6,603 research outputs found
Axiomatizing the lexicographic products of modal logics with linear temporal logic
Given modal logics L1 and L2, their lexicographic product L1 x L2 is a new logic whose frames are the Cartesian products of an L1-frame and an L2-frame, but with the new accessibility relations reminiscent of a lexicographic ordering. This article considers the lexicographic products of several modal logics with linear temporal logic (LTL) based on ``next'' and ``always in the future''. We provide axiomatizations for logics of the form L x LTL and define cover-simple classes of frames; we then prove that, under fairly general conditions, our axiomatizations are sound and complete whenever the class of L-frames is cover-simple. Finally, we prove completeness for several concrete logics of the form L x LTL
The decision problem of modal product logics with a diagonal, and faulty counter machines
In the propositional modal (and algebraic) treatment of two-variable
first-order logic equality is modelled by a `diagonal' constant, interpreted in
square products of universal frames as the identity (also known as the
`diagonal') relation. Here we study the decision problem of products of two
arbitrary modal logics equipped with such a diagonal. As the presence or
absence of equality in two-variable first-order logic does not influence the
complexity of its satisfiability problem, one might expect that adding a
diagonal to product logics in general is similarly harmless. We show that this
is far from being the case, and there can be quite a big jump in complexity,
even from decidable to the highly undecidable. Our undecidable logics can also
be viewed as new fragments of first- order logic where adding equality changes
a decidable fragment to undecidable. We prove our results by a novel
application of counter machine problems. While our formalism apparently cannot
force reliable counter machine computations directly, the presence of a unique
diagonal in the models makes it possible to encode both lossy and
insertion-error computations, for the same sequence of instructions. We show
that, given such a pair of faulty computations, it is then possible to
reconstruct a reliable run from them
Linear Time Logics - A Coalgebraic Perspective
We describe a general approach to deriving linear time logics for a wide
variety of state-based, quantitative systems, by modelling the latter as
coalgebras whose type incorporates both branching behaviour and linear
behaviour. Concretely, we define logics whose syntax is determined by the
choice of linear behaviour and whose domain of truth values is determined by
the choice of branching, and we provide two equivalent semantics for them: a
step-wise semantics amenable to automata-based verification, and a path-based
semantics akin to those of standard linear time logics. We also provide a
semantic characterisation of the associated notion of logical equivalence, and
relate it to previously-defined maximal trace semantics for such systems.
Instances of our logics support reasoning about the possibility, likelihood or
minimal cost of exhibiting a given linear time property. We conclude with a
generalisation of the logics, dual in spirit to logics with discounting, which
increases their practical appeal in the context of resource-aware computation
by incorporating a notion of offsetting.Comment: Major revision of previous version: Sections 4 and 5 generalise the
results in the previous version, with new proofs; Section 6 contains new
result
Strongly Complete Logics for Coalgebras
Coalgebras for a functor model different types of transition systems in a
uniform way. This paper focuses on a uniform account of finitary logics for
set-based coalgebras. In particular, a general construction of a logic from an
arbitrary set-functor is given and proven to be strongly complete under
additional assumptions. We proceed in three parts. Part I argues that sifted
colimit preserving functors are those functors that preserve universal
algebraic structure. Our main theorem here states that a functor preserves
sifted colimits if and only if it has a finitary presentation by operations and
equations. Moreover, the presentation of the category of algebras for the
functor is obtained compositionally from the presentations of the underlying
category and of the functor. Part II investigates algebras for a functor over
ind-completions and extends the theorem of J{\'o}nsson and Tarski on canonical
extensions of Boolean algebras with operators to this setting. Part III shows,
based on Part I, how to associate a finitary logic to any finite-sets
preserving functor T. Based on Part II we prove the logic to be strongly
complete under a reasonable condition on T
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
Dual-Context Calculi for Modal Logic
We present natural deduction systems and associated modal lambda calculi for
the necessity fragments of the normal modal logics K, T, K4, GL and S4. These
systems are in the dual-context style: they feature two distinct zones of
assumptions, one of which can be thought as modal, and the other as
intuitionistic. We show that these calculi have their roots in in sequent
calculi. We then investigate their metatheory, equip them with a confluent and
strongly normalizing notion of reduction, and show that they coincide with the
usual Hilbert systems up to provability. Finally, we investigate a categorical
semantics which interprets the modality as a product-preserving functor.Comment: Full version of article previously presented at LICS 2017 (see
arXiv:1602.04860v4 or doi: 10.1109/LICS.2017.8005089
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