# Bunched logic

## Idea

In logic, and more specifically in sequent calculus/natural deduction, a bunched logic is a logic in which the formulas in the context are not just a list or set but have some additional, usually tree-like, structure.

This can be indicated syntactically by the use of two or more punctuation symbols, such as comma and semicolon, along with parentheses for grouping. Thus for instance a sequent with bunches might be written like

$A,B,(C;(D,E);F),(G;H) \vdash K$

The contexts put together with both commas and semicolons are called bunches. The general phrase bunched logic is not entirely standard, although the word “bunches” has been used with more than one logic of this form (the original is “bunched implication (BI)”, below).

Each type of punctuation also comes with a nullary version. The punctuation symbols like comma and semicolon are sometimes called structural connectives, since they are part of the judgmental structure (like structural rules) but are closely related to the logical connectives such as conjunction and disjunction.

Usually the reason for using a bunched logic is that the different structural connectives obey different structural rules. For instance, the semicolon might allow the contraction rule and/or the weakening rule, while the comma does not.

## Examples

### Bunched implications

In bunched implications logic (BI), the semicolon allows contraction and weakening while the comma does not. This allows defining both an additive conjunction and a multiplicative conjunction, internalizing the semicolon and the comma respectively, such that both distribute over the additive disjunction and moreover both come with a corresponding implication: the additive implication? and the multiplicative implication.

### Relevance logic

Some forms of relevance logic can be presented with a bunched sequent calculus that is similar to BI, but in which the comma also has contraction (though not weakening) and there is no additive implication. See for instance (Mints).

### Classical bunched implication

A logic of classical bunched implication is like BI but with arbitrary bunches on the right as well as on the left. On the right, the semicolon represents the additive disjunction and the comma represents a multiplicative disjunction, and there are both an additive and a multiplicative negation that move formulas back and forth. Both negations are “classical” with respect to their corresponding connectives, e.g. we have $\sim\sim A \multimap A$ (where $\sim$ is the multiplicative negation and $\multimap$ the multiplicative implication) and also $\neg\neg A \to A$ (where $\neg$ is the additive negation and $\to$ the additive implication). See Pym 2002 and this discussion.

## Categorical semantics

Bunched logics naturally have semantics in categories with more than one monoidal structure, so that a bunch such as $(A,(B;C),((D,E);F))$ can be interpreted as $A \otimes (B\boxtimes C) \otimes ((D\otimes E)\boxtimes F)$. Frequently (e.g. if one kind of bunch admits contraction and weakening) one of the two monoidal structures is a cartesian one. A typical and motivating example of a model for BI is the category of presheaves $[C^{op},Set]$ over a monoidal category $C$, which comes equipped both with the ordinary ccc structure on presheaves as well as the closed monoidal structure given by Day convolution.

• Peter W. O’Hearn and David J. Pym, The Logic of Bunched Implications. PDF
• G. E. Mints. Cut-elimination theorem for relevant logics, Zap. Nauchn. Sem. LOMI, 1972, Volume 32, Pages 90–97. (math-net.ru). An English translation appears in the Journal of Soviet Mathematics 6 (1976) pp.422-8. (doi)
• Bodil Biering, On the logic of bunched implications - and its relation to separation logic, masters thesis

• David Pym, The Semantics and Proof Theory of the Logic of Bunched Implications, Google books

• Brotherston and Calcagno, Classical BI: Its Semantics and Proof Theory, arxiv