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 G¨odel 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 ex- change (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 | α, Γ ⇒ Π G | β, Γ ⇒ Π G | Γ ⇒ α G | Γ ⇒ β G | Γ ⇒ αi (∨l) (∧r) (∨r) (⊤r) G | α ∨ β, Γ ⇒ Π G | Γ ⇒ α ∧ β G | Γ ⇒ α1 ∨ α2 G | Γ ⇒⊤ 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).