On Noetherian Spaces Jean Goubault-Larrecq∗ LSV, ENS Cachan, CNRS, INRIA Futurs 61, avenue du président-Wilson, F-94235 Cachan, France [email protected] Abstract For any subset A of X, let Pre∃δ(A) be the preimage x X y A x δ y . The commutation property { ∈ |∃ ∈ · } ∃ A topological space is Noetherian iff every open is com- ensures that the preimage Pre δ(V ) of any upward-closed pact. Our starting point is that this notion generalizes that subset V is again upward-closed (V is upward-closed iff of well-quasi order, in the sense that an Alexandroff-discrete whenever x V and x x′, then x′ V ). Standard ∈ ≤ ∈ ∃∗ space is Noetherian iff its specialization quasi-ordering is arguments then show that one may compute Pre δ(V ), well. For more general spaces, this opens the way to ver- the set of states in X from which we can reach some state ifying infinite transition systems based on non-well quasi in V in finitely many steps: Compute the set Vi of states ordered sets, but where the preimage operator satisfies an from which we can reach some state in V in at most i steps, ∃ additional continuity assumption. The technical develop- backwards, by V0 = V , Vi+1 = Vi Pre δ(Vi): this ∪ ∃∗ ment rests heavily on techniques arising from topology and stabilizes at some stage i, where Vi = Pre δ(V ). domain theory, including sobriety and the de Groot dual of This provides an algorithm for coverability: given two a stably compact space. We show that the category Nthr ′ states x,x X, is there a trace x = x0 δ x1 δ ... δxk of Noetherian spaces is finitely complete and finitely cocom- ′ ∈ ∃∗ ′ ′ such that x xk? Just check x Pre ( x ), where x plete. Finally, we note that if X is a Noetherian space, then is the upward-closed≤ set y X ∈x′ y .↑ ↑ the set of all (even infinite) subsets of X is again Noethe- { ∈ | ≤ } Outline. We generalize this by replacing quasi- rian, a result that fails for well-quasi orders. orderings by topologies. We shall definitely rest on the rich relationship between theories of order and topology. We recapitulate what we need in two sections, Section 2 1. Introduction for basic notions, and Section 5 for more advanced con- cepts such as Stone duality, sobriety, and stable compact- ness which we don’t need in earlier sections. The Zariski A topological space X is Noetherian iff every open sub- topology on spectra of Noetherian rings was the first known set of X is compact [13, chapitre 0, § 2]. We shall explain Noetherian topology; we discuss it only in Section 8, in the how this generalizes the theory of well quasi-orders. light of the rest of our paper. Our contribution occupies the Recall that a well quasi-ordering is a quasi-ordering other sections. We first show the tight relationship between (a reflexive and transitive relation) that is not only well- well-quasi orders and Noetherian spaces in Section 3, and founded, i.e., has no infinite descending chain, but also has show a few easy constructions of new Noetherian spaces no infinite antichain (a set of incomparable elements). One from given Noetherian spaces in Section 4. This culminates use of well quasi-orderings is in verifying well-structured in showing that the category Nthr of Noetherian spaces transition systems [2, 4, 11, 14]. These are transition sys- is finitely cocomplete. Section 6 is technically more chal- tems, usually infinite-state, with two ingredients. lenging, and characterizes those Noetherian spaces that are First, a well quasi-ordering on the also sober. This is the cornerstone of the theory. E.g., this ≤ / ′ set X of states. Second, the transi- x ≤ x (1) is instrumental to show that Nthr is finitely complete, and tion relation δ commutes with , i.e., that the Hoare space of a Noetherian space is again Noethe- ′ ≤ δ δ if x δ y and x x , then there is a rian. We show the latter in Section 7. We then prove the ′ ≤′ ′ ′ / state y such that x δ y and y y : y ≤ / y′ unexpected result that the set of all subsets of a Noetherian ≤ Examples include Petri nets, VASS [15], lossy channel sys- space X (even infinite ones) has a topology that makes it tems [3], timed Petri nets [6] to cite a few. Noetherian. This would be wrong in a pure theory of or- ders; topology makes the difference. Finally, our theory of ∗Partially supported by the INRIA ARC ProNoBis. Noetherian sober spaces suggests an alternative algorithm for coverability based on computing downward-closed sets, K Ui, then K Ui for some i I already. (A ⊆ Si∈I ⊆ ∈ which we describe in Section 9. We conclude in Section 10. family (xi)i∈I of elements quasi-ordered by is a non- We stress that this paper is not specifically geared to- empty family such that for every i, j I there≤ is k I ∈ ∈ wards applications. Its aim is rather to lay the theoretical such that xi xk and xj xk.) ≤ ≤ basis for Noetherian topological spaces. Write E = x X y E y x , E = x Related Work. If is a quasi-ordering on X then let X y ↑E x { y ∈. If |∃K is∈ compact,· ≤ then} ↓ K is,{ too,∈ ≤ ♯ |∃ ∈ · ≤ } ↑ Pfin(X) be the set of finite subsets of X, and order it by , and is also saturated. We shall usually reserve the letter Q ≤ where A ♯ B iff for every y B there is an x A such for saturated compacts. When E is finite, E is compact ≤ ∈ ∈ that x y. It is well-known that ♯ needs not be well even saturated: call these the finitary compacts. Similarly,↑ E is when ≤ is well. This is a shortcoming,≤ among others, of closed: call these the finitary closed subsets. ↓ ≤ the theory of well quasi-orderings. Such shortcomings led We have gone one direction, from topology to quasi- Nash-Williams [23] to invent better quasi-orderings (bqos). orderings. There are in general many return paths. The Bqos have a rather unintuitive definition but a wonderful finest topology having as specialization quasi-ordering is theory, see [19]. The only application of bqos we know of to the Alexandroff topology≤ of . Its opens are the upward- verification problems is by Abdulla and Nylén [5], where it closed subsets of X with respect≤ to . The coarsest is is used to show the termination of the backward reachability the upper topology, generated by the complements≤ of sets iteration, using disjunctive constraints. x , x X. Its closed sets are the unions of subsets This paper is not on bqos, and in fact not specifically of↓ { the} form∈ E, E finite. An intermediate topology is the on well quasi-orderings. While bqos are restrictions of well Scott topology↓ , whose opens are those upward-closed sub- quasi-orderings, Noetherian spaces generalize the latter. We sets U such that every directed family (xi)i∈I that has a hope that Noetherian spaces will be valuable in verifica- least upper bound in U meets U. The latter crops up in do- tion in the future. The fact that Pfin(X), with the upper main theory, where a cpo is a partially ordered set where topology of ♯, and that P(X), with another topology, are every directed family has a least upper bound. ≤ Noetherian whenever X is (Section 7) is a promising result. A topological space is Alexandroff-discrete iff every in- Our work is more connected to topology, and in partic- tersection of opens is again open. Equivalently, iff its topol- ular to topology as it is practiced in domain theory. As we ogy is the Alexandroff topology of its specialization quasi- shall see later, the notions of specialization quasi-ordering ordering. While every finitary compact is compact satu- of a topological space, of upper, Scott and Alexandroff rated, the converse holds in Alexandroff-discrete spaces. topologies, of sober space, of sobrification of a space, and A map f from X to Y is continuous iff f −1(V ) is of stably compact spaces are central to our work. Topology open in X for every open V of Y . Any continuous func- and domain theory form another wonderful piece of mathe- tion is monotonic with respect to the specialization quasi- matics, and one may consult [12, 7, 18, 21]. orderings of X and Y . The converse holds when X Last but not least, Noetherian spaces arise from algebraic is Alexandroff-discrete: while continuity is usually seen geometry [13]: we discuss this briefly in Section 8. as stronger than monotonicity, continuity also generalizes monotonicity, in the sense that monotonicity is just conti- 2. Preliminaries I: Order and Topology nuity with respect to Alexandroff topologies. When X and Y are equipped with Scott topologies, f : X Y is continuous iff f is Scott-continuous, i.e., A topology on a set X is a collection of subsets (the f is monotonic→ and, for every directed family (x ) in X opens) of X thatO is closed under arbitrary unions and finite i i∈I having a least upper bound x, the family (f(x )) (which intersections. We say that X itself is a topological space, i i∈I is directed in Y ) admits f(x) as least upper bound. Conti- leaving implicit. The complements of opens are closed. nuity notions extend to binary relations. A relation R from The largestO open contained in A is its interior, the smallest X to Y is a subset of X Y . It is lower semi-continuous iff closed subset cl(A) containing it is its closure.