Cut-Simulation and Impredicativity

We investigate cut-elimination and cut-simulation in impredicative (higher-order) logics. We illustrate that adding simple axioms such as Leibniz equations to a calculus for an impredicative logic -- in our case a sequent calculus for classical type theory -- is like adding cut. The phenomenon equally applies to prominent axioms like Boolean- and functional extensionality, induction, choice, and description. This calls for the development of calculi where these principles are built-in instead of being treated axiomatically.


Introduction
One of the key questions of automated reasoning is the following: "When does a set Φ of sentences have a model?"In fact, given reasonable assumptions about calculi, most inference problems can be reduced to determining (un)-satisfiability of a set Φ of sentences.Since building models for Φ is hard in practice, much research in computational logic has concentrated on finding sufficient conditions for satisfiability, e.g.whether there is a Hintikka set H extending Φ.
Of course in general the answer to the satisfiability question depends on the class of models at hand.In classical first-order logic, model classes are well-understood.In impredicative higher-order logic, there is a whole landscape of plausible model classes differing in their treatment of functional and Boolean extensionality.Satisfiability then strongly depends on these classes, for instance, the set Φ : = {a, b, qa, ¬qb} is unsatisfiable in a model class where the universes of Booleans are required to have at most two members (see property b below), but satisfiable in the class without this restriction.
In [5] we have shown that certain (i.e.saturated) Hintikka sets always have models and have derived syntactical conditions (so-called saturated abstract consistency properties) for satisfiability from this fact.The importance of abstract consistency properties is that one can check completeness for a calculus C by verifying proof-theoretic conditions (checking that C-irrefutable sets of formulae have the saturated abstract consistency property) instead of performing model-theoretic analysis (for historical background of the abstract consistency method in first-order logic, cf.[11,16,17]).Unfortunately, the saturation condition (if Φ is abstractly consistent, then for all sentences A one of Φ ∪ {A} or Φ ∪ {¬A} is as well) is very difficult to prove for machine-oriented calculi (indeed as hard as cut elimination as we will show).
In this paper we investigate further the relation between the lack of the subformula property in the saturation condition (we need to "guess" whether to extend Φ by A or ¬A on our way to a Hintikka set) and the cut rule (where we have to "guess," i.e. "search for" in an automated reasoning setting the cut formula A).An important result is the insight that there exist "cut-strong" formulae which support the effective simulation of cut in calculi for impredicative logics.Prominent examples of cut-strong formulae are Leibniz equations and the axioms for comprehension, extensionality, induction, description and choice.The naive addition of any of these cut-strong formulae to any calculus for an impredicative logic is a strong threat for effective automated proof search, since these formulae in a way introduce the cut rule through the backdoor (even if the original calculus is cut-free and thus appears appropriate for proof automation at first sight).Cut-strong formulae thus introduce additional sources for breaking the subformula property and therefore they should either be avoided completely or treated with great care in calculi designed for automated proof search.
Consider the following formula of higher-order logic representing Boolean extensionality: For a theorem prover to make use of this formula, it must instantiate A and B with terms of type o.In other words, the theorem prover must synthesize two arbitrary formulas.Requiring a theorem prover to synthesize these formulas is just as hard (and unrealistic) as requiring a theorem prover to synthesize cut formulas.An alternative to including the formula for Boolean extensionality is to include a rule in the search procedure which allows the theorem prover to reduce proving A .
= o B to the subgoal of proving A ⇔ B. Using this rule does not require the prover to synthesize any terms.Simply adding such a rule is not enough to obtain a complete calculus.We will explore what additional rules are required to obtain completeness and argue that these rules are appropriate for mechanized proof search.
In Section 2, we will fix notation and review the relevant results from [5].We define in Section 3 a basic sequent calculus and study the correspondence between saturation in abstract consistency classes and cut-elimination.In Section 4 we introduce the notion of "cut-strong" formulae and sequents and show that they support the effective simulation of cut.In Section 5 we demonstrate that the pertinent extensionality axioms are cut-strong.We develop alternative extensionality rules which do not suffer from this problem.Further rules are needed to ensure Henkin completeness for this calculus with extensionality.These new rules correspond to the acceptability conditions we propose in Section 6 to ensure the existence of models and the existence of saturated extensions of abstract consistency classes.

Higher-Order Logic
In [5] we have re-examined the semantics of classical higher-order logic with the purpose of clarifying the role of extensionality.For this we have defined eight classes of higher-order models with respect to various combinations of Boolean extensionality and three forms of functional extensionality.We have also developed a methodology of abstract consistency (by providing the necessary model existence theorems) needed for instance, to analyze completeness of higher-order calculi with respect to these model classes.We now briefly summarize the main notions and results of [5] as required for this paper.Our impredicative logic of choice is Church's classical type theory.
2.1.Syntax: Church's Simply Typed λ-Calculus.As in [9], we formulate higherorder logic (HOL) based on the simply typed λ-calculus.The set of simple types T is freely generated from basic types o and ι using the function type constructor →.
For formulae we start with a set V of (typed) variables (denoted by X α , Y, Z, X 1 β , X 2 γ . ..) and a signature Σ of (typed) constants (denoted by c α , f α→β , . ..).We let V α (Σ α ) denote the set of variables (constants) of type α.The signature Σ of constants includes the logical constants ¬ o→o , ∨ o→o→o and Π α (α→o)→o for each type α; all other constants in Σ are called parameters.As in [5], we assume there is an infinite cardinal ℵ s such that the cardinality of Σ α is ℵ s for each type α (cf.[5](3.16)).The set of HOL-formulae (or terms) are constructed from typed variables and constants using application and λ-abstraction.We let wff α (Σ) be the set of all terms of type α and wff(Σ) be the set of all terms.
We use vector notation to abbreviate k-fold applications and abstractions as AU k and λX k A, respectively.We also use Church's dot notation so that stands for a (missing) left bracket whose mate is as far to the right as possible (consistent with given brackets).We use infix notation A ∨ B for ((∨A)B) and binder notation ∀X α A for (Π α (λX α A o )).We further use A ∧ B, A ⇒ B, A ⇔ B and ∃X α A as shorthand for formulae defined in terms of ¬, ∨ and Π α (cf.[5]).Finally, we let (A α .= α B α ) denote the Leibniz equation ∀P α→o (P A) ⇒ P B.
Each occurrence of a variable in a term is either bound by a λ or free.We use f ree(A) to denote the set of free variables of A (i.e., variables with a free occurrence in A).We consider two terms to be equal if the terms are the same up to the names of bound variables (i.e., we consider α-conversion implicitly).A term A is closed if f ree(A) is empty.We let cwff α (Σ) denote the set of closed terms of type α and cwff(Σ) denote the set of all closed terms.Each term A ∈ wff o (Σ) is called a proposition and each term A ∈ cwff o (Σ) is called a sentence.
We denote substitution of a term A α for a variable X α in a term B β by [A/X]B.Since we consider α-conversion implicitly, we assume the bound variables of B avoid variable capture.
Two common relations on terms are given by β-reduction and η-reduction.A β-redex (λX A)B β-reduces to [B/X]A.An η-redex (λX CX) (where X / ∈ f ree(C)) η-reduces to C. For A, B ∈ wff α (Σ), we write A≡ β B to mean A can be converted to B by a series of β-reductions and expansions.Similarly, A≡ βη B means A can be converted to B using both β and η.For each A ∈ wff(Σ) there is a unique β-normal form (denoted A↓ β ) and a unique βη-normal form (denoted A↓ βη ).From this fact we know A≡ β B (A≡ βη B) iff A↓ β ≡ B↓ β ( A↓ βη ≡ B↓ βη ).
A non-atomic formula in wff o (Σ) is any formula whose β-normal form is of the form [cA n ] where c is a logical constant.An atomic formula is any other formula in wff o (Σ).
Given an applicative structure (D, @), an assignment ϕ is a (typed) function from V to D. An evaluation function E maps an assignment ϕ and a term A α ∈ wff α (Σ) to an element E ϕ (A) ∈ D α .Evaluations E are required to satisfy four properties (cf.[5](3.18)): (1) E ϕ V ≡ ϕ.
If A is closed, then we can simply write E(A) since the value E ϕ (A) cannot depend on ϕ.
For each * ∈ {β, βη, βξ, βf, βb, βηb, βξb, βfb} (the latter set will be abbreviated by 8 in the remainder) we define M * to be the class of all Σ-models M such that M satisfies property q and each of the additional properties {η, ξ, f, b} indicated in the subscript * (cf.[5](3.49)).We always include β in the subscript to indicate that β-equal terms are always interpreted as identical elements.We do not include property q as an explicit subscript; q is treated as a basic, implicit requirement for all model classes.See [5](3.52)for a discussion on why we require property q.Since we are varying four properties, one would expect to obtain 16 model classes.However, we showed in [5] that f is equivalent to the conjunction of ξ and η.Hence we obtain the eight model classes depicted as a cube in Figure 1.There are example models constructed in [5] to demonstrate that each of the eight model classes is distinct.For instance, Example 5.6 of [5] describes how to construct a model without η by attaching labels to functions.
Special cases of Σ-models are Henkin models and standard models (cf.[5](3.50 and 3.51)).A Henkin model is a model in M βfb such that the applicative structure (D, @) is a frame, i.e.D α→β is a subset of the function space (D β ) Dα for each α, β ∈ T and @ is function application.A standard model is a Henkin model in which D α→β is the full function space (D β ) Dα .Every model in M βfb is isomorphic to a Henkin model (see the discussion following [5](3.68)).

Saturated Abstract Consistency Classes and Model
Existence.Finally, we review the model existence theorems proved in [5].There are three stages to obtaining a model in our framework.First, we obtain an abstract consistency class Γ Σ (usually defined as the class of irrefutable sets of sentences with respect to some calculus).Second, given a (sufficiently pure) set of sentences Φ in the abstract consistency class Γ Σ we construct a Hintikka set H extending Φ.Third, we construct a model of this Hintikka set (and hence a model of Φ).
A Σ-abstract consistency class Γ Σ is a class of sets of Σ-sentences.An abstract consistency class is always required to be closed under subsets (cf.[5](6.1)).Sometimes we require the stronger property that Γ Σ is compact, i.e. a set Φ is in Γ Σ iff every finite subset of Φ is in Γ Σ (cf.[5](6.1,6.2)).
To describe further properties of abstract consistency classes, we use the notation S * a for S ∪ {a} as in [5].The following is a list of properties a class Γ Σ of sets of sentences can satisfy with respect to arbitrary Φ ∈ Γ Σ (cf.[5](6.5)): We say Γ Σ is an abstract consistency class if it is closed under subsets and satisfies ∇ c , ∇ ¬ , ∇ β , ∇ ∨ , ∇ ∧ , ∇ ∀ and ∇ ∃ .We let Acc β denote the collection of all abstract consistency classes.For each * ∈ 8 we refine Acc β to a collection Acc * where the additional properties {∇ η , ∇ ξ , ∇ f , ∇ b } indicated by * are required (cf.[5](6.7)).We say an abstract consistency class Γ Σ is saturated if ∇ sat holds.
Using ∇ c (atomic consistency) and the fact that there are infinitely many parameters at each type, we can show every abstract consistency class satisfies non-atomic consistency.That is, for every abstract consistency class Γ Σ , A ∈ cwff o (Σ) and Φ ∈ Γ Σ , we have either A / ∈ Φ or ¬A / ∈ Φ (cf.[5](6.10)).In [5](6.32)we show that sufficiently Σ-pure sets in saturated abstract consistency classes extend to saturated Hintikka sets.(A set of sentences Φ is sufficiently Σ-pure if for each type α there is a set P α of parameters of type α with cardinality ℵ s and such that no parameter in P occurs in a sentence in Φ.A Hintikka set is a maximal element in an abstract consistency class.) In the Model Existence Theorem for Saturated Sets [5](6.33)we show that these saturated Hintikka sets can be used to construct models M which are members of the corresponding model classes M * .Then we conclude (cf.[5] In [5] we apply the abstract consistency method to analyze completeness for different natural deduction calculi.Unfortunately, the saturation condition is very difficult to prove for machine-oriented calculi (indeed as we will see in Section 3 it is equivalent to cut elimination), so Theorem [5](6.34)cannot be easily used for this purpose directly.
In Section 6 we therefore motivate and present a set of extra conditions for Acc βfb we call acceptability conditions.The new conditions are sufficient to prove model existence.

Sequent Calculi, Cut and Saturation
We will now study cut-elimination and cut-simulation with respect to (one-sided) sequent calculi.
3.1.Sequent Calculi G.We consider a sequent to be a finite set ∆ of β-normal sentences from cwff o (Σ).A sequent calculus G provides an inductive definition for when ⊢ ⊢ G ∆ holds.We say a sequent calculus rule For any natural number k ≥ 0, we call an admissible rule r k-admissible if any instance of r can be replaced by a derivation with at most k additional proof steps.Given a sequent ∆, a model M, and a class M of models, we say ∆ is valid for M (or valid for M), if M |= D for some D ∈ ∆ (or ∆ is valid for every M ∈ M).As for sets in abstract consistency classes, we use the notation ∆ * A to denote the set ∆ ∪ {A} (which is simply ∆ if A ∈ ∆). Figure 2 introduces several sequent calculus rules.Some of these rules will be used to define sequent calculi, while others will be shown admissible (or even k-admissible).

Basic Rules
A atomic (and

(Alternative Formulations
).There are many kinds of sequent calculi given in the literature.We could have chosen to work with two sided sequents.This choice would have allowed us to generalize many of our results to the intuitionistic case.The notion of cut-strong formulae could still be defined and many of our examples of cut-strong formulae would also be cut-strong in the intuitionistic case.On the other hand, assuming we only treat the classical case, we could restrict to negation normal forms in the same way that we restrict to β-normal forms.This would eliminate the need to consider the rules G(¬) and G(Inv ¬ ).Both of these alternatives are reasonable.The choices we have made are for ease of presentation and to make the connection with [5] as simple as possible.
In a straightforward manner, one can prove the following results (see the Appendix).
Lemma 3.3.Let G be a sequent calculus such that G(Inv ¬ ) is admissible.For any finite sets Φ and ∆ of sentences, if We can furthermore show the following relationship between saturation and cut (see the Appendix).
Since saturation is equivalent to admissibility of cut, we need weaker conditions than saturation.A natural condition to consider is the existence of saturated extensions.Definition 3.6 (Saturated Extension).Let * ∈ 8 and Γ Σ , Γ ′ Σ ∈ Acc * be abstract consistency classes.We say In the first case, applying ∇ ∀ with the constant q, ∇ ∨ and ∇ c contradicts (qa), ¬(qb) ∈ Φ.In the second case, ∇ b and Existence of any saturated extension of a sound sequent calculus G implies admissibility of cut.The proof uses the model existence theorem for saturated abstract consistency classes (cf.[5](6.34)).The proof is in the Appendix.Theorem 3.8.Let G be a sequent calculus which is sound for

3.3.
Sequent Calculus G β .We now study a particular sequent calculus G β defined by the rules G(init), G(¬), G(∨ − ), G(∨ + ), G(Π C − ) and G(Π c + ) (cf. Figure 2).It is easy to show that G β is sound for the eight model classes and in particular for class M β .
The reader may easily prove the following Lemma.
Lemma 3.9.Let A ∈ cwff o (Σ) be an atom, B ∈ cwff α (Σ), and ∆ be a sequent. ( The proof of the next Lemma is by induction on derivations and is given in the Appendix.Andrews proves admissibility of cut for a sequent calculus similar to G β in [1].The proof in [1] contains the essential ingredients for showing completeness. While G(cut) is admissible in G β the next theorem shows that G(cut) is not k-admissible in G β for any k ∈ N, which means G β is not only superficially cut-free and that by adding G(cut) to G β we can achieve significantly shorter proofs.Theorem 3.12.G(cut) is not k-admissible in G β for any k ∈ N.
Proof.The proof is not formally worked out here; we only sketch the argumentation: The main idea is to show that the hyper-exponential speed-up results known for first-order logic do transfer to (the first-order fragment of) our calculus.For this, we compare our sequent calculus G β with a standard first-order variant of it which we call G F O β (this only requires appropriate modifications of the rules G(Π C − ) and G(Π c + )).Clearly, any first-order sequent which can be derived in G F O β can be derived in G β with the same number of steps (using essentially the same derivation).More interestingly, one can show that for any derivation D in G β of a first-order sequent ∆ there is a derivation D ′ in G F O β of ∆ with the same number of rule applications.(During the induction, one collapses higher-order terms to first-order terms in such a way that first-order terms collapse to themselves.)Thus no speedup with respect to first-order provability can be achieved by using G β instead of the cut-free first-order sequent calculus G F O β .Finally we refer to the following results: • Theorem 5.2.13 in [19] shows that for a classical first-order sequent calculus there is at least an exponential speed-up of proofs with cut.Furthermore, Propositions 6.11.3 and 6.11.4 there show a related hyper-exponential speed-up result.
• An example for hyper-exponential speed-up is also given in [18,13].
• In higher-order logic the speed-up should be faster than any primitive recursive function according to the "curious inference" George Boolos presents in [7].
We will now show that G(cut) actually becomes k-admissible in G β if certain formulae are available in the sequent ∆ we wish to prove.Our examples below illustrate that cut-strength of a formula usually only weakly depends on the calculus G: it only presumes standard ingredients such as β-normalization, weakening, and rules for the logical connectives.
We present some simple examples of cut-strong formulae for our sequent calculus G β .A corresponding phenomenon is observable in other higher-order calculi, for instance, for the calculi presented in [1,4,8,12].
Example 4.2.The Formula ∀P o P : = Π o (λP o P ) is 3-cut-strong in G β .This is justified by the following derivation which actually shows that rule G(cut A ) for this specific choice of A is derivable in G β by maximally 3 additional proof steps.The only interesting proof step is the instantiation of P with formula D : = ¬C ∨ C in rule G(Π D − ).(Note that C must be β-normal; sequents such as ∆ * C by definition contain only β-normal formulae.) Clearly, ∀P o P is not a very interesting cut-strong formula since it implies falsehood, i.e. inconsistency. 1 Here, we could alternatively use (k-)derivability (see [10]) to give a stronger but less general notion of k-cut-strongness.In fact, all axioms we discuss in this paper would remain k-cut-strong.From a proof theoretic point of view one may argue that this alternative notion leads to a more interesting result although it may generally apply to fewer axioms.Example 4.5.The original formulation of higher-order logic (cf.[15]) contained comprehension axioms of the form is arbitrary with P / ∈ f ree(B).Church eliminated the need for such axioms by formulating higher-order logic using typed λ-calculus.We will now show that the instance . This motivates building-in comprehension principles instead of treating comprehension axiomatically.
As we will show later, many prominent axioms for higher-order logic also belong to the class of cut-strong formulae.4.2.Cut-Simulation.The cut-simulation theorem is a main result of this paper.It says that cut-strong sequents support an effective simulation (and thus elimination) of cut in G β .Effective means that the size of cut-free derivation grows only linearly for the number of cut rule applications to be eliminated.Proof.Note that the rules G(cut) and G(cut A ) coincide whenever ¬A ∈ ∆.Intuitively, we can replace each occurrence of G(cut) in D by G(cut A ) in order to obtain a D ′ of same size.Technically, in the induction proof one must weaken to ensure ¬A stays in the sequent and carry out a parameter renaming to make sure the eigenvariable condition is satisfied.
This means that G(cut A ) can be eliminated in D and each single elimination of G(cut A ) introduces maximally k new proof steps.Now the assertion can be easily obtained by a simple induction over n.
Corollary 4.9.Let ∆ be a k-cut-strong sequent.For each derivation D : ⊢ ⊢ G cut β ∆ with d proof steps and n applications of rule G(cut) there exists an alternative cut-free derivation D ′ : ⊢ ⊢ G β ∆ with maximally d + nk proof steps.

The Extensionality Axioms are Cut-Strong
We have shown comprehension axioms can be cut-strong (cf.Example 4.5).Further prominent examples of cut-strong formulae are the Boolean and functional extensionality axioms.The Boolean extensionality axiom (abbreviated as B o in the remainder) is The infinitely many functional extensionality axioms (abbreviated as F αβ ) are parameterized over α, β ∈ T .
= α→β G These axioms usually have to be added to higher-order calculi to reach Henkin completeness, i.e. completeness with respect to model class M βfb .For example, Huet's constrained resolution approach as presented in [12] is not Henkin complete without adding extensionality axioms.The need for adding Boolean extensionality to this calculus is actually illustrated by the set of unit literals Φ : = {a, b, (qa), ¬(qb)} from Example 3.7.As the reader may easily check, this clause set Φ, which is inconsistent for Henkin semantics, cannot be proven by Huet's system without, e.g., adding the Boolean extensionality axiom.By relying on results in [1], Huet essentially shows completeness with respect to model class M β as opposed to Henkin semantics.
We will now investigate whether adding the extensionality axioms to a machine-oriented calculus in order to obtain Henkin completeness is a suitable option.
Theorem 5.1.The Boolean extensionality axiom B o is a 14-cut-strong formula in G β .
Proof.The following derivation justifies this theorem (a o is a parameter).
Proof.The following derivation justifies this theorem (f α→β is a parameter).
In [4] and [8] we have already argued that the extensionality principles should not be treated axiomatically in machine-oriented higher-order calculi and there we have developed resolution and sequent calculi in which these principles are built-in.Here we have now developed a strong theoretical justification for this work: Corollary 4.9 along with Theorems 5.2 and 5.1 tell us that adding the extensionality principles B o and F αβ as axioms to a calculus is like adding a cut rule.
In Figure 3 we show rules that add Boolean and functional extensionality in an axiomatic manner to G β .More precisely we add rules G(F αβ ) and G(B) allowing to introduce the axioms for any sequent ∆; this way we address the problem of the infinitely many possible instantiations of the type-schematic functional extensional axiom In Figure 4 we define alternative extensionality rules which correspond to those developed for resolution and sequent calculi in [4] and [8].
Soundness of G(f) and G(b) for Henkin semantics is again easy to show.
Our aim is to develop a machine-oriented sequent calculus for automating Henkin complete proof search.We argue that for this purpose G(f) and G(b) are more suitable rules than G(F αβ ) and G(B).
Our next step now is to show Henkin completeness for G E β .This will be relatively easy since we can employ cut-simulation.Then we analyze whether calculus G − βfb has the same deductive power as G E β .First we extend Theorem 3.4.The proof is given in the Appendix.In order to reach Henkin completeness and to show cut-elimination we thus need to add further rules.Our example motivates the two rules presented in Figure 5. G(Init .= ) introduces Leibniz equations such as qa .
= o qb as is needed in our example and G(d) realizes the required decomposition into a .Is G βfb complete for Henkin semantics?We will show in the next Section that this indeed holds (cf.Theorem 6.3).
With G E and G βfb we have thus developed two Henkin complete calculi and both calculi are cut-free.However, as our exploration shows, "cut-freeness" is not a well-chosen criterion to differentiate between their suitability for proof search automation: G E inherently supports effective cut-simulation and thus cut-freeness is meaningless.
The next claim, which is analogous to Theorem 3.12, has not been formally proven yet.It claims that, in contrast to G E , the cut-freeness of G βfb is meaningful.
The proof idea is similar to that of Theorem 3.12, however, the two additional rules G(Init .= ) and G(d) do introduce additional technicalities which we have not fully worked out yet.
The criterion we propose for the analysis of calculi in impredicative logics is "freeness of effective cut-simulation".The idea behind this notion is to capture also hidden sources (such as the extensionality axioms) where the subformula property may break and where the cut rule may creep in through the backdoor.5.2.Other Rules for Other Model Classes.In [6] we developed respective complete and cut-free sequent calculi not only for Henkin semantics but for five of the eight model classes.In particular, no additional rules are required for the β, βη and βξ case.Meanwhile, the βf case requires additional rules allowing η-conversion.We do not present and analyze these cases here.

Acceptability Conditions
We now turn our attention again to the existence of saturated extensions of abstract consistency classes.
As illustrated by Example 3.7, we need some extra abstract consistency properties to ensure the existence of saturated extensions.We call these extra properties acceptability conditions.They actually closely correspond to additional rules G(Init .= ) and G(d).
Definition 6.1 (Acceptability Conditions).Let Γ Σ be an abstract consistency class in Acc βfb .We define the following properties: We now replace the strong saturation condition used in [5] by these acceptability conditions.Definition 6.2 (Acceptable Classes).An abstract consistency class Γ Σ ∈ Acc βfb is called acceptable in Acc βfb if it satisfies the conditions ∇ m and ∇ d .
One can show a model existence theorem for acceptable abstract consistency classes in Acc βfb (cf.[6](8.1)).From this model existence theorem, one can conclude G βfb is complete for M βfb (hence for Henkin models) and that cut is admissible in G βfb .Theorem 6.3.The sequent calculus G βfb is complete for Henkin semantics and the rule G(cut) is admissible.
Proof.The argumentation is similar to Theorem 3.11 but here we employ the acceptability conditions ∇ m and ∇ d .
One can further show the Saturated Extension Theorem (cf.[6](9.3)):Theorem 6.4.There is a saturated abstract consistency class in Acc βfb that is an extension of all acceptable Γ Σ in Acc βfb .
Given Theorem 3.8, one can view the Saturated Extension Theorem as an abstract cut-elimination result.
The proof of a model existence theorem employs Hintikka sets and in the context of studying Hintikka sets we have identified a phenomenon related to cut-strength which we call the Impredicativity Gap.That is, a Hintikka set H is saturated if any cut-strong formula A (e.g. a Leibniz equation C .= D) is in H. Hence we can reasonably say there is a "gap" between saturated and unsaturated Hintikka sets.Every Hintikka set is either saturated or contains no cut-strong formulae.

Conclusion
We have shown that adding cut-strong formulae to a calculus for an impredicative logic is like adding cut.For machine-oriented automated theorem proving in impredicative logics -such as classical type theory -it is therefore not recommendable to naively add cut-strong axioms to the search space.In addition to the comprehension principle and the functional and Boolean extensionality axioms as elaborated in this paper the list of cut-strong axioms includes: = α is λX α λY α ∀Q α→α→o (∀Z α (Q Z Z)) ⇒ (Q X Y ) (cf. [3]).The argument is similar to 4.2-4.5;here we the crucial step is to instantiate Q with λX α λY α C. Example 7.2 (Axiom of Induction.).The axiom of induction for the naturals ∀P ι→o P 0 ∧ (∀X ι P X ⇒ P (sX)) ⇒ ∀X ι P X is 18-cut-strong in G β .(Other well-founded ordering axioms are analogous.)The crucial step in the proof is to instantiate P with λX ι a .
As we have shown in Example 4.5, comprehension axioms can be cut-strong.Church's formulation of type theory (cf.[9]) used typed λ-calculus to build comprehension principles into the language.One can view Church's formulation as a first step in the program to eliminate the need for cut-strong axioms.For the extensionality axioms a start has been made by the sequent calculi in this paper (and [6]), for resolution in [4] and for sequent calculi and extensional expansion proofs in [8].The extensional systems in [8] also provide a complete method for using primitive equality instead of Leibniz equality.For improving the automation of higher-order logic our exploration thus motivates the development of higher-order calculi which directly include reasoning principles for equality, extensionality, induction, choice, description, etc., without using cut-strong axioms.

Example 3 . 7 .
saturated and an extension of Γ Σ .There exist abstract consistency classes Γ in Acc βfb which have no saturated extension.Let a o , b o , q o→o ∈ Σ and Φ : = {a, b, (qa), ¬(qb)}.We construct an abstract consistency class Γ Σ from Φ by first building the closure Φ ′ of Φ under relation ≡ β and then taking the power set of Φ ′ .It is easy to check that this Γ Σ is in Acc βfb .Suppose we have a saturated extension Γ

Lemma 3 . 10 .
The rules G(Inv ¬ ) and G(weak ) are 0-admissible in G β .Theorem 3.11.The sequent calculus G β is complete for the model class M β and the rule G(cut) is admissible.Proof.By Theorem 3.4 and Lemma 3.10, Γ G β Σ ∈ Acc β .Suppose ⊢ ⊢ G β ∆ does not hold.Then ¬∆ ∈ Acc β by Lemma 3.3.By the model existence theorem for Acc β (cf.[6](8.1))there exists a model for ¬∆ in M β .This gives completeness of G β .We can use completeness to conclude cut is admissible in G β .

4. Cut-Simulation 4 . 1 .Definition 4 . 1 . 1 ∆
Cut-Strong Formulae and Sequents.k-cut-strong formulae can be used to effectively simulate cut.Effectively means that the elimination of each application of a cut-rule introduces maximally k additional proof steps, where k is constant.Given an arbitrary but fixed number k > 0. We call formulaA ∈ cwff o (Σ) k-cut-strong for G (or simply cut-strong) if the following cut rule variant is k-admissible in G: * C ∆ * ¬C G(cut A ) ∆ * ¬AWe can alternative state the condition for A to be k-cut-strong for G as follows: For all ∆ and C, if ⊢ ⊢ G ∆ * C in n steps and ⊢ ⊢ G ∆ * ¬C in m steps, then ⊢ ⊢ G ∆ * ¬A in at most n + m + k steps.

Example 4 . 3 .
The formula ∀P o P ⇒ P : = Π o (λP o ¬P ∨P ) is 3-cut-strong in G β .This is an example of a tautologous cut-strong formula.Now P is simply instantiated with D : = C in rule G(Π D − ).Except for this first step the derivation is identical to the one for Example 4.2.

Example 4 . 4 .
Leibniz equations M .= α N : = Π α (λP ¬P M ∨ P N) (for arbitrary formulae M, N ∈ cwff α (Σ) and types α ∈ T ) are 3-cut-strong in G β .This includes the special cases M .= α M. Now P is instantiated with D : = λX α C in rule G(Π D − ).Except for this first step the derivation is identical to the one for Example 4.2.

Definition 4 . 6 .
A sequent ∆ is called k-cut-strong (or simply cut-strong) if there exists a k-cut-strong formula A ∈ cwff o (Σ) such that ¬A ∈ ∆.We call A the k-realizer of ∆.We first fix the following calculi: Calculus G cut β extends G β by the rule G(cut) and calculus G cut A β extends G β by the rule G(cut A ) for some arbitrary but fixed cut-strong formula A.

Theorem 4 . 7 .
Let ∆ be a k-cut-strong sequent with realizer A. For each derivation D : ⊢ ⊢ G cut β ∆ with d proof steps there is an alternative derivation D ′ : ⊢ ⊢ G cut A β ∆ with d proof steps.

Theorem 4 . 8 .
Let ∆ be a k-cut-strong sequent with realizer A. For each derivation D : ⊢ ⊢ G cut A β ∆ with d proof steps and with n applications of rule G(cut) there exists an alternative derivation D

Figure 3 :Figure 4 : Proper Extensionality Rules 5 . 1 .
Figure 3: Axiomatic Extensionality Rules Calculus G β enriched by the new rules G(F αβ ) and G(B) is called G E β .Soundness of the the new rules is easy to verify: In [5](4.3)we show that G(F αβ ) and G(B) are valid for Henkin models.

Theorem 5 . 3 .
Let G be a sequent calculus such that G(Inv ¬ ) and G(¬) are admissible.(1) If G(f) and G(Π c + ) are admissible, then Γ G Σ satisfies ∇ f .(2) If G(b) is admissible, then Γ G Σ satisfies ∇ b .Theorem 5.4.The sequent calculus G E β is Henkin complete and the rule G(cut) is 12admissible.Proof.G(cut) can be effectively simulated and hence eliminated in G E β by combining rule G(F αβ ) with the 11-step derivation presented in the proof of Theorem 5.2.Let Γ G E β Σ be defined as in Definition 3.2.We prove Henkin completeness of G E β by showing that the class Γ G E β Σ is a saturated abstract consistency class in Acc βfb .We here only analyze the crucial conditions ∇ b , ∇ f and ∇ sat .For the other conditions we refer to Theorem 3.4.Note that 0-admissibility of G(Inv ¬ ) and G(weak ) can be shown for G E β by a suitable induction on derivations as in Lemma 3.10.∇ f : G(Π c + ) is a rule of G E β and thus admissible.According to Theorem 5.3 it is thus sufficient to ensure admissibility of rule G(f) to show ∇ f .This is justified by the following derivation where N : = A .= α→β B and M : = (∀X α AX .= β BX)  β (for β-normal A, B).

∇ b :Theorem 5 . 5 .Example 5 . 6 .
With a similar derivation using G(B) we can show that G(b) is admissible.We conclude ∇ b by Theorem 5.3.∇ sat : Since G(cut) is admissible we get saturation by Theorem 3.5.Does G − βfb have the same deductive strength as G E β ?I.e., is G − βfb Henkin complete?We show this is not yet the case.The sequent calculus G − βfb is not complete for Henkin semantics.We illustrate the problem by a counterexample.Consider the sequent ∆ : = {¬a, ¬b, ¬(qa), (qb)} where a o , b o , q o→o ∈ Σ are parameters.For any M ≡ (D, @, E, υ) ∈ M βfb , either υ(E(a)) ≡ F, υ(E(b)) ≡ F or E(a) ≡ E(b) by property b.Hence sequent ∆ is valid for every M ∈ M βfb .However, ⊢ ⊢ G − βfb ∆ does not hold.By inspection, ∆ cannot be the conclusion of any rule.

Figure 5 :
Figure 5: Additional Rules G(Init .= ) and G(d) We thus extend the sequent calculus G − βfb to G βfb by adding the decomposition rule G(d) and the rule G(Init .= ) which generally checks if two atomic sentences of opposite polarity are provably equal (as opposed to syntactically equal).Is G βfb complete for Henkin semantics?We will show in the next Section that this indeed holds (cf.Theorem 6.3).With G E and G βfb we have thus developed two Henkin complete calculi and both calculi are cut-free.However, as our exploration shows, "cut-freeness" is not a well-chosen criterion to differentiate between their suitability for proof search automation: G E inherently supports effective cut-simulation and thus cut-freeness is meaningless.The next claim, which is analogous to Theorem 3.12, has not been formally proven yet.It claims that, in contrast to G E , the cut-freeness of G βfb is meaningful.
(6.34)): Model Existence Theorem for Saturated Abstract Consistency Classes: For all * ∈ 8 , if Γ Σ is a saturated abstract consistency class in Acc * and Φ ∈ Γ Σ is a sufficiently Σ-pure set of sentences, then there exists a model M ∈ M * that satisfies Φ.Furthermore, each domain of M has cardinality at most ℵ s .