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 ordinal 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 ordered 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 signal 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. signal tuple s is an element of the set (T * V ar * V). In fact, the two definitions are equivalent in the sense that Proposition 2.2. Two signals s1, s2 ∈ S have an upper for any s ∈ (V ar * S), there exists a unique s0 ∈ (T * bound if and only if they are comparable. V ar * V) i ∈ V ar τ ∈ T , such that for all and , either Proof. If s1 and s2 are comparable, then they trivially share s(i)(τ) s0(τ)(i) and are both undefined, or they are both an upper bound, moreover s1 t s2 ∈ {s1, s2}. s(i)(τ) = s0(τ)(i) defined and , and vice versa. In the other direction, if s1 = ∅ or s2 = ∅, then s1 With this definitional equivalence in mind, we will write and s2 are trivially comparable. Otherwise, assume that s1 I s1(τ) ' s2(τ) for all tuples s1, s2 ∈ S and any tag τ, iff and s2 are incomparable. Then, and without loss of gen- s1(i)(τ) ' s2(i)(τ) for all i ∈ I. Similarly, for any signal erality, there exist tags τ1 ∈ T (s1) and τ2 ∈ T (s2), such tuple s and any set of tags T ⊆ T , we will let s T denote that s (τ ) 6' s (τ ) and τ < τ . For any signal s w s , 1 1 2 1 1 2 1 the signal tuple (i, si T ) | i ∈ dom s . s ⊇ s1, and we deduce that s(τ1) 6' s2(τ1). Hence s2 is incomparable to any upper bound of s . Definition 2.5 (Natural signal tuple). A signal tuple s ∈ SI 1 is natural iff there exists an order-embedding from its tag- Proposition 2.3. If C ⊆ S is a non-empty set of signals, set T (s) to the set of all natural numbers ω. then the following are equivalent: We denote the set of all natural I-tuples of signals by (i) C is totally ordered, I [S ]ω. Note that this is different from the set of all I-tuples I (ii) C is directed, of natural signals [Sω] . In fact, it is easy to verify that I I [S ]ω ⊆ [Sω] . We denote the set of all natural tuples of (iii) C is consistent. signals by (V ar * S)ω. Proof. If C is totally ordered, then for any pair of signals Definition 2.6 (Ordinal signal tuple). A signal tuple s ∈ SI s1, s2 ∈ C, either s1 v s2 or s2 @ s1 implying s1 t s2 ∈ is ordinal iff its tag-set T (s) is well ordered. {s1, s2} ⊆ C. Hence C is directed. If C is directed, then every pair of signals in C has an Equivalently, a tuple of signals is ordinal iff there exists upper bound in C, and by straightforward induction, any an order-isomorphism from its tag-set to some ordinal. As finite subset of C will have an upper bound in C. Hence C in the case of natural and ordinal signals, any natural tuple is consistent. of signals is ordinal. We denote the set of all ordinal I- I If C is consistent, then any pair of signals s1, s2 ∈ C has tuples of signals by [S ]Ω. Again, this is different from the I an upper bound, and s1 and s2 need to be comparable by set of all I-tuples of ordinal signals [SΩ] , and it is easy to I I Proposition 2.2. Hence C is totally ordered. verify that [S ]Ω ⊆ [SΩ] . We denote the set of all ordinal tuples of signals by (V ar * S)Ω. Proposition 2.4. The least upper bound F C of a set of signals C ⊆ S exists if and only if C is totally ordered, in Proposition 2.1. If I ⊆ V ar is a finite set of variables, then F S I I which case C = C. [S ]Ω = [SΩ] . Proof. Let C be totally ordered and consider the set S C ⊆ I Proof. Consider an arbitrary tuple of signals s ∈ S . If E. We first need to show that S C is a signal. Assume T (s) is well ordered, then T (si) will be well ordered for to the contrary that S C/∈ S. Then there exist events any i ∈ I, since T (s ) ⊆ T (s). Hence, [SI ] ⊆ [S ]I . S i Ω Ω (τ1, v1), (τ2, v2) ∈ C with v1 6= v2 and τ1 = τ2. Con- For the other inclusion, notice that any non-empty subset sequently, there need to be signals s1, s2 ∈ C such that of tags T ⊆ T (s) can be expressed as S T ∩T (s ). If i∈I i (τ1, v1) ∈ s1 and (τ2, v2) ∈ s2, which cannot be com- T (si) is well ordered for any i ∈ I, then T ∩T (si) will have parable, contradicting the fact that C is totally ordered. a least tag for any i ∈ I. The set of these least tags will be Now for any signal s ∈ C, s ⊆ S C and (S C) \ s = totally ordered and finite by assumption. Therefore it will S 0 0 0 (s \ s). Since C is totally ordered, s \ s 6= ∅ S s ∈C have a least element that is also least in i∈I T ∩ T (si) . if and only if s s0. Hence, by Definition 2.7, for all I I @ Hence T (s) will be well ordered, and [S ]Ω ⊇ [SΩ] . tags τ ∈ T (s) and τ 0 ∈ T (s0 \ s), τ < τ 0. And since S S 0 S 0 T (( C) \ s) = T ( s0∈C (s \ s)) = s0∈C T (s \ s), we 2.3. Ordering Signals and Tuples of Signals conclude that s v S C. Therefore S C is an upper bound of C. If u ∈ S is another upper bound of C, then for any S Definition 2.7 (Signal prefix). A signal s1 ∈ S is a prefix signal s ∈ C, s ⊆ u implying that C ⊆ u. Suppose there S 0 S 0 of a signal s2 ∈ S, and we write s1 v s2, iff s1 ⊆ s2 and are tags τ ∈ C and τ ∈ T (u \ C) such that τ < τ. for all tags τ1 ∈ T (s1) and τ2 ∈ T (s2) \T (s1), τ1 < τ2. Then there needs to be a signal s ∈ C with τ ∈ T (s) and

3 0 I τ ∈/ T (s), which cannot be comparable with u, in contra- Proposition 2.7. The ordered set ([S ]ω, v) is a consis- diction to the fact that u is an upper bound of C. Hence tently complete cpo. S C v u, establishing that S C is a least upper bound of C. Proof. Similar to Proposition 2.5. For the reverse implication, notice that if C is not to- Proposition 2.8. The ordered set ([SI ] , v) is a consis- tally ordered, then by Proposition 2.3 it is not consistent Ω tently complete cpo. and therefore cannot have a least upper bound. It is evident that the set of all signals S is a consistently Proof. Similar to Proposition 2.6. complete cpo under the signal-preﬁx relation. The situation becomes less clear in the case of natural and ordinal signals. 3. Functions over Tuples of Signals

Proposition 2.5. The ordered set (Sω, v) is a consistently complete cpo. 3.1. Composition

Proof. The set (Sω, v) is a pointed partially ordered set For sets I, J, K ⊆ V ar and functions f : SI → SJ and with bottom element the empty signal ∅. Let D ⊆ Sω g : SJ → SK , the sequential composition of f and g is be a directed set. Then by Proposition 2.3, D will be to- I K I F S the function g ◦ f : S → S such that for any s ∈ S , tally ordered, and by Proposition 2.4, D = D. We (g ◦f)(s) = g(f(s)). The n-fold composition of a function need to show that S D is a natural signal. Observe that I I n S S f : S → S with itself is denoted by f and deﬁned T ( D) = s∈D T (s). For any signal s ∈ D we can recursively as f ◦ f n−1, with f 0 being the identity function ﬁnd an order-embedding ϕs : T (s) ,→ ω from its tag- on SI . For sets I, J, K, L ⊆ V ar, with J ∩ L = ∅, and set to the set of all natural numbers. Consider the map I J K L S S functions f : S → S and g : S → S , the parallel ϕ : T ( D) → ω, such that for any τ ∈ T ( D): composition of f and g is the function f ⊕ g : SI∪K → SJ∪L such that for any s ∈ SI∪K , (f ⊕ g)(s) = f(s ϕ(τ) = min ϕs(τ) | s ∈ D and τ ∈ T (s) I) ⊕ g(s K). It is easy to verify that ϕ is an order-embedding. Hence S D ∈ Sω and (Sω, v) is a cpo. 3.2. Causality Let C ⊆ Sω be a consistent set. Then by Proposition 2.3, F C will be directed. Hence C ∈ Sω and (Sω, v) is con- Let A be a non-empty set and (Γ, ≤) a pointed totally sistently complete. ordered set with a zero element 0 ∈ Γ, such that for any γ ∈ Γ, 0 ≤ γ. A function d : A × A → Γ is an ultrametric Proposition 2.6. The ordered set (SΩ, v) is a consistently complete cpo. distance function iff for all a1, a2, a3 ∈ A:

Proof. Similar to Proposition 2.5. (under construction...) (i) d(a1, a2) = 0 ⇔ a1 = a2,

(ii) d(a1, a2) = d(a2, a1), It is a simple exercise to show that the cpo of natural signals, and similarly the cpo of ordinal signals, is algebraic, (iii) d(a1, a3) ≤ max d(a1, a2), d(a2, a3) . though we do not pursuit this here. Let (Γ, ≤) be a complete totally ordered set with a zero Let I ⊆ V ar be a set of variables. Perhaps surprisingly, element 0 ∈ Γ, and ϕ : T ,→ Γ an order-embedding from we do not use the induced pointwise order to order the set the set (T , ≤) to the set (Γ \ 0 , ≥). We define the gen- of signal tuples SI . eralized Cantor metric dC : S × S → Γ such that for all I Definition 2.8 (Signal-tuple prefix). A signal tuple s1 ∈ S signals s1, s2 ∈ S: I is a prefix of a signal tuple s2 ∈ S , and we write s1 v s2, _ iff s1(i) ⊆ s2(i) for all i ∈ I, and for all tags τ1 ∈ T (s1) dC(s1, s2) = ϕ(τ) | τ ∈ T and s1(τ) 6' s2(τ) Γ and τ2 ∈ T (s2) \T (s1), τ1 < τ2. It is easy to verify that d is an ultrametric. The signal-tuple prefix relation v ⊂ SI × SI is a partial C We can extend the generalized Cantor metric directly to order, and for any signal tuple s ∈ SI , ∅I v s. Notice that tuples of signals with consideration to their alternative de- for all tuples s , s ∈ SI , s v s implies s (i) v s (i) 1 2 1 2 1 2 finition. Hence, for any set of variables I and all tuples for all i ∈ I. s , s ∈ SI : We can immediately deduce that (SI , v) is a consis- 1 2 tently complete cpo. This is less obvious in the case of _ dC(s1, s2) = ϕ(τ) | τ ∈ T and s1(τ) 6' s2(τ) natural and ordinal tuples of signals. Γ

4 Definition 3.1 (Causal function). A function f : SI → SJ It is interesting to note that the class of all ordinal re- I is causal iff for all tuples s1, s2 ∈ S : tractions is a set, since for any ordinal α, ψα ∈ (V ar * S)Ω → (V ar * S)Ω . dC(f(s1), f(s2)) ≤ dC(s1, s2) Proposition 3.1. There exists an ordinal β such that for all Definition 3.2 (Strictly causal function). A function f : 0 + 0 0 ordinals α, α ∈ β , α 6= α implies ψα 6= ψα, and for any SI → SJ is strictly causal iff for all tuples s , s ∈ SI , 1 2 ordinal γ 3 β, ψγ = ψβ = id. s1 6= s2 implies: Proof. Omitted. dC(f(s1), f(s2)) < dC(s1, s2) We define the ordinal approximation ΦΩ as a mapping Clearly, any strictly causal function is causal. Causal from the ordinal functions to the restrictions of ordinal func- functions are closed under both sequential and parallel com- tions to the set of all ordinal tuples of signals. In particular, position. A sequential composition of causal functions is for any ordinal function f : SI → SI and any ordinal tuple I strictly causal if at least one of the composed functions is s ∈ [S ]Ω: strictly causal. A parallel composition, however, is strictly causal iff all of the composed functions are strictly causal. ΦΩ(f)(s) = ψα+ (f(ψα(s)))

3.3. Naturality and Ordinality where α is the ordinal number of T (s u f(s)). Notice that the greatest lower bound of s and f(s) exists and is ordinal since the cpo ([SI ] , v) is consistently complete. The set Deﬁnition 3.3 (Natural function). A function f : SI → SJ Ω I J T (s u f(s)) coincides with T (s) iff s v f(s), in which is natural iff for any tuples s ∈ [S ]ω, f(s) ∈ [S ]ω. case ψα(s) = s. Similarly, T (s u f(s)) coincides with Natural functions, while closed under sequential compo- T (f(s)) iff s w f(s), in which case ψα(f(s)) = f(s). sition, are not closed under parallel composition. In any case, ψα(s) = ψα(f(s)) and both T (ψα(s)) and

I J T (ψα(f(s))) have ordinal number α. Definition 3.4 (Ordinal function). A function f : S → S I I I I It is important to note that if f : S → S is a is ordinal iff for any tuples s ∈ [S ]Ω, f(s) ∈ [S ]Ω. strictly causal ordinal function, then for any ordinal tuple I Ordinal functions are closed under sequential composi- s ∈ [S ]Ω, f(ψα(s)) w ψα(s), with α being the ordinal tion and finite parallel composition. That is, if A is a finite number of T (s u f(s)). This follows directly by the defin- Ia Ja set, fa : S → S is an ordinal function for any a ∈ A, ition of strict causality. 0 0 and a 6= a implies Ja ∩ Ja0 = ∅ for all a, a ∈ A, then Lemma 3.1. If f : SI → SI is a strictly causal ordi- ⊕a∈Afa is an ordinal function. Establishing this is easy. I I nal function, then the function ΦΩ(f):[S ]Ω → [S ]Ω is order-preserving. 3.4. Fixed-point Properties I Proof. Consider two ordinal tuples s1, s2 ∈ [S ]Ω such We define an ordinal retraction ψα for each ordinal α, that s1 v s2. Let α1 be the ordinal number of T (s1 u such that for any ordinal tuple s ∈ (V ar * S)Ω: f(s1)), and α2 be the ordinal number of T (s2 u f(s2)). If s1 = s2, then trivially ΦΩ(f)(s1) = ΦΩ(f)(s2). ( ∼ 0 0 s seg τ if α = τ ∈ T (s) | τ < τ If s1 @ s2, then dC(s1, s2) = ϕ(τ), where τ is ψα(s) = s otherwise. the least tag of T (s2) \T (s1). Since f is strictly causal, dC(f(s1), f(s2)) < dC(s1, s2), which implies 0 0 0 for some τ ∈ T (s). Notice that for any ordinal α, ψα is f(s1)(τ ) ' f(s2)(τ ) for any τ ≤ τ. Moreover, 00 00 a well defined function. For any two well ordered sets, ei- ψα1 (f(s1)) = f(s1) seg τ for some tag τ ≤ 00 ther they are order-isomorphic, or one is order-isomorphic τ, with T (f(s1) seg τ ) being order-isomorphic to 00 00 to a unique initial segment of the other. Hence, there is no α1. However, f(s1) seg τ = f(s2) seg τ ambiguity in the choice of τ when the ordinal α is order- and therefore ψα1 (f(s2)) = ψα1 (f(s1)). Clearly, isomorphic to an initial segment of T (s). ψα1 (s2) = ψα1 (s1) and ψα1 (f(s1)) = ψα1 (s1), and

Ordinal retractions are easily seen to be continuous func- thus ψα1 (s2) = ψα1 (f(s2)) = ψα1 (s1). Hence, tions. For all ordinals α and β, ψα ◦ ψβ = ψγ , with γ = α ψα1 (s1) v ψα2 (s2). If ψα1 (s1) = ψα2 (s2), then trivially if α ∈ β and γ = β otherwise. For any ordinal α and any ΦΩ(f)(s1) = ΦΩ(f)(s2). Otherwise, ψα1 (s1) @ ψα2 (s2) 000 000 ordinal tuple of signals s, ψα(s) v s. For all ordinal tuples and dC(ψα1 (s1), ψα2 (s2)) = ϕ(τ ), where τ is the of signals s1 and s2 such that s1 @ s2, if α1 is the ordinal least tag of T (ψα2 (s2)) \T (ψα1 (s1)). Strict causality 0 0 number of T (s1), then ψα1 (s2) = s1, and for any ordinal of f implies f(ψα1 (s1))(τ ) ' f(ψα2 (s2))(τ ) for any 0 000 β 3 α1, ψβ(s2) A s1. τ ≤ τ . However, ψα2 (s2) v f(ψα2 (s2)) and since

5 000 0 0 τ ∈ T (ψα2 (s2)), f(ψα1 (s1))(τ ) ' ψα2 (s2)(τ ) for [5] R. J. van Glabbeek. Denotational semantics. Lecture any τ 0 ≤ τ 000. Now τ 000 is easily seen to be the great- notes, School of Computer Science & Engineering, The

+ est tag of T (ψ (ψα2 (s2)), which is order-isomorphic to University of New South Wales, Sydney 2052, Aus- α1 + + + tralia, Apr. 2003. α . Consequently, ψ (f(ψα1 (s1))) = ψ (ψα2 (s2)). 1 α1 α1 + Since α1 ∈ α2, ψ (ψα2 (s2)) v ψα2 (s2) and therefore α1 + + ψ (f(ψα1 (s1))) v ψα2 (s2) v ψ (f(ψα2 (s2))), yield- α1 α2 ing ΦΩ(f)(s1) v ΦΩ(f)(s2). Lemma 3.2. If f : SI → SI is a strictly causal ordinal function, then any ﬁxed point of ΦΩ(f) is a ﬁxed point of f.

I Proof. Consider an ordinal tuple s ∈ [S ]Ω such that s 6= f(s). Let α be the ordinal number of T (s u f(s)), and β be the ordinal number of T (s). If α = β, then ψα(s) = s and s @ f(s). Hence, ΦΩ(f)(s) = ψα+ (f(s)) A s. + If α = β, then ψα(s) @ s and dC(ψα(s), s) = ϕ(τ), where τ is the greatest tag of T (s). Notice that T (s) has a greatest element in this case since β is a successor ordinal. Since f is strictly causal, dC(f(ψα(s)), f(s)) < 0 0 dC(ψα(s), s), which implies f(ψα(s))(τ ) ' f(s)(τ ) for any τ 0 ≤ τ. However, f(s)(τ 0) 6' s(τ 0) for some τ 0 ≤ τ 0 0 and therefore f(ψα(s))(τ ) 6' s(τ ). Hence, ΦΩ(f)(s) = ψα+ (f(ψα(s))) 6= s. + If α ∈ β, then trivially ΦΩ(f)(s) = ψα+ (f(ψα(s))) 6= s. Theorem 3.1. Let f : SI → SI be a strictly causal ordinal function. Then µx.ΦΩ(f)(x) is the unique ﬁxed point of f.

Proof. The function ΦΩ(f) will be order-preserving, by Lemma 3.1, and will therefore have a least ﬁxed point µx.ΦΩ(f)(x). By Lemma 3.2, µx.ΦΩ(f)(x) will be a ﬁxed point of f. To establish its uniqueness, consider an ordinal I tuple s ∈ [S ]Ω such that s 6= µx.ΦΩ(f)(x), and suppose to the contrary that s = f(s). Since f is strictly causal, dC(f(µx.ΦΩ(f)(x)), f(s)) < dC(µx.ΦΩ(f)(x), s), in contradiction to dC(f(µx.ΦΩ(f)(x)), f(s)) = dC(µx.ΦΩ(f)(x), s) as derived by direct substitution.

References

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