Investigations into Linear Logic with Fixed-Point Operators
Francesco Gavazzo
Abstract:
Linear logic is a substructural logic that refines both classical and intuitionistic logic. In fact, linear logic is characterized by several dualities (which derive from the presence of a de Morgan negation), but at the same time has a strong constructive flavor. From a proof-theoretical perspective, classical (resp. intuitionistic) linear logic is obtained from classical (risp. intuitionistic) sequent calculus by dropping the structural rules of weakening and contraction. This makes the use of hypothesis in a proof linear, in the sense that each hypothesis must be used exactly once. Linear logic has two modalities, ! and ?, called exponential modalities, that allow to restore weakening and contraction in a controlled form. Having these modalities, both intuitionistic and classical logic can be encoded into linear logic.
Despite being interested per se, linear logic has several applications. In fact, linearity of hypothesis allows to look at formulas as resources or pieces of information, that cannot be neither freely duplicated nor deleted. Moreover, the absence of weakening and contraction leads to a finer distinction between classical (risp. intuitionistic) connectives, thus obtaining a new stock of connectives which capture in a natural way several operations between computational processes.
Categorical Quantum Mechanics studies quantum processes as special computational processes. The underlying mathematical framework is given by (enrichments of) monoidal categories. One of the main feature of monoidal categories is that the notion of categorical product is replaced with the weaker notion of tensor product. Tensor products allow to describe a rudimentary form of parallel composition and thus make monoidal categories suitable for an abstract description of physical and computational processes. It is well known that the underlying logic of monoidal categories is the multiplicative tensorial fragment of intuitionistic linear logic, so that the latter can be thought of as the logic describing the abstract structure of quantum processes.
For these reasons, it is useful to have a framework that allows to study and define processes (both physical and computational) that are characterized by infinite and iterative behaviors. This thesis deals with extensions of (specific enrichments of) monoidal categories with initial algebras and final coalgebras for a class of functors generalizing polynomial functors over the monoidal signature, as well as their underlying logics. The latter are nothing but (fragments of) linear logic extended with least and greatest fixed point operators. Categories are mostly defined and studied equationally, according to Lambek’s methodology. This allows to easily design syntactical systems for such categories, which can then be made into logical systems. We provide sequent calculi for all the logics investigated, and a deep inference system for the extension of classical linear logic with least and greatest fixed point operators. We define exponential, relevant and affine modalities as least and greatest fixed point of specific functors. This leads to a finer analysis of such modalities and their proof-theoretical properties, as well as their relationship.
Finally, some possible applications of the logics investigated are sketched, in particular in the direction of modal (especially epistemic) logics over a linear base.