32 research outputs found
Powerset residuated algebras
We present an algebraic approach to canonical embeddings of arbitrary residuated algebras into powerset residuated algebras. We propose some construction of powerset residuated algebras and prove a representation theorem for symmetric residuated algebras
Vector spaces as Kripke frames
In recent years, the compositional distributional approach in computational
linguistics has opened the way for an integration of the \emph{lexical} aspects
of meaning into Lambek's type-logical grammar program. This approach is based
on the observation that a sound semantics for the associative, commutative and
unital Lambek calculus can be based on vector spaces by interpreting fusion as
the tensor product of vector spaces.
In this paper, we build on this observation and extend it to a `vector space
semantics' for the \emph{general} Lambek calculus, based on \emph{algebras over
a field} (or -algebras), i.e. vector spaces endowed
with a bilinear binary product. Such structures are well known in algebraic
geometry and algebraic topology, since they are important instances of Lie
algebras and Hopf algebras. Applying results and insights from duality and
representation theory for the algebraic semantics of nonclassical logics, we
regard -algebras as `Kripke frames' the complex algebras of which
are complete residuated lattices.
This perspective makes it possible to establish a systematic connection
between vector space semantics and the standard Routley-Meyer semantics of
(modal) substructural logics
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
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
Completeness via Canonicity for Distributive Substructural Logics: A Coalgebraic Perspective
We prove strong completeness of a range of substructural logics with respect to their relational semantics by completeness-via-canonicity. Specifically, we use the topological theory of canonical (in) equations in distributive lattice expansions to show that distributive substructural logics are strongly complete with respect to their relational semantics. 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