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 th...

The connection between stochastic differential equations and associated Fokker-Planck equations is elucidated by the full functional calculus. One-variable equations with either additive or multiplicative noise are considered. The central focus is on approximate Fokker-Planck equations which describe the consequences of using "colored" noise, which has an exponential correlation function and a ...

In the Böhm theorem workshop on Crete island, Zoran Petric called Statman’s “Typical Ambiguity theorem” typed Böhm theorem. Moreover, he gave a new proof of the theorem based on set-theoretical models of the simply typed lambda calculus. In this paper, we study the linear version of the typed Böhm theorem on a fragment of Intuitionistic Linear Logic. We show that in the multiplicative fragment ...

We extend multiplicative exponential linear logic (MELL) by a noncommutative, self-dual logical operator. The extended system, called NEL, is defined in the formalism of the calculus of structures, which is a generalisation of the sequent calculus and provides a more refined analysis of proofs. We are then able to extend the expressiveness of MELL by modelling a broad notion of sequentiality. W...

We present the logic of linear processes (i.e. LLP), a calculus for the connectives of multiplicative additive linear logic (without ⊕) that defines the meaning of connectives in the style of a process calculus. 1 A transition-based view of linear logic Because we want to represent states as linear (and, more generally, as substructural) contexts, we want to define a logic in such a way that th...

The inverse method, due to Maslov, is a forward theorem provingmethod for cut-free sequent calculi that relies on the subformula property. The Logic of Bunched Implications (BI), due to Pym and O’Hearn, is a logic which freely combines the familiar connectives of intuitionistic logic with multiplicative linear conjunction and its adjoint implication. We present the first formulation of an inver...

In the Böhm theorem workshop on Crete island, Zoran Petric called Statman’s “Typical Ambiguity theorem” typed Böhm theorem. Moreover, he gave a new proof of the theorem based on set-theoretical models of the simply typed lambda calculus. In this paper, we study the linear version of the typed Böhm theorem on a fragment of Intuitionistic Linear Logic. We show that in the multiplicative fragment ...

Proof theory for a logic with categorical semantics can be developed by the following methodology: define a sound and complete display calculus for an extension of the logic with additional adjunctions; translate this calculus to a shallow inference nested sequent calculus; translate this calculus to a deep inference nested sequent calculus; then prove this final calculus is sound with respect ...

The calculus of structures is a proof theoretical formalism which generalizes the sequent calculus with the feature of deep inference: in contrast to the sequent calculus, inference rules can be applied at any depth inside a formula, bringing shorter proofs than all other formalisms supporting analytical proofs. However, deep applicability of inference rules causes greater nondeterminism than i...

In this short paper we introduce a variant of intuitionistic multiplicative linear logic that features both commutative and non-commutative connectives. This new logic extends conservatively both Girard’s multiplicative linear logic [2] and Lambek’s syntactic calculus [3]. Ultimately, our goal will be to fulfil the programme achieved by Girard in [2], i.e., to provide our system with: (1) a seq...