ' a, hb ' fy `∀ y ' ghb · {fy � hb, gfy � y, fy ' fy}

(rrbs)(er)

`∀ y ' y · {fy � hb, gfy � y, fy ' fy, y ' ghb}

This answer constraint can be simplified to {y ' ghb}.Theorem 3.9 yields the following.

THEOREM 3.10. Any rigid equation has a finite number of answer constraints.There is an algorithm giving for any rigid equation R the set of all answerconstraints for R.

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Γ1, ϕ ∧ ψ | . . . | ΓnΓ1, ϕ ∧ ψ,ϕ, ψ | . . . | Γn (α)

Γ1, ϕ ∨ ψ | . . . | ΓnΓ1, ϕ | Γ1, ψ | . . . | Γn (β)

Γ1,∀xϕ(x) | . . . | ΓnΓ1,∀xϕ(x), ϕ(y) | . . . | Γn (γ)

In the rules (γ) the variable y does not occur in the premise.

Figure 1. Tableau expansion rules.

4. Answer Constraints and the Tableau Method

In this section we consider how to use the system BSE for theorem proving bythe tableau method. Since we consider only Skolemized formulas, we have noδ-rules in tableau calculi.

DEFINITION 4.1 (Branch and tableau). A branch is a finite multiset of formu-las. A tableau is a finite multiset of branches. A tableau with branches Γ1, . . . ,Γnwill be denoted by Γ1 | . . . | Γn. The tableau with n = 0 is called the emptytableau and is denoted by #.

Often, tableaux are represented in tree form. Representation of tableaux asmultisets of branches is more convenient for us for several reasons. For thisrepresentation we introduce a logical system allowing to expand tableaux:

DEFINITION 4.2 (Tableau expansion rules). The rules (α), (β) and (γ) of Fig-ure 1 are called tableau expansion rules.

DEFINITION 4.3. Let Γ be a branch of a tableau. The set of rigid equations onΓ is defined in the following way. A rigid equation E `∀ s ' t is on Γ if E isthe multiset of all equations in Γ and (s 6' t) ∈ Γ.

We extend the notion of answer constraints to tableau branches.

DEFINITION 4.4. A constraint C is an answer constraint for a tableau branchΓ if C is an answer constraint for some rigid equation on Γ.

By Theorem 3.10, we obtain the following.

THEOREM 4.5. Any tableau branch has a finite number of answer constraints.There is an algorithm that for any tableau branch Γ computes the set of allanswer constraints for Γ.

The following theorem states soundness and completeness of the tableaumethod with answer constraints:

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THEOREM 4.6 (Soundness and completeness). Let ξ be a sentence in Skolemnegation normal form. Then ξ is unsatisfiable if and only if there is a tableauΓ1 | . . . | Γn obtained from ξ by tableau expansion rules and answer constraintsC1, . . . , Cn for Γ1, . . . ,Γn, respectively, such that C1 ∪ . . . ∪ Cn is satisfiable.

Proof. Soundness follows from soundness of BSE .The proof of completeness is quite lengthy and is given in Appendix A. It

is based on the completeness of the equality elimination method (Degtyarev andVoronkov, 1994; 1995a; 1996b).

To illustrate this theorem, consider the formula of Example 1.1. Assume thatwe want to prove the formula ξ = ∃xyzu((a ' b ⊃ g(x, u, v) ' g(y, fc, fd)) ∧(c ' d ⊃ g(u, x, y) ' g(v, fa, fb))). The negation normal form of ¬ξ is∀xyzu((a ' b∧ g(x, u, v) 6' g(y, fc, fd))∨ (c ' d∧ g(u, x, y) 6' g(v, fa, fb))).

The ordering � is the lexicographic path ordering (see, e.g., (Nieuwenhuisand Rubio, 1995)) based on the precedence g > f > a > b > c > d. Forpurely illustrative purposes, we shall display tableaux in the tree form. After onequantifier duplication (application of a γ-rule) and some other tableau expansionrules, we obtain the following tableau:

��

PPPPPPPa ' b c ' d

g(x, u, v) 6' g(y, fc, fd) g(u, x, y) 6' g(v, fa, fb)

There is one rigid equation on each branch of the tableau:

a ' b `∀ g(x, u, v) ' g(y, fc, fd) (1)

c ' d `∀ g(u, x, y) ' g(v, fa, fb) (2)

Rigid basic superposition is applicable to none of these rigid equations. Rigidequation (1) has one answer constraint {g(x, u, v) ' g(y, fc, fd)} obtained byan application of the equality resolution rule:

a ' b `∀ g(x, u, v) ' g(y, fc, fd) · ∅`∀ g(x, u, v) ' g(x, u, v) · {g(x, u, v) ' g(y, fc, fd)} (er)

Similarly, rigid equation (2) has one answer constraint {g(u, x, y) ' g(v, fa, fb)}.The union of these constraints {g(x, u, v) ' g(y, fc, fd), g(u, x, y) ' g(v, fa, fb)}is unsatisfiable. Thus, our method does not find solution after one quantifier dupli-cation. After three quantifier duplications and some other tableau expansion steps,we obtain the following tableau:

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��

PPPPPPP

��

QQQQ

a ' b c ' d

g(x1, u1, v1) 6' g(y1, fc, fd) g(u1, x1, y1) 6' g(v1, fa, fb)��QQQ

a ' b c ' d

g(x2, u2, v2) 6' g(y2, fc, fd)

g(u2, x2, y2) 6' g(v2, fa, fb)

��QQQ

c ' da ' b

g(u3, x3, y3) 6' g(v3, fa, fb)

g(x3, u3, v3) 6' g(y3, fc, fd)

It has four branches:

Γ1 : {a ' b, a ' b, g(x1, u1, v1) 6' g(y1, fc, fd), g(x2, u2, v2) 6' g(y2, fc, fd)}Γ2 : {a ' b, c ' d, g(x1, u1, v1) 6' g(y1, fc, fd), g(u2, x2, y2) 6' g(v2, fa, fb)}Γ3 : {a ' b, c ' d, g(u1, x1, y1) 6' g(v1, fa, fb), g(x3, u3, v3) 6' g(y3, fc, fd)}Γ4 : {c ' d, c ' d, g(u1, x1, y1) 6' g(v1, fa, fb), g(u3, x3, y3) 6' g(v3, fa, fb)}

Consider the following rigid equations R1–R4 on the branches Γ1–Γ4, respec-tively:

R1 : a ' b, a ' b `∀ g(x2, u2, v2) ' g(y2, fc, fd)

R2 : a ' b, c ' d `∀ g(x1, u1, v1) ' g(y1, fc, fd)

R3 : a ' b, c ' d `∀ g(u1, x1, y1) ' g(v1, fa, fb)

R4 : c ' d, c ' d `∀ g(u3, x3, y3) ' g(v3, fa, fb)

We can apply the following BSE-derivations to R1–R4:

a ' b, a ' b `∀ g(x2, u2, v2) ' g(y2, fc, fd) · ∅`∀ g(x2, u2, v2) ' g(x2, u2, v2) · {g(x2, u2, v2) ' g(y2, fc, fd)} (er)

a ' b, c ' d `∀ g(x1, u1, v1) ' g(y1, fc, fd) · ∅(rrbs)

a ' b, c ' d `∀ g(x1, u1, v1) ' g(y1, fd, fd)·{c � d, g(y1, fc, fd) � g(x1, u1, v1), c ' c}

(er)`∀ g(x1, u1, v1) ' g(x1, u1, v1)

·{c � d, g(y1, fc, fd) � g(x1, u1, v1), c ' c, g(x1, u1, v1) ' g(y1, fd, fd)}a ' b, c ' d `∀ g(u1, x1, y1) ' g(v1, fa, fb) · ∅ (rrbs)a ' b, c ' d `∀ g(u1, x1, y1) ' g(v1, fb, fb)·{a � b, g(v1, fa, fb) � g(u1, x1, y1), a ' a}

(er)`∀ g(u1, x1, y1) ' g(u1, x1, y1)

·{a � b, g(v1, fa, fb) � g(u1, x1, y1), a ' a, g(u1, x1, y1) ' g(v1, fb, fb)}

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c ' d, c ' d `∀ g(u3, x3, y3) ' g(v3, fa, fb) · ∅`∀ g(u3, x3, y3) ' g(u3, x3, y3) · {g(u3, x3, y3) ' g(v3, fa, fb)}

(er)

The union of the answer constraints of these derivations is

{g(x2, u2, v2) ' g(y2, fc, fd),

c � d, g(y1, fc, fd) � g(x1, u1, v1), c ' c, g(x1, u1, v1) ' g(y1, fd, fd),

a � b, a ' a, g(v1, fa, fb) � g(u1, x1, y1), g(u1, x1, y1) ' g(v1, fb, fb),

g(u3, x3, y3) ' g(v3, fa, fb)}

This constraint is satisfiable. To check this, we can consider the following sub-stitution:

{fb/x1, fb/y1, fd/u1, fd/v1, b/x2, b/y2, fc/u2, fd/v2,

d/u3, d/v3, fa/x3, fb/y3}.

5. Tableau Basic Superposition

As a simple consequence of our results, we prove a completeness result for aparamodulation rule working on tableaux. A paramodulation rule working direct-ly on tableaux was proposed in (Loveland, 1978) in the context of model elim-ination and later in (Fitting, 1990). However, their formulations have all thedisadvantages of the early paramodulation rule of (Robinson and Wos, 1969):

(1) Functional reflexivity rule is used.(2) Paramodulation into variables is allowed.(3) Increasing applications of paramodulation are allowed (for example, x can

be rewritten to f(x)).

As a consequence, for a given tableau expansion there may be an infinite sequenceof paramodulations, in particular due to the use of functional reflexivity orincreasing applications of paramodulation. Since the publication of the book(Loveland, 1978), no improvements of the paramodulation-based tableau calculihave been described except for (Plaisted, 1995), who has shown how to trans-form derivations with resolution and paramodulation to tableaux by introducinga tableau factoring rule.

Here we show that paramodulation is complete under the following restric-tions:

(1) No functional reflexivity is needed.(2) Paramodulation into variables is not allowed.(3) Orderings are used so that there are no increasing applications of paramod-

ulation.(4) Basic restriction on paramodulation prohibits paramodulation into terms

introduced by unification.

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Γ1, ϕ ∧ ψ | . . . | Γn · CΓ1, ϕ ∧ ψ,ϕ, ψ | . . . | Γn · C (α)

Γ1, ϕ ∨ ψ | . . . | Γn · CΓ1, ϕ | Γ1, ψ | . . . | Γn · C (β)

Γ1,∀xϕ(x) | . . . | Γn · CΓ1,∀xϕ,ϕ(y) | . . . | Γn · C (γ)

Γ1, s 6' t | Γ2 | . . . | Γn · CΓ2 | . . . | Γn · C ∪ {s ' t} (er)

Γ1, l ' r, s[p] ' t | Γ2 | . . . | Γn · CΓ1, l ' r, s[r] ' t | Γ2 | . . . | Γn · C ∪ {l � r, s[p] � t, l ' p} (lrbs)

Γ1, l ' r, s[p] 6' t | Γ2 | . . . | Γn · CΓ1, l ' r, s[r] 6' t | Γ2 | . . . | Γn · C ∪ {l � r, s[p] � t, l ' p} (rrbs)

In the rules (γ) the variable y does not occur in the premise. The conditionson the rules (lrbs) and (rrbs) are the same as for the corresponding rulesof BSE .

Figure 2. Calculus T BSE .

All these refinements are a consequence of our main result (Theorem 4.6).To formalize the basic strategy, we keep the substitution condition as a set of

constraints, as before. Thus, we work with constraint tableaux.

DEFINITION 5.1 (Constraint tableau). A constraint tableau is a pair consistingof a tableau T and a constraint C, denoted T · C.

Now we adapt the tableau rules of (Fitting, 1990) to the case of constrainttableaux. For simplicity, we consider only signatures whose sole predicate symbolis '. When we prove a formula ϕ, we construct the Skolem negation normalform ψ of ¬ϕ and, starting with the constraint tableau ψ · ∅, try to derive theempty tableau # with some satisfiable constraint.

DEFINITION 5.2 (Calculus T BSE). The free-variable tableau calculus T BSEis shown in Figure 2.

DEFINITION 5.3 (Constraint tableau expansion rules). The rules (α), (β) and(γ) of T BSE are called constraint tableau expansion rules.

The calculus T BSE has the required completeness property:

THEOREM 5.4 (Soundness and completeness). Let ϕ be a formula in the Skolemnegation normal form. It is unsatisfiable if and only if there is a derivation fromthe constraint tableau ϕ · ∅ of a constraint tableau # · C.

Proof. Straightforward from Theorem 4.6 by noting that the rules of BSEcan be simulated by the corresponding tableau rules.

This logical system has one more pleasant property:

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��

PPPPPPP

��

QQQQ

a ' b c ' d

g(x1, u1, v1) 6' g(y1, fc, fd) g(u1, x1, y1) 6' g(v1, fa, fb)��QQQ

a ' b c ' d

g(x2, u2, v2) 6' g(y2, fc, fd)

g(u2, x2, y2) 6' g(v2, fa, fb)g(x1, u1, v1) 6' g(y1, fd, fd)

��QQQ

c ' da ' b

g(u3, x3, y3) 6' g(v3, fa, fb)

g(x3, u3, v3) 6' g(y3, fc, fd)g(u1, x1, y1) 6' g(v1, fb, fb)

Figure 3. Tableau after rigid basic superposition applications.

THEOREM 5.5 (Termination). For any constraint tableau T · C there are onlya finite number of derivations from T · C not using constraint tableau expansionrules.

Proof. Similar to that of Theorem 3.9.

This means, that for a given amplification, we cannot have infinite search. Infinitesearch without any expansion steps is possible in Fitting’s system.

To illustrate the connection between the tableau rigid basic superposition ruleand rules of BSE , we reconsider the example of Section 4. On the branch con-taining the literal g(x1, u1, v1) 6' g(y1, fc, fd) and the equation c ' d, wecan apply rigid basic superposition that adds g(x1, u1, v1) 6' g(y1, fd, fd) to thebranch. Similarly, we can apply rigid basic superposition to the branch containingg(u1, x1, y1) 6' g(v1, fa, fb) and a ' b, obtaining g(u1, x1, y1) 6' g(v1, fb, fb).This results in the tableau given in Figure 3 (the picture does not include theconstraint; it is discussed below).

After four applications, of the (er) rules all branches of this tableau becomeclosed. The resulting constraint of this derivation is the same as the union of theanswer constraints shown at the end of Section 4.

6. Related Work

The problem of extending tableaux with equality rules is crucial for enrich-ing the deductive capabilities of the tableau method. Despite the fact that thisproblem has been attacked by a growing number of researchers during the pastfew years, known solutions are not yet convincing. At the same time, tableaumethods of automated deduction play an important role in various areas of arti-ficial intelligence and computer science – see, for example, the special issues of

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the Journal of Automated Reasoning vol. 13, no. 2,3, 1994. These issues con-tain a survey (Schumann, 1994) of implementations of tableau-based provers.Among 24 systems mentioned in the survey, only two are able to handle equal-ity.

The systems PROTEIN (Baumgartner and Furbach, 1994) and KoMeT (Bibelet al., 1995) implement the modification method of (Brand, 1975). This methodtransforms a set of clauses with equality into a set of clauses without equality.This transformation usually yields a considerably larger set of clauses. In partic-ular, the symmetry and the transitivity axioms must be explicitly applied to allpositive occurrences of the equality predicate. Recently, we proposed (Degtyarevand Voronkov, 1996c) a new translation method based on the so-called basicfolding demonstrated for Horn clauses.

According to (Schumann, 1994), the system 3TAP uses the method of (Beck-

ert and Hahnle, 1992). The paper (Beckert and Hahnle, 1992) claims the com-pleteness of the method, but this claim is not true. The method expands thetableau by using the standard tableau rules, including γ-rules. For finding a clos-ing substitution, an analogue of linear paramodulation without function reflex-ivity has been proposed. As is well known, linear paramodulation is incompletewithout function reflexivity. The same is true for the method of (Beckert andHahnle, 1992), as the following example shows. Suppose that we prove the for-mula ∃x(a ' b∧ g(fa, fb) ' h(fa, fb) ⊃ g(x, x) ' h(x, x)). To prove it usingparamodulation, we need to paramodulate a ' b into g(fa, fb) ' h(fa, fb).The method of (Beckert and Hahnle, 1992) allows for paramodulation only intocopies of g(x, x) ' h(x, x) obtained by applications of the γ-rule. Thus, this(provable) formula cannot be proved using the method of (Beckert and Hahnle,1992).

Consider now approaches based on simultaneous rigid E-unifiability by (Gal-lier et al., 1987; Gallier et al., 1992) and related methods. We do not con-sider numerous works dedicated to (nonsimultaneous) rigid E-unifiability. Thisproblem is NP-complete and there exist a number of complete algorithms forits solution (Gallier et al., 1988; Gallier et al., 1990; Goubault, 1993; Bech-er and Petermann, 1994; De Kogel, 1995; Plaisted, 1995). Since simultaneousrigid E-unification is undecidable (Degtyarev and Voronkov, 1996d; Degtyarevand Voronkov, 1996d), these completeness results are useless from the view-point of general-purpose theorem proving as proposed in (Gallier et al., 1987;Gallier et al., 1992). Our system BSE can easily be extended to a calculus com-plete for rigid E-unifiability, but such completeness is not our aim. We tried torestrict the number of possible BSE-derivations preserving completeness of thegeneral-purpose method of Section 4.

It is not known whether the procedure described in (Gallier et al., 1992) iscomplete for theorem proving.? Even if it is complete, our procedure based? For example, the completeness of Gallier et al.’s procedure does not follow from our method

because, as noted above, our calculus BSE can give more solutions to some rigid equations.

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on BSE has some advantages over Gallier et al.’s procedure. For example,for every tableau branch with p equations and q disequations, we considerq rigid equations, while Gallier et al.’s procedure checks q · 2p rigid equa-tions.

In (Gallier et al., 1988; Gallier et al., 1990), the authors introduced the notionof a complete set of solutions for rigid E-unification, proved finiteness of suchsets, and gave an algorithm computing finite complete set of solutions. Accord-ing to this result, (Goubault, 1994) proposed to solve simultaneous rigid E-unifiability by using finite complete sets of solutions to the components of thesimultaneous problem. The paper (Goubault, 1994) contained faulty results. Theundecidability of simultaneous rigid E-unification shows that finite complete setsof solutions do not give a solution to the simultaneous problem. The reason is thatsubstitutions belonging to complete sets of solutions for different rigid equationsare minimal modulo different congruences.

In (Petermann, 1994), the author introduces a ‘complete connection calcu-lus with rigid E-unification’. Here the completeness is achieved by changingthe notion of a complete set of unifiers so that solutions to all subproblems arecompared modulo the same congruence (generated by the empty equality the-ory). In this case, a nonsimultaneous problem can have an infinite number ofsolutions and no finite complete set of solutions. For example, for the rigid E-unification problem f(a) ' a `∀ x ' a the complete set of solutions in thesense of (Gallier et al., 1992) consists of one substitution {a/x} (and there isonly one answer constraint {x ' a} obtained by our method), but the completeset of solutions in the sense of (Petermann, 1994) is infinite and consists ofsubstitutions {fn(a)/x}, for all n ∈ {0, 1, . . .}. This implies that proof-searchby the method of (Petermann, 1994) can be nonterminating even for a limitednumber of applications of the γ-rule (i.e., for a particular tableau), unlike algo-rithms based on the finite complete sets of unifiers in the sense of (Gallier et al.,1992) or based on minus-normalization (Kanger, 1983). The implementation ofthe method of (Petermann, 1994) uses a completion-based procedure by (Beck-ert, 1994) of generation of complete sets of rigid E-unifiers. This procedure isdeveloped with the aim of solving a more general problem – so-called mixedE-unification – and has been implemented as part of the tableau-based theoremprover 3T

AP . Complete sets of unifiers both in the sense of (Gallier et al., 1992)and in the sense of (Petermann, 1994) can be computed by this procedure inthe case when all variables are treated as rigid. However, the termination is notguaranteed even for complete sets of rigid E-unifiers in the sense of (Gallieret al., 1992).

In (Plaisted, 1995), the author gives ‘techniques for incorporating equali-ty into theorem proving; these techniques have a rigid flavor’. His method ofpath paramodulation guarantees termination for a given amplification and, inthe case of success, ‘solves the simultaneous rigid E-unification problem, ina sense’. However, this does not solve the problem attacked by a number of

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researchers: extend the method of matings to languages with equality by rigidE-unification. First, unlike (Gallier et al., 1992), the search for solutions fora given amplification is not incremental (the method does not allow ‘branch-wise’ computation of solutions to rigid E-unification for separate branches).Second, within a given amplification Plaisted uses factoring rules that involvestwo branches (paths). As a consequence, even when the original formula containsno equality, his method results in the standard tableau calculus plus the factoringrule.

In fact, path paramodulation of (Plaisted, 1995) simulates resolution-para-modulation inference in a connection-like calculus. Although it is not noted in(Plaisted, 1995), this technique has been demonstrated for resolution in manypapers, for example, in (Bibel, 1981; Eder, 1988; Eder, 1991; Mints, 1990;Baumgartner and Furbach, 1993; Avron, 1993). The generalization of this simu-lation to paramodulation is straightforward.

However, this simulation technique is insufficient for proving the results ofour paper since, in particular, it gives no insight on how to avoid factoring intableaux with equality. The use of factoring prevents not only the independentsearch for solutions for tableau branches, but even the incremental solving ofrigid equations on tableau branches as proposed by Gallier et al.

Our equality elimination method (Degtyarev and Voronkov, 1996b; Degtyarevand Voronkov, 1995b; Degtyarev and Voronkov, 1995a) is based on extend-ing a tableau prover by a bottom-up equation solver using basic superposition.Solutions to equations are generated by this solver and used to close branch-es of a tableau. Thus, the method combines (nonlocal) tableau proof searchwith the (local) equation solving. Only completely solved equations are used inthe tableau part of the proof, thus reducing nondeterminism created by applica-tions of the MGU replacement rule of (Fitting, 1990). The equation solution iseven more restricted by the use of orderings, basic simplification and subsump-tion.

A similar idea combination of proof-search in tableaux with a bottom-upequality saturation of the original formula, is used in (Moser et al., 1995) forconstructing a goal-directed version of model elimination and paramodulation.

One of advantages of the tableau method is its applicability to nonclassicallogics. However, handling equality in nonclassical logics seems to be a muchmore difficult problem than that in classical logic. For example, it is shown in(Voronkov, 1996) that procedures for intuitionistic logic with equality must handlesimultaneous rigid E-unification. This implies that our method based on BSEdoes not give a complete procedure for intuitionistic logic with equality. Otherresults on relations between simultaneous rigid E-unification and intuitionisticlogic are considered in (Degtyarev and Voronkov, 1996a; Degtyarev et al., 1996a;Degtyarev et al., 1996b; Veanes, 1997).

An extensive discussion of equality reasoning in sequent-based systems maybe found in (Degtyarev and Voronkov, 1997).

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Appendix

A. Proof of the Completeness Theorem

This appendix proves the completeness part of Theorem 4.6. It is based onthe completeness theorem for the equality elimination method (Degtyarev andVoronkov, 1994; Degtyarev and Voronkov, 1995b; Degtyarev and Voronkov,1995a; Degtyarev and Voronkov, 1996b). We shall introduce several notions.

DEFINITION A.1 (Clause). A clause is a finite multiset of literals {L1, . . . , Ln},denoted L1, . . . , Ln. If L is a literal and C a clause, then L,C denotes the clause{L} ∪C . The empty clause is denoted by 2.

DEFINITION A.2 (Constraint clause). A constraint clause is a pair consistingof a clause C and a constraint C. Such a constraint clause will be denoted C · C.

A.1. THE EQUALITY ELIMINATION METHOD

We assume that all different occurrences of quantifiers in any formula bind dif-ferent variables. For example, a formula cannot have the form ∀xA ∧ ∀xB. Forthe rest of this section we assume that ξ denotes a closed formula in Skolemnegation normal form to be checked for unsatisfiability. When we speak aboutsubformulas of ξ, we mean concrete occurrences of these subformulas in ξ. Forexample, in the formula ξ of the form A ∧ (A ∨B) the second occurrence of Ais considered a subformula of (A ∨B), but the first occurrence of A is not.

DEFINITION A.3 (Subformula and fresh-variable subformulas). The set of sub-formulas and fresh-variable subformulas of a formula in Skolem negation normalform is defined inductively as follows.

(1) Any formula is a subformula and a fresh-variable subformula of itself.(2) Both formulas ϕ1 and ϕ2 are subformulas and fresh-variable subformulas of

ϕ1 ∧ ϕ2, and similar for ∨ instead of ∧.(3) The formula ϕ(x) is a subformula of ∀xϕ(x); any formula ϕ(y), where

y 6∈ var(∀xϕ(x)), is a fresh-variable subformula of ∀xϕ(x).(4) If ϕ1 is a subformula of ϕ2 and ϕ2 is a subformula of ϕ3, then ϕ1 is a

subformula of ϕ3, and similar for fresh-variable subformulas.

Note that we do not consider an atom A to be a subformula of ¬A. The reasonis that we want to restrict ourselves to positive subformulas only. Also note thata fresh-variable subformula is not necessarily a subformula.

We shall deal only with some subformulas of ξ called disjunctive subformulas.

DEFINITION A.4 (Disjunctive subformula). The occurrence of a subformula ϕof ξ is called disjunctive if it is an occurrence in a subformula ϕ∨ψ or in ψ∨ϕ.

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A disjunctive superformula of ϕ is a superformula? ψ of ϕ that is disjunctive.The least disjunctive superformula of ϕ is the disjunctive superformula ψ of ϕsuch that any other disjunctive superformula of ϕ is a superformula of ψ.

Let us note that disjunctive superformulas do not necessarily exist. For exam-ple, ξ has no disjunctive superformula. Any formula having a disjunctive super-formula has the unique least disjunctive superformula.

We can enumerate all disjunctive subformulas ξ1, . . . , ξn of ξ, for example,in the order of their occurrences in ξ. Thus we can unambiguously use ‘the kthdisjunctive (sub)formula’ ξk of ξ.

Let A1, . . . ,An be predicate symbols not occurring in ξ.

DEFINITION A.5 (ξ-name of a subformula). An atomic formulaAk(x1, . . . , xm)is the ξ-name of a subformula ϕ of ξ if

(1) The least disjunctive superformula of ϕ is ξk;(2) x1, . . . , xm are all free variables of ξk in the order of their first occurrences

in ξk.

If a ξ-name of a formula ϕ exists, then it is unique. Note that different formulasmay have the same ξ-names. Also note that some subformulas of ξ do not haveξ-names. We can use the set of ξ-names of a subformula. The set of ξ-names ofa formula ϕ is either ∅ or a singleton {Ak(x1, . . . , xm)}.

Table A.1 illustrates least disjunctive superformulas and ξ-names.

LEMMA A.6. Let ϕ,ψ be subformulas of ξ whose sets of ξ-names coincide, ϕbe a proper subformula of ψ, and ψ′ = ψη′, where η′ is a substitution withdom(η′) = var(ψ). Let ϕ′ be a fresh-variable subformula of ψ′ that is also avariant of ϕη′.

Let T = Γ1, ψ′ | Γ2 | . . . | Γn such that var(T ) ∩ (var(ϕ′) \ var(ψ′)) = ∅.

Then there is a derivation

Γ1, ψ′ | Γ2 | . . . | Γn

...

Γ′1, ψ′, ϕ′ | Γ2 | . . . | Γn

consisting only of applications of α- and γ-rules of Figure 1 such that Γ1 v Γ′1.Proof. Induction by ψ′. Note that ψ′ cannot be a disjunction because then

the set of ξ-names of ϕ and ψ would be different. Hence, there are two possiblecases: either ψ′ is a conjunction or it begins with a universal quantifier. Let ϕ′ =ϕη′{z1/x1, . . . , zm/xm}, where {x1, . . . , xm} = var(ϕη′) \ var(ψη′). Considerthe two cases mentioned above.

? ϕ is a superformula of ψ if ψ is a subformula of ϕ (not necessarily proper).

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Table A.1. Least disjunctive superformulas and sets of ξ-names of subformulas of the formula(∀x(F (x)∨ (B(x) ∧ ∀yC(x, y))) ∨ ∀zD(z)) ∧E.

Subformula Least Disjunctive Superformula

(∀x(F (x) ∨ (B(x) ∧ ∀yC(x, y))) no ∅∨∀zD(z)) ∧ E

∀x(F (x) ∨ (B(x) ∧ ∀yC(x, y))) no ∅∨∀zD(z)

∀x(F (x) ∨ (B(x) ∧ ∀yC(x, y))) ∀x(F (x) ∨ (B(x) ∧ ∀yC(x, y))) {A1}F (x) ∨ (B(x) ∧ ∀yC(x, y)) ∀x(F (x) ∨ (B(x) ∧ ∀yC(x, y))) {A1}F (x) F (x) {A2(x)}B(x) ∧ ∀yC(x, y) B(x) ∧ ∀yC(x, y) {A3(x)}B(x) B(x) ∧ ∀yC(x, y) {A3(x)}∀yC(x, y) B(x) ∧ ∀yC(x, y) {A3(x)}C(x, y) B(x) ∧ ∀yC(x, y) {A3(x)}∀zD(z) ∀zD(z) {A4}D(z) ∀zD(z) {A4}E no ∅

(1) ψ′ has the form ∀xχ(x). If for some i ∈ {1, . . . ,m} we have x = xi, thenwe apply the rule

. . . ∀xiχ(xi) . . .

. . . ∀xiχ(xi), χ(zi) . . .(γ).

Otherwise, choose any fresh variable z and apply the rule

. . . ∀xχ(x) . . .

. . . ∀xχ(x), χ(z) . . .(γ).

If χ(zi) (respectively, χ(z)) coincides with ϕ′, then the required derivationis found. Otherwise, apply the induction hypothesis to χ(zi) (respectively,χ(z)).

(2) ψ′ has the form χ1 ∧ χ2. Then we apply the rule

. . . χ1 ∧ χ2 . . .

. . . χ1 ∧ χ2, χ1, χ2 . . .(α).

If ϕ′ coincides with χ1 or χ2 then the required derivation is found. Otherwise,ϕ′ is a fresh-variable proper subformula of some χi, where i ∈ {1, 2}. Thenapply the induction hypothesis to χi.

LEMMA A.7. Let T = Γ1 | . . . | Γn be a tableau obtained from ξ by tableauexpansions rules and ϕ a proper subformula of ξ whose set of γ-names is empty.

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Let ϕ′ be a fresh-variable subformula of ξ that is also a variant of ϕ such thatvar(ϕ′) ∩ var(T ) = ∅. Then there is a derivation

Γ1 | Γ2 | . . . | Γn...

Γ′1, ϕ′ | Γ2 | . . . | Γn

consisting only of applications of α- and γ-rules of Figure 1 such that Γ1 v Γ′1.Proof. If ξ ∈ Γ1, then the claim is immediate by Lemma A.6, where ψ = ξ

and η′ is the empty substitution. Suppose (by contradiction) ξ 6∈ Γ1. Evidently,ξ is a disjunction. But then ξ has no proper subformulas whose sets of ξ-namesare empty. Contradiction.

Theorem 4.6 that we prove in this section connects two calculi: (i) calculusBSE of rigid equations and (ii) tableau expansion rules for tableaux. The maintheorem on the completeness of the equality elimination method ((Degtyarev andVoronkov, 1994; Degtyarev and Voronkov, 1996b), Theorem A.15 below) alsoconnects two calculi, both introduced below: (i) the calculus CBSE of constraintclauses and (ii) ξ-expansion rules for named ξ-tableaux. Both calculi depend onthe goal formula ξ.

DEFINITION A.8 (Calculus CBSE). The calculus CBSE of constraint clausesconsists of the following inference rules. As usual, we assume that premises ofrules have disjoint variables that can be achieved by renaming variables.

Left rigid basic superposition:

l ' r,C1 · C1 s[p] ' t, C2 · C2

s[r] ' t, C1, C2 · C1 ∪ C2 ∪ {l � r, s[p] � t, l ' p}(lrbs).

Right rigid basic superposition:

l ' r,C1 · C1 s[p] 6' t, C2 · C2

s[r] 6' t, C1, C2 · C1 ∪ C2 ∪ {l � r, s[p] � t, l ' p}(rrbs).

Equality resolution:

s 6' t, C · CC · C ∪ {s ' t} (er).

Application of all the rules is restricted to the following conditions:

(1) The constraint at the conclusion of the rule is satisfiable;(2) In the basic superposition rules, the term p is not a variable.(3) In the left basic superposition rule, s[r] 6= t.

DEFINITION A.9 (Named ξ-tableau). A named ξ-tableau is a tableau all ofwhose formulas are atomic formulas of the form Ai(t1, . . . , tm).

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We shall identify named ξ-tableaux obtained by renaming of variables.

DEFINITION A.10 (Initial named ξ-tableau). The initial named ξ-tableau is thetableau 2 that has one branch consisting of the empty set of formulas.

DEFINITION A.11 (ξ-expansion rules). Let {Ai(xi)}, {Aj(xj)} and D be thesets of ξ-names of ϕ,ψ and ϕ ∨ ψ, where ϕ ∨ ψ is a subformula of ξ. Then thefollowing are ξ-expansion rules.

(1) If D = ∅, then

C1 | C2 | . . . | CmC1,Ai(xi) | C1,Aj(xj) | C2 | . . . | Cm

(2) If D = {Ak(xk)}, then

C1,Ak(y) | C2 | . . . | Cm(C1,Ai(xi),Ak(y) | C1,Aj(xj),Ak(y) | C2 | . . . | Cm){xk/y}

We assume that the variables of the premises are disjoint from the variables ofxi, xj , xk.

DEFINITION A.12 (Initial constraint ξ-clause). Initial constraint ξ-clauses aredefined as follows:

(1) Whenever a literal s 6' t occurs in ξ and D is the set of ξ-names of thisoccurrence of s 6' t, the constraint clause s 6' t,D ·{} is an initial constraintξ-clause.

(2) Whenever a literal s ' t occurs in ξ and D is the set of ξ-names of thisoccurrence of s ' t, the constraint clause s ' t,D ·{} is an initial constraintξ-clause.

DEFINITION A.13 (Solution constraint ξ-clause). A solution constraint ξ-clauseis any constraint clause C · C that is derivable from initial constraint ξ-clausesusing rules (lrbs), (rrbs) and (er), and such that C does not contain ' (i.e., Ccontains only atoms built from Ai).

DEFINITION A.14 (Answer constraint). Let C ′ · C′ be a solution constraint ξ-clause and C be a branch of a named ξ-tableau, so that var(C) ∩ (var(C ′) ∪var(C′)) = ∅. If η is a substitution such that (i) dom(η) = var(C ′), (ii) C ′η v C,and (iii) C′η is satisfiable, then C′η is an answer constraint for C.

The following theorem is a reformulation of the soundness and completenesstheorems of (Degtyarev and Voronkov, 1994) for named ξ-tableaux and constraintclauses.

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THEOREM A.15. The formula ξ is unsatisfiable if and only if there exists anamed ξ-tableau C1 | . . . | Cn obtained from the initial named ξ-tableau byξ-expansion rules with the following property. There exist answer constraintsC1, . . . , Cn for C1, . . . , Cn, respectively, such that C1 ∪ . . . ∪ Cn is satisfiable.?

A.2. THE PROOF OF THE MAIN THEOREM

DEFINITION A.16 (Image). Let the atomic formula Ak(x1, . . . , xm) be the ξ-name of a disjunctive subformula ϕ(x1, . . . , xm) of ξ. For any terms t1, . . . , tm,the image of the formula Ak(t1, . . . , tm) is the formula ϕ(t1, . . . , tm). The imageof a named ξ-tableau C1 | . . . | Cn is the tableau Γ1 | . . . | Γn obtained fromC1 | . . . | Cn by replacing every atomic formula Ak(t1, . . . , tm) by its image.

Note that the image of a named ξ-tableau is uniquely defined.

LEMMA A.17. Let C1 | . . . | Cn be any named ξ-tableau obtained from 2 byξ-expansion rules and Γ1 | . . . | Γn be its image. Then there exists a tableauΓ′1 | . . . | Γ′n obtained from ξ by tableau expansion rules such that Γi v Γ′i, forall i ∈ {1, . . . , n}.

Proof. By induction on the number of applications of expansion rules. Thebasic case is obvious: we can take ξ as the required tableau. Consider the induc-tion step. The following two cases are possible.

(1) The expansion rule has the form

C1 | C2 | . . . | CnC1,Ai(xi) | C1,Aj(xj) | C2 | . . . | Cn

.

Denote the image of C1 | . . . | Cn by Γ1 | . . . | Γn. By the inductionhypothesis, there is a tableau Γ′1 | . . . | Γ′n obtained from ξ by tableau

? This theorem is a reformulation of the theorems in (Degatyarev and Voronkov, 1994; 1996b).We shall briefly explain the correspondence between the formulations.

The formalization of basic superposition in (Degtyarev and Voronkov, 1994) used closures (see(Bachmair et al., 1995)), i.e., pairs C · σ, where C is a clause and σ is a substitution. In thispaper we use constraint clauses (see (Nieuwenhuis and Rubio, 1995)), i.e., pairs C · C, whereC is a clause and C is a constraint. Closures and constraint clauses are two ways to formalizethe basic restriction on superposition. In (Degtyarev and Voronkov, 1994) we could have usedconstraint clauses instead of closures, but in this paper we have to use constraint clauses to ensuretermination (Theorem 3.9).

Another distinction is that in (Degtyarev and Voronkov, 1994) we used sets of literals insteadof multisets. Consequently, instead of the multiset inclusion used in Definition A.14 of answerconstraint, (Degtyarev and Voronkov, 1994) uses a set inclusion formalized by subset unification.Again, we have to use multisets to ensure termination. The completeness proof of (Degtyarev andVoronkov, 1994) works for multisets without any special changes.

Therefore, we decided to refer to the modified version of the completeness theorem of (Degtyarevand Voronkov, 1994) instead of reproducing the whole proof here. We are preparing a journalversion of (Degtyarev and Voronkov, 1994), where the completeness theorem will be formulatedprecisely as here.

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expansion rules such that Γi v Γ′i, for all i ∈ {1, . . . , n}. Let ξi(xi) be theimage of Ai(xi) and ξj(xj) be the image of Aj(xj). Then ξi(xi)∨ξj(xj) hasthe empty set of ξ-names, and it is a fresh-variable subformula of ξ. By thedefinition of ξ-expansion rules, variables xi, xj do not occur in C1 | . . . | Cn.Evidently, we can assume that xi, xj do not occur in Γ′1 | . . . | Γ′n (we canrename variables in Γ′k to make them distinct from variables of C1, . . . , Cn).Applying Lemma A.7 to the tableau T = Γ′1 | . . . | Γ′n and formula ϕ =ξi(xi)∨ξj(xj) we see that there is a tableau Γ′′1 , ξi(xi)∨ξj(xj) | Γ′2 | . . . | Γ′nobtained from Γ′1 | . . . | Γ′n by tableau expansion rules such that Γ′1 v Γ′′1 .Applying β-rule

Γ′′1 , ξi(xi) ∨ ξj(xj) | Γ′2 | . . . | Γ′nΓ′′1 , ξi(xi) | Γ′′1 , ξj(xj) | Γ′2 | . . . | Γ′n

,

we obtain the required tableau.(2) The expansion rule has the form

C1,Ak(y) | C2 | . . . | Cn(C1,Ai(xi),Ak(y) | C1,Aj(xj),Ak(y) | C2 | . . . | Cn){xk/y}

.

Denote the image of C1 | . . . | Cn by Γ1 | . . . | Γn and the image of Ak(y)by ξk(y). By the induction hypothesis, there is a tableau Γ′1, ξk(y) | Γ′2 |. . . | Γ′n obtained from ξ by tableau expansion rules such that Γi v Γ′i, forall i ∈ {1, . . . , n}. Since ξ contains no free variables, any variant of thistableau can also be obtained from ξ by tableau expansion rules, in particularthe tableau (Γ′1, ξk(y) | Γ′2 | . . . | Γ′n){xk/y}. The formula ξi(xi)∨ξj(xj) is asubformula of ξk(xk), and their sets of ξ-names coincide. Then ξi(xi)∨ξj(xj)either coincides with, or is a proper subformula of ξk(xk). Consider the twocorresponding cases:

(a) ξi(xi) ∨ ξj(xj) coincides with ξk(xk). Applying β-rule

(Γ′1, ξi(xi) ∨ ξj(xj) | Γ′2 | . . . | Γ′n){xk/y}(Γ′1, ξi(xi) | Γ′1, ξj(xj) | Γ′2 | . . . | Γ′n){xk/y}

,

we obtain the required tableau.(b) ξi(xi)∨ξj(xj) is a proper subformula of ξk(xk). It is also a fresh-variable

subformula. To apply Lemma A.6 (where η′ is the empty substitution,ψ = ψ′ = ξk(xk) and ϕ = ϕ′ = ξi(xi) ∨ ξj(xj)), we have to guaranteethe condition

var((Γ′1, ξk(y) | Γ′2 | . . . | Γ′n){xk/y}) ∩ ((xi ∪ xj) \ xk) = ∅. (3)

By the conditions of this lemma, xi, xj do not occur in Γ1, ξk(y) | Γ2 |. . . | Γn. Since in the derivation of Γ′1, ξk(y) | Γ′2 | . . . | Γ′n we couldintroduce fresh variables different from xi, xj , we have

(xi ∪ xj) ∩ var((Γ′1, ξk(y) | Γ′2 | . . . | Γ′n){xk/y}) ⊆ xk.

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Thus, Condition (3) can be guaranteed. By Lemma A.6, there is arequired derivation of the form

ξ

...

(Γ′1, ξk(xk) | Γ′2 | . . . | Γ′n){xk/y}...

(Γ′′1 , ξk(xk), ξi(xi) ∨ ξj(xj) | Γ′2 | . . . | Γ′n){xk/y}(Γ′′1 , ξk(xk), ξi(xi) | Γ′′1 , ξk(xk), ξj(xj) | Γ′2 | . . . | Γ′n){xk/y}

.

LEMMA A.18. Let C be an answer constraint for a rigid equation R and η bea substitution such that Cη is satisfiable. Then there exists C′ v C such that C′ηis an answer constraint for Rη.

Proof. Consider the derivation

R·∅ = E0 `∀ s0 ' t0 ·C0 ; E1 `∀ s1 ' t1 ·C1 ; · · · ; En `∀ sn ' tn ·Cn(4)

such that Cn = C. Consider the figure

(E0 `∀ s0 ' t0 · C0)η(E1 `∀ s1 ' t1 · C1)η

...

(En `∀ sn ' tn · Cn)η

(5)

The figure

(Ei `∀ si ' ti · Ci)η(Ei+1 `∀ si+1 ' ti+1 · Ci+1)η

is not a correct inference rule of BSE in one of the following cases.

(1) Ei+1 \ Ei contains an equation s[r] ' t such that s[r]η = tη. In this caseCondition (4) on the rules of BSE is violated. Note that s[r] ' t cannot beused in the inference rules of (4). Indeed, otherwise C would contain eithers[r] � t or t � s[r], which contradicts that the condition Cη is satisfiable.Thus, we can assume without loss of generality that (4) does not containapplications of left basic superposition giving such equations s[r] ' t. Then(5) is a correct BSE-derivation satisfying the conditions.

(2) si+1η = ti+1η, where i + 1 6= n. In this case Condition (2) on the rules ofBSE is violated. This case is considered similarly.

Let τ be a tree derivation of a solution constraint ξ-clause C · C. Without lossof generality we can assume that the leaves of τ have disjoint variables. Denoteby L(τ) the multiset of the leaves of τ .

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We shall associate with τ a rigid equation Rτ such that C is an answerconstraint for Rτ .

DEFINITION A.19. A rigid equation E `∀ s ' t is associated with τ if E isthe multiset of all equations occurring in L(τ), and s 6' t is the only disequationoccurring in L(τ).

We recall that any element of L(τ) has either the form p ' q,D · {}, or the formp 6' q,D · {}, where D does not contain equations or disequations.

LEMMA A.20. Let E `∀ s ' t be a rigid equation associated with a derivationτ of a solution constraint clause C · C. Then C is an answer constraint forE `∀ s ' t.

Proof. Straightforward.

Combining Lemmas A.18 and A.20, we obtain the following.

LEMMA A.21. Let E `∀ s ' t is a rigid equation associated with a derivationτ of a solution constraint clause C ′ · C′ and η be a substitution such that C′ · ηis satisfiable. Then there exists a constraint C′′ v C′ such that C′′η is an answerconstraint for (E `∀ s ' t)η.

DEFINITION A.22 (αγ-expansion). Let a tableau T ′ be obtained from a tableauT by a sequence of α- and γ-rules. Then T ′ is called an αγ-expansion of T .

LEMMA A.23. Let Γ′1 | . . . | Γ′n be an αγ-expansion of Γ1 | . . . | Γn. ThenΓi v Γ′i, for all i ∈ {1, . . . , n}.

Proof. Obvious.

Let us introduce a technical definition.

DEFINITION A.24. Let E `∀ s ' t be a rigid equation. Its clause form is theclause E, s 6' t.

Finally, we come to the proof of the main theorem.

THEOREM 4.6. Let ξ be a sentence in Skolem negation normal form. Thenξ is unsatisfiable if and only if there is a tableau Γ1 | . . . | Γn obtained fromξ by tableau expansion rules and answer constraints C1, . . . , Cn for Γ1, . . . ,Γn,respectively, such that C1 ∪ . . . ∪ Cn is satisfiable.

Proof. The soundness part is proven in Section 4. We prove only the com-pleteness part. Let ξ be unsatisfiable. By Theorem A.15, there exist a namedξ-tableau C1 | . . . | Cn obtained from the initial named ξ-tableau by ξ-expansionrules and answer constraints C0

1 , . . . , C0n for C1, . . . , Cn, respectively, such that

C01 ∪ . . . ∪ C0

n is satisfiable. By Definition A.14 of answer constraints, there existsolution constraint ξ-clauses C ′1 ·C′1, . . . , C ′n ·C′n and substitutions η1, . . . , ηn suchthat

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(1) var(Ci) ∩ var(C ′i · C′i) = ∅;(2) dom(ηi) = var(C ′i);(3) C ′iηi v Ci;(4) C′iηi = C0

i .

Without loss of generality we can assume that var(C ′i · C′i) ∩ var(C ′j · C′j) = ∅,whenever i 6= j and the variables of C1 | . . . | Cn are disjoint from boundvariables of ξ. Hence, we can introduce the substitution η = η1 ∪ . . . ∪ ηn.

Let Γ′1 | . . . | Γ′n be the image of C1 | . . . | Cn. By Lemma A.17, there isa tableau Γ′′1 | . . . | Γ′′n obtained from ξ by tableau expansion rules such thatΓ′i v Γ′′i , for all i ∈ {1, . . . , n}.

Let τ1, . . . , τn be derivations of C ′1 · C′1, . . . , C ′n · C′n, respectively in the treeform. Let τ be the multiset τ1, . . . , τn of trees and L(τ) be the multiset of leavesin τ . We assume any two members of L(τ) have disjoint sets of variables. SinceΓ′′1 | . . . | Γ′′n is derived from the closed formula ξ, we can also assume that

var(τ) ∩ var(Γ′′1 | . . . | Γ′′n) = ∅. (6)

Let for all i ∈ {1, . . . , n} the rigid equation Ei `∀ si ' ti be associated with τiand Ei, si 6' ti be its clause form.

Now we shall construct an αγ-expansion Γ1 | . . . | Γn of Γ′′1 | . . . | Γ′′n suchthat (Ei, si 6' ti)η v Γi, for all i ∈ {1, . . . , n}.

Let L1, . . . , Ll be the multiset of all equations and disequations in the leavesof τ . We shall construct the required Γ1 | . . . | Γn using a sequence of αγ-expansions

Γ′′1 | . . . | Γ′′n = Γ(0)1 | . . . | Γ(0)

n

Γ(1)1 | . . . | Γ(1)

n

· · ·

Γ(l)1 | . . . | Γ(l)

n = Γ1 | . . . | Γn.During the construction we shall satisfy the following conditions:

(1) For every i, j with 1 6 i 6 l and 1 6 j 6 n, if Li is in the leaf of τj , then

Liη is on the branch Γ(i)j ;

(2) For every i with 1 6 i < l we have

var(Γ(i)1 | . . . | Γ(i)

n ) ∩ var(Li+1, . . . , Ll) = ∅.(Initially, this condition holds by (6).)

It is straightforward to check that the first condition implies that Γ1, . . . ,Γn isthe required tableau.

Suppose that we have already constructed the tableau Γ(k)1 | . . . | Γ

(k)n for

k < l. We show how to construct Γ(k+1)1 | . . . | Γ(k+1)

n . Suppose that Lk+1 is in

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the leaf of the tree τi. Then this leaf has the form Lk+1,D · {} such that D v C ′iand C ′iη v Ci.

There are two possible cases: either D 6= ∅ or D = ∅.(1) Case D 6= ∅. Let y = var(Lk+1), x = var(D) and Lk+1(y),D(x) · {}

be a variant of an initial constraint ξ-clause Lk+1(z),Am(xm) · {}. ThenDη = {Am(xη)} v Ci. Let ξm(xη) be the image of Am(xη). By theconstruction of Γ′′i , we have ξm(xη) ∈ Γ′′i since Am(xη) ∈ Ci.There are two possible cases:

(a) Lk+1(z) = ξm(xm). In this case we define Γ(k+1)1 | . . . | Γ

(k+1)n as

Γ(k)1 | . . . | Γ(k)

n .(b) Lk+1(z) is a proper subformula of ξm(xm) (whose ξ-name is Am(xm)).

Then apply Lemma A.6. To this end we let

ψ = ξm(xm),

ϕ = Lk+1(z),

ψ′ = ξm(xη) = ξm(xm){xη/xm},

η′ = {xη/xm},

ϕ′ = Lk+1(yη) = Lk+1(z)η′,

T = Γ(k)1 | . . . | Γ(k)

n .

Let us check that the condition of Lemma A.6

var(T ) ∩ (var(ϕ′) \ var(ψ′)) = ∅is satisfied. Let v be any variable such that v ∈ var(yη) and v 6∈ var(xη).If v belongs to y, then v 6∈ var(T ), since by the construction of T wehave var(T )∩var(Lk+1) = ∅. Hence, it is enough to prove that v belongsto y.Suppose, by contradiction, that v does not belong to y. Using v ∈var(yη), we obtain v ∈ var(xη), since dom(η) ∩ (x ∪ y) = x. Thiscontradicts v 6∈ var(xη).

Thus, we can apply Lemma A.6 and deduce the tableau Γ(k+1)1 | . . . |

Γ(k+1)n .

To guarantee the condition

var(Γ(k+1)1 | . . . | Γ(k+1)

n ) ∩ var(Lk+2, . . . , Ll) = ∅,we can require that variables introduced by γ-rules when we come fromΓ

(k)1 | . . . | Γ

(k)n to Γ

(k+1)1 | . . . | Γ

(k+1)n do not belong to the set

var(Lk+2, . . . , Ll).

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(2) Case D = ∅. Analogous, but using Lemma A.7 instead of Lemma A.6.

Thus, we have constructed an αγ-expansion Γ1 | . . . | Γn of Γ′′1 | . . . | Γ′′n suchthat (Ei, si 6' ti)η v Γi, for all i ∈ {1, . . . , n}. Then the set of rigid equationson Γi contains a rigid equation E′i, Eiη `∀ (si ' ti)η, where E′i is a multisetof equations. Evidently, the set of answer constraints for E′i, Eiη `∀ (si ' ti)ηcontains all answer constraints for Eiη `∀ (si ' ti)η. By Lemma A.20, C′i is ananswer constraint for Ei `∀ si ' ti. Since C′iη is satisfiable, by Lemma A.21there exists a constraint C′′i v C′i such that C′′i η is an answer constraint for(Ei `∀ si ' ti)η. Hence, C′′i η is an answer constraint for Γi. Since

⋃i∈{1,...,n} C′iη

is satisfiable, then⋃i∈{1,...,n} C′′i η is also satisfiable. Evidently, the constraints

C1 = C′′1 η, . . . , Cn = C′′nη satisfy the claim of the theorem.

Acknowledgments

We thank Gerard Becher and Uwe Petermann for helpful discussions, anonymousreferees for their valuable comments, and the guest editors Jose Julio Alferes andLuis Monız Pereira for their patience.

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