An Abstract Approach to Consequence Relations

We generalise the Blok-J\'onsson account of structural consequence relations, later developed by Galatos, Tsinakis and other authors, in such a way as to naturally accommodate multiset consequence. While Blok and J\'onsson admit, in place of sheer formulas, a wider range of syntactic units to be manipulated in deductions (including sequents or equations), these objects are invariably aggregated via set-theoretical union. Our approach is more general in that non-idempotent forms of premiss and conclusion aggregation, including multiset sum and fuzzy set union, are considered. In their abstract form, thus, deductive relations are defined as additional compatible preorderings over certain partially ordered monoids. We investigate these relations using categorical methods, and provide analogues of the main results obtained in the general theory of consequence relations. Then we focus on the driving example of multiset deductive relations, providing variations of the methods of matrix semantics and Hilbert systems in Abstract Algebraic Logic

Stone-Type Dualities for Separation Logics

Stone-type duality theorems, which relate algebraic and relational/topological models, are important tools in logic because -- in addition to elegant abstraction -- they strengthen soundness and completeness to a categorical equivalence, yielding a framework through which both algebraic and topological methods can be brought to bear on a logic. We give a systematic treatment of Stone-type duality for the structures that interpret bunched logics, starting with the weakest systems, recovering the familiar BI and Boolean BI (BBI), and extending to both classical and intuitionistic Separation Logic. We demonstrate the uniformity and modularity of this analysis by additionally capturing the bunched logics obtained by extending BI and BBI with modalities and multiplicative connectives corresponding to disjunction, negation and falsum. This includes the logic of separating modalities (LSM), De Morgan BI (DMBI), Classical BI (CBI), and the sub-classical family of logics extending Bi-intuitionistic (B)BI (Bi(B)BI). We additionally obtain as corollaries soundness and completeness theorems for the specific Kripke-style models of these logics as presented in the literature: for DMBI, the sub-classical logics extending BiBI and a new bunched logic, Concurrent Kleene BI (connecting our work to Concurrent Separation Logic), this is the first time soundness and completeness theorems have been proved. We thus obtain a comprehensive semantic account of the multiplicative variants of all standard propositional connectives in the bunched logic setting. This approach synthesises a variety of techniques from modal, substructural and categorical logic and contextualizes the "resource semantics" interpretation underpinning Separation Logic amongst them

A linear algebra approach to linear metatheory

Linear typed λ-calculi are more delicate than their simply typed siblings when it comes to metatheoretic results like preservation of typing under renaming and substitution. Tracking the usage of variables in contexts places more constraints on how variables may be renamed or substituted. We present a methodology based on linear algebra over semirings, extending McBride's kits and traversals approach for the metatheory of syntax with binding to linear usage-annotated terms. Our approach is readily formalisable, and we have done so in Agda

Sequent and Hypersequent Calculi for Abelian and Lukasiewicz Logics

We present two embeddings of infinite-valued Lukasiewicz logic L into Meyer and Slaney's abelian logic A, the logic of lattice-ordered abelian groups. We give new analytic proof systems for A and use the embeddings to derive corresponding systems for L. These include: hypersequent calculi for A and L and terminating versions of these calculi; labelled single sequent calculi for A and L of complexity co-NP; unlabelled single sequent calculi for A and L.Comment: 35 pages, 1 figur

Quantitative Equality in Substructural Logic via Lipschitz Doctrines

Substructural logics naturally support a quantitative interpretation of formulas, as they are seen as consumable resources. Distances are the quantitative counterpart of equivalence relations: they measure how much two objects are similar, rather than just saying whether they are equivalent or not. Hence, they provide the natural choice for modelling equality in a substructural setting. In this paper, we develop this idea, using the categorical language of Lawvere's doctrines. We work in a minimal fragment of Linear Logic enriched by graded modalities, which are needed to write a resource sensitive substitution rule for equality, enabling its quantitative interpretation as a distance. We introduce both a deductive calculus and the notion of Lipschitz doctrine to give it a sound and complete categorical semantics. The study of 2-categorical properties of Lipschitz doctrines provides us with a universal construction, which generates examples based for instance on metric spaces and quantitative realisability. Finally, we show how to smoothly extend our results to richer substructural logics, up to full Linear Logic with quantifiers

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.Comment: 42 pages, 8 figure

Substructural Simple Type Theories for Separation and In-place Update

This thesis studies two substructural simple type theories, extending the "separation" and "number-of-uses" readings of the basic substructural simply typed lambda-calculus with exchange. The first calculus, lambda_sep, extends the alpha lambda-calculus of O'Hearn and Pym by directly considering the representation of separation in a type system. We define type contexts with separation relations and introduce new type constructors of separated products and separated functions. We describe the basic metatheory of the calculus, including a sound and complete type-checking algorithm. We then give new categorical structure for interpreting the type judgements, and prove that it coherently, soundly and completely interprets the type theory. To show how the structure models separation we extend Day's construction of closed symmetric monoidal structure on functor categories to our categorical structure, and describe two instances dealing with the global and local separation. The second system, lambda_inplc, is a re-presentation of substructural calculus for in-place update with linear and non-linear values, based on Wadler's Linear typed system with non-linear types and Hofmann's LFPL. We identify some problems with the metatheory of the calculus, in particular the failure of the substitution rule to hold due to the call-by-value interpretation inherent in the type rules. To resolve this issue, we turn to categorical models of call-by-value computation, namely Moggi's Computational Monads and Power and Robinson's Freyd-Categories. We extend both of these to include additional information about the current state of the computation, defining Parameterised Freyd-categories and Parameterised Strong Monads. These definitions are equivalent in the closed case. We prove that by adding a commutativity condition they are a sound class of models for lambda_inplc. To obtain a complete class of models for lambda_inplc we refine the structure to better match the syntax. We also give a direct syntactic presentation of Parameterised Freyd-categories and prove that it is soundly and completely modelled by the syntax. We give a concrete model based on Day's construction, demonstrating how the categorical structure can be used to model call-by-value computation with in-place update and bounded heaps