150 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
Uniform Interpolation in Coalgebraic Modal Logic
A logic has uniform interpolation if its formulas can be projected down to given subsignatures, preserving all logical consequences that do not mention the removed symbols; the weaker property of (Craig) interpolation allows the projected formula - the interpolant - to be different for each logical consequence of the original formula. These properties are of importance, e.g., in the modularization of logical theories. We study interpolation in the context of coalgebraic modal logics, i.e. modal logics axiomatized in rank 1, restricting for clarity to the case with finitely many modalities. Examples of such logics include the modal logics K and KD, neighbourhood logic and its monotone variant, finite-monoid-weighted logics, and coalition logic. We introduce a notion of one-step (uniform) interpolation, which refers only to a restricted logic without nesting of modalities, and show that a coalgebraic modal logic has uniform interpolation if it has one-step interpolation. Moreover, we identify preservation of finite surjective weak pullbacks as a sufficient, and in the monotone case necessary, condition for one-step interpolation. We thus prove or reprove uniform interpolation for most of the examples listed above
Limits in categories of Vietoris coalgebras
Motivated by the need to reason about hybrid systems, we study limits in
categories of coalgebras whose underlying functor is a Vietoris polynomial one
- intuitively, the topological analogue of a Kripke polynomial functor. Among
other results, we prove that every Vietoris polynomial functor admits a final
coalgebra if it respects certain conditions concerning separation axioms and
compactness. When the functor is restricted to some of the categories induced
by these conditions the resulting categories of coalgebras are even complete.
As a practical application, we use these developments in the specification and
analysis of non-deterministic hybrid systems, in particular to obtain suitable
notions of stability, and behaviour.publishe
Characteristic Logics for Behavioural Hemimetrics via Fuzzy Lax Extensions
In systems involving quantitative data, such as probabilistic, fuzzy, or
metric systems, behavioural distances provide a more fine-grained comparison of
states than two-valued notions of behavioural equivalence or behaviour
inclusion. Like in the two-valued case, the wide variation found in system
types creates a need for generic methods that apply to many system types at
once. Approaches of this kind are emerging within the paradigm of universal
coalgebra, based either on lifting pseudometrics along set functors or on
lifting general real-valued (fuzzy) relations along functors by means of fuzzy
lax extensions. An immediate benefit of the latter is that they allow bounding
behavioural distance by means of fuzzy (bi-)simulations that need not
themselves be hemi- or pseudometrics; this is analogous to classical
simulations and bisimulations, which need not be preorders or equivalence
relations, respectively. The known generic pseudometric liftings, specifically
the generic Kantorovich and Wasserstein liftings, both can be extended to yield
fuzzy lax extensions, using the fact that both are effectively given by a
choice of quantitative modalities. Our central result then shows that in fact
all fuzzy lax extensions are Kantorovich extensions for a suitable set of
quantitative modalities, the so-called Moss modalities. For nonexpansive fuzzy
lax extensions, this allows for the extraction of quantitative modal logics
that characterize behavioural distance, i.e. satisfy a quantitative version of
the Hennessy-Milner theorem; equivalently, we obtain expressiveness of a
quantitative version of Moss' coalgebraic logic. All our results explicitly
hold also for asymmetric distances (hemimetrics), i.e. notions of quantitative
simulation
Bisimulation for Labelled Markov Processes
AbstractIn this paper we introduce a new class of labelled transition systems—labelled Markov processes— and define bisimulation for them. Labelled Markov processes are probabilistic labelled transition systems where the state space is not necessarily discrete. We assume that the state space is a certain type of common metric space called an analytic space. We show that our definition of probabilistic bisimulation generalizes the Larsen–Skou definition given for discrete systems. The formalism and mathematics is substantially different from the usual treatment of probabilistic process algebra. The main technical contribution of the paper is a logical characterization of probabilistic bisimulation. This study revealed some unexpected results, even for discrete probabilistic systems. •Bisimulation can be characterized by a very weak modal logic. The most striking feature is that one has no negation or any kind of negative proposition.•We do not need any finite branching assumption, yet there is no need of infinitary conjunction.
We also show how to construct the maximal autobisimulation on a system. In the finite state case, this is just a state minimization construction. The proofs that we give are of an entirely different character than the typical proofs of these results. They use quite subtle facts about analytic spaces and appear, at first sight, to be entirely nonconstructive. Yet one can give an algorithm for deciding bisimilarity of finite state systems which constructs a formula that witnesses the failure of bisimulation
Epistemic Modality, Mind, and Mathematics
This book concerns the foundations of epistemic modality. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality relates to the computational theory of mind; metaphysical modality; the types of mathematical modality; to the epistemic status of large cardinal axioms, undecidable propositions, and abstraction principles in the philosophy of mathematics; to the modal profile of rational intuition; and to the types of intention, when the latter is interpreted as a modal mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal cognitivism and modal expressivism. Chapter \textbf{3} provides an abstraction principle for epistemic intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal -calculus in order to account for the iteration of epistemic states, by contrast to availing of modal axiom 4 (i.e. the KK principle). Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's "criterial" identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapter \textbf{4} is availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. Chapter \textbf{8} examines the modal commitments of abstractionism, in particular necessitism, and epistemic modality and the epistemology of abstraction. Chapter \textbf{9} examines the modal profile of -logic in set theory. Chapter \textbf{10} examines the interaction between epistemic two-dimensional truthmaker semantics, epistemic set theory, and absolute decidability. Chapter \textbf{11} avails of modal coalgebraic automata to interpret the defining properties of indefinite extensibility, and avails of epistemic two-dimensional semantics in order to account for the interaction of the interpretational and objective modalities thereof. The hyperintensional, topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapter \textbf{2} is applied in chapters \textbf{7}, \textbf{8}, \textbf{10}, and \textbf{11}. Chapter \textbf{12} provides a modal logic for rational intuition and provides four models of hyperintensional semantics. Chapter \textbf{13} examines modal responses to the alethic paradoxes. Chapter \textbf{14} examines, finally, the modal semantics for the different types of intention and the relation of the latter to evidential decision theory
Characteristic Logics for Behavioural Hemimetrics via Fuzzy Lax Extensions
In systems involving quantitative data, such as probabilistic, fuzzy, or
metric systems, behavioural distances provide a more fine-grained comparison of
states than two-valued notions of behavioural equivalence or behaviour
inclusion. Like in the two-valued case, the wide variation found in system
types creates a need for generic methods that apply to many system types at
once. Approaches of this kind are emerging within the paradigm of universal
coalgebra, based either on lifting pseudometrics along set functors or on
lifting general real-valued (fuzzy) relations along functors by means of fuzzy
lax extensions. An immediate benefit of the latter is that they allow bounding
behavioural distance by means of fuzzy (bi-)simulations that need not
themselves be hemi- or pseudometrics; this is analogous to classical
simulations and bisimulations, which need not be preorders or equivalence
relations, respectively. The known generic pseudometric liftings, specifically
the generic Kantorovich and Wasserstein liftings, both can be extended to yield
fuzzy lax extensions, using the fact that both are effectively given by a
choice of quantitative modalities. Our central result then shows that in fact
all fuzzy lax extensions are Kantorovich extensions for a suitable set of
quantitative modalities, the so-called Moss modalities. For nonexpansive fuzzy
lax extensions, this allows for the extraction of quantitative modal logics
that characterize behavioural distance, i.e. satisfy a quantitative version of
the Hennessy-Milner theorem; equivalently, we obtain expressiveness of a
quantitative version of Moss' coalgebraic logic. All our results explicitly
hold also for asymmetric distances (hemimetrics), i.e. notions of quantitative
simulation
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