In mathematical logic, the hypersequent framework is an extension of the proof-theoretical framework of sequent calculi used in structural proof theory to provide analytic calculi for logics that are not captured in the sequent framework. A hypersequent is usually taken to be a finite multiset of ordinary sequents, written

${\displaystyle \Gamma _{1}\Rightarrow \Delta _{1}\mid \cdots \mid \Gamma _{n}\Rightarrow \Delta _{n}}$

The sequents making up a hypersequent are called components. The added expressivity of the hypersequent framework is provided by rules manipulating different components, such as the communication rule for the intermediate logic LC (Gödel–Dummett logic)

${\displaystyle {\frac {\Gamma _{1}\Rightarrow \Delta _{1}\mid \dots \mid \Gamma _{n}\Rightarrow \Delta _{n}\mid \Sigma \Rightarrow A\qquad \Omega _{1}\Rightarrow \Theta _{1}\mid \dots \mid \Omega _{m}\Rightarrow \Theta _{m}\mid \Pi \Rightarrow B}{\Gamma _{1}\Rightarrow \Delta _{1}\mid \dots \mid \Gamma _{n}\Rightarrow \Delta _{n}\mid \Omega _{1}\Rightarrow \Theta _{1}\mid \dots \mid \Omega _{m}\Rightarrow \Theta _{m}\mid \Sigma \Rightarrow B\mid \Pi \Rightarrow A}}}$

or the modal splitting rule for the modal logic S5:^[1]

${\displaystyle {\frac {\Gamma _{1}\Rightarrow \Delta _{1}\mid \dots \mid \Gamma _{n}\Rightarrow \Delta _{n}\mid \Box \Sigma ,\Theta \Rightarrow \Box \Pi ,\Omega }{\Gamma _{1}\Rightarrow \Delta _{1}\mid \dots \mid \Gamma _{n}\Rightarrow \Delta _{n}\mid \Box \Sigma \Rightarrow \Box \Pi \mid \Theta \Rightarrow \Omega }}}$

Hypersequent calculi have been used to treat modal logics, intermediate logics, and substructural logics. Hypersequents usually have a formula interpretation, i.e., are interpreted by a formula in the object language, nearly always as some kind of disjunction. The precise formula interpretation depends on the considered logic.

Formal definitions and propositional rules

Formally, a hypersequent is usually taken to be a finite multiset of ordinary sequents, written

${\displaystyle \Gamma _{1}\Rightarrow \Delta _{1}\mid \dots \mid \Gamma _{n}\Rightarrow \Delta _{n}}$

The sequents making up a hypersequent consist of pairs of multisets of formulae, and are called the components of the hypersequent. Variants defining hypersequents and sequents in terms of sets or lists instead of multisets are also considered, and depending on the considered logic the sequents can be classical or intuitionistic. The rules for the propositional connectives usually are adaptions of the corresponding standard sequent rules with an additional side hypersequent, also called hypersequent context. E.g., a common set of rules for the functionally complete set of connectives ${\displaystyle \{\bot ,\to \}}$ for classical propositional logic is given by the following four rules:

${\displaystyle {\frac {}{{\mathcal {G}}\mid \Gamma ,p\Rightarrow p,\Delta }}}$

${\displaystyle {\frac {}{{\mathcal {G}}\mid \Gamma ,\bot \Rightarrow \Delta }}}$

${\displaystyle {\frac {{\mathcal {G}}\mid \Gamma ,B\Rightarrow \Delta \qquad {\mathcal {G}}\mid \Gamma \Rightarrow A,\Delta }{{\mathcal {G}}\mid \Gamma ,A\to B\Rightarrow \Delta }}}$

${\displaystyle {\frac {{\mathcal {G}}\mid \Gamma ,A\Rightarrow B,\Delta }{{\mathcal {G}}\mid \Gamma \Rightarrow A\to B,\Delta }}}$

Due to the additional structure in the hypersequent setting the structural rules are considered in their internal and external variants. The internal weakening and internal contraction rules are the adaptions of the corresponding sequent rules with an added hypersequent context:

${\displaystyle {\frac {{\mathcal {G}}\mid \Gamma \Rightarrow \Delta }{{\mathcal {G}}\mid \Gamma ,\Sigma \Rightarrow \Delta ,\Pi }}}$

${\displaystyle {\frac {{\mathcal {G}}\mid \Gamma ,A,A\Rightarrow \Delta }{{\mathcal {G}}\mid \Gamma ,A\Rightarrow \Delta }}}$

${\displaystyle {\frac {{\mathcal {G}}\mid \Gamma \Rightarrow A,A,\Delta }{{\mathcal {G}}\mid \Gamma \Rightarrow A,\Delta }}}$

The external weakening and external contraction rules are the corresponding rules on the level of hypersequent components instead of formulae:

${\displaystyle {\frac {\mathcal {G}}{{\mathcal {G}}\mid \Gamma \Rightarrow \Delta }}}$

${\displaystyle {\frac {{\mathcal {G}}\mid \Gamma \Rightarrow \Delta \mid \Gamma \Rightarrow \Delta }{{\mathcal {G}}\mid \Gamma \Rightarrow \Delta }}}$

Soundness of these rules is closely connected to the formula interpretation of the hypersequent structure, nearly always as some form of disjunction. The precise formula interpretation depends on the considered logic, see below for some examples.

Main examples

History

The hypersequent structure seem to have appeared first in^[2] under the name of cortege, to obtain a calculus for the modal logic S5. It seems to have been developed independently in,^[3] also for treating modal logics, and in the influential,^[1] where calculi for modal, intermediate and substructural logics are considered, and the term hypersequent is introduced.

References