arXiv subjects
Publications and source records attributed to Jean Goubault-Larrecq.
We define and study Noetherian topologies for spaces of infinite sets, and infinite words. In each case, we also obtain S-representations, namely, computable presentations of the sobrifications of those spaces.
The probabilistic powerdomain $\mathbf V X$ on a space $X$ is the space of all continuous valuations on $X$. We show that, for every quasi-continuous domain $X$, $\mathbf V X$ is again a quasi-continuous domain, and that the Scott and weak topologies then agree on $\mathbf V X$. This also applies to the subspaces of probability and subprobability valuations on $X$. We also show that the Scott and weak topologies on the $\mathbf V X$ may differ when $X$ is not quasi-continuous, and we give a simple, compact Hausdorff counterexample.
This paper is a sequel of "Forward Analysis for WSTS, Part I: Completions" [STACS 2009, LZI Intl. Proc. in Informatics 3, 433-444] and "Forward Analysis for WSTS, Part II: Complete WSTS" [Logical Methods in Computer Science 8(3), 2012]. In these two papers, we provided a framework to conduct forward reachability analyses of WSTS, using finite representations of downward-closed sets. We further develop this framework to obtain a generic Karp-Miller algorithm for the new class of very-WSTS. This allows us to show that coverability sets of very-WSTS can be computed as their finite ideal decompositions. Under natural effectiveness assumptions, we also show that LTL model checking for very-WSTS is decidable. The termination of our procedure rests on a new notion of acceleration levels, which we study. We characterize those domains that allow for only finitely many accelerations, based on ordinal ranks.
It is well-known that the higher-order language PCF is not fully abstract: there is a program - the so-called parallel or tester, meant to test whether its input behaves as a parallel or - which never terminates on any input, operationally, but is denotationally non-trivial. We explore a probabilistic variant of PCF, and ask whether the parallel or tester exhibits a similar behavior there. The answer is no: operationally, one can feed the parallel or tester an input that will fool it into thinking it is a parallel or. We show that the largest probability of success of such would-be parallel ors is exactly $8/27$. The bound is reached by a very simple probabilistic program. The difficult part is to show that that bound cannot be exceeded.
Elegant and general algorithms for handling upwards-closed and downwards-closed subsets of WQOs can be developed using the filter-based and ideal-based representation for these sets. These algorithms can be built in a generic or parameterized way, in parallel with the way complex WQOs are obtained by combining or modifying simpler WQOs.
There is no known way of giving a domain-theoretic semantics to higher-order probabilistic languages, in such a way that the involved domains are continuous or quasi-continuous - the latter is required to do any serious mathematics. We argue that the problem naturally disappears for languages with two kinds of types, where one kind is interpreted in a Cartesian-closed category of continuous dcpos, and the other is interpreted in a category that is closed under the probabilistic powerdomain functor. Such a setting is provided by Paul B. Levy's call-by-push-value paradigm. Following this insight, we define a call-by-push-value language, with probabilistic choice sitting inside the value types, and where conversion from a value type to a computation type involves demonic non-determinism. We give both a domain-theoretic semantics and an operational semantics for the resulting language, and we show that they are sound and adequate. With the addition of statistical termination testers and parallel if, we show that the language is even fully abstract - and those two primitives are required for that.
We investigate the Eilenberg-Moore algebras of the extended probabilistic powerdomain monad $\mathcal V_w$ over the category $\mathbf{TOP}_0$ of $T_0$ topological spaces and continuous maps. We prove that every $\mathcal V_w$-algebra in our setting is a weakly locally convex sober topological cone, and that a map is the structure map of a $\mathcal V_w$-algebra if and only if it is continuous and sends every continuous valuation to its unique barycentre. Conversely, for locally linear sober cones (a strong form of local convexity), the mere existence of barycentres entails that the barycentre map is the structure map of a $\mathcal V_w$-algebra; moreover the algebra morphisms are exactly the linear continuous maps in that case. We also examine the algebras of two related monads, the simple valuation monad $\mathcal V_{\mathrm f}$ and the point-continuous valuation monad $\mathcal V_{\mathrm p}$. In $\mathbf{TOP}_0$ their algebras are fully characterised as weakly locally convex topological cones and weakly locally convex sober topological cones, respectively. In both cases, the algebra morphisms are continuous linear maps between the corresponding algebras.
We study $G_δ$ subspaces of continuous dcpos, which we call domain-complete spaces, and $G_δ$ subspaces of locally compact sober spaces, which we call LCS-complete spaces. Those include all locally compact sober spaces-in particular, all continuous dcpos-, all topologically complete spaces in the sense of Čech, and all quasi-Polish spaces-in particular, all Polish spaces. We show that LCS-complete spaces are sober, Wilker, compactly Choquet-complete, completely Baire, and $\odot$-consonant-in particular, consonant; that the countably-based LCS-complete (resp., domain-complete) spaces are the quasi-Polish spaces exactly; and that the metrizable LCS-complete (resp., domain-complete) spaces are the completely metrizable spaces. We include two applications: on LCS-complete spaces, all continuous valuations extend to measures, and sublinear previsions form a space homeomorphic to the convex Hoare powerdomain of the space of continuous valuations.
We introduce a pointfree theory of convergence on lattices and coframes. A convergence lattice is a lattice $L$ with a monotonic map $\lim_L$ from the lattice of filters on $L$ to $L$, meant to be an abstract version of the map sending every filter of subsets to its set of limits. This construction exhibits the category of convergence spaces as a coreflective subcategory of the opposite of the category of convergence lattices. We extend this construction to coreflections between limit spaces and the opposite of so-called limit lattices and limit coframes, between pretopological convergence spaces and the opposite of so-called pretopological convergence coframes, between adherence spaces and the opposite of so-called adherence coframes, between topological spaces and the opposite of so-called topological coframes. All of our pointfree categories are complete and cocomplete, and topological over the category of coframes. Our final pointfree category, that of topological coframes, shares with the category of frames the property of being in a dual adjunction with the category of topological spaces. We show that the latter arises as a retract of the former, and that this retraction restricts to a reflection between frames and so-called strong topological coframes.
We show analogues of the Daniell-Kolmogorov and Prohorov theorems on the existence of projective limits of measures, in the setting of continuous valuations on $T_0$ topological spaces.
Given a poset $P$, the set, $Γ(P)$, of all Scott closed sets ordered by inclusion forms a complete lattice. A subcategory $\mathbf{C}$ of $\mathbf{Pos}_d$ (the category of posets and Scott-continuous maps) is said to be $Γ$-faithful if for any posets $P$ and $Q$ in $\mathbf{C}$, $Γ(P) \cong Γ(Q)$ implies $P \cong Q$. It is known that the category of all continuous dcpos and the category of bounded complete dcpos are $Γ$-faithful, while $\mathbf{Pos}_d$ is not. Ho & Zhao (2009) asked whether the category $\mathbf{DCPO}$ of dcpos is $Γ$-faithful. In this paper, we answer this question in the negative by exhibiting a counterexample. To achieve this, we introduce a new subcategory of dcpos which is $Γ$-faithful. This subcategory subsumes all currently known $Γ$-faithful subcategories. With this new concept in mind, we construct the desired counterexample which relies heavily on Johnstone's famous dcpo which is not sober in its Scott topology.
Using the notion of formal ball, we present a few new results in the theory of quasi-metric spaces. With no specific order: every continuous Yoneda-complete quasi-metric space is sober and convergence Choquet-complete hence Baire in its $d$-Scott topology; for standard quasi-metric spaces, algebraicity is equivalent to having enough center points; on a standard quasi-metric space, every lower semicontinuous $\bar{\mathbb{R}}_+$-valued function is the supremum of a chain of Lipschitz Yoneda-continuous maps; the continuous Yoneda-complete quasi-metric spaces are exactly the retracts of algebraic Yoneda-complete quasi-metric spaces; every continuous Yoneda-complete quasi-metric space has a so-called quasi-ideal model, generalizing a construction due to K. Martin. The point is that all those results reduce to domain-theoretic constructions on posets of formal balls.
This document gives semantics to programs written in a C-like programming language, featuring interactions with an external environment with noisy and imprecise data.
We describe a simple, conceptual forward analysis procedure for infinity-complete WSTS S. This computes the so-called clover of a state. When S is the completion of a WSTS X, the clover in S is a finite description of the downward closure of the reachability set. We show that such completions are infinity-complete exactly when X is an omega-2-WSTS, a new robust class of WSTS. We show that our procedure terminates in more cases than the generalized Karp-Miller procedure on extensions of Petri nets and on lossy channel systems. We characterize the WSTS where our procedure terminates as those that are clover-flattable. Finally, we apply this to well-structured counter systems.
Is there any Cartesian-closed category of continuous domains that would be closed under Jones and Plotkin's probabilistic powerdomain construction? This is a major open problem in the area of denotational semantics of probabilistic higher-order languages. We relax the question, and look for quasi-continuous dcpos instead. We introduce a natural class of such quasi-continuous dcpos, the omega-QRB-domains. We show that they form a category omega-QRB with pleasing properties: omega-QRB is closed under the probabilistic powerdomain functor, under finite products, under taking bilimits of expanding sequences, under retracts, and even under so-called quasi-retracts. But... omega-QRB is not Cartesian closed. We conclude by showing that the QRB domains are just one half of an FS-domain, merely lacking control.
Well-structured transition systems provide the right foundation to compute a finite basis of the set of predecessors of the upward closure of a state. The dual problem, to compute a finite representation of the set of successors of the downward closure of a state, is harder: Until now, the theoretical framework for manipulating downward-closed sets was missing. We answer this problem, using insights from domain theory (dcpos and ideal completions), from topology (sobrifications), and shed new light on the notion of adequate domains of limits.
Logical relations and their generalizations are a fundamental tool in proving properties of lambda-calculi, e.g., yielding sound principles for observational equivalence. We propose a natural notion of logical relations able to deal with the monadic types of Moggi's computational lambda-calculus. The treatment is categorical, and is based on notions of subsconing, mono factorization systems, and monad morphisms. Our approach has a number of interesting applications, including cases for lambda-calculi with non-determinism (where being in logical relation means being bisimilar), dynamic name creation, and probabilistic systems.