8,590 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
Disjunctive bases: normal forms and model theory for modal logics
We present the concept of a disjunctive basis as a generic framework for
normal forms in modal logic based on coalgebra. Disjunctive bases were defined
in previous work on completeness for modal fixpoint logics, where they played a
central role in the proof of a generic completeness theorem for coalgebraic
mu-calculi. Believing the concept has a much wider significance, here we
investigate it more thoroughly in its own right. We show that the presence of a
disjunctive basis at the "one-step" level entails a number of good properties
for a coalgebraic mu-calculus, in particular, a simulation theorem showing that
every alternating automaton can be transformed into an equivalent
nondeterministic one. Based on this, we prove a Lyndon theorem for the full
fixpoint logic, its fixpoint-free fragment and its one-step fragment, a Uniform
Interpolation result, for both the full mu-calculus and its fixpoint-free
fragment, and a Janin-Walukiewicz-style characterization theorem for the
mu-calculus under slightly stronger assumptions.
We also raise the questions, when a disjunctive basis exists, and how
disjunctive bases are related to Moss' coalgebraic "nabla" modalities. Nabla
formulas provide disjunctive bases for many coalgebraic modal logics, but there
are cases where disjunctive bases give useful normal forms even when nabla
formulas fail to do so, our prime example being graded modal logic. We also
show that disjunctive bases are preserved by forming sums, products and
compositions of coalgebraic modal logics, providing tools for modular
construction of modal logics admitting disjunctive bases. Finally, we consider
the problem of giving a category-theoretic formulation of disjunctive bases,
and provide a partial solution
Propositional Logics Complexity and the Sub-Formula Property
In 1979 Richard Statman proved, using proof-theory, that the purely
implicational fragment of Intuitionistic Logic (M-imply) is PSPACE-complete. He
showed a polynomially bounded translation from full Intuitionistic
Propositional Logic into its implicational fragment. By the PSPACE-completeness
of S4, proved by Ladner, and the Goedel translation from S4 into Intuitionistic
Logic, the PSPACE- completeness of M-imply is drawn. The sub-formula principle
for a deductive system for a logic L states that whenever F1,...,Fk proves A,
there is a proof in which each formula occurrence is either a sub-formula of A
or of some of Fi. In this work we extend Statman result and show that any
propositional (possibly modal) structural logic satisfying a particular
formulation of the sub-formula principle is in PSPACE. If the logic includes
the minimal purely implicational logic then it is PSPACE-complete. As a
consequence, EXPTIME-complete propositional logics, such as PDL and the
common-knowledge epistemic logic with at least 2 agents satisfy this particular
sub-formula principle, if and only if, PSPACE=EXPTIME. We also show how our
technique can be used to prove that any finitely many-valued logic has the set
of its tautologies in PSPACE.Comment: In Proceedings DCM 2014, arXiv:1504.0192
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