The G4i Analogue of a G3i Sequent Calculus

This paper provides a method to obtain terminating analytic calculi for a large class of intuitionistic modal logics. For a given logic L with a cut-free calculus G that is an extension of G3ip the method produces a terminating analytic calculus that is an extension of G4ip and equivalent to G. G4ip was introduced by Roy Dyckhoff in 1992 as a terminating analogue of the calculus G3ip for intuitionistic propositional logic. Thus this paper can be viewed as an extension of Dyckhoff’s work to intuitionistic modal logic.


Introduction
One of the standard calculi without structural rules for IPC is G3ip (Fig. 1), which is the propositional part of the calculus G3i from [23]. This is an elegant analytic calculus, but it has the unfortunate feature that unrestricted proof search is not terminating in it. The reason for this lies in its left implication rule, in which the principal formula occurs in one of the premises. In [5] Roy Dyckhoff introduced a calculus G4ip 1 (Fig. 2) that is the result of replacing the single left implication rule in G3ip by four left implication rules, each corresponding to the outermost logical symbol of the antecedent of the principal implication. This system was shown to be terminating and equivalent to G3ip, where a calculus is terminating if there exists an order on sequents under which in all rules the premises come before the conclusion in that order.
The modest aim in this short note is to extend Dyckhoff's result to intuitionistic modal logics. For any extension G3iX of G3ip by modal rules R, a calculus, called G4iX, is defined that is an extension of G4ip by R and several additional rules determined by R. It is shown that under some mild closure conditions G3iX and G4iX are equivalent. Moreover, if the rules in R are  terminating in an order that is an extension (to modal logic) of Dyckhoff's original order on sequents, then G4iX terminates in that order too. Thus for a logic that has a calculus G3iX that is an extension of G3ip by terminating rules, establishing the equivalence of G3iX and G4iX indeed is a method to obtain a terminating sequent calculus for that logic.
The interest in terminating calculi lies in the fact that they can be a useful tool in establishing certain properties of a logic, such as decidability, or uniform interpolation, where the syntactic approach developed in [15,16] defines interpolants on the basis of the rules of a calculus, for which it is essential that the calculus is terminating. In fact, its potential use in syntactic proofs of uniform interpolation was the main motivation for developing the method in this paper. This method to obtain a terminating calculus for a given logic uses a (usually nonterminating) calculus based on G3ip. Since many intuitionistic modal logics have a calculus based on G3ip, we therefore hope that this paper will be a convenient tool in settings where terminating calculi are required.
The general result presented in this paper grew out of the research carried out in [8,9,15], in which G4ip based calculi are developed for four concrete logics, namely the intuitionistic versions of the logics K, KD, Gödel-Löb Logic GL and Strong Löb Logic SL. The results for the first two logics follow from the results in this note, while the results for the other two logics require a slight adjustment of the method presented in this note, as explained in Section 4.
The paper is built-up as follows. Section 2 introduces G3iX and G4iX, the order on sequents based on Dyckhoff's weight function, and the further notions needed in the paper. Section 3 contains the main theorem, the equivalence of G3iX and G4iX. Section 4 summarizes the results and discusses possible extensions.
ranges over the modal operators and a formula of the form ϕ is a pure modal formula. We use the convention that ∧ and ∨ bind stronger than →.
We denote finite multisets of formulas by Γ, Π, Δ, Σ. We denote by Γ ∪ Π the multiset that contains only formulas ϕ that belong to Γ or Π and the number of occurrences of ϕ in Γ ∪ Π is the sum of the occurrences of ϕ in Γ and in Π. Furthermore (a for antecedent, s for succedent): We only consider single-conclusion sequents, which are expressions (Γ ⇒ Δ), where Δ contains at most one formula, that are interpreted as I(Γ ⇒ Δ) = ( Γ → Δ). In a sequent, Γ, Π is short for Γ ∪ Π. When sequents are used in the setting of formulas, we often write S for I(S), such as in In the setting of G4iX systems we need an order on sequents based on a weight function, which is a function w(·) that assigns positive numbers to formulas in such a way that all atoms and ⊥ have weight 1 and all other formulas have a weight above 1. With a weight function w we associate the following order on sequents: is the order on multisets determined by w as in [3] (where they in fact define ): for multisets Γ, Δ we have Δ w Γ if Δ is the result of replacing one or more formulas in Γ by zero or more formulas of lower weight.

Calculi G3iX and G4iX
In this paper, a right modal rule is a rule of the following form, where is one of the modal operators: With such a rule we associate the following implication rule If G3iX (X a finite string) is the name of a calculus that consists of G3ip plus a set of modal rules R, then G4iX is the calculus G4ip extended by the rules in R plus the rules R → for those R ∈ R that are right modal rules.
Then G4iX consists of G4ip, R X , and the rule A rule is nonflat if its conclusion contains at least one connective or modal operator, and it is not an axiom, i.e it has nonempty premises. A set of rules or a calculus is nonflat if all of its rules that are not axioms are nonflat. Note that a calculus G3iX or G4iX is nonflat if all rules in R are nonflat.
Given a sequent calculus G and a sequent S, G S denotes that S is derivable in G.

Terminating Calculi
Given an order on sequents, a rule is terminating in if its premises come before the conclusion in order . A calculus is terminating if there exists a weight function w such that all rules of G terminate in the order w on sequents (see the beginning of Sect. 2 for the definition of w ).
Example 2.2. A natural weight function based on the weight function from [5] is inductively defined as follows: the weight of an atom and the constant ⊥ is 1, w D (ϕ • ψ) = w D (ϕ) + w D (ψ) + i, where i = 1 in case • ∈ {∨, →} and i = 2 otherwise, and w D ( i ϕ) = w D (ϕ) + 1. Let D denote w D and call it the Dyckhoff order. It is not hard to see that all rules of G4ip terminate in D . The following are examples of well-known modal rules that terminate in D (the standard notation for these modalities is , here we use to be consistent with the notation in the rest of the paper).
If the rules in R are terminating in the Dyckhoff order, then G4iX is terminating in the Dyckhoff order.

Structural Rules
A calculus G is closed under weakening if the following two rules are admissible: It is closed under contraction if the following rule is admissible: G has cut-elimination or is closed under cut if the Cut rule is admissible: G is closed under the structural rules if it is closed under weakening and contraction and has cut-elimination. G is closed under Implication Inversion if the following rule is admissible:

Equivalence of G3iX and G4iX
In this section, let G3iX be a calculus that consists of G3ip plus a set of modal rules R. Recall that G4iX is the calculus G4ip extended by the rules in R plus the rules R → for those R ∈ R that are right modal rules. We show that G4iX and G3iX are equivalent, for which we first have to prove a normal form theorem (Lemma 3.3) for derivations in G3iX.

Strict proofs in G3iX
A multiset is irreducible if it has no element that is a disjunction or a conjunction or falsum and for no atom p does it contain both p → ψ and p. A sequent S is irreducible if S a is. A proof is sensible if its last inference does not have a principal formula on the left of the form p → ψ for some atom p and formula ψ. 2 A proof in G3iX is strict if in the last inference, in case it is an instance of L → with principal formula ϕ → ψ, the left premise is an axiom or the conclusion of an application of a right modal rule.
Observe that "strict" and "sensible" are properties of proofs and depend only on the lowest inferences of a proof. Namely, the end of a proof always has one of the following two forms: . . .
Here any of the rules R i may be an axiom, in which case there are no sequents above its conclusion in the picture above. Now whether the proof is strict or sensible depends only on the sequents S i and the rules R i . It is necessarily strict and sensible if R 0 is not L →. In the case that it is, the principal formula must be of the form ϕ → ψ and the proof looks as follows: . . .
In case ϕ is not atom or a pure modal formula, the proof is strict and sensible. Only when ϕ is an atom, the proof is not sensible, but it is strict. If ϕ is a pure modal formula, it is sensible but may not be strict, depending on the rule R 1 . The proof is strict once R 1 is an axiom or a right modal rule, as in the following two examples.
Example 3.1. Two strict proofs in G3iK , where the left premise of the last inference is the right modal rule R K in the leftmost example and an axiom in the other example: The rightmost example is of course a nonsensical derivation, but strict proofs of this form are such. The important strict proofs are the ones where the left premise of the last inference in the conclusion of a right modal rule, as in the leftmost example.

Remark 3.2.
If R is nonflat, then in any strict proof ending with an instance of L → with principal formula ϕ → ψ and conclusion S, because the formula in the succedent of the left premise is ϕ, in case the left premise is an instance of an axiom it can only be an instance of L⊥. This implies that if S is irreducible, the left premise cannot be an instance of an axiom and thus is required to be the conclusion of an application of a right modal rule. Proof. This is proved in the same way as the corresponding lemma (Lemma 1) in [5]. Arguing by contradiction, assume that among all provable irreducible sequents that have no sensible strict proofs, S is such a sequent with the shortest proof, D, where the length of a proof is the length of its leftmost branch. Thus the last inference in the proof is an application where ϕ is an atom or a pure modal formula. Since S a is irreducible, ⊥ ∈ S a and if ϕ is an atom, ϕ ∈ S a . Therefore the left premise cannot be an instance of an axiom and hence is the conclusion of a rule, say R. Since the succedent of the conclusion of R consists of an atom or a pure modal formula, R is a left rule or a right modal rule. The latter case cannot occur, since the proof then would be strict and sensible. Thus R is a left rule.
We proceed as in [5]. Sequent (Γ, ϕ → ψ ⇒ ϕ) is irreducible and has a shorter proof than S. Thus its subproof D 1 is strict and sensible. Since the sequent is irreducible and ϕ is an atom or a pure modal formula, the last inference of D 1 is L→ with a principal formula ϕ → ψ such that ϕ is not an atom. Let D be the proof of the left premise (Γ, ϕ → ψ ⇒ ϕ ). Thus the last part of D looks as follows, where Π, ϕ → ψ = Γ.
The existence of D follows from the assumption of Implication Inversion and the existence of D 2 . The obtained proof is strict and sensible: In case ϕ is not a pure modal formula, this is straightforward. In case ϕ is a pure modal formula, it follows from the fact that was observed above, namely that D 1 is strict and sensible.

Theorem 3.4. If G3iX is nonflat and closed under the structural rules, and
G4iX is terminating and closed under weakening and contraction, then G3iX and G4iX are equivalent (derive exactly the same sequents).
Proof. The proof is an adaptation of the proof of Theorem 1 in [5] and a generalization of the proofs of similar statements in [13]. Under the assumptions in the theorem we have to show that for all sequents S: G3iX S if and only if G4iX S.
The proof of the direction from right to left is straightforward because G3iX is closed under the structural rules, but let us fill in some of the details. We use induction to the height of the proof of a sequent in G4iX, where the height of a derivation is the length of its longest branch, where the length of a branch is the number of nodes it contains. Thus under this convention proofs that consist of a single axiom have height one.
Suppose G4iX S. If S is an instance of an axiom, then clearly G3iX S as well. Suppose S is not an instance of an axiom and consider the last inference of the proof of S. We distinguish according to the rule R of which the last inference is an instance.
If R is a rule that belongs to G3iX we reason as follows. The premises of the instance are both derivable in G3iX by the induction hypothesis. Since R belongs to G3iX, this implies that S is derivable in G3iX as well.
It remains to treat the case that R does not belong to G3iX, which means that it is one of the rules L∧ →, L∨ →, L →→ or Lp →.
In the case of L∧ →, S is of the form Γ, ϕ ∧ ψ → γ ⇒ Δ and the premise of the rule is Γ, ϕ → (ψ → γ) ⇒ Δ. The premise is derivable in G3iX by the induction hypothesis. Since ϕ ∧ ψ → γ ⇒ ϕ → (ψ → γ) is clearly derivable in G3ip, the closure under cut implies that S is derivable in G3iX. The rule L∨ → can be treated in the same manner.
If the rule is Lp →, then S is of the form Γ, p, p → ϕ ⇒ Δ. The premise is Γ, p, ϕ ⇒ Δ, which, by the induction hypothesis, is derivable in G3iX. It is not hard to show that Γ, p, p → ϕ ⇒ ϕ is also derivable in G3iX. Closure under Cut and Contraction shows that so is S.
The other direction, left to right, is proved by induction on the order with respect to which G4iX is terminating. So suppose G3iX S. Sequents lowest in the order do not contain connectives or modal operators by definition of the weight function underlying . Since the calculi are nonflat, such sequents have to be instances of axioms, and since G3iX and G4iX have the same axioms, S is provable in G4iX.
We turn to the case that S is not the lowest in the order. If S a contains a conjunction, S = (Γ, ϕ 1 ∧ϕ 2 ⇒ Δ), then S = (Γ, ϕ 1 , ϕ 2 ⇒ Δ) is provable in G3iX by Lemma 2.4. As G4ip contains L∧ and G4iX is terminating, S S follows. Hence S is provable in G4iX by the induction hypothesis. Thus so is (Γ, ϕ 1 ∧ ϕ 2 ⇒ Δ). A disjunction in S a as well as the case that both p and p → ϕ belong to S a , can be treated in the same way.
Thus only the case that S is irreducible remains, and by Lemmas 2.4 and 3.3 we may assume its proof in G3iX to be sensible and strict. The irreducibility of S implies that the last inference of the proof is an application of a rule, R, that is either a nonmodal right rule, a modal rule or L →. In the first two cases, R belongs to both calculi and the fact that G4iX is terminating implies that the premise(s) of R is lower in the order than S. Thus the induction hypothesis implies that the premise(s) is derivable in G4iX, and since R belongs to G4iX, the conclusion S is derivable in G4iX as well.
We turn to the third case. Suppose that the principal formula of the last inference is (γ → ψ) and S = (Γ, γ → ψ ⇒ Δ). Since the proof is sensible, γ is not atomic. We distinguish according to the main connective of γ.
If γ = ⊥, then Γ ⇒ Δ is derivable in G3iX because of the closure under cut: G3ip derives ( ⇒ ⊥ → ψ), and so the cut S, it follows that Γ ⇒ Δ is derivable in G4iX by the induction hypothesis. As G4iX is closed under weakening, S is derivable in G4iX too.
If γ = ϕ 1 → ϕ 2 , then because γ → ψ is the principal formula, both premises S 1 = (Γ, ψ ⇒ Δ) and Γ, γ → ψ ⇒ γ are derivable in G3iX. Thus so is Γ, γ → ψ, ϕ 1 ⇒ ϕ 2 by the invertibility of R → (Lemma 2.4). Consider the following inference steps, where χ is short for ϕ 2 → ψ: As just observed, the right premise of the cut is derivable in G3iX. It is clear that the left premise of the cut can be extended into a derivation in G3ip (we have provided the last steps and leave the prove that Π, ϕ ⇒ ϕ is derivable in G3ip, for any Π and ϕ, to the reader). Thus both premises of the cut are derivable in G3iX, and therefore the conclusion Γ, χ , ϕ 1 , ϕ 1 ⇒ ϕ 2 is too. As G3iX is closed under contraction and contains R →, the last sequent S 2 = (Γ, ϕ 2 → ψ ⇒ γ) is also derivable in G3iX. Since G4iX is terminating and S 1 and S 2 are the premises of L →→, they both are lower in the order than S. Therefore they are derivable in G4iX by the induction hypothesis. And thus so is S by an application of L →→.
If γ = ϕ, then Remark 3.2 and the fact that the proof is strict and S is irreducible implies that the left premise is the conclusion of an application of a right modal rule R with premises S 1 , . . . , S n . Thus the derivation looks as follows: Since G4iX is terminating, it follows that (Γ, ψ ⇒ Δ) S and S i S. By the induction hypothesis, the S i as well as Γ, ψ ⇒ Δ are derivable in G4iX, say with derivations D i and D 0 , respectively. Since R → belongs to the calculus, the following is a proof of Γ, ϕ → ψ, ϕ → ψ ⇒ Δ in G4iX: Closure under contraction implies that Γ, ϕ → ψ ⇒ Δ is derivable in G4iX.

Conclusion
It has been shown that for any calculus G3iX that consists of G3ip plus a set of nonflat (multi-)modal rules R, there exists a calculus G4iX that is equivalent to G3iX, provided G3iX is nonflat and closed under the structural rules and G4iX is terminating and closed under weakening and contraction. In the setting of intuitionistic modal logics, one usually requires one's sequent calculi to be closed under the structural rules, so that requirement on G3iX is relatively innocent. Likewise for the closure under weakening and contraction of G4iX. The requirement that G4iX be terminating is of a different kind. Although many common modal rules are terminating in the Dyckhoff order, some well-known rules are not. Examples are the standard rules for transitivity, Gödel-Löb Logic and Strong Löb Logic (writing for ): Therefore, to apply the method in this paper to such logics, an order different from the Dyckhoff order has to be found with respect to which G4iX is terminating. Gödel-Löb Logic and Strong Löb Logic are examples for which that can be done, as shown in [8,9] using an ingenious order introduced by Bílková [1]. For reasons that are out of the scope of this paper we conjecture that for G4iK4 and G4iS4 such an order does not exist but leave it for future work to settle that conjecture.
Acknowledgements. I thank Iris van der Giessen for the pleasant collaboration that lead to two papers [8,9] that were the inspiration for the material presented in this paper. I thank an anonymous referee for the useful remarks on an earlier draft of this paper. Support by the Netherlands Organisation for Scientific Research under grant 639.073.807 is gratefully acknowledged.
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