1 Introduction
In this paper we consider the following two questions, which are closely related to P. Pudlák’s Problem 22 listed in [5]:
Given a cutfree proof of a formula from the axiom , where is a new function symbol, find a cutfree proof of the formula from the axiom . What is the complexity of the elimination of ?
Given a cutfree proof of a formula , where is a new function symbol, find a cutfree proof of the formula . What is the complexity of the elimination of ?
The Skolem functions and are called a witnessing function for and a counterexample function to , respectively.
Skolem functions play an important role both in proof theory and in automatic theorem proving. Skolemization with witnessing functions is a process which transforms a firstorder formula to an equisatisfiable universal formula by replacing all its existential quantifiers with Skolem witnessing functions. We are interested here in the reverse process: given a proof of a formula from the skolemization of an axiom, what is the length of the shortest proof of the same formula from the (original) axiom? Skolemization with counterexample functions is used for transforming formulas to validityequivalent existential formulas; the process, sometimes called the Herbrandization, is dual to the previous one.
The complexity of general methods for eliminating Skolem functions from proofs is at least exponential (see [7, 8, 10, 3]), some of them are even superexponential. A partial positive solution is given by Avigad in [2]: theories strong enough to code finite functions can eliminate Skolem functions in polynomial time. But the general problem for predicate calculus either with or without equality is still an open question.
In this paper we consider the problem of deskolemization for firstorder logic without equality. We show that elimination of a single Skolem function from cutfree proofs increases the length of such proofs only linearly. Our result is based on adapting Maehara’s method [10, Lemma 8.11] for a cutfree variant of sequent calculus without eigenvariable condition.
In our method of proof system, called here , we use Henkin constants instead of free variables in quantifier inferences. (We borrowed the idea from [6].) Strong quantifier rules of are of the form: . The Henkin constants and are called a witness for and a counterexample to , respectively. If is true then the constant denotes an element from the domain of discourse which, when assigned to the variable , makes true. If is false then the constant denotes an element from the domain which, when assigned to the variable , makes false.
This intended meaning of Henkin constants is expressed by the Henkin witnessing and counterexample axioms. These are formulas of the form:
The sequent calculus is sound in the following sense: every formula provable in is true in any structure which is a model for the set consisting of all Henkin axioms. If the formula is pure, i.e. if it does not contain any Henkin constant, then this implies that the formula is logically valid (see Thm. 2.4).
In [6], we have already examined the feasibility of using Henkin constants in a firstorder tableaux. Our tableaux are dual to the tableaux of Smullyan [9] as they demonstrate logical consequence. By a direct translation of the arguments used for our tableaux we obtain soundness and completeness of the sequent calculus for granted (see Thm. 3.6).
At the heart of our results, as it will be demonstrated in the proofs of the elimination lemmas 6.1 and 6.3, there is the following useful structural property of inferences and a novel transformation of derivations:

Strong locality property. The validity of each inference step depends only on the form of its principal and minor formulas, it does not depend on its side formulas.

Deep replacement of terms. (Deep occurrences in an expression include also all deep occurrences in the indices of each Henkin constant occurring in the expression. See 5.6 for details.) The replacement lemma 5.8 characterizes proofs invariant under deep replacement of terms. Deep replacement turns an invariant strong quantifier rule into another valid inference of the same kind.
Both the property and the transformation are specific to proofs; they cannot be readily adapted to proof calculi with eigenvariable condition.
2 Firstorder logic
We start with a quick review of basic notions from firstorder logic. The reader is referred to [10, 4] for more details. We wish to stress here that under ‘terms’ and ‘formulas’ we understand ‘closed terms’ and ‘closed formulas’ (sentences). Our use of ‘semiterms’ and ‘semiformulas’ comes from Takeuti [10].
2.1 Firstorder languages.
Logical symbols of firstorder languages include the full set of propositional connectives , , , , , , and quantifiers , ; binary connectives are right associative and listed in the order of decreasing precedence. Each firstorder language is fully given by the set of its nonlogical symbols: constants, function symbols and predicate symbols.
Semiterms are built from variables and constants by application of function symbols. Terms are closed semiterms; they do not contain variables. Semiformulas are built from atomic semiformulas by application of propositional connectives and quantifiers. Formulas are closed semiformulas; all their variables are bound. We will use letters and to stand for semiterms and semiformulas, respectively.
By we denote the substitution of a variable by a term for every free occurrence of the variable in a semiformula . The simultaneous substitution is defined analogously. We write to indicate every free occurrence of (pairwise distinct) variables in , and as an abbreviation for . A similar notation convention holds also for semiterms.
The notions of subsemiterm and subsemiformula have a standard definition. By subformula we mean to be that of a subformula in the sense of Gentzen. For instance, if or is a subformula of , then so is each (closed) instance of .
By expression of a firstorder language (expression for short) we mean either semiterm or semiformula of . We will use as the syntactic identity over expressions of the same kind.
2.2 Henkin constants and witnessing expansion.
Let be a firstorder language. Let further be an infinite sequence of sets of firstorder constants defined inductively as follows
The set is thus obtained from the set by addition of a new constant for each existential formula of the firstorder language provided it is not already in . The set of Henkin (witnessing) constants for is defined as their union:
We refer to the formula as the index of the witnessing constant . We will use the lowercase letters to denote Henkin constants.
We obtain the witnessing expansion of by adding all Henkin witnessing constants to the firstorder language :
The rank of a semiterm or semiformula of is the minimal number such that the expression belongs to the firstorder language . Pure semiterms and pure semiformulas have ranks equal to ; they do not contain any Henkin constants.
2.3 Henkin and quantifier axioms.
Let be a firstorder language and its witnessing expansion. Henkin axioms for are formulas of the form
The first formula is called witnessing axiom and the second counterexample axiom. The Henkin constants and are called a witness for and a counterexample to , respectively. The informal idea behind the constants is obvious.
Quantifier axioms for are formulas of the form ( is arbitrary term)
It is obvious that quantifier axioms are logically valid formulas.
By a Henkin structure we mean an structure which is a model for the set consisting of all Henkin axioms for . As we will see in Section 3, formulas provable in are true in every Henkin structure. As a straightforward consequence of the next theorem we immediately obtain that provable pure formulas are logically valid. This proves the soundness of the sequent calculus .
2.4 Theorem (Pure formula reduction).
A pure formula is logically valid if and only if it is true in every Henkin structure.
Proof.
It follows from [4, Lemma 4.7]. ∎
2.5 Remark.
The following theorem reduces the problem of recognizing logical validity to conceptually much simpler problem of recognizing a certain propositional tautology. From the theorem one gets without difficulty the completeness for ordinary firstorder proof calculi. Smullyan [9] calls it the fundamental theorem of quantification theory. A modern and very readable presentation is by Barwise in [4]
2.6 Theorem (The reduction to propositional logic).
A pure formula is logically valid if and only if there are Henkin and quantifier axioms such that the implication is a (propositional) tautology.
Proof.
It follows from [4, Main Lemma 4.8]. ∎
3 Sequent calculus
The axiom system is a variant of the sequent calculus G3c. The distinguished feature of G3c is that sequents are pairs of multisets of formulas. Weak structural rules such as contraction and weakening are thus absorbed into logical inference rules and axioms (see [11] for details). The main difference between G3c and our method of sequent calculus is that uses Henkin constants instead of free variables. We also wish to stress here that derivations are cutfree.
Throughout the section we assume that a firstorder language and its witnessing expansion is fixed. All formulas and terms, unless otherwise specified, are firstorder expressions of .
3.1 Finite multisets.
By finite multisets we mean finite unordered collections of elements with repetitions. So a finite multiset is like an ordinary set only it may contain some elements with multiple occurrences. We will use Greek capitals to stand for finite multisets.
For finite multisets we adopt the following notation conventions. If is an element, then by the same symbol we denote the finite multiset containing as its only element. By we denote the union of finite multisets and . Multisets and are thus the result of adding the element to the finite multiset .
3.2 Sequents.
By a sequent
we mean the ordered pair
of finite (possibly empty) multisets of formulas, The sequent arrow separates the antecedent of the sequent from its succedent . Together they are called cedents. We will use the uppercase letter , possibly subscripted, as a syntactic variable ranging over sequents.The meaning of a sequent of the form is equivalent in the meaning to the formula . An empty conjunction () is defined to be and an empty disjunction () is defined to be . The sequent has thus the same meaning as the formula and the empty sequent is logically unsatisfiable.
3.3 Inference rules.
The rules of inference of are of the form , where are sequents. Premises (hypotheses) of a rule are its upper sequents, the conclusion of a rule is its lower sequent. Nullary rules, i.e. rules with no premises, are called axioms. The following is the list of valid rules of inference of :

Axioms
Ax ( is atomic) .

Propositional rules
.

Quantifier rules
.
The quantifier rules and are called strong quantifier rules, the other two quantifier rules are called weak. We say that the Henkin constants and belong to the strong quantifier rules and , respectively.
For every rule, the new formula introduced into the rule’s conclusion is called the principal (main) formula, formulas from which the principal formula is derived are called the minor (auxiliary) formulas, and all the remaining formulas are called the side formulas (context). In the axiom Ax both occurrences of are principal, in and both occurrences of the propositional constants are principal.
Note that all inference rules of are local in the following strong sense: the validity of each inference step depends only on the form the principal and minor formulas, it does not depend on the context.
3.4 Proofs.
A proof in (proof) is a rooted labeled finite tree with sequents as its nodes. The root of the tree, written at the bottom, is called endsequent and it is the sequent to be proved. The remaining nodes of the tree are built by inference rules. The leaves, at the top of the tree, are initial sequents inferred by axiom rules. The inner nodes of the tree are inferred by the remaining inference rules.
We write if is an proof of , and if there is such a proof. By an proof of a formula we mean any proof of its corresponding sequent . The length of a proof is the number of sequents in . We write if the length of the proof is , and if there is such a proof. We write if the length of the proof is , and if there is such a proof. The notation and is defined analogously.
3.5 Remark.
Our usage of Henkin constants in strong quantifier rules is rather nonstandard; therefore, we give here an outline of the proof of the soundness and completeness theorem for . It is an adaptation of the proof for a formal system based on firstorder tableaux with Henkin constants instead of free variables (see [6] for details). But first, we need to introduce some new notation and terminology:

By H and Q we denote the set of all Henkin and quantifier axioms, respectively.

By PK we denote the propositional sequent calculus containing all inference rules of except for quantifier rules.

The cut rule is an inference rule of the form Cut .
We follow closely the proof of the soundness and completeness theorem for the method of tableaux as presented in [6]. By a straightforward translation of the arguments for tableaux to sequents we obtain soundness and completeness of .
3.6 Soundness and completeness theorem.
A pure formula is logically valid if and only if it is provable.
Proof outlined.
A pure formula is logically valid iff, by reduction to propositional logic (see Thm. 2.6), for some the formula is a tautology iff, by the soundness and completeness theorem for PK (see [6, Thm. 7]), for some the formula has a PKproof with cuts iff, by the soundness and completeness theorem for nonanalytic calculus (see [6, Thm. 11]), the formula has an proof with cuts iff, by the cutelimination theorem (see [6, Thm. 18]), the formula has a (cutfree) proof. ∎
4 Examples of proofs
We give here several examples of derivations so that the reader gets familiar with the usage of Henkin constants in formal proofs. The reader interested primarily in the main result of the paper may skip and go directly to Section 5.
4.1 Example.
The following tree is an proof Ax () ) of the formula . The defining abbreviations for the Henkin constants and are shown on the right next to the rule they belong. The inferences in question are the following strong quantifier rules shown in the full form: . When the tree is read as a standard LKproof, then both free variables and satisfy the standard eigenvariable condition.
4.2 Example (Smullyan’s drinker paradox).
Consider now an proof Ax () of the formula . The Henkin constant belongs to the strong quantifier inference (shown in unabbreviated form): . When the tree is read as a standard LKproof, then the proper variable of the strong quantifier rule violates the standard eigenvariable condition. This is because occurs in the side formula of the conclusion of the inference. This kind of violation is still sound because it satisfies the prerequisites of the Smullyan’s liberalized form of eigenvariable condition (see [9] for details).
4.3 Example.
The following is an proof Ax () () of the formula . The Henkin constants and belong to the strong quantifier rule and , respectively. Note that the constant occurs deeply in the constant .
When the tree is read as a standard LKproof, then the proper variable of the inference violates even the Smullyan’s liberalized form of eigenvariable condition. Namely, the principal formula of the rule contains the proper variable of a strong quantifier inference occurring below ; in this case, it contains the free variable which belongs to the rule. Even this kind of violation is sound because it satisfies the prerequisites of a very liberalized form of eigenvariable condition for the doublet of sequent calculi / introduced in [1]. This is because the rank of the constant is less than the rank of the constant . Hence, the dependencies between proper variables of the proof do not form a “cycle” (see [1] for details).
4.4 Example.
Finally, consider an proof Ax Ax of the formula . Again, for the sake of convenience, we have used an abbreviation; in this case . The Henkin constant belongs to two strong quantifier rules (shown in the full form) with different principal formulas: and .
5 Properties of proofs
In this section we investigate the properties of derivations. Some of them are vital in the proof of our main result in the last section: the inversion lemmas 5.4 and 5.5 for and quantifier inferences, and the replacement lemma 5.8. They are specific to proofs as they cannot be easily adapted to sequent calculi with eigenvariable condition.
Throughout the whole section we assume that a firstorder language and its witnessing expansion is fixed. We also extend the notion of expression to include sequents, inference rules, and proofs.
5.1 Subformula property.
If is an proof, then every formula of is a subformula of some formula from the endsequent of .
Proof.
By a straightforward induction on the structure of the proof . ∎
5.2 Weakening lemma.
If then .
Proof.
By induction on the structure of the proof . We consider several cases according to the last rule applied in . Suppose, for example, that the proof ends with an inference of the form . We apply the inductive hypothesis to the subderivation and find a proof such that . We now use a similar rule and obtain a derivation . We are done because we clearly have .
The remaining cases are proved similarly. ∎
5.3 Inversion lemma for .
If then .
Proof.
By a straightforward induction on the structure of proofs. ∎
5.4 Inversion lemma for .
Let . Let further … be a sequence of all inferences applied in with as principal formula. Then
Proof.
By induction on the structure of the proof . We distinguish several cases according to the last rule applied in .
The case, when the last inference of is an axiom, is straightforward. Note that in this case it must be .
Suppose the proof ends with an inference with as principal formula. We may assume w.l.o.g. that the term in the minor formula is : . The subproof contains the remaining inferences with as principal formula. We apply the inductive hypothesis to and, since , we can deduce that .
Suppose now the proof ends with an inference of the form . We apply the inductive hypothesis to the subderivation and find a proof such that . We now use a similar inference and construct a proof . We are done because we clearly have .
Suppose, for example, the proof ends with an inference of the form , where and for some numbers such that . We may assume w.l.o.g. that the subproof contains the first rules with as principal formula and the subproof the remaining . We apply the inductive hypothesis to both subderivations and obtain
By the weakening lemma 5.2 we find proofs and such that
Finally, we use a similar rule and obtain a proof . From the identity we now conclude that and we are done.
The remaining cases are proved similarly. ∎
5.5 Inversion lemma for .
Let . Let further … be a sequence of all inferences applied in with as principal formula. Then
Proof.
By induction on the structure of the proof . We distinguish several cases according to the last rule applied in .
The case, when the last inference of is an axiom, is straightforward. Note that in this case it must be .
Suppose the proof ends with an inference with as principal formula. We may assume w.l.o.g. that the term in the minor formula is : . The subproof contains the remaining inferences with as principal formula. We apply the inductive hypothesis to and, since , we can deduce that .
Suppose now the proof ends with an inference of the form . We apply the inductive hypothesis to the subderivation and find a proof such that . We now use a similar inference and construct a proof . We are done because we clearly have .
Suppose, for example, the proof ends with an inference of the form , where and for some numbers such that . We may assume w.l.o.g. that the subproof contains the first rules with as principal formula and the subproof the remaining . We apply the inductive hypothesis to both subderivations and obtain
By the weakening lemma 5.2 we find proofs and such that
Finally, we use a similar rule and obtain a proof . From the identity we now conclude that and we are done.
The remaining cases are proved similarly. ∎
5.6 Deep replacement of terms.
We say that a term occurs deeply in an expression if the term has an (ordinary) occurrence in the expression or else there is a Henkin constant , which have an (ordinary) occurrence in the expression , such that the term occurs deeply in its index . We then say that the term is a deep subterm of the expression . The subterm is proper if .
By a deep replacement of a term by a term in an expression , written , we mean the replacement of every deep occurrence of the term in the expression by the term . The replacement is nontrivial if .
Example.
Consider the Henkin witnessing axiom . The constant has three occurrences in it. The first two are ordinary occurrences; the third one — that in the index of the Henkin constant – is deep. The deep replacement of by applied to the axiom yields another Henkin witnessing axiom:
5.7 Inference rules under deep replacement.
We now study the effect of deep replacement on proofs. An rule is invariant under a deep replacement, if the replacement applied to the rule yields an inference of the same kind: Axiom Axiom Rule
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