447 research outputs found
On kleene algebras for weighted computation
Kleene algebra with tests (KAT) was introduced as an alge-
braic structure to model and reason about classic imperative programs,
i.e. sequences of discrete actions guarded by Boolean tests.
This paper introduces two generalisations of this structure able to ex-
press programs as weighted transitions and tests with outcomes in a not
necessary bivalent truth space, namely graded Kleene algebra with tests
(GKAT) and Heyting Kleene algebra with tests (HKAT).
On these contexts, in analogy to Kozen's encoding of Propositional Hoare
Logic (PHL) in KAT [10], we discuss the encoding of a graded PHL in
HKAT and of its while-free fragment in GKAT.This work is financed by the 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 - Fundacao para a Ciencia e a Tecnologia, within projects POCI-01-0145-FEDER-016692 and UID/MAT/04106/2013. The second author is also supported by the individual grant SFRH/BPD/103004/2014
MV-algebras freely generated by finite Kleene algebras
If V and W are varieties of algebras such that any V-algebra A has a reduct
U(A) in W, there is a forgetful functor U: V->W that acts by A |-> U(A) on
objects, and identically on homomorphisms. This functor U always has a left
adjoint F: W->V by general considerations. One calls F(B) the V-algebra freely
generated by the W-algebra B. Two problems arise naturally in this broad
setting. The description problem is to describe the structure of the V-algebra
F(B) as explicitly as possible in terms of the structure of the W-algebra B.
The recognition problem is to find conditions on the structure of a given
V-algebra A that are necessary and sufficient for the existence of a W-algebra
B such that F(B) is isomorphic to A. Building on and extending previous work on
MV-algebras freely generated by finite distributive lattices, in this paper we
provide solutions to the description and recognition problems in case V is the
variety of MV-algebras, W is the variety of Kleene algebras, and B is finitely
generated--equivalently, finite. The proofs rely heavily on the Davey-Werner
natural duality for Kleene algebras, on the representation of finitely
presented MV-algebras by compact rational polyhedra, and on the theory of bases
of MV-algebras.Comment: 27 pages, 8 figures. Submitted to Algebra Universali
An Algebraic Framework for Compositional Program Analysis
The purpose of a program analysis is to compute an abstract meaning for a
program which approximates its dynamic behaviour. A compositional program
analysis accomplishes this task with a divide-and-conquer strategy: the meaning
of a program is computed by dividing it into sub-programs, computing their
meaning, and then combining the results. Compositional program analyses are
desirable because they can yield scalable (and easily parallelizable) program
analyses.
This paper presents algebraic framework for designing, implementing, and
proving the correctness of compositional program analyses. A program analysis
in our framework defined by an algebraic structure equipped with sequencing,
choice, and iteration operations. From the analysis design perspective, a
particularly interesting consequence of this is that the meaning of a loop is
computed by applying the iteration operator to the loop body. This style of
compositional loop analysis can yield interesting ways of computing loop
invariants that cannot be defined iteratively. We identify a class of
algorithms, the so-called path-expression algorithms [Tarjan1981,Scholz2007],
which can be used to efficiently implement analyses in our framework. Lastly,
we develop a theory for proving the correctness of an analysis by establishing
an approximation relationship between an algebra defining a concrete semantics
and an algebra defining an analysis.Comment: 15 page
Towards a Uniform Theory of Effectful State Machines
Using recent developments in coalgebraic and monad-based semantics, we
present a uniform study of various notions of machines, e.g. finite state
machines, multi-stack machines, Turing machines, valence automata, and weighted
automata. They are instances of Jacobs' notion of a T-automaton, where T is a
monad. We show that the generic language semantics for T-automata correctly
instantiates the usual language semantics for a number of known classes of
machines/languages, including regular, context-free, recursively-enumerable and
various subclasses of context free languages (e.g. deterministic and real-time
ones). Moreover, our approach provides new generic techniques for studying the
expressivity power of various machine-based models.Comment: final version accepted by TOC
Sound and complete axiomatizations of coalgebraic language equivalence
Coalgebras provide a uniform framework to study dynamical systems, including
several types of automata. In this paper, we make use of the coalgebraic view
on systems to investigate, in a uniform way, under which conditions calculi
that are sound and complete with respect to behavioral equivalence can be
extended to a coarser coalgebraic language equivalence, which arises from a
generalised powerset construction that determinises coalgebras. We show that
soundness and completeness are established by proving that expressions modulo
axioms of a calculus form the rational fixpoint of the given type functor. Our
main result is that the rational fixpoint of the functor , where is a
monad describing the branching of the systems (e.g. non-determinism, weights,
probability etc.), has as a quotient the rational fixpoint of the
"determinised" type functor , a lifting of to the category of
-algebras. We apply our framework to the concrete example of weighted
automata, for which we present a new sound and complete calculus for weighted
language equivalence. As a special case, we obtain non-deterministic automata,
where we recover Rabinovich's sound and complete calculus for language
equivalence.Comment: Corrected version of published journal articl
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