From Axioms to Analytic Rules in Nonclassical Logics

From axioms to analytic rules in nonclassical logics

Agata Ciabattoni∗ Nikolaos Galatos Vienna University of Technology University of Denver [email protected] [email protected] Kazushige Terui National Institute of Informatics, Japan Laboratoire d’Informatique de Paris Nord, CNRS [email protected]

Abstract generalization of Gentzen sequents to multisets of sequents. Since then hypersequent calculi have been discovered for a We introduce a systematic procedure to transform large wide range of nonclassical logics, e.g. [2, 3, 4, 5, 11]. This classes of (Hilbert) axioms into equivalent inference rules is traditionally done by (i) looking for the “right” inference in sequent and hypersequent calculi. This allows for the rule(s) formalizing the particular properties of the logic un- automated generation of analytic calculi for a wide range der consideration (e.g., Avron introduced the rule (com), of propositional nonclassical logics including intermedi- corresponding to the prelinearity axiom) and (ii) proving ate, fuzzy and substructural logics. Our work encom- cut-elimination (or cut-admissibility) to show that the re- passes many existing results, allows for the definition of sulting calculus is analytic. These two steps are usually new calculi and contains a uniform semantic proof of cut- tailored to the particular logic at hand, and each calculus elimination for hypersequent calculi. needs its own proof of soundness, completeness and cut- elimination. This holds even when adding the same rules to different base calculi (e.g. IL + (com) dealt with in [2], IL 1. Introduction + (com) - contraction in [4], and IL + (com) - weakening - contraction in [11]), which might cause a combinatorial explosion on the number of the papers to be produced. Nonclassical logics are often presented by extending In this paper we introduce a systematic procedure for with suitable axioms the (Hilbert) calculi of well known performing step (i) and a uniform (semantic) proof for step systems. The applicability/usefulness of these logics, how- (ii) for a wide range of logics extending FLe1, i.e., in- ever, strongly depends on the availability of analytic calculi. tuitionistic linear logic without exponentials. This allows Such calculi, where proof search proceeds by step-wise de- for the automated generation of analytic calculi for a wide composition of the formula to be proved, are not only useful range of nonclassical logics including intermediate, sub- in establishing important properties of corresponding log- structural and fuzzy logics. ics, such as decidability or the Herbrand theorem, but also the key to develop automated reasoning methods. More precisely, we define a hierarchy – analogous to Since its introduction by Gentzen in [7], sequent calculus the arithmetical hierarchy Σn, Πn – over the formulas of has been one of the preferred frameworks to define analytic FLe and show how to translate the axioms at levels up to calculi. This framework is however not capable of handling N2 (resp. up to P3) into equivalent structural sequent rules all interesting and useful logics. A large range of variants (resp. hypersequent rules). See Figure 2 for examples of and extensions of sequent calculus have been indeed intro- axioms, considered in the literature of intermediate, sub- duced in the last few decades to define analytic calculi for structural and fuzzy logics, that fall into these two classes. logics that seemed to escape a (cut-free) sequent formalisa- When the generated rules satisfy an additional condition or tion; a prominent example being Gödel logic, obtained by the base calculus admits weakening, these are further trans- extending intuitionistic logic IL with the prelinearity axiom formed (completed) into equivalent analytic rules, i.e., they (α → β) ∨ (β → α). An analytic calculus for this logic preserve cut-elimination once added to FLe. The analyt- was defined by Avron using hypersequents [2] – a simple icity of the generated calculi is proved once and for all by

∗Research supported by FWF Project P18731. 1FLe stands for Full Lambek calculus with exchange, see e.g. [8]. extending Okada’s semantic proof of cut-elimination [13] to Notice that a metavariable is used in two ways: as a nota- hypersequent calculi. The completion procedure sheds also tion that stands for (possibly compound) concrete formulas light on the expressive power and limitations of structural and as an (atomic) building block for defining axioms and (hyper)sequent rules. rules. We do not make a rigorous distinction between them, Our work accounts uniformly for many existing results, relying on the standard practice in our field. and new ones can be generated in an automated way. For The notion of proof in FLe is defined as usual. Let R be instance, by applying our procedure a first analytic calculus a set of rules. If there is a proof in FLe extended with R is found for Weak Nilpotent Minimum Logic WNM [6] – (FLe + R, for short) of a sequent S0 from a set of sequents 2 the logic of left-continuous t-norms satisfying the weak S, we say that S0 is derivable from S in FLe + R and write nilpotent minimum property (Corollary 8.9). S ⊢FLe+R S0. We write ⊢FLe+R α if ∅⊢FLe+R ⇒ α. The logical connectives of FLe are classified into two 2. Preliminaries groups according to their polarities [1]: 1, ⊥, ·, ∨ (resp. 0, ⊤, →, ∧) are positive (resp. negative) connectives for which the left (resp. right) logical rule is invertible, i.e., the The base calculus we will deal with is the sequent sys- conclusion implies the premises. E.g. we have for (∨l): tem FLe i.e., Full Lambek calculus FL extended with exchange (see e.g. [8]). Roughly speaking, FLe is obtained • ⊢FLe α ∨ β, Γ ⇒ Π iff ⊢FLe α, Γ ⇒ Π and ⊢FLe by dropping the structural rules of weakening (w) and con- β, Γ ⇒ Π. traction (c) from the sequent calculus LJ (FLewc, in our Connectives of the same polarity interact well with each terminology) for intuitionistic logic. Also, FLe is the same other: as intuitionistic linear logic without exponentials. The formulas of FLe are built from propositional vari- (P) ⊢FLe α · 1 ↔ α, α ∨ ⊥ ↔ α, (α ·⊥) ↔⊥, ables p,q,r,... and the 0-ary connectives (constants) 1 α · (β ∨ γ) ↔ (α · β) ∨ (α · γ). (unit), ⊥ (false), ⊤ (true) and 0 by using the binary logi- (N) ⊢FLe α ∧ ⊤ ↔ α, (1 → α) ↔ α, (α →⊤) ↔⊤, cal connectives · (fusion), → (implication), ∧ (conjunction) (α → (β ∧γ)) ↔ (α → β)∧(α → γ), (⊥ → α) ↔⊤, and ∨ (disjunction). ¬α and α ↔ β will be used as ab- ((α ∨ β) → γ) ↔ (α → γ) ∧ (β → γ). breviations for α → 0 and (α → β) ∧ (β → α). (Our notation should not be confused with that of linear logic, (Notice that polarity is reversed on the left hand side of an where symbol 0 is used for ⊥ and vice versa.) implication. For instance, the ∨ on the left-hand side (LHS) Henceforth metavariables α,β,... will denote formulas, of → in the last equivalence is considered negative.) Π, Θ will stand for stoups, i.e., either a formulaor the empty Since connectives ∧, ∨, · have units ⊤, ⊥, 1 respectively, set, and Γ, ∆,... for finite (possibly empty) multisets of we adopt a natural convention: β1 ∨···∨βm (resp. β1 ∧···∧ formulas. In this paper we will only consider sequents in βm and β1 · · · βm) stands for ⊥ (resp. ⊤ and 1) if m =0. the language of FLe that are single-conclusion, i.e., whose We say that two formulas α and β are (externally) equiv- right-hand side (RHS) contains at most one formula. As alent in FLe if α ⊢FLe β and β ⊢FLe α. Obviously usual, axioms and inference rules are specified by using ⊢FLe α ↔ β implies that α and β are equivalent, while metavariables together with metaformulas, i.e., expressions the converse does not hold due to lack of a deduction theo- built from metavariables α,β,... by using the logical con- rem. A counterexample is that α and α ∧ 1 are equivalent nectives of FLe. See Fig. 1 for the inference rules of FLe. whereas 6⊢FLe α ↔ α ∧ 1. In contrast with internal equiv- An axiom (scheme) is a metaformula ψ, which we iden- alence (i.e. ⊢FLe α ↔ β), external equivalence is not a tify with a rule ⇒ ψ with 0 premises. congruence; indeed, α ∨ β and (α ∧ 1) ∨ β are not equiva- By structural rule we mean any sequent rule, with the lent. If we are allowed to use the modality !α of linear logic, exception of (init) and (cut), of the form (n ≥ 0) external equivalence can be internalized: ⊢FLe!α ↔!β. Two rules (r0) and (r1) are equivalent (in FLe) if the Υ ⇒ Ψ · · · Υ ⇒ Ψ relations ⊢FLe and ⊢FLe coincide. In particular, 1 1 n n (r) +(r0) +(r1) Υ0 ⇒ Ψ0 when the conclusion of (r0) (and resp. of (r1)) is derivable from its premises in FLe+(r1) (resp. FLe+(r0)) then (r0) where each Υi is a specific multiset of metavariables al- and (r1) are equivalent. The definition naturally extends to lowed to be of both types: metavariables for formulas (α)or sets of rules. for multisets of formulas (Γ), and each Ψi is either empty, a metavariable for formulas, or a metavariable for stoups (Π). 3. Substructural Hierarchy Examples of structural rules are found in Figure 3.

2T -norms are the main tool in fuzzy logic to combine vague informa- To address systematically the problem of translating ax- tion. ioms into equivalent structural rules in an automated way

2 G | Γ ⇒ α G | α, ∆ ⇒ Π G | Γ ⇒ Π (cut) (init) (1l) (1r) G | Γ, ∆ ⇒ Π G | α ⇒ α G | 1, Γ ⇒ Π G | ⇒ 1

G | α, β, Γ ⇒ Π G | Γ ⇒ α G | ∆ ⇒ β G | Γ ⇒ (· l) (· r) (0r) (0l) G | α · β, Γ ⇒ Π G | Γ, ∆ ⇒ α · β G | Γ ⇒ 0 G | 0 ⇒

G | Γ ⇒ α G | β, ∆ ⇒ Π G | α, Γ ⇒ β G | αi, Γ ⇒ Π (→ l) (→ r) (∧l) (⊥l) G | Γ, α → β, ∆ ⇒ Π G | Γ ⇒ α → β G | α1 ∧ α2, Γ ⇒ Π G |⊥, Γ ⇒ Π

G | Γ ⇒ Π | Γ ⇒ Π G (EW ) (EC) G | Γ ⇒ Π G | Γ ⇒ Π The inference rules of FLe are obtained by dropping ‘G | ’ and removing (EW ), (EC).

Figure 1. Inference Rules of HFLe

we introduce below a hierarchy on the FLe formulas based It is easy to see that formulas in each class admit the on their polarities, which is analogous to the arithmetical following normal forms: hierarchy Σn, Πn. Lemma 3.4. Deﬁnition 3.1. For each n ≥ 0, the sets Pn, Nn of (positive and negative) formulas are deﬁned as follows: (P) If α ∈Pn+1, then we have ⊢FLe α ↔ β1 ∨···∨ βm, where each βi is a fusion of formulas in Nn. (0) P0 = N0 = the set of propositional variables.

(N) If , then FLe , (P1) 1, ⊥ and any formula α ∈ Nn belong to Pn+1. α ∈ Nn+1 ⊢ α ↔ (V1≤i≤m γi → βi) where each βi is either 0 or a formula in Pn, and each (P2) If α, β ∈Pn+1, then α ∨ β, α · β ∈Pn+1. γi is a fusion of formulas in Nn.

(N1) 0, ⊤ and any formula α ∈Pn belong to Nn+1. A formula α is said to be N2-normal if it is of the form where (N2) If α, β ∈ Nn+1, then α ∧ β ∈ Nn+1. α1 · · · αn → β

(N3) If α ∈Pn+1 and β ∈ Nn+1, then α → β ∈ Nn+1. • β =0 or β1 ∨···∨ βk with each βi a fusion of propositional variables and An axiom ψ belongs to Pn (resp. Nn) if this holds when its metavariables are instantiated with propositional variables. each is of the form j j , where j • αi V1≤j≤mi γi → βi βi = j Example 3.2. ¬(α · β) ∨ (α ∧ β → α · β) (weak nilpotent 0 or is a propositional variable, and γi is a fusion of minimum [6]) ∈P3 while Łukasiewicz axiom ((α → β) → propositional variables. β) → ((β → α) → α) ∈ N3. See Figure 2 for further examples. As a consequence of the above lemma, every formula α in N2 can be transformed into a ﬁnite conjunction V i n αi Theorem 3.3. 1≤ ≤ of N2-normal formulas such that ⊢FLe α ↔ V1≤i≤n αi. FLe 1. Every FLe formula belongs to some Pn and Nn. The lack of weakening in makes it hard to deal with ′ formulas in P3. We therefore introduce a subclass P3 of P3 2. Pn ⊆Pn+1 and Nn ⊆ Nn+1 for every n. deﬁned as: Hence the classes P , N constitute a hierarchy, which n n 1. 1, ⊥∈P′ we call substructural hierarchy, of the following form: 3 pppppppp P0 - P1 - P2 - P3 - ′ 2. If α ∈ N2 then α ∧ 1 ∈P3 @ @ @ ′ ′ @ @ @ 3. If α, β ∈P3 then α · β, α ∨ β ∈P3. @R @R @@R - - - pppppppppp- N0 N1 N2 N3 Henceforth we write (α)∧1 for α ∧ 1.

3 Class Axiom Name Rule (cf. Fig. 3) Reference ′ N2 α → 1, 0 → α weakening (w), (w ) [7] α → α · α contraction (c) [7] α · α → α expansion (mingle) [12] n m n ∗2 α → α knotted axioms (n,m ≥ 0) (knotm) [9, 16] ¬(α ∧ ¬α) weak contraction (wc) P2 α ∨ ¬α excluded middle (em) [3] (α → β) ∨ (β → α) prelinearity (com)∗1 [2] ′ ∗3 P3 ((α → β) ∧ 1) ∨ ((β → α) ∧ 1) linearity (com) [2, 4, 11] ∗1 P3 ¬α ∨ ¬¬α weak excluded middle (lq) [5] k ∗1 Wi=0(pi → Wj6=i pj) Kripke models with width ≤ k (Bwk) [5] ∗1 p0 ∨ (p0 → p1) ∨···∨ (p0 ∧···∧ pk−1 → pk) Kripke models with k worlds (Bck) [5] ∗1: The rule is equivalent to the axiom in FLew. n ∗2: The rules (knotm) arise by applying the completion procedure in [18] to the rules in [16, 9]. ∗3: The rule (com) is due to [2] (added to IL). Later added to FLew by [4] and to FLe by [11]. Figure 2. Axioms vs (hyper)sequent rules

′ Lemma 3.5. Every formula in P3 is equivalent to a ﬁnite Proof. The left-to-right direction follows by (cut). For in- set of formulas (α1)∧1 ∨···∨ (αn)∧1 where each αi is N2- stance, assume the premises of (r2). Then ψ1,...,ψn ⇒ ξ normal. follows from S~ by (r). One can then apply (cut) to obtain the conclusion of (r2). For the right-to-left direction we Proof. We may assume that any formula ψ ∈ P′ is built 3 instantiate αi with ψi (i =1,...,n) and β with ξ. from conjunctions of N2-normal formulas by clauses 2 and 3. ψ can be transformed into the desired sets by applying the equivalences between Theorem 4.2. Every axiom in N2 is equivalent to a ﬁnite set of structural rules. • (α∧1 · β∧1) ∨ γ and the set {α∧1 ∨ γ, β∧1 ∨ γ},

Proof. By Lemma 3.4, it sufﬁces to consider N2-normal ax- • (α ∧ β)∧1 ∨ γ and {α∧1 ∨ γ, β∧1 ∨ γ}. ioms . Let be with j ψ ψ ψ1 · · · ψn → ξ ψi = V1≤j≤mi φi → j ′ ξ for i =1,...,n. By the invertibility of (→ r), (· l), (1l) where γ is any formula in P3. To prove these equivalences i we use the fact that α∧1 ∨ γ,β∧1 ∨ γ ⇒ (α∧1 · β∧1) ∨ γ and and Lemma 4.1 follows that ψ1 · · · ψn → ξ is equivalent to (α∧1 · β∧1) ∨ γ ⇒ (α ∧ β)∧1 ∨ γ are provable in FLe. the rule α1 ⇒ ψ1 · · · αn ⇒ ψn Notice that in presence of weakening (i.e., in FLew) α1,...,αn ⇒ ξ Lemma 3.5 holds for every formula in P3, as ⊢FLew α ↔ with α1,...,αn fresh metavariables. The invertibility of α∧1. (∧r), (→ r), (· l), (1l) and (0r) allows us to replace the premises with an equivalent set S of sequents that consist 4. N2 and sequent rules of metavariables only. If ξ = 0, then remove it from the conclusion to obtain a structural rule. Otherwise, ξ = ξ1 ∨ We provide a systematic procedure for transforming any ···∨ξk. By Lemma 4.1 and the invertibility of (∨l) follows that the above rule is equivalent to axiom in N2 into a set of equivalent structural rules. The key observation being S ξ1 ⇒ β · · · ξk ⇒ β S1 · · · Sm α1,...,αn ⇒ β Lemma 4.1. The rule (r) is equivalent to ψ1,...,ψn ⇒ ξ each of the rules with β fresh metavariable. The claim follows by the invertibility of (· l) and (1l). S~ α1 ⇒ ψ1 · · · αn ⇒ ψn S~ ξ ⇒ β (r1) (r2) α1,...,αn ⇒ ξ ψ1,...,ψn ⇒ β Example 4.3. Using the algorithm described in the proof of ~ where S = S1 · · · Sm and α1,...,αn,β are fresh metavari- the theorem above, axiom ¬(α∧¬α) (weak contraction, see ables for formulas. Figure 2) is transformed into an equivalent structural rule as

4 follows: act on several components of one or more hypersequents. It is this type of rule which increases the expressive power α α α α ⇒ ¬( ∧ ¬ ) −→ ( ∧ ¬ ) ⇒ of hypersequent calculi compared to ordinary sequent cal- −→ β ⇒ α ∧ ¬α culi. Examples of these rules are Avron’s communication β ⇒ rule (com) and (lq) in Figure 3. E.g., extending the hyper- −→ β ⇒ α α, β ⇒ sequent version of LJ, that is HFLewc, by β ⇒ • (com) we get a cut-free calculus for Godel¨ logic, ax- A further transformation (called completion), described in iomatized by (any Hilbert system for) intuitionistic Section 6, makes it into the analytic rule (wc) in Fig. 3. logic IL + (prelinearity), see [2].

5. P and hypersequent rules • (lq) we get a cut-free calculus for the intermediate 3 logic LQ axiomatized by IL + (weak excluded middle), see [5]. Consider some axiom beyond N2 such as weak excluded middle ¬α ∨ ¬¬α and prelinearity (α → β) ∨ (β → α), More formally, by a hypersequent structural rule (hyper- see Fig. 2. Since they are neither valid nor contradictory in structural rule, for short) we mean (EW ), (EC) and any intuitionistic logic, Corollary 7.2 ensures that no structural rule, except for (init) and (cut), of the form(n ≥ 0) rule added to FLe is equivalent to them. These axioms ′ ′ ′ ′ G | Υ1 ⇒ Ψ1 · · · G | Υn ⇒ Ψn have been instead formalized in a natural way by structural (hr) rules in hypersequent calculus – a simple generalization of G | Υ1 ⇒ Ψ1 | ··· | Υm ⇒ Ψm sequent calculus due to Avron (see [3] for an overview). ′ where G is a metavariable that stands for hypersequents, We show below that this holds for all axioms in the class P3 ′ ′ and Υi, Ψi, Υ , Υ are as in structural rules. Observe that, (P , in presence of weakening)and we providean algorithm j j 3 with the notable exceptions of (EC) and (EW ), each for transforming any such axiom into a set of equivalent premise of a hypersequent structural rule contains exactly structural hypersequent rules. ′ ′ one active component Υi ⇒ Ψi. Definition 5.1. A hypersequent G is a multiset S1 | . . . | Sn Two hypersequent rules (hr0) and (hr1) are equiva- where each Si for i =1,...,n is a sequent, called a compo- lent in HFLe if the derivability relations ⊢HFLe+(hr0) 3 nent of the hypersequent. A hypersequent is called single- and ⊢HFLe+(hr1) coincide when restricted to sequents : conclusion if all its components are single-conclusion,it is S ⊢HFLe+(hr0) S0 iff S ⊢HFLe+(hr1) S0 for any set called multiple-conclusion otherwise. S∪{S0} of sequents. We introduce below an algorithm to transform axioms in The symbol “|” is intended to denote disjunction at the subclass P′ of P into equivalent hyperstructural rules. the meta-level. (This will be made precise in Defini- 3 3 Let us begin with establishing a connection between the two tion 5.2 below.) As sequents are assumed here to be derivability relations ⊢FLe and ⊢HFLe. single-conclusion, hypersequents are likewise assumed to be single-conclusion. Definition 5.2. We define the interpretation function ( )I Like sequent calculi, hypersequent calculi consist of ini- as follows: tial hypersequents, logical rules, cut and structural rules. I Initial hypersequents, cut and logical rules are the same as 1. (α1,...,αn ⇒ β) = α1 · · · αn → β. those in sequent calculi, except that a “side hypersequent” I 2. (α1,...,αn ⇒ ) = α1 · · · αn → 0. may occur, denoted here by the variable G. Structural rules I I I are divided into two categories: internal and external rules. 3. (S1|···|Sn) = (S1 )∧1 ∨···∨ (Sn)∧1. The former deal with formulaswithin sequents as in sequent Note that our interpretation of “|” differs from that of calculi. External rules manipulate whole sequents. For ex- [11] due to lack of weakening and linearity. ample, external weakening and contraction rules (EW ) and We obviously have S ⊢FLe S if and only if (EC) in Figure 1 add and contract componentsrespectively. +ψ 0 S ⊢HFLe S for any axiom ψ and any set S∪{S } Henceforth we will consider HFLe the hypersequent +ψ 0 0 of sequents. However, the next proposition gives us a cru- version of FLe (Figure 1). Let ⊢HFLe indicate the deriv- cial step to “unfold” an axiom ψ to a hyperstructural rule in ability relation in HFLe. Note that the “hyperlevel” HFLe. of HFLe is in fact redundant, in the sense that ⊢HFLe 3 S1 | ··· | Sn if and only if for some i ∈ {1,...,n}, al- The reason for this restriction is that we are primarily interested in derivability relations on formulas (or at most sequents); hypersequents are ready ⊢FLe S . i merely a convenient means to obtain analytic calculi. Note that our main In hypersequent calculi it is however possible to define theorem in this section (Theorem 5.6) holds only with this restricted notion additional external structural rules which simultaneously of equivalence.

5 ∆, ∆, Γ ⇒ Π Γ, Γ ⇒ ∆ , Γ ⇒ Π ∆ , Γ ⇒ Π Γ ⇒ Π (w) Γ ⇒ (w′) (c) (wc) 1 2 (mingle) ∆, Γ ⇒ Π Γ ⇒ Π ∆, Γ ⇒ Π Γ ⇒ ∆1, ∆2, Γ ⇒ Π

{∆ 1 ,..., ∆ , Γ ⇒ Π} {G | Γ , Γ ⇒ Π } i im i1,...,im∈{1,...,n} n G | Γ, ∆ ⇒ Π i j i 0≤i,j≤k,i6=j (knotm) (em) (Bwk) ∆1,..., ∆n, Γ ⇒ Π G | Γ ⇒ | ∆ ⇒ Π G | Γ0 ⇒ Π0 | . . . | Γk ⇒ Πk

G | Γ, ∆ ⇒ G | ∆1, Γ1 ⇒ Π1 G | ∆2, Γ2 ⇒ Π2 {G | Γi, Γj ⇒ Πi}0≤i≤k−1; i+1≤j≤k (lq) (com) (Bck) G | Γ ⇒ | ∆ ⇒ G | ∆2, Γ1 ⇒ Π1 | ∆1, Γ2 ⇒ Π2 G | Γ0 ⇒ Π0 | . . . | Γk−1 ⇒ Πk−1 | Γk ⇒

Figure 3. Some Known (Hyper) structural rules

Proposition 5.3. For any hypersequent G and any set S ∪ is equivalent to each of the rules I HFLe −−−→ −−−→ {S0} of sequents, we have {G}∪S ⊢ S0 iff {⇒ G }∪ G | Φ G | Υ1 ⇒ ψ1 ··· G | Υn ⇒ ψn G | Φ G | ξ, Υ ⇒ Ψ I HFLe FLe S ⊢ S0 iff {⇒ G }∪S⊢ S0. G | Φ | Υ1,..., Υn ⇒ ξ G | Φ | ψ1,...,ψn, Υ ⇒ Ψ I I −−−→ Proof. {⇒ G }∪S ⊢FLe S0 obviously implies {⇒ G }∪ where G | Φ = (G | Φ1, · · · , G | Φm), Υi is a fresh S ⊢HFLe S0, which in turn implies {G}∪S ⊢HFLe S0 by metavariable αi or Γi, and Υ ⇒ Ψ is either ⇒ β or Σ ⇒ Π I G ⊢HFLe⇒ G . For instance when G = (⇒ α | ⇒ β), with β, Σ, Π fresh. we have Proof. Proceed as the proof of Lemma 4.1. To see that the (1r) third rule implies the first one, instantiate Υ ⇒ Ψ with ⇒ ξ. ⇒ 1 (EW ) (1r) The converse direction follows by (cut). ⇒ α | ⇒ β ⇒ 1 | ⇒ β ⇒ 1 (∧r) (EW ) ⇒ α∧1 | ⇒ β ⇒ α∧1 | ⇒ 1 Theorem 5.6. (∧r) ⇒ α∧1 | ⇒ β∧1 ′ (∨r) 1. Every axiomin P3 is equivalent to a finite set of hyper- ⇒ α∧1 ∨ β∧1 | ⇒ α∧1 ∨ β∧1 structural rules in HFLe. (EC) ⇒ α∧1 ∨ β∧1 . 2. Every axiomin P3 is equivalent to a finite set of hyper- I To show that {G}∪S ⊢HFLe S0 implies {⇒ G }∪S ⊢FLe structural rules in HFLew. S0, we prove by induction on the length of derivation that Proof. 1. By Lemma 3.5, every axiom in P′ is equiva- I I 3 {G}∪S ⊢HFLe G0 implies {⇒ G }∪S ⊢FLe G0 for any lent to a finite set of axioms (ψ ) ∨···∨ (ψ ) where I 1 ∧1 n ∧1 hypersequent G0. The claim then follows since S implies 0 ψ1,...,ψn are N2-normal. By Corollary 5.4, the latter is S0 in FLe. equivalent to ⇒ ψ1 |···| ⇒ ψn in HFLe, which is in The base case being easy. For the inductive case it is turn equivalent to Φ = (G | ⇒ ψ1 |···| ⇒ ψn) with G a enough to observe that for each inference rule in HFLe metavariable for hypersequents, by (EW ) and instantiation with premises G | S1,..., G | Sn and conclusion G | S0, the of G with the empty hypersequent. By applying the pro- I I I sequent (G | S1) ,..., (G | Sn) ⇒ (G | S0) is provable cedure described in the proof of Theorem 4.2 to each com- in FLe. For instance we have for the ∧ right rule: (assume ponent of Φ we obtain a finite set of hyperstructural rules, for simplicity that G consists of a single component) which is equivalent to Φ by Lemma 5.5. I I I I ⊢FLe (G )∧1 ∨ (Γ ⇒ α)∧1, (G )∧1 ∨ (Γ ⇒ β)∧1 2. Follows by ⊢FLew α ↔ α ∧ 1. I I ⇒ (G )∧1 ∨ (Γ ⇒ α ∧ β)∧1. ′ Example 5.7. The P3 version of the weak nilpotent minimum axiom (see Example 3.2), i.e., Corollary 5.4. For any hypersequent G and any set (¬(α · β))∧1 ∨ (α ∧ β → α · β)∧1 S ∪{S0} of sequents, we have S ⊢HFLe+G S0 iff is transformed into the hyperstructural rule (wnm0) as fol- S ⊢HFLe I S iff S ⊢FLe I S . +(⇒G ) 0 +(⇒G ) 0 lows: The key Lemma 4.1 naturally extends to hypersequents. −→ G | ⇒ ¬(α · β) | ⇒ α ∧ β → α · β We state it in a slightly generalized form for later use. −→ G | α, β ⇒ | α ∧ β ⇒ α · β Lemma 5.5. Let be (meta)hypersequents Φ, Φ1,..., Φm G | τ ⇒ α ∧ β G | α · β ⇒ σ consisting of metavariables. The hypersequent rule −→ G | α, β ⇒ | τ ⇒ σ G | Φ1 · · · G | Φm G | τ ⇒ α G | τ ⇒ β G | α, β ⇒ σ −→ (wnm0) G | Φ | ψ1,...,ψn ⇒ ξ G | α, β ⇒ | τ ⇒ σ

6 ′ Linearity and the P3 version of the weak law of excluded Clearly the original rule implies the new one. The con- middle (see Fig. 2) are transformed into verse also holds because any multiset Γ = α1,...,αn of formulas (resp. the empty stoup Π= ∅) can be turnedinto a G | β ⇒ γ G | α ⇒ δ G | γ ⇒ α G | β, α ⇒ (com0) (lq0) single formula ◦Γ = α1 · · · αn (resp. 0). Hence given con- G | α ⇒ γ | β ⇒ δ G | γ ⇒ | β ⇒ crete instances of the premises of the original rule, one can first replace a multiset Γ (resp. the empty stoup) with for- 6. Rule Completion mula ◦Γ (resp. 0), apply the new rule and later on recover the multiset Γ (resp. the empty stoup) by the invertibility of ′ As seen in the previous section, all axioms in P3 (and in (· l), (1l) and (0r). presence of weakening in P3) can be transformed into hy- This step is not needed when the given hyperstructural perstructural rules. These rules however do not always pre- rule contains neither Γ nor Π, as in the case of the rules serve cut admissibility once added to HFLe. For instance, generated by the algorithms in Theorems 4.2 and 5.6. No- HFLewc extended with tice that this step preserves acyclicity, i.e. if the original rule G | Γ, ∆ ⇒ α is acyclic, so is the rule after applying the preliminary step. (SI ) G | Γ ⇒ α | ∆ ⇒ α Example 6.3. Applied to the rule (SI ) above the prelimi- does not enjoy cut admissibility, see [3]. Nevertheless, the nary yields G | βΓ, β∆ ⇒ α above rule can be transformed into an equivalent one (in (S′) HFLe) which does preserve cut admissibility. The same G | βΓ ⇒ α | β∆ ⇒ α applies to any hyperstructural rule when we admit weaken- 2. Restructuring. Given any hyperstructural rule only ing (see Section 7), or when the rule satisfies the acyclicity containing metavariables for formulas. We replace each condition below. The purpose of this section is to describe component (α ,...,α ⇒ β) in its conclusion with this transformation (we refer to it as completion), which ex- 1 n (Γ ,..., Γ , Σ ⇒ Π ) and add n+1 premises (G | Γ ⇒ tends a similar procedure in [18] that works for suitable se- 1 n β β 1 α ), ..., (G | Γ ⇒ α ), (G | β, Σ ⇒ Π ), quent structural rules in FL, and is analogous to the princi- 1 n n β β where Γ ,..., Γ , Σ , Π are fresh and mutually distinct ple of reflection in [17]. 1 n β β metavariables. Likewise, we replace each component Definition 6.1. Given a hyperstructural rule in its conclusion of the form (α1,...,αn ⇒ ) with

′ ′ ′ ′ (Γ1,..., Γn ⇒ ) and add n premises (G | Γ1 ⇒ α1), G | Υ1 ⇒ Ψ1 · · · G | Υn ⇒ Ψn (hr) ..., (G | Γn ⇒ αn). As a result, we obtain a new rule in G | Υ1 ⇒ Ψ1 | ··· | Υm ⇒ Ψm which we build its dependency graph D(hr) as follows: (linear-conclusion) each metavariable occurs (at most) • The vertices of D(hr) are the metavariables for once in the conclusion formulas occurring in the premises G | Υ′ ⇒ 1 (separation) no metavariable occurring on the LHS (resp. Ψ′ , · · · , G | Υ′ ⇒ Ψ′ (we do not distinguish occur- 1 n n RHS) of a component of the conclusion does occur on rences). the RHS (resp. LHS) of a premise, and • There is a directed edge α −→ β in D(r) if and only ′ ′ (coupling) any pair (Σβ, Πβ) of metavariables associated if there is a premise G | Υi ⇒ Ψi such that α occurs ′ ′ to the same occurrence of β always occur together, in Υi and β =Ψ . namely Σβ occurs in a premise iff Πβ does. A hyperstructural rule (hr) is said to be acyclic if D(hr) is acyclic. Example 6.4. Applied to (wnm0) in Example 5.7, the restructuring step yields a new rule (wnm1) Example 6.2. The rules (wnm0), (com0) and (lq0) in Ex- ample 5.7 and (SI ) above are acyclic, while this is not the G | τ ⇒ α G | τ ⇒ β G | α, β ⇒ σ case for the rule G | Γ ⇒ α G | ∆ ⇒ β G | Λ ⇒ τ G | σ, Σ ⇒ Π G | γ, α ⇒ β G | β ⇒ α G | Γ, ∆ ⇒ | Λ, Σ ⇒ Π G | γ ⇒ β while applied to the rule (S′) in Example 6.3 this step yields 1. Preliminary step. Given any hyperstructural rule, we replace each metavariable Γ for multisets of formulas (resp. G | βΓ, β∆ ⇒ α G | Γ1 ⇒ βΓ each metavariable Π for stoups), if any, by a fresh metavari- G | α, Λ1 ⇒ Π1 G | Γ2 ⇒ β∆ G | α, Λ2 ⇒ Π2 G 1, 1 1 2, 2 2 able βΓ (resp. βΠ) for formulas. | Γ Λ ⇒ Π | Γ Λ ⇒ Π

7 When the original rule is acyclic, so is the resulting rule as (linear-conclusion) and (coupling) together with a strength- we add only metavariables for multisets and stoups. The ened form of (separation): equivalence with the original rule is ensured by Lemma (strong subformula property) every metavariable occur- 5.5. ring on the LHS (resp. RHS) in a premise also occurs on the LHS (resp. RHS) of the conclusion. 3. Cutting. Given any acyclic hyperstructural rule. We eliminate from the set G of its premises all metavariables Example 6.6. The completion of the rule (SI ) leads to the not occurring in the conclusion (we call these variables re- rule dundant). The procedure is as follows. G | Γ1, Γ2, Λ1 ⇒ Π1 G | Γ1, Γ2, Λ2 ⇒ Π2 G | Γ1, Λ1 ⇒ Π1 | Γ2, Λ2 ⇒ Π2 Let α be such a redundant metavariable and G1 = ′ {G | Υi ⇒ α :1 ≤ i ≤ k} be the subset of premises which suggested by Mints (see [3]) and equivalent to (com) in have α on the RHS, and G2 = {G | Υj ,α,...,α ⇒ Ψj : presence of weakening and contraction. 1 ≤ j ≤ m} be the set of those which have one or more Example 6.7. By applying to the axioms in Figure 2 the occurrences of α on the LHS, where Υ does not contain α. j translation of Sections 4 and 5 followed by the completion By acyclicity, α does not appear in Υ′ and Ψ . i j procedure, we obtain the known rules in Figure 3 (up to If k = 0 (resp. m = 0), then remove G (resp. G ) from 2 1 contraction (c) for (Bwk) and (Bck)). G. The resulting rule implies the original one, by instantiat- ing α with ⊥ (resp. ⊤). (This remains true even if ⊥, ⊤ are not in the language.) 7 On the power of (hyper)structural rules Otherwise, let G3 be the set of all hypersequents of the ′ ′ If we admit weakening the completion procedure de- form G | Υj, Υi1 ,..., Υip ⇒ Ψj , where 1 ≤ j ≤ m and 1 ≤ i1,...,ip ≤ k. We replace G1 ∪G2 with G3 thus obtain- scribed in Section 6 does not need the acyclicity condition ing a new hyperstructural rule. It is clear that the acyclicity anymore and hence all (hyper)structural rule can be com- is preserved by this transformation and the number of re- pleted. This observation leads to two results (Corollary 7.2 dundantvariables decreases by one. Hence by repeating this and Corollary 7.3) that shed light on the expressive power process, we obtain a hyperstructural rule without redundant of single-conclusion (hyper)sequent calculi. variables. Theorem 7.1. Example 6.5. Applied to the metavariables τ and σ of (a) Any acyclic hyperstructural rule can be transformed the rule (wnm1) in the previous example, the cutting step into a completed rule which is equivalent in HFLe; yields (b) Any hyperstructural rule can be transformed into a G | Λ ⇒ α G | Λ ⇒ β completed rule which is equivalent in HFLew. G | Γ ⇒ α G | ∆ ⇒ β G | α, β, Σ ⇒ Π (wnm2) G | Γ, ∆ ⇒ | Λ, Σ ⇒ Π Proof. (a) Follows by results in the preceeding section. (b) Steps 1 and 2 in Section 6 can be applied to any hy- and applied further on α and β, perstructural rule. As for step 3 (Cutting), all premises in G , i = 1, 2, 3 of the form Υ, α ⇒ α can be simply re- G | Γ, ∆, Σ ⇒ Π, G | Λ, ∆, Σ ⇒ Π i G | Γ, Λ, Σ ⇒ Π, G | Λ, Λ, Σ ⇒ Π moved, being already derivable by weakening in HFLew. (wnm) It is easy to see that the resulting rule is equivalent to the G , , | Γ ∆ ⇒ | Λ Σ ⇒ Π original rule in HFLew. To see that this step preserves equivalence, we show that The completion procedure for hyperstructural rules out- the two rules above ((wnm ) and (wnm)) are equivalent. 2 lined in Section 6 subsumes completion of structural rules It is clear that the conclusion G | Γ, ∆ ⇒ | Λ, Σ ⇒ Π in sequent calculi. Hence is derivable from the premises of (wnm2) by using (cut) and (wnm). Conversely, consider concrete instances of the Corollary 7.2. Any structural rule is either derivable in premises of (wnm). Let ◦Λ be the fusion of all formulas in Gentzen’s LJ or derives every formula in LJ. Λ, and α = ◦Λ ∨◦Γ, β = ◦Λ ∨◦∆. Since (G | Λ ⇒ α), Proof. Given a structural rule (r), we apply the completion (G | Λ ⇒ β), (G | Γ ⇒ α) and (G | ∆ ⇒ β) are provable procedure to obtain, by Theorem 7.1(b), a completed rule and G | α, β, Σ ⇒ Π is derivable from the premises of (r′) equivalent to (r) in LJ. If (r′) has no premises, by (wnm), we obtain the conclusion by (wnm ). 2 linear-conclusion any formula is provable in LJ + (r′) and We call completed any hyperstructural rule obtained by hence in LJ + (r). Otherwise, the conclusion of (r′) is applying the above completion procedure (steps 1-3). derivable from any of its premises by weakening and con- Any completed hyperstructural rule satisﬁes the properties traction due to the strong subformula property.

8 Let HSM be the hypersequent calculus HFLewc + metavariables in Υ0 and those for stoups with α, and all (com) + (Bc2) (see Figure 3) introduced in [5] for three- others with the empty multiset. Then all the premises of valued G¨odel logic SM – the strongest intermediate logic, (hr)′ are of the form α,...,α ⇒ α and hence are provable semantically characterized by linearly ordered Kripke mod- in HFLew. By the linear-conclusion property, the conclu- els containing two worlds. sion is of the form ⇒ α |···| ⇒ α | α,...,α ⇒, from which α ∨ ¬αn is easily derivable in HFLew. Corollary 7.3. Any hyperstructural rule is either derivable n in HSM or derives α ∨ ¬α in HFLew, for some n ∈ N. Corollaries 7.2 and 7.3 can be used to establish negative Proof. First note that both rules results on the transformation of axioms into inference rules.

G | Υ1, Υ2 ⇒ Π1 G | Υ0, Υ2 ⇒ Π2 Example 7.4. No (hyper)structural rule is equivalent to (Bc2′) G | Υ1 ⇒ Π1 | Υ2 ⇒ Π2 | Υ0 ⇒ ((α → β) → β) → ((β → α) → α) (Łukasiewicz axiom, see Example 3.2). Indeed this axiom is not valid in and SM (take an evaluation v in SM, i.e., v : SM-formulas

1 n 1 0 1 n i 2 i n → {0, 1/2, 1} with v(α → β)=1 if v(α) ≤ v(β) and G | Υ ,..., Υ ⇒ Π {G | Υ , Υ ,..., Υ ⇒ Π } ≤ ≤ ′ (Bcn ) v(α → β) = v(β) otherwise, and assign v(α) =1/2 G | Υ1 ⇒ Π1 | ··· | Υn ⇒ Πn | Υ0 ⇒ and v(β) =0). Moreover for no finite n, is α ∨ ¬αn are derivable in HSM. Indeed (Bc2′) follows by derivable from HFLew extended with Łukasiewicz ax- (com), (Bc2), (c), (EW ) and (EC), while (Bcn′) is deriv- iom. This follows from the fact that Corollary 5.4 also able in HSM by n − 1 applications of both (w) and holds for HFLew and FLew and from the non validity ′ n (Bc2 ) with premises G | Υ1,..., Υn ⇒ Π1 and of α ∨ ¬α in infinite-valued Łukasiewicz logic L, which G | Υ0,..., Υn ⇒ Πi (2 ≤ i ≤ n), followed by several is obtained by adding Łukasiewicz axiom to FLew (take applications of (com), (c), (EW ) and (EC). an evaluation v in L, i.e., v : L-formulas → [0, 1] with Given any (hyper)structural rule. By Theorem 7.1(b) it v(¬α)=1 − v(α), v(α · β) = max{0, v(α)+ v(β) − 1} is equivalent in HFLew to a completed rule, say (hr). and assign v(α)= n/(n + 1)). If at least one active component in the premises of (hr) has empty RHS then (hr) is derivable in HSM (use 8. Cut Elimination (com), (w), (w′), (c) and (EW )). Otherwise, we can assume that (hr) has the form We introduce a uniform (and first semantic) proof of cut- ′ ′ G | Ξ1 ⇒ Π1 ... G | Ξk ⇒ Πk admissibility for any (hyper)sequentcalculus defined by ex- ′ ′ G | Υ1 ⇒ Π1 | ··· | Υn ⇒ Πn | Υ1 ⇒ | . . . | Υm ⇒ tending HFLe with any set of completed rules. Our result is obtained by extending to hypersequents a powerful se- with m ≥ 0. We can also assume Π1,..., Πn ⊆ mantic technique introduced by Okada (see e.g. [13]) which ′ ′ ′ ′ {Π1,..., Πk} as if e.g. Π1 6∈ {Π1,..., Πk} then (hr) is proved cut-elimination for (higher order) linear, intuitionis- equivalent in HFLew to the rule with premises G | Ξ1 ⇒ tic and classical logics. ′ ′ Π1,...,G | Ξk ⇒ Πk and conclusion G | Υ2 ⇒ Traditional (syntactic) proofs of cut-elimination start ′ ′ Π2 | ··· | Υn ⇒ Πn | Υ1 ⇒ | . . . | Υm ⇒ | Υ1 ⇒. with derivations containing cuts and generate derivations ′ ′ ′ Let Υ0 =Υ1,..., Υm and consider the rule (hr) without cuts. Semantic proofs go instead in the opposite di- ′ ′ rection; they start with a notion of cut-free provability and G | Ξ1 ⇒ Π1 ... G | Ξk ⇒ Πk build a model in which cuts are valid. G | Υ1 ⇒ Π1 | ··· | Υn ⇒ Πn | Υ0 ⇒ The latter step is analogous to the process of obtain- obtained applying to the conclusion of (hr): (EW ), when ing the field of reals from the field of rationals Q = m = 0 and both (w) and (EC), when m > 1. Note that (Q, +, ·, 0, 1) via the Dedekind-MacNeille completion. For (hr)′ is derivable from (hr) in HFLew and satisfies linear- any X ⊆ Q, define: conclusion. Two cases can occur: £ ′ ′ 1. There is a premise G | Ξj ⇒ Πj in (hr) such that X = {y : ∀x ∈ X. x ≤ y} ′ ′ ¡ Ξj ∩ Υ0 = ∅. As Π1,..., Πn ⊆{Π1,..., Πk}, the conclu- X = {y : ∀x ∈ X. y ≤ x} sion of (hr)′ is derivable from (some of) its premises using (Bcn′) and (w), from which the conclusion of (hr) follows Then the set R = {X ⊆ Q : X = X£¡} can be thought by several applications of (com) and (c). Hence (hr) is of as the set of real numbers extended with ±∞ and or- derivable in HSM. dered by inclusion ⊆, and one can naturally embed Q into ′ 2. Assume otherwise that each premise G | Ξi ⇒ Πi R = (R, +, ·, 0, 1), where +, ·, 0, 1 are suitably defined, ′ • £¡ ¡ of (hr) involves a metavariable in Υ0. We instantiate all by mapping r ∈ Q to r = r = r .

9 This construction yields a continuous structure out of a • (C, ∩, ⊕, MH, ∅£¡) is a lattice with greatest element discontinuous one. In our case, we start with an ‘intransi- MH and least element ∅£¡; tive’ structure (as ⇒ of a sequent is intransitive in the ab- sence of the cut rule), and obtain a ‘transitive’ one in which • (C, ⊗, 1) is a commutative monoid; the cut rule is valid. We refer to [15] for further algebraic • for any X , Y, Z∈C, X⊗Y⊆Z⇐⇒X ⊆Y−◦Z. account. Let R be a set of completed hyperstructural rules. We 0 is just a point and there is no condition on it. Bounded cf write ⊢HFLe+R G if G is cut-free provable in HFLe + pointed commutative residuated lattices, also known as cf R. Fora set G of hypersequents, we write ⊢HFLe+R G if FLe-algebras, give rise to an algebraic semantics for FLe cf (see [8]). Hence we can interpret our formulas in A. ⊢HFLe+R G for every G ∈ G. We denote the set of multisets of formulas by M, the A valuation on A is a function ( )• that maps each set of sequents by S and the set of hypersequents by H. We propositional variable p to a closed set p• ∈ C. It can be write MH and SH for M×H and S×H, respectively. The naturally extended to arbitrary formulas. If Γ= α1,...,αn • • • • • empty hypersequent in H and the empty multiset in M are (resp. Π = β), then Γ = α1 ◦···◦ αn (resp. Π = β ). respectively denoted by ∅ and ε. Given (Γ; G), (∆; H) ∈ If Γ (resp. Π) is empty, then Γ• = 1 (resp. Π• = 0). We £ MH and (Σ ⇒ Π; F ) ∈ SH, we deﬁne: interpret a sequent S = Γ ⇒ Π by S• = Γ•@(Π• ), and • • • a hypersequent G = S1 | ··· | Sn by G = S1 | ··· | Sn. (Γ; G) ◦ (∆; H) = (Γ, ∆; G | H) ∈ MH If Γ is empty, then G• = {∅}. Notice that S• and G• are (Γ; G)@(Σ ⇒ Π; F ) = Γ, Σ ⇒ Π | G | F ∈ H subsets of H. Our model supports ‘focusing’ of a component in a hy- Then (MH, ◦, (ε; ∅)) forms a commutative monoid. In persequent: the sequel, we write X , Y, Z . . . for subsets of MH, and cf • U, V,... for those of SH. The binary operations ◦, @ and Lemma 8.2. ⊢HFLe+R (Γ ⇒ Π | G) if and only if | are naturally extended: X ◦ Y = {x ◦ y : x ∈ X ,y ∈ Y} (ǫ; G•) ⊆ Γ• −◦Π•, where (ǫ; G•) denotes the set {(ǫ; H) : and similarly for X @U and G | G′. H ∈ G•}. In particular when G is empty, we have Furthermore, we deﬁne: cf • • • ⊢HFLe+R (Γ ⇒ Π) if and only if (ǫ; ∅) ∈ Γ −◦ Π .

£ cf • • •£ X = {u ∈ SH : ∀x ∈ X . ⊢HFLe+R x@u} Proof. (⇒) Let (ǫ; H) ∈ (ǫ; G ), x ∈ Γ and u ∈ Π . ¡ cf Then x@u | H belongs to (Γ ⇒ Π | G)•, and so is cut- U = {x ∈ MH : ∀u ∈U. ⊢HFLe x@u} +R free provable. Since x@u | H = ((ǫ; H) ◦ x)@u, we have £¡ Notice in particular that if (Γ; G) ∈ X (resp. ∈ U ¡) and (ǫ; H) ◦ x ∈ Π• = Π•, and hence (ǫ; H) ∈ Γ• −◦ Π•. (∆ ⇒ Π; H) ∈ X £ (resp. ∈U), the hypersequent Γ, ∆ ⇒ The converse direction is also easy. Π | G | H is cut-free provable in HFLe + R. • The two operations ( )£ and ( )¡ form a so-called Theorem 8.3 (Soundness). Let ( ) be a valuation on A. cf Galois connection between P(MH) and P(SH): X ⊆ For any hypersequent G, ⊢HFLe+R G implies ⊢HFLe+R ¡ £ £¡ • • • U ⇐⇒ U ⊆ X , inducing a closure operator ( ) G . Hence ⊢HFLe+R Γ ⇒ Π implies (ǫ; ∅) ∈ Γ −◦ Π . on P(MH). We have Proof. By induction on the length of derivation. The iden- 1. X ⊆ X £¡, U⊆U ¡£. tity axiom, cut and logical rules are dealt with by Lemmas

£ £ ¡ ¡ 8.1 and 8.2. For instance, when the derivation ends with an 2. X ⊆ Y =⇒ Y ⊆ X , U⊆V =⇒V ⊆U . instance of (cut): 3. £¡£ £, ¡£¡ ¡. X = X U = U G | Γ ⇒ α G | α, ∆ ⇒ Π 4. X £¡ ◦ Y£¡ ⊆ (X ◦ Y)£¡. G | Γ ⇒ Π £¡ The last property makes ( ) a nucleus [8]. Let us denote the induction hypothesis together with Lemma 8.2 yields £¡ by C the set of all closed sets w. r. t. ( ) and deﬁne (ǫ; G•) ⊆ Γ• −◦ α• and (ǫ; G•) ⊆ α• ◦ ∆• −◦ Π•. Hence (ǫ; G•) ◦ Γ• ⊆ α• and (ǫ; G•) ◦ α• ⊆ ∆• −◦ Π•. From X ⊕ Y = (X ∪ Y)£¡, 0 = {( ⇒ ; ∅)}¡, this, we derive (ǫ; G•) ◦ (ǫ; G•) ⊆ Γ• ◦ ∆• −◦ Π•. Since X ⊗ Y = (X ◦ Y)£¡, 1 = {(ǫ; ∅)}£¡, (ǫ; G•) ◦ (ǫ; G•) = (ǫ; G•|G•) = (ǫ; (G|G)•), Lemma 8.2 X −◦Y = {y ∈ MH : ∀x ∈ X .x ◦ y ∈ Y} . cf • yields ⊢HFLe+R (G | G | Γ, ∆ ⇒ Π) . By (EC), we £¡ cf • Lemma 8.1 ([14]). A = (C, ∩, ⊕, ⊗, −◦, MH, ∅ , 1, 0) obtain ⊢HFLe+R (G | Γ, ∆ ⇒ Π) . is a bounded pointed commutative residuated lattice. Suppose now that the derivation ends with an instance of Namely, a completed hyperstructural rule (r) ∈ R. For simplicity,

10 we assume that it is of the form: Proof. For any sequent S = (Γ ⇒ Π), we have (Γ; ∅) ∈ Γ• and Π• ⊆ ( ⇒ Π; ∅)¡ by Lemma 8.4 (and by the G | S1 · · · G | Sn deﬁnition of 0 when Π is empty). The latter implies ( ⇒ G | Γ ,..., Γ , Λ ⇒ Π | Γ ,..., Γ ⇒ 1 n n+1 m Π; ∅) ∈ Π•£, hence Γ ⇒ Π ∈ S•, and so G ∈ G• for any

By the coupling and strong subformula properties, each Si hypersequent G. must be either of the form (1) Γi1 ,..., Γi , Λ ⇒ Π, or of k Corollary 8.6 (Uniform cut-elimination). Let R be a set of the form (2) Γi1 ,..., Γik ⇒ with 1 ≤ i1,...,ik ≤ m. • • completed hyperstructural rules. If ⊢HFLe+R G, then G is Now, let F0 ∈ G , (∆1; H1) ∈ Γ1, ..., (∆m; Hm) ∈ • • •£ cut-free provable in HFLe + R. Γm, (Σ1; F1) ∈ Λ and (Σ2 ⇒ Θ; F2) ∈ Π . Our purpose is to show that Proof. Follows by Theorems 8.3 and 8.5.

H | ∆1,..., ∆n, Σ ⇒ Θ | ∆n+1,..., ∆m ⇒ Remark 8.7. The lattice reduct of the algebra A is com- is cut-free provable in HFLe + R, where H = plete. Hence Corollary 8.6 can be easily extended to predi- (F0 | F1 | F2 | H1 | ··· | Hm) and Σ=Σ1, Σ2. cate logics. Extensions to higher order logics and noncom- ∗ mutative ones are also easy. For each 1 ≤ i ≤ n, let Si = ∆i1 ,..., ∆ik , Σ ⇒ Θ and H∗ = (F | F | F | H | ··· | H ) if S is of i 0 1 2 i1 ik i Corollary 8.8 (Uniform algebraic completeness). Suppose type (1) above, and S∗ = ∆ ,..., ∆ ⇒ and H∗ = i i1 ik i that R is equivalent in HFLe to a set K of axioms. A for- (F | H | ··· | H ) if S is of type (2). It is not hard 0 i1 ik i mula α is valid in every FL -algebra satisfying K if and to see that e only if α is provable in HFLe + R. ∗ ∗ H | S1 · · · H | Sn Proof. (⇒) Since ⊢HFLe+R K, Theorem 8.3 implies that H | ∆1,..., ∆n, Σ ⇒ Θ | ∆n+1,..., ∆m ⇒ A satisfies K. Hence by assumption (ǫ; ∅) ∈ α•. The is a correct instance of (r); notice in particular that there claim follows by completeness. (⇐) ⊢HFLe+R α implies is no matching constraint for the conclusion because of the ⊢FLe+K α. The claim then follows by the soundness of 4 ∗ ∗ • linear-conclusion property. Since Hi | Si ∈ (G | Si) for FLe. every 1 ≤ i ≤ n, the induction hypothesis and (EW ) imply that the conclusion is cut-free provable. To illustrate the use of our results, let WNM be the fuzzy logic defined in [6] as FLew + (prelinearity) + (weak nilpo- £¡ Let us now consider a valuation given by p• = (p; ∅) . tent minimum) (see Example 3.2). Theorems 5.6, 7.1(b) Under this specific valuation, we have the following form and Corollary 8.6 automatically yield: of Okada’s lemma [13]: Corollary 8.9. The hypersequent calculus obtained by ex- • ¡ Lemma 8.4. For any formula α, (α; ∅) ∈ α ⊆ (⇒ α; ∅) . tending HFLew with (com) and the rule (wnm) of Exam- Proof. By induction on the structure of α. The case α = p ple 6.5 is a cut-free calculus for WNM. follows by the identity axiom. Suppose that α = β → γ. To show that (β → γ; ∅) ∈ 9. Conclusion β• −◦ γ•, let (Γ; G) ∈ β• and (∆ ⇒ Π; H) ∈ γ•£. The induction hypotheses • ¡ and • β ⊆ ( ⇒ β; ∅) (γ; ∅) ∈ γ We introduce an algorithm that generates equivalent imply that and are cut-free Γ ⇒ β | G γ, ∆ ⇒ Π | H structural rules, in sequent and hypersequent calculi, from provable. Hence so is . This Γ,β → γ, ∆ ⇒ Π | G | H a large class of (Hilbert) axioms. The key idea for deter- proves • •£¡ • •. (β → γ; ∅) ∈ β −◦ (γ )= β −◦ γ mining when this is possible is the identification of a hierar- To show that β• −◦γ• ⊆ ( ⇒ β → γ; ∅)¡, let (Γ; G) ∈ chy of formulas Pn, Nn– similar to the arithmetic hierarchy β• −◦ γ•. The induction hypothesis (β; ∅) ∈ β• implies Σ , Π – which keeps track of polarity alternation (cf. [1]). • ¡. Hence is cut- n n (β, Γ; G) ∈ γ ⊆ (⇒ γ; ∅) β, Γ ⇒ γ | G We show how to transform free provable and so is Γ ⇒ β → γ | G. This proves the claim. The other cases are similar. 1. any axiom in N2 into an equivalent set of (sequent) structural rules, and Theorem 8.5 (Completeness). For any hypersequent G, we £¡ have G ∈ G• under the valuation p• = (p; ∅) . Hence ′ 2. any axiom in P3 (⊆ P3) into an equivalent set of hy- if (ǫ; ∅) ∈ Γ• −◦ Π•, then Γ ⇒ Π is cut-free provable in perstructural rules. HFLe + R. If the generated rules are acyclic, they are further trans- 4It is instructive to try to prove soundness for HFLe + (SI ) (see Sec- tion 6). The argument would break down precisely at this point, due to formed (completed) into equivalent analytic rules. This also lack of the linear-conclusion and coupling properties. holds when the base calculus contains weakening, in which

11 case the automated transformation of axioms into equiv- [12] M. Ohnishi and K. Matsumoto. A system for strict impli- alent sets of analytic hyperstructural rules applies to the cation. Annals of the Japan Association for Philosophy of whole class P3. Science, 2:183–188, 1964. Every hypersequent calculus defined by extending [13] M. Okada. A uniform semantic proof for cut-elimination HFLe with a set of completed rules is shown to enjoy cut- and completeness of various first and higher order logics. Theoretical Computer Science, 281:471–498, 2002. admissibility, via a uniform and semantic proof (the first [14] H. Ono. Semantics for Substructural Logics. In P. Schroeder- such for any hypersequent calculus). Heister and K. Doˇsen ed. Substructural Logics, 259–291. Although some particular formulas beyond N (resp. P ) 1993. 2 3 [15] F. Belardinelli, H. Ono and P. Jipsen. Algebraic aspects of can be captured by multiple-conclusion (hyper)sequent cal- cut elimination. Studia Logica, 68:1–32, 2001. culi, as in the case of weak excluded middle in Gentzen LK [16] A. Prijately. Bounded Contraction and Gentzen style Formu- calculus for classical logic or of Łukasiewicz axiom (see lation of Łukasiewicz Logics. Studia Logica, 57: 437–456, Example 7.4) in the hypersequent calculus for Łukasiewicz 1996. logic in [10], we conjecture that the expressive power of [17] G. Sambin, G. Battilotti and C. Faggian, Basic logic: reflec- single-conclusion sequent (resp. hypersequent) structural tion, symmetry, visibility. Journal of Symbolic Logic, 65: rules is limited to N2 (resp. P3) formulas. 979–1013, 2000. We conclude the paper by stating the challenging ques- [18] K. Terui. Which Structural Rules Admit Cut Elimination? tion of identifying the level of generality, beyond hyperse- An Algebraic Criterion. Journal of Symbolic Logic, 72(3): 738–754, 2007. quents, appropriate for dealing, in a uniform way, with axioms at levels higher than P3.

References

[1] J.-M. Andreoli. Logic programming with focusing proofs in linear logic. Journal of Logic and Computation, 2(3):297– 347, 1992. [2] A. Avron. Hypersequents, logical consequence and intermediate logics for concurrency. Annals of Mathematics and Ar- tiﬁcial Intelligence, 4: 225–248, 1991. [3] A. Avron. The method of hypersequents in the proof theory of propositional non-classical logics. In W. Hodges et al. ed- itors, Logic: from foundations to applications. Proc. Logic Colloquium, Keele, UK, 1993, pages 1–32, 1996. [4] M. Baaz, A. Ciabattoni, and F. Montagna. Analytic calculi for monoidal t-norm based logic. Fundamenta Informaticae, 59(4):315–332, 2004. [5] A. Ciabattoni and M. Ferrari. Hypersequent Calculi for some Intermediate Logics with Bounded Kripke Models. Journal of Logic and Computation, 11(2): 283-294. 2001. [6] F. Esteva and L. Godo. Monoidal t-norm based Logic: to- wards a logic for left-continuous t-norms. Fuzzy Sets and Systems, 124: 271–288, 2001. [7] G. Gentzen. Untersuchungen ¨uber das logische schliessen. Math. Zeitschrift, 39:176 –210, 405–431, 1935. [8] N. Galatos, P. Jipsen, T. Kowalski and H. Ono. Residuated Lattices: an algebraic glimpse at substructural logics. Stud- ies in Logics and the Foundations of Mathematics, Elsevier, 2007. [9] R. Hori, H. Ono, and H. Schellinx. Extending intuitionistic linear logic with knotted structural rules. Notre Dame Journal of Formal Logic, 35(2): 219–242, 1994. [10] G. Metcalfe, N. Olivetti and D. Gabbay. Sequent and Hyper- sequent Calculi for Abelian and Łukasiewicz Logics. ACM TOCL. 6(3): 578-613, 2005. [11] G. Metcalfe and F. Montagna. Substructural Fuzzy Logics. Journal of Symbolic Logic 72(3): 834–864, 2007.