419 research outputs found
On CSP and the Algebraic Theory of Effects
We consider CSP from the point of view of the algebraic theory of effects,
which classifies operations as effect constructors or effect deconstructors; it
also provides a link with functional programming, being a refinement of Moggi's
seminal monadic point of view. There is a natural algebraic theory of the
constructors whose free algebra functor is Moggi's monad; we illustrate this by
characterising free and initial algebras in terms of two versions of the stable
failures model of CSP, one more general than the other. Deconstructors are
dealt with as homomorphisms to (possibly non-free) algebras.
One can view CSP's action and choice operators as constructors and the rest,
such as concealment and concurrency, as deconstructors. Carrying this programme
out results in taking deterministic external choice as constructor rather than
general external choice. However, binary deconstructors, such as the CSP
concurrency operator, provide unresolved difficulties. We conclude by
presenting a combination of CSP with Moggi's computational {\lambda}-calculus,
in which the operators, including concurrency, are polymorphic. While the paper
mainly concerns CSP, it ought to be possible to carry over similar ideas to
other process calculi
Natural Factors of the Medvedev Lattice Capturing IPC
Skvortsova showed that there is a factor of the Medvedev lattice which
captures intuitionistic propositional logic (IPC). However, her factor is
unnatural in the sense that it is constructed in an ad hoc manner. We present a
more natural example of such a factor. We also show that for every non-trivial
factor of the Medvedev lattice its theory is contained in Jankov's logic, the
deductive closure of IPC plus the weak law of the excluded middle. This answers
a question by Sorbi and Terwijn
On lattices and their ideal lattices, and posets and their ideal posets
For P a poset or lattice, let Id(P) denote the poset, respectively, lattice,
of upward directed downsets in P, including the empty set, and let
id(P)=Id(P)-\{\emptyset\}. This note obtains various results to the effect that
Id(P) is always, and id(P) often, "essentially larger" than P. In the first
vein, we find that a poset P admits no "<"-respecting map (and so in
particular, no one-to-one isotone map) from Id(P) into P, and, going the other
way, that an upper semilattice S admits no semilattice homomorphism from any
subsemilattice of itself onto Id(S).
The slightly smaller object id(P) is known to be isomorphic to P if and only
if P has ascending chain condition. This result is strengthened to say that the
only posets P_0 such that for every natural number n there exists a poset P_n
with id^n(P_n)\cong P_0 are those having ascending chain condition. On the
other hand, a wide class of cases is noted here where id(P) is embeddable in P.
Counterexamples are given to many variants of the results proved.Comment: 8 pages. Copy at http://math.berkeley.edu/~gbergman/papers may be
updated more frequently than arXiv copy. After publication, updates, errata,
etc. may be noted at that pag
Automaton semigroups: new construction results and examples of non-automaton semigroups
This paper studies the class of automaton semigroups from two perspectives:
closure under constructions, and examples of semigroups that are not automaton
semigroups. We prove that (semigroup) free products of finite semigroups always
arise as automaton semigroups, and that the class of automaton monoids is
closed under forming wreath products with finite monoids. We also consider
closure under certain kinds of Rees matrix constructions, strong semilattices,
and small extensions. Finally, we prove that no subsemigroup of arises as an automaton semigroup. (Previously, itself was
the unique example of a finitely generated residually finite semigroup that was
known not to arise as an automaton semigroup.)Comment: 27 pages, 6 figures; substantially revise
Entailment systems for stably locally compact locales
The category SCFrU of stably continuous frames and preframe ho-momorphisms (preserving ¯nite meets and directed joins) is dual to the Karoubi envelope of a category Ent whose objects are sets and whose
morphisms X ! Y are upper closed relations between the ¯nite powersets FX and FY . Composition of these morphisms is the \cut composition" of Jung et al. that interfaces disjunction in the codomains with conjunctions in the domains, and thereby relates to their multi-lingual sequent
calculus. Thus stably locally compact locales are represented by \entailment systems" (X; `) in which `, a generalization of entailment relations,is idempotent for cut composition.
Some constructions on stably locally compact locales are represented
in terms of entailment systems: products, duality and powerlocales.
Relational converse provides Ent with an involution, and this gives a simple treatment of the duality of stably locally compact locales. If A and B are stably continuous frames, then the internal preframe hom A t B is isomorphic to e A Â B where e A is the Hofmann-Lawson dual.
For a stably locally compact locale X, the lower powerlocale of X is shown to be the dual of the upper powerlocale of the dual of X
Type-Based Termination, Inflationary Fixed-Points, and Mixed Inductive-Coinductive Types
Type systems certify program properties in a compositional way. From a bigger
program one can abstract out a part and certify the properties of the resulting
abstract program by just using the type of the part that was abstracted away.
Termination and productivity are non-trivial yet desired program properties,
and several type systems have been put forward that guarantee termination,
compositionally. These type systems are intimately connected to the definition
of least and greatest fixed-points by ordinal iteration. While most type
systems use conventional iteration, we consider inflationary iteration in this
article. We demonstrate how this leads to a more principled type system, with
recursion based on well-founded induction. The type system has a prototypical
implementation, MiniAgda, and we show in particular how it certifies
productivity of corecursive and mixed recursive-corecursive functions.Comment: In Proceedings FICS 2012, arXiv:1202.317
Using models to model-check recursive schemes
We propose a model-based approach to the model checking problem for recursive
schemes. Since simply typed lambda calculus with the fixpoint operator,
lambda-Y-calculus, is equivalent to schemes, we propose the use of a model of
lambda-Y-calculus to discriminate the terms that satisfy a given property. If a
model is finite in every type, this gives a decision procedure. We provide a
construction of such a model for every property expressed by automata with
trivial acceptance conditions and divergence testing. Such properties pose
already interesting challenges for model construction. Moreover, we argue that
having models capturing some class of properties has several other virtues in
addition to providing decidability of the model-checking problem. As an
illustration, we show a very simple construction transforming a scheme to a
scheme reflecting a property captured by a given model.Comment: Long version of a paper presented at TLCA 201
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