Classical BI: Its Semantics and Proof Theory

We present Classical BI (CBI), a new addition to the family of bunched logics which originates in O’Hearn and Pym’s logic of bunched implications BI. CBI differs from existing bunched logics in that its multiplicative connectives behave classically rather than intuitionistically (including in particular a multiplicative version of classical negation). At the semantic level, CBI-formulas have the normal bunched logic reading as declarative statements about resources, but its resource models necessarily feature more structure than those for other bunched logics; principally, they satisfy the requirement that every resource has a unique dual. At the proof-theoretic level, a very natural formalism for CBI is provided by a display calculus `a la Belnap, which can be seen as a generalisation of the bunched sequent calculus for BI. In this paper we formulate the aforementioned model theory and proof theory for CBI, and prove some fundamental results about the logic, most notably completeness of the proof theory with respect to the semantics.

💡 Research Summary

The paper introduces Classical BI (CBI), a new member of the family of bunched logics that extends O’Hearn and Pym’s original logic of bunched implications (BI) by making the multiplicative fragment classical rather than intuitionistic. The authors begin by motivating the need for a classical treatment of the multiplicative connectives: in standard BI the “” (multiplicative conjunction) and its associated implication are intuitionistic, which limits the ability to express dualities such as debt versus credit, consumption versus replenishment, or any situation where a resource has a unique opposite. CBI remedies this by adding a multiplicative negation (¬₍₎) and a corresponding falsum (⊥₍*₎), thereby allowing statements that say “no resource together with its dual can satisfy A”.

Syntax. The language of CBI contains the usual additive connectives (∧, ∨, →) and the multiplicative connectives (, ⟹). In addition it introduces a multiplicative negation ¬₍₎ and a multiplicative falsum ⊥₍*₎. Formulas are built from atomic propositions using a “bunch” structure, i.e., a tree whose internal nodes are labeled either by additive (;) or multiplicative (,) separators, exactly as in BI.

Semantics. The semantic foundation is an involutive monoid (R,·,e, (·)⁻¹). Here (R,·,e) is a commutative monoid representing resource combination, and (·)⁻¹ is a bijective involution that assigns to each resource a unique dual. The defining equations are a·a⁻¹ = e and (a⁻¹)⁻¹ = a for every a∈R. Truth of a formula at a resource a is defined inductively as in BI, except for the multiplicative negation:

 a ⊨ ¬₍*₎A iff for every b∈R, if a·b ⊨ A then b = ⊥,

which captures the idea that no dual resource can turn a·b into a witness for A. The involution guarantees that each resource has exactly one “opposite”, making the semantics of ¬₍₎ well‑behaved. The authors give concrete examples, such as a monetary model where “+” is ordinary addition and “−” (the involution) represents debt; in this setting ¬₍₎A expresses that a given amount cannot be paired with a debt to satisfy A.

Proof Theory. To match the richer semantics, the authors develop a display calculus for CBI, following Belnap’s methodology. The calculus distinguishes structural expressions (the “contexts” that collect resources) from logical formulas, and provides a family of display rules that can always bring any sub‑structure to the top‑level of a sequent. This property is crucial for handling the duality operator uniformly. The logical rules for ∧, ∨, →, , ⟹ are the usual introduction and elimination rules, while the rules for ¬₍₎ and ⊥₍₎ are designed to reflect the involutive monoid equations. For instance, the left‑introduction of ¬₍₎ requires that the antecedent be displayed together with its dual, mirroring the semantic clause.

A central metatheoretic result is cut‑elimination: every CBI proof can be transformed into a cut‑free proof that respects the subformula property. The proof proceeds by the standard reduction steps for display calculi, exploiting the fact that the structural rules are invertible and that the involution behaves like a logical symmetry. Consequently, the calculus is consistent, and proof search can be confined to a finite space of subformulas.

Soundness and Completeness. Soundness is proved by a straightforward induction on the height of derivations, showing that each rule preserves truth in every involutive monoid model. Completeness is more involved. The authors construct a canonical CBI frame from maximal consistent sets of formulas (a Henkin‑style construction). Each equivalence class of such a set becomes a resource element, and the involution is defined by mapping a set to the set of formulas whose multiplicative negations belong to the original. The monoid operation is induced by the multiplicative conjunction *. They then verify that this canonical frame satisfies the involutive monoid axioms and that every formula provable in the display calculus holds in the canonical model. By the usual Lindenbaum‑Henkin argument, any formula valid in all models is derivable, establishing completeness.

Comparison and Applications. The paper contrasts CBI with ordinary BI, Boolean BI, and other bunched logics. Unlike BI, CBI can express classical reasoning about resources, making it suitable for domains where resources have natural opposites (e.g., financial accounting, energy consumption vs. generation, or security permissions vs. revocations). The display calculus also generalises the bunched sequent calculus of BI, offering a uniform framework that could be extended to other resource‑sensitive logics. Potential applications mentioned include type systems for languages with linear/affine features, automated verification of programs that manipulate dual resources, and modelling of quantum resources where creation and annihilation are dual operations.

Conclusion and Future Work. The authors conclude that CBI fills a conceptual gap in the landscape of bunched logics by providing a classical multiplicative fragment together with a robust proof system. They outline several directions for further research: integrating CBI into programming language semantics, developing decision procedures or proof assistants based on the display calculus, exploring richer algebraic structures (e.g., groups, rings) as resource models, and investigating connections with categorical semantics (e.g., *-autonomous categories). Overall, the paper delivers a thorough treatment of both the semantic foundations and the proof‑theoretic machinery of Classical BI, establishing it as a promising tool for reasoning about resources that possess a natural dual.