Towards a Denotational Semantics for Discrete-Event Systems Eleftherios Matsikoudis University of California at Berkeley Berkeley, CA, 94720, USA ematsi@eecs. berkeley.edu Abstract For any two well ordered sets, either they are order- isomorphic or one is order-isomorphic to an initial segment This work focuses on establishing a semantic interpreta- of the other. tion for discrete-event systems. In this paper we describe A set A is called transitive iff a ∈ b ∈ A implies a ∈ A some of the basic mathematical structures, building on the for all sets a, b. tagged-signal model of [4, 3], and the term-based formal- An ordinal is a transitive set that is well ordered by set ism of [5]. containment ∈. We denote the class of all ordinals by Ω. Every well ordered set is order-isomorphic to a unique ordinal, called the ordinal number of the set. Two well or- 1. Prerequisites dered sets are order-isomorphic iff they have the same ordi- nal number. For any set A, the set A ∪ {A} is called the successor of In this section we introduce some terminology and re- A and is denoted by A+. view some fundamental concepts and results from set the- For all ordinals α and β, either α ∈ β, or α = β, or ory and order theory [2, 1]. β ∈ α. Any transitive set of ordinals is an ordinal. The empty 1.1. Isomorphisms set ∅ is an ordinal. If α is an ordinal, then α+ is an ordinal. If A is a set of ordinals, then S A is an ordinal. + Let (P, ≤P ), (Q, ≤Q), (R, ≤R) be ordered sets. For all ordinals α and β, α ∈ β if and only if α ∈ β or A map ϕ : P → Q is an order-embedding, and we write α+ = β. + ϕ : P,→ Q, iff for all p1, p2 ∈ P : An ordinal α is a successor ordinal iff α = β for some ordinal β. Otherwise α = S α, and either α = ∅ or α is a p1 ≤P p2 ⇔ ϕ(p1) ≤Q ϕ(p2) limit ordinal. The class of all ordinals Ω is not a set (Burtali-Forti para- An order-embedding ϕ : P,→ Q is an order- dox). isomorphism iff it maps P onto Q. The ordered sets P and A set A is dominated by a set B iff there exists a one-to- ∼ Q are order-isomorphic, and we write P = Q, iff there ex- one function from A into B. ists an order-isomorphism from P to Q. For any set A, there exists an ordinal α not dominated by The concept of isomorphism obeys laws of reflexivity, A (Hartogs’ theorem). ∼ ∼ symmetry, and transitivity. In particular, P = P , P = Q Note that we adopt von Neumann’s approach to the con- ∼ ∼ ∼ ∼ implies Q = P , and P = Q = R implies P = R. struction of the natural numbers. Under this approach, each natural number is the set of all smaller natural numbers. 1.2. Well Orderings and Ordinals Hence, every natural number is a finite ordinal and the set of all natural numbers ω = {0, 1, 2,...} coincides with the Let (P, ≤) be an ordered set. least limit ordinal. The ordered set (P, ≤) is well ordered iff it is totally or- dered and every non-empty subset of P has a least element. 1.3. Complete Partial Orders If p ∈ P , then the set p0 | p0 < p is called the initial segment up to p and is denoted by seg p. Let (P, ≤) be a partially ordered set. 1 An element ⊥ ∈ P such that ⊥ ≤ p for any p ∈ P is Definition 2.2 (Signal). A signal s is a partial function from called a bottom or zero element. A partially ordered set is the set of tags T to the set of values V, that is s ∈ (T * V). pointed iff it has a bottom element. A subset D of P is directed iff it is non-empty and every Alternatively, a signal s can be defined as a subset of pair of elements in D has an upper bound in D. E, such that for all events (τ1, v1), (τ2, v2) ∈ s, v1 6= v2 τ 6= τ A pointed partially ordered set in which every directed implies 1 2. The two definitions are equivalent and will subset has a least upper bound is called a complete partial be used interchangeably according to context. We denote S S = (T * V) ⊂ E order or simply cpo. the set of all signals by , that is P . The tag-set T (s) of a signal s ∈ S is defined to be the A subset C of P is consistent iff it is non-empty and domain of s, that is T (s) = dom s. every finite subset of C has an upper bound in P . For notational convenience, we will write s (τ) ' s (τ) A cpo (P, ≤) is consistently complete iff every consistent 1 2 iff the signals s and s are either both defined, or both set C ⊆ P has a least upper bound in P . 1 2 undefined at tag τ, and if defined s (τ) = s (τ). An element p of a cpo (P, ≤) is finite iff whenever p ≤ 1 2 W D for a directed set D ⊆ P , p ≤ d for some d ∈ D. Definition 2.3 (Natural Signal). A signal s ∈ S is natural A cpo (P, ≤) is algebraic iff for any p ∈ P , there exists iff there exists an order-embedding from its tag-set T (s) to a directed set D ⊆ P of finite elements such that p = W D. the set of all natural numbers ω. An algebraic cpo is called ω-algebraic iff the set of all finite elements is denumerable. We denote the set of all natural signals by Sω. Definition 2.4 (Ordinal signal). A signal s ∈ S is ordinal 1.4. Least Fixed Points iff its tag-set T (s) is well ordered. Equivalently, a signal is ordinal iff there exists an order- Let (P, ≤P ), (Q, ≤Q) be complete partial orders. A function f : P → Q is order-preserving iff for all isomorphism from its tag-set to some ordinal. A natural sig- nal s ∈ Sω is clearly ordinal, since the existence of an order- p1, p2 ∈ P such that p1 ≤P p2, f(p1) ≤Q f(p2). T (s) A function f : P → Q is continuous iff for every di- embedding from its tag-set to the set of all natural ω rected set D ⊆ P , the set f(d) | d ∈ D is a directed numbers implies the existence of an order-isomorphism from T (s) to some ordinal α, where in particular α ∈ ω or subset of Q, and f(W D) = W f(d) | d ∈ D . Every P Q α = ω. We denote the set of all ordinal signals by S . Note continuous function is order-preserving. Ω that there is no ambiguity in terming the class of all ordinal An element p ∈ P is a fixed point of the function f : signals a set, since S ⊆ S. P → P iff f(p) = p. A fixed point p of f is the least fixed Ω point of f, and we use the expression µx.f(x) to denote it, 2.2. Tuples of Signals iff for any fixed point p0 of f, p ≤ p0. If f : P → P is a continuous function, then f has a least V ar x, y, z, x ,... fixed point and µx.f(x) = W f n(⊥). Let be an infinite set of variables. Let i n∈ω range over V ar and I, J, K, . over V ar. If f : P → P is an order-preserving function, then f P α Given sets I and A, the set of all functions from I to A has a least fixed point and µx.f(x) = f (⊥) for some or- I I + is denoted by (I → A) or A . An element of A can be dinal α, where f β (⊥) = f(f β(⊥)) for any ordinal β, and γ W β thought of as an I-tuple of elements of A, and we usually f (⊥) = f (⊥) | β ∈ γ if γ is a limit ordinal. I I write ai instead of a(i) for a ∈ A , i ∈ I. If a ∈ A and b ∈ AJ with I ∩ J = ∅, then a ⊕ b ∈ AI∪J denotes the 2. Signals and Tuples of Signals (I ∪ J)-tuple satisfying: ( a if i ∈ I, 2.1. Signals (a ⊕ b)(i) = i bi if i ∈ J. Let V be a non-empty set of possible values, and T a A tuple of signals or signal tuple s is an I-tuple of signals non-empty set of tags. While we impose no structure on the in S for some set of variables I ⊆ V ar. Notice that the set of values V, we require that the set of tags be a totally set of all tuples of signals is essentially the set of all partial ordered set (T , ≤). functions from the set of all variables to the set of all signals, Definition 2.1 (Event). An event e is a tuple (τ, v) with τ ∈ that is (V ar * S). I T and v ∈ V. The tag-set T (s) of a tuple s ∈ S is defined as the union of the tag-sets of the tupled signals, that is T (s) = S We denote the set of all events by E, that is E = T × V. i∈I T (si). 2 Occasionally, it will be convenient to consider an alter- The signal-prefix relation v ⊂ S × S is a partial order, native definition for tuples of signals, according to which a and for any signal s ∈ S, ∅ v s.