363 research outputs found
Cyclic proof systems for modal fixpoint logics
This thesis is about cyclic and ill-founded proof systems for modal fixpoint logics, with and without explicit fixpoint quantifiers.Cyclic and ill-founded proof-theory allow proofs with infinite branches or paths, as long as they satisfy some correctness conditions ensuring the validity of the conclusion. In this dissertation we design a few cyclic and ill-founded systems: a cyclic one for the weak Grzegorczyk modal logic K4Grz, based on our explanation of the phenomenon of cyclic companionship; and ill-founded and cyclic ones for the full computation tree logic CTL* and the intuitionistic linear-time temporal logic iLTL. All systems are cut-free, and the cyclic ones for K4Grz and iLTL have fully finitary correctness conditions.Lastly, we use a cyclic system for the modal mu-calculus to obtain a proof of the uniform interpolation property for the logic which differs from the original, automata-based one
Multi-graded Featherweight Java
Resource-aware type systems statically approximate not only the expected
result type of a program, but also the way external resources are used, e.g.,
how many times the value of a variable is needed. We extend the type system of
Featherweight Java to be resource-aware, parametrically on an arbitrary grade
algebra modeling a specific usage of resources. We prove that this type system
is sound with respect to a resource-aware version of reduction, that is, a
well-typed program has a reduction sequence which does not get stuck due to
resource consumption. Moreover, we show that the available grades can be
heterogeneous, that is, obtained by combining grades of different kinds, via a
minimal collection of homomorphisms from one kind to another. Finally, we show
how grade algebras and homomorphisms can be specified as Java classes, so that
grade annotations in types can be written in the language itself
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
A Categorical Framework for Program Semantics and Semantic Abstraction
Categorical semantics of type theories are often characterized as
structure-preserving functors. This is because in category theory both the
syntax and the domain of interpretation are uniformly treated as structured
categories, so that we can express interpretations as structure-preserving
functors between them. This mathematical characterization of semantics makes it
convenient to manipulate and to reason about relationships between
interpretations. Motivated by this success of functorial semantics, we address
the question of finding a functorial analogue in abstract interpretation, a
general framework for comparing semantics, so that we can bring similar
benefits of functorial semantics to semantic abstractions used in abstract
interpretation. Major differences concern the notion of interpretation that is
being considered. Indeed, conventional semantics are value-based whereas
abstract interpretation typically deals with more complex properties. In this
paper, we propose a functorial approach to abstract interpretation and study
associated fundamental concepts therein. In our approach, interpretations are
expressed as oplax functors in the category of posets, and abstraction
relations between interpretations are expressed as lax natural transformations
representing concretizations. We present examples of these formal concepts from
monadic semantics of programming languages and discuss soundness.Comment: MFPS 202
Expressiveness Results for Timed Modal Mu-Calculi
This paper establishes relative expressiveness results for several modal
mu-calculi interpreted over timed automata. These mu-calculi combine modalities
for expressing passage of (real) time with a general framework for defining
formulas recursively; several variants have been proposed in the literature. We
show that one logic, which we call , is strictly more
expressive than the other mu-calculi considered. It is also more expressive
than the temporal logic TCTL, while the other mu-calculi are incomparable with
TCTL in the setting of general timed automata
On when the union of two algebraic sets is algebraic
In universal algebraic geometry, an algebra is called an equational domain if
the union of two algebraic sets is algebraic. We characterize equational
domains, with respect to polynomial equations, inside congruence permutable
varieties, and with respect to term equations, among all algebras of size two
and all algebras of size three with a cyclic automorphism. Furthermore, for
each size at least three, we prove that, modulo term equivalence, there is a
continuum of equational domains of that size.Comment: 50 pages, 1 figure, 1 tabl
Catoids and modal convolution algebras
We show how modal quantales arise as convolution algebras QX
of functions from catoids X, multisemigroups equipped with source and target maps, into modal quantales value or weight quantales Q. In the tradition of boolean algebras with operators we study modal correspondences between algebraic laws in X, Q and QX. The catoids introduced generalise Schweizer and Sklar’s function systems and single-set categories to structures isomorphic to algebras of ternary relations, as they are used for boolean algebras with operators and substructural logics. Our correspondence results support a generic construction of weighted modal quantales from catoids. This construction is illustrated by many examples. We also relate our results to reasoning with stochastic matrices or probabilistic predicate transformers
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