22 research outputs found
Convolution, Separation and Concurrency
A notion of convolution is presented in the context of formal power series
together with lifting constructions characterising algebras of such series,
which usually are quantales. A number of examples underpin the universality of
these constructions, the most prominent ones being separation logics, where
convolution is separating conjunction in an assertion quantale; interval
logics, where convolution is the chop operation; and stream interval functions,
where convolution is used for analysing the trajectories of dynamical or
real-time systems. A Hoare logic is constructed in a generic fashion on the
power series quantale, which applies to each of these examples. In many cases,
commutative notions of convolution have natural interpretations as concurrency
operations.Comment: 39 page
Deriving real-time action systems with multiple time bands using algebraic reasoning
The verify-while-develop paradigm allows one to incrementally develop programs from their specifications using a series of calculations against the remaining proof obligations. This paper presents a derivation method for real-time systems with realistic constraints on their behaviour. We develop a high-level interval-based logic that provides flexibility in an implementation, yet allows algebraic reasoning over multiple granularities and sampling multiple sensors with delay. The semantics of an action system is given in terms of interval predicates and algebraic operators to unify the logics for an action system and its properties, which in turn simplifies the calculations and derivations
Non-smooth and zeno trajectories for hybrid system algebra
Hybrid systems are heterogeneous systems characterised by the interaction of discrete and continuous dynamics. In this paper we
compare a slightly extended version of our earlier algebraic approach
to hybrid systems with other approaches. We show that hybrid automata,
which are probably the standard tool for describing hybrid systems, can
conveniently be embedded into our algebra. But we allow general transition functions, not only smooth ones. Moreover we embed other models and point out some important advantages of the algebraic approach. In particular, we show how to easily handle Zeno effects, which are excluded by most other authors. The development of the theory is illustrated by a running example and a larger case study
Semiring neighbours
In 1996 Zhou and Hansen proposed a first-order interval logic called Neighbourhood Logic (NL) for specifying liveness and fairness of computing systems and also defining notions of real analysis in terms of expanding modalities. After that, Roy and Zhou presented a sound and relatively complete Duration Calculus as an extension of NL. We present an embedding of NL into an idempotent semiring of intervals. This embedding allows us to extend NL from single intervals to sets of intervals as well as to extend the approach to arbitrary idempotent semirings. We show that most of the required properties follow directly from Galois connections, hence we get the properties for free. As one important result we get that some of the axioms which were postulated for NL can be dropped since they are theorems in our generalisation. Furthermore, we present some possible interpretations for neighbours beyond intervals. Here we discuss for example reachability in graphs and applications to hybrid systems. At the end of the paper we add finite and infinite iteration to NL and extend idempotent semirigs to Kleene algebras and omega algebras. These extensions are useful for formulating repetitive properties and procedures like loops
Probabilistic Demonic Refinement Algebra
We propose an abstract algebra for reasoning about probabilistic programs in a total-correctness framework. In contrast to probablisitic Kleene algebra it allows genuine reasoning about total correctness and in addition to Kleene star also has a strong iteration operator. We define operators that determine whether a program is enabled, has certain failure or does not have certain failure, respectively. The algebra is applied to the derivation of refinement rules for probabilistic action systems
Generalising KAT to verify weighted computations
Kleene algebra with tests (KAT) was introduced as an algebraic
structure to model and reason about classic imperative programs, i.e.
sequences of discrete transitions guarded by Boolean tests. This paper
introduces two generalisations of this structure able to express programs
as weighted transitions and tests with outcomes in non necessarily
bivalent truth spaces: graded Kleene algebra with tests (GKAT) and a
variant where tests are also idempotent (I-GKAT). In this context, and
in analogy to Kozen's encoding of Propositional Hoare Logic (PHL) in
KAT we discuss the encoding of a graded PHL in I-GKAT and of its
while-free fragment in GKAT. Moreover, to establish semantics for these
structures four new algebras are de ned: FSET (T ), FREL(K; T )
and FLANG(K; T ) over complete residuated lattices K and T , and
M(n;A) over a GKAT or I-GKAT A. As a nal exercise, the paper
discusses some program equivalence proofs in a graded context.POCI-01-0145-FEDER-03094, NORTE-01-0145-FEDER-000037. ERDF – European Regional Development Fund
through the Operational Programme for Competitiveness and Internationalisation - COMPETE 2020 Programme and by National Funds through the Portuguese funding agency, FCT - Fundação para a Ciência e a Tecnologia, within project POCI-01-0145-FEDER-030947. This paper is also a result of the project SmartEGOV, NORTE-01-0145-FEDER-000037. The second author is supported in the scope of the framework contract foreseen in the numbers 4, 5 and 6 of the article 23, of the Decree-Law 57/2016, of August 29, changed by Portuguese Law 57/2017, of July 19, at CIDMA (Centro de Investigação e Desenvolvimento em Matemática e Aplicações) UID/MAT/04106/2019
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An Interval Logic for Stream-Processing Functions: A Convolution-Based Construction
We develop an interval-based logic for reasoning about systems consisting of components speci ed using stream-processing functions, which map streams of inputs to streams of outputs. The construction is algebraic and builds on a theory of convolution from formal power series. Using these algebraic foundations, we uniformly (and systematically) de ne operators for time- and space-based (de)composition. We also show that Banach's xed point theory can be incorporated into the framework, building on an existing theory of partially ordered monoids, which enables a feedback operator to be de ned algebraically.This research is supported by EPSRC Grant EP/N016661/1
An ω-Algebra for Real-Time Energy Problems
International audienceWe develop a *-continuous Kleene ω-algebra of real-time energy functions. Together with corresponding automata, these can be used to model systems which can consume and regain energy (or other types of resources) depending on available time. Using recent results on *-continuous Kleene ω-algebras and computability of certain manipulations on real-time energy functions, it follows that reachability and Büchi acceptance in real-time energy automata can be decided in a static way which only involves manipulations of real-time energy functions