50 research outputs found
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
Semantic cut elimination for the logic of bunched implications, formalized in Coq
The logic of bunched implications (BI) is a substructural logic that forms
the backbone of separation logic, the much studied logic for reasoning about
heap-manipulating programs. Although the proof theory and metatheory of BI are
mathematically involved, the formalization of important metatheoretical results
is still incipient. In this paper we present a self-contained formalized, in
the Coq proof assistant, proof of a central metatheoretical property of BI: cut
elimination for its sequent calculus.
The presented proof is *semantic*, in the sense that is obtained by
interpreting sequents in a particular "universal" model. This results in a more
modular and elegant proof than a standard Gentzen-style cut elimination
argument, which can be subtle and error-prone in manual proofs for BI. In
particular, our semantic approach avoids unnecessary inversions on proof
derivations, or the uses of cut reductions and the multi-cut rule.
Besides modular, our approach is also robust: we demonstrate how our method
scales, with minor modifications, to (i) an extension of BI with an arbitrary
set of \emph{simple structural rules}, and (ii) an extension with an S4-like
modality.Comment: 15 pages, to appear in CPP 202
Weakening Relation Algebras and FL\u3csup\u3e2\u3c/sup\u3e-algebras
FL2-algebras are lattice-ordered algebras with two sets of residuated operators. The classes RA of relation algebras and GBI of generalized bunched implication algebras are subvarieties of FL2-algebras. We prove that the congruences of FL2-algebras are determined by the congruence class of the respective identity elements, and we characterize the subsets that correspond to this congruence class. For involutive GBI-algebras the characterization simplifies to a form similar to relation algebras.
For a positive idempotent element p in a relation algebra A, the double division conucleus image p/A/p is an (abstract) weakening relation algebra, and all representable weakening relation algebras (RWkRAs) are obtained in this way from representable relation algebras (RRAs). The class S(dRA) of subalgebras of {p/A/p∶ A ϵ RA; 1 ≤ p2 = p ϵ A} is a discriminator variety of cyclic involutive GBI-algebras that includes RA. We investigate S(dRA) to find additional identities that are valid in all RWkRAs. A representable weakening relation algebra is determined by a chain if and only if it satisfies 0 ≤ 1, and we prove that the identity 1 ≤ 0 holds only in trivial members of S(dRA).https://digitalcommons.chapman.edu/scs_books/1050/thumbnail.jp
Varieties of unary-determined distributive -magmas and bunched implication algebras
A distributive lattice-ordered magma (-magma)
is a distributive lattice with a binary operation that preserves joins
in both arguments, and when is associative then is an
idempotent semiring. A -magma with a top is unary-determined if
. These
algebras are term-equivalent to a subvariety of distributive lattices with
and two join-preserving unary operations . We obtain simple
conditions on such that is
associative, commutative, idempotent and/or has an identity element.
This generalizes previous results on the structure of doubly idempotent
semirings and, in the case when the distributive lattice is a Heyting algebra,
it provides structural insight into unary-determined algebraic models of
bunched implication logic. We also provide Kripke semantics for the algebras
under consideration, which leads to more efficient algorithms for constructing
finite models. We find all subdirectly irreducible algebras up to cardinality
eight in which is a closure operator, as well as all finite
unary-determined bunched implication chains and map out the poset of
join-irreducible varieties generated by them
Coalgebraic completeness-via-canonicity for distributive substructural logics
We prove strong completeness of a range of substructural logics with respect
to a natural poset-based relational semantics using a coalgebraic version of
completeness-via-canonicity. By formalizing the problem in the language of
coalgebraic logics, we develop a modular theory which covers a wide variety of
different logics under a single framework, and lends itself to further
extensions. Moreover, we believe that the coalgebraic framework provides a
systematic and principled way to study the relationship between resource models
on the semantics side, and substructural logics on the syntactic side.Comment: 36 page
Distributive Residuated Frames and Generalized Bunched Implication Algebras
We show that all extensions of the (non-associative) Gentzen system for distributive full Lambek calculus by simple structural rules have the cut elimination property. Also, extensions by such rules that do not increase complexity have the finite model property, hence many subvarieties of the variety of distributive residuated lattices have decidable equational theories. For some other extensions, we prove the finite embeddability property, which implies the decidability of the universal theory, and we show that our results also apply to generalized bunched implication algebras. Our analysis is conducted in the general setting of residuated frames
Intuitionistic Layered Graph Logic: Semantics and Proof Theory
Models of complex systems are widely used in the physical and social sciences, and the concept of layering, typically building upon graph-theoretic structure, is a common feature. We describe an intuitionistic substructural logic called ILGL that gives an account of layering. The logic is a bunched system, combining the usual intuitionistic connectives, together with a non-commutative, non-associative conjunction (used to capture layering) and its associated implications. We give soundness and completeness theorems for a labelled tableaux system with respect to a Kripke semantics on graphs. We then give an equivalent relational semantics, itself proven equivalent to an algebraic semantics via a representation theorem. We utilise this result in two ways. First, we prove decidability of the logic by showing the finite embeddability property holds for the algebraic semantics. Second, we prove a Stone-type duality theorem for the logic. By introducing the notions of ILGL hyperdoctrine and indexed layered frame we are able to extend this result to a predicate version of the logic and prove soundness and completeness theorems for an extension of the layered graph semantics . We indicate the utility of predicate ILGL with a resource-labelled bigraph model
Bunched logics: a uniform approach
Bunched logics have found themselves to be key tools in modern computer science, in particular through the industrial-level program verification formalism Separation Logic. Despite this—and in contrast to adjacent families of logics like modal and substructural logic—there is a lack of uniform methodology in their study, leaving many evident variants uninvestigated and many open problems unresolved. In this thesis we investigate the family of bunched logics—including previously unexplored intuitionistic variants—through two uniform frameworks. The first is a system of duality theorems that relate the algebraic and Kripke-style interpretations of the logics; the second, a modular framework of tableaux calculi that are sound and complete for both the core logics themselves, as well as many classes of bunched logic model important for applications in program verification and systems modelling. In doing so we are able to resolve a number of open problems in the literature, including soundness and completeness theorems for intuitionistic variants of bunched logics, classes of Separation Logic models and layered graph models; decidability of layered graph logics; a characterisation theorem for the classes of bunched logic model definable by bunched logic formulae; and the failure of Craig interpolation for principal bunched logics. We also extend our duality theorems to the categorical structures suitable for interpreting predicate versions of the logics, in particular hyperdoctrinal structures used frequently in Separation Logic
An Algebraic Approach to Inquisitive and DNA-Logics
This article provides an algebraic study of the propositional system InqB of inquisitive logic. We also investigate the wider class of DNA-logics, which are negative variants of intermediate logics, and the corresponding algebraic structures, DNA -varieties. We prove that the lattice of DNA-logics is dually isomorphic to the lattice of DNA -varieties. We characterise maximal and minimal intermediate logics with the same negative variant, and we prove a suitable version of Birkhoff's classic variety theorems. We also introduce locally finite DNA -varieties and show that these varieties are axiomatised by the analogues of Jankov formulas. Finally, we prove that the lattice of extensions of InqB is dually isomorphic to the ordinal omega + 1 and give an axiomatisation of these logics via Jankov DNA -formulas. This shows that these extensions coincide with the so-called inquisitive hierarchy of [9].(1)Peer reviewe