Weak bisimulations for coalgebras over ordered functors

Transcription

1 Weak bisimulations or coalgebras over ordered unctors Tomasz Brengos Faculty o Mathematics and Inormation Sciences Warsaw University o Technology Koszykowa Warszawa, Poland t.brengos@mini.pw.edu.pl Abstract. The aim o this paper is to introduce a coalgebraic setting in which it is possible to generalize and compare the two known approaches to deining weak bisimulation or labelled transition systems. We introduce two deinitions o weak bisimulation or coalgebras over ordered unctors, show their properties and give suicient conditions or them to coincide. We ormulate a weak coinduction principle. Keywords: coalgebra, bisimulation, saturator, weak bisimulation, weak coinduction 1 Introduction The notion o a strong bisimulation or dierent transition systems plays an important role in theoretical computer science. A weak bisimulation is a relaxation o this notion by allowing silent, unobservable transitions. It is a well established notion or many deterministic and probabilistic transition systems (see [1], [4], [7], [8]). For many state-based systems one can equivalently introduce weak bisimulation in two dierent ways one o which has computational advantages over the other. To be more precise we will demonstrate this phenomenon on labelled transition systems. By a labelled transition system (or LTS in short) we mean a tuple A, Σ,, where A is a set o states, Σ is a non-empty set called an alphabet and is a subset o A Σ A and is called a transition. For an LTS A, Σ, and s Σ we deine a relation on A by s : {(a, a ) A 2 (a, s, a ) }. For a ixed alphabet letter τ Σ, representing a silent, unobservable transition label, and an LTS A, Σ, let τ be the relexive and transitive closure o the relation. τ The ollowing deinition o a weak bisimulation or LTS can be ound in [4]. This work has been supported by the European Union in the ramework o European Social Fund through the Warsaw University o Technology Development Programme and the grant o Warsaw University o Technology no. 504M or young researchers.

2 Deinition 1. A relation R A A is a weak bisimulation i it satisies the ollowing conditions. For (a, b) R and σ τ i a τ σ τ a then there is b A such that b τ σ τ b with (a, b ) R and conversely, or b τ σ τ b there is a A such that a τ σ τ a and (a, b ) R. Moreover, or σ τ i a τ a then b τ b or some b B with (a, b ) R and conversely, i b τ b then a τ a or some a A and (a, b ) R. It is easily shown we can equivalently restate the deinition o a weak bisimulation as ollows. Deinition 2. A relation R A A is a weak bisimulation i it satisies the ollowing condition. I (a, b) R then or σ τ i a σ a then b τ σ τ b or some b A and (a, b ) R, or σ τ i a τ a then b τ b or some b A and (a, b ) R, moreover or σ τ i b σ b then a τ σ τ a or some a A and (a, b ) R, or σ τ i b τ b then a τ a or some a A and (a, b ) R. From the point o view o computation and automatic reasoning the latter approach to deining weak bisimulation is better since, unlike the ormer, it does not require the knowledge o the ull saturated transition. Indeed, in order to show that two states a, b A o a labelled transition system A, Σ, are weakly bisimilar in the sense o Deinition 1 one needs to consider all states a A reachable rom a via the saturated transitions τ σ τ or τ and compare them with similar states reachable rom b. Whereas, to prove that two states a, b A are weakly bisimilar in the sense o Deinition 2 one needs to consider all states reachable rom a via single step transitions σ and compare them with some states reachable rom b via the saturated transitions. The notion o a strong bisimulation, unlike the weak bisimulation, has been well captured coalgebraically (see e.g. [2],[11]). Dierent approaches to deining weak bisimulations or coalgebras have been presented. The earliest paper is [10], where the author studies weak bisimulations or while programs. In [5] the author introduces a deinition o weak bisimulation or coalgebras by translating a coalgebraic structure into an LTS. This construction works or coalgebras over a large class o unctors but does not cover the distribution unctor, hence it is not applicable to dierent types o probabilistic systems. In [6], weak bisimulations are introduced via weak homomorphisms. As noted in [9] this construction does not lead to intuitive results or probabilistic systems. Finally, in [9] the authors present a deinition o weak bisimulation or classes o coalgebras over unctors obtained rom biunctors. Here, weak bisimulation o a system is deined as a strong bisimulation o a transormed system. First o all, it is worth noting that, although very interesting, neither o the approaches cited above expresses coalgebraically the computational advantages o Deinition 2 over Deinition 1. Secondly, all o them require to explicitly work with observable and unobservable part o the coalgebraic structure. The method o deining weak bisimulation presented in this paper only requires that a saturator is given and no explicit knowledge o silent and visible part o computation is neccessary.

3 The aim o this paper is to introduce a coalgebraic setting in which we can deine weak bisimulation in two ways generalizing Deinition 1 and Deinition 2 and compare them. Additionally, we ormulate a weak coinduction principle. The paper is organized as ollows. In Section 2 we present basic deinitions and properties rom known universal coalgebra. In Section 3 we present a deinition o a saturator and present some natural and well-known examples o saturators. In Section 4 we give two approaches to deining weak bisimulation via saturators and show their properties. Finally, Section 5 is devoted to ormulation o a weak coinduction rule. 2 Basic notions and properties Let Set be the category o all sets and mappings between them. Let F : Set Set be a unctor. An F -coalgebra is a tuple A, α, where A is a set and α is a mapping α: A F A. The set A is called a carrier and the mapping α is called a structure o the coalgebra A, α. A homomorphism rom an F -coalgebra A, α to a F -coalgebra B, β is a mapping : A B such that T () α β. An F -coalgebra S, σ is said to be a subcoalgebra o an F -coalgebra A, α whenever there is an injective homomorphism rom S, σ into A, α. This act is denoted by S, σ A, α. We denote the disjoint union o a amily {X j } j J o sets by Σ j J X j. Let { A i, α i } i I be a amily o F -coalgebras. The disjoint sum Σ i I A i, α i o the amily { A i, α i } i I o F -coalgebras is an F -coalgebra deined as ollows. The carrier set o the disjoint sum Σ i I A i, α i is the disjoint union o the carriers o A i, α i, i.e. A : Σ i I A i. The structure α o the disjoint sum Σ i I A i, α i is deined as α : A F A; A i a F (e i ) α i (a), where e i : A i A; a (a, i) or any i I. A unctor F : Set Set preserves pullbacks i or any mappings : A B and g : C B and their pullback P (, g) {(a, c) A C (a) g(c)} with and π 2 the ollowing diagram is a pullback diagram. F A F F B F [P (, g)] F C We say that F weakly preserves pullbacks i the diagram above is a weak pullback. A unctor F (weakly) preserves kernel pairs i it (weakly) preserves pullbacks P (, ),, π 2 or any mapping : A B. For a detailed analysis o pullback and kernel pair preservations the reader is reerred to [2]. Let Pos be the category o all posets and monotonic mappings. Note that there is a orgetul unctor U : Pos Set assigning to each poset (X, ) the underlying set X and to each monotonic map : (X, ) (Y, ) the map : X Y. F g

4 From now on we assume that a unctor F we work with is F : Set Pos. We may naturally assign to F its composition F U F with the orgetul unctor U : Pos Set. For the sake o simplicity o notation most o the times we will identiy the unctor F : Set Pos with the Set-endounctor F U F and write F to denote both F and F. Considering set-based coalgebras over unctors whose codomain category is a concrete category dierent rom Set is not a new approach. A similar one has been adopted by e.g. J. Hughes and B. Jacobs in [3] when deinining simulations or coalgebras. Example 1. The powerset endounctor P : Set Set can be considered a unctor P : Set Pos which assigns to any set X the poset (P(X), ) and to any map : X Y the order preserving map P(). Example 2. For any unctor H : Set Set the composition PH may be regarded as a unctor PH : Set Pos with a natural ordering given by inclusion. In this paper we will ocus our attention on coalgebras over the ollowing unctors: P(Σ Id), P(Σ + Id), P(Σ D), where D is the distribution unctor, i.e. a unctor which assigns to any set X the set DX : {µ : X [0, 1] x X µ(x) 1} o discrete measures and to any mapping : X Y a mapping D : DX DY, which assigns to any measure µ DX the measure D(µ) : Y [0, 1] such that D(µ)(y) (x)y µ(x) or any y Y. The coalgebras over the irst unctor are exactly labelled transition systems. The coalgebras or P(Σ + Id) expand a class o coalgebras studied by J. Rutten in [10]. Finally, the P(Σ D)-coalgebras generalize the class o simple Segala systems introduced and thoroughly studied in [7],[8]. For a unctor F : Set Pos and or any sets X, Y we introduce an order on the set Hom(X, F Y ) as ollows. For, g Hom(X, F Y ) put g de (x) F Y g(x) or any x X. Given : X Y, α : X F Z, g : Z U and β : Y F U an inequality F g α β will be denoted by a diagram on the let and an equality F g α β will be denoted by a diagram on the right: X α Y β F Z F g F U X α Y β F Z F g F U

5 Lemma 1. Let α, β Hom(X, F Y ) and let : Z X be an epimorphism in Set. I α β in Hom(Z, F Y ) then α β in Hom(X, F Y ). Proo. Since α β then or any z Z we have α((z)) F Y β((z)). Because (Z) X we have α(x) F Y β(x) or any x X. Hence, α β in Hom(X, F Y ). 3 Coalgebraic operators and saturators Deinition 3. Let U : Set F Set be the orgetul unctor and let C be ull subcategory o the category o F -coalgebras and homomorphisms between them which is closed under taking inverse images o homomorphisms, i.e. i B, β C and there is a homomorphism : A, α B, β or A, α Set F then A, α C. A coalgebraic operator o with respect to a class C is a unctor o : C Set F such that the ollowing diagram commutes: o C Set F U U Set In other words, i : A B is a homomorphism between two F -coalgebras A, α and B, β belonging to C then is a homomorphism between A, oα and B, oβ, i.e. A α F B F F B B β A oα B oβ F A F F B We say that a coalgebraic operator s with respect to a class C is a saturator i or any two F -coalgebras A, α, B, β rom C and any mapping : A B the inequality F α sβ is equivalent to F sα sβ. We may express the property in diagrams as ollows: α A B sβ F A F F B A sα B sβ F A F F B Lemma 2. Let s : C Set F be an operator w.r.t. a ull subcategory C o Set F and additionally let s(c) C. Then s is a saturator i and only i it satisies the ollowing three properties: α sα or any coalgebra A, α C (extensivity), s s s (idempotency),

6 i F α β then F sα sβ or any : X Y (monotonicity): A α F A F F B B β A sα B sβ F A F F B The intuition behind the notion o a saturator is the ollowing. Given a coalgebraic structure α : A F A it contains some inormation about observable and unobservable single step transitions. The process o saturating a structure intuitively boils down to adding additional inormation to α about multiple compositions o unobservable steps and a single composition o observable transitions (see examples below). Since or any set A the set F A is intuitively considered as the set o all possible outcomes o a computation, the partial order on F A compares those outcomes. In particular, the property o extensivity o a saturator means the saturated structure sα contains at least the same inormation about single-step transitions as α (in the sense o the partial order ). Idempotency means that the process o adding new inormation to α by saturating it ends ater one iteration. Finally, monotonicity is sel-explanatory. Example 3. Let Σ be a non-empty set. Any LTS A, Σ, may be represented as a P(Σ Id)-coalgebra A, α as ollows. We deine α : A P(Σ A) by: (σ, a ) α(a) a σ a. Let τ Σ be a silent transition label. For a coalgebra structure α : A P(Σ A) we deine its saturation sα : A P(Σ A) as ollows. For any element a A put sα(a) : α(a) {(τ, a ) a τ a } {(σ, a ) a τ σ τ a or σ τ}. Veriying that s : Set P(Σ Id) Set P(Σ Id) ; A, α A, sα is a coalgebraic saturator with respect to the class o all P(Σ Id)-coalgebras is let to the reader. Example 4. Consider the unctor F P(Σ + Id). Let α : A P(Σ + A) be a structure o an F -coalgebra A, α. For the sake o simplicity o notation or any a A let η(a) : α(a) A and θ(a) : α(a) Σ. Put η (a) : {a} n N ηn (a), where η n (a) : η(η n 1 (a)) or n > 1 and θ (a) : θ(η (a)). Deine the saturation sα : A F A as ollows: sα(a) : η (a) θ (a) or any a A. The assignment s is a coalgebraic saturator with respect to the class o all F - coalgebras. Example 5. For the unctor F P(Σ D), an F -coalgebra A, α, a state a A and σ Σ we write a σ µ i (σ, µ) α(a). For a state a A and a measure

7 ν D(Σ A) a pair (a, ν) is called a step in A, α only i there is σ A and µ DA such that a σ µ and ν(σ, a ) µ(a ) or any a A. A combined step in A, α is a pair (a, ν), where a A and ν D(Σ A) or which there is a countable amily o non-negative numbers {p i } i I such that i I p i 1 and a countable amily o steps {(a, ν i )} i I in A, α such that ν i I p i ν i. The deinition o a combined step is a slight modiication o a similar deinition σ presented in [7]. The notion o weak arrows P remains the same regardless o the small dierence between the two deinitions. Let τ Σ be the invisible σ transition. As in [7] or any σ Σ we write a P µ whenever σ τ and µ DA or which µ(a) 1 or there is a combined step (a, ν) in A, α such that i (σ, a ) / {σ, τ} A then ν(σ, a ) 0 and µ (σ,a ) {σ,τ} A ν(σ, a ) µ (σ,a ) and i σ σ then a τ P µ (σ,a ) otherwise σ τ and a σ P µ (σ,a ). Now, σ deine sα : A F A by putting sα(a) : {(σ, µ) a P µ} or any a A. A proo that s is a coalgebraic saturator is let to the reader. 4 Two approaches to deining weak bisimulation In this section we assume that A, α and B, β are members o the class C. Deinition 4. A relation R A B is called a weak bisimulation provided that there is a structure γ 1 : R F R and a structure γ 2 : R F R or which: α γ 1 and γ 1 sβ π 2, β π 2 γ 2 and γ 2 sα. π A 2 R B α F A γ 1 F R sβ sα π A 2 R B γ 2 F B F A F R β F B Example 6. Consider the LTS unctor F P(Σ Id) and the saturator s introduced in Example 3. Let A, α be an F -coalgebra. Consider a relation R A A which satisies the assumptions o Deinition 4. This means that i (a, b) R then there is γ 1 : R P(Σ R) such that α(a) F ( )(γ 1 (a, b)) and F (π 2 )(γ 1 (a, b)) sα(b). In other words, there is a subset S Σ R such that γ 1 (a, b) S and F ( )(S) α(a) and F (π 2 )(S) sα(b). This means that or any (σ, a ) α(a) there is b A such that (σ, b ) sα(b) and (a, b ) R. Hence, i σ τ then a τ a implies b τ b and (a, b ) R otherwise a σ a implies b τ στ b and (a, b ) R. The second condition rom Deinition 4 gives us the ollowing assertion. I (a, b) R and b τ b then there is a A such that a τ a and (a, b ) R. Moreover, i or σ τ we have b σ b then there is a A such that a τ στ a and (a, b ) R. We see that this is exactly the condition presented in Deinition 2.

8 Example 7. Consider the unctor F P(Σ + Id) and the saturator s rom Example 4. Since the unctor Σ + Id is a subunctor o F take an F -coalgebra A, α which is a Σ + Id-coalgebra, i.e. the structure α is a mapping α : A Σ + A. Now take a relation R A A which satisies the assumptions presented in Deinition 4 and let (a, b) R. This means that there is a structure γ 1 : R P(Σ + R) such that F ( )(γ 1 (a, b)) α(a) and F (π 2 )(γ 1 (a, b)) sα(a). In other words there is a pair X, S o subsets, where X R and S Σ, and F ( )(X S) ( (X) S) α(a) and F (π 2 )(X S) (π 2 (X) S) sα(a). I α(a) a A then this means that (X) {a } and S. Hence, there is b A such that (a, b ) X R and b η (b) {b b b }. I α(a) σ Σ then (X), and hence X, S {σ}. Thereore, σ θ (b). It ollows that there is b B such that b b and b σ. By the second condition o Deinition 4 we iner that i b b then there is a A such that a a and (a, b ) R. Otherwise i b σ Σ then there is a A such that a a and a σ. This deinition coincides with a deinition o weak bisimulation between Σ + Id-coalgebras presented in [10]. Example 8. Let F P(Σ D) and consider the saturator s : Set F Set F deined in Example 5. It is easy to see that or two simple Segala systems A, α, B, β a relation R A B is a weak bisimulation provided that the ollowing condition holds. I (a, b) R and a σ σ µ then b P µ and (µ, µ ) (, )(DR). Moreover, i (a, b) R and b σ µ σ then a P µ and (µ, µ ) (, )(DR). This deinition coincides with the one presented in [7],[8]. Proposition 1. Let R A B be a standard bisimulation between A, α and B, β. Then R is also a weak bisimulation between A, α and B, β. Theorem 1. I a relation R A B is a weak bisimulation between A, α and B, β then R 1 {(b, a) (a, b) R} is a weak bisimulation between B, β and A, α. Theorem 2. I all members o a amily {R i } i I o relations R i A B are weak bisimulations between A, α and B, β then i I R i is also a weak bisimulation between A, α and B, β. Proo. Let {R i } i I together with γ i 1 : R i F R i and γ i 2 : R i F R i be a amily o weak bisimulations between A, α and B, β. π A 2 B α F A R i γ i 1 F R i sβ sα π A 2 B R i γ i 2 F B F A F R i β F B First consider the disjoint sum i I Ri, γ1 i i I R i, γ 1. We will prove that given τ : i I R i A B; (r, i) r the mappings p 1 τ and

9 p 2 π 2 τ satisy: α p 1 F (p 1 ) γ 1, sβ p 2 F (p 2 ) γ 1. Note that or any i I we have α p 1 e i α F ( ) γ1 i F (p 1 ) F (e i ) γ1 i F (p 1 ) γ 1 e i. Hence, α p 1 F (p 1 ) γ 1. Moreover, or any i I we have sβ p 2 e i sβ π 2 F (π 2 ) γ1 i F (p 2 ) F (e i ) γ1 i F (p 2 ) γ 1 e i. By Lemma 1 it ollows that sβ p 2 F (p 2 ) γ 1. Σ i I R i γ 1 e i p 1 R i γ i 1 A F e i F Σ i I R i F R i F A F p 1 α Σ i I R i γ 1 F Σ i I R i e i F e i p 2 R i γ i 1 F p 2 π 2 B sβ F R i F B Note that the image o i I R i under the map τ : i I R i A B is equal to i I R i A B. Put γ 1 : i I R i F ( i I R i) so that, or any r i I R i there is (r, i) i I R i such that γ 1(r) F (τ)(γ 1 (r, i)). Observe that α( (r)) α( (τ(r, i))) α(p 1 (r, i)) F (p 1 )(γ 1 (r, i)) F ( )(F (τ)(γ 1 (r, i))) F ( ) γ 1(r), sβ(π 2 (r)) sβ(π 2 (τ(r, i))) sβ p 2 (r, i) F (p 2 ) γ 1 (r, i) F (π 2 ) F (τ) γ 1 (r, i) F (π 2 ) γ 1(r). Hence, α F ( ) γ 1 and sβ π 2 F (π 2 ) γ 1. Similarily we prove existence o γ 2 : i I R i F ( i I R i) possessing the desired properties and making i I R i a weak bisimulation. The ollowing lemma is an analogue o a similar result or standard bisimulations presented in e.g. [11] (Lemma 5.3). Moreover, the proo o Lemma 3 is a direct translation o the proo o the analogous result. Hence, we leave the ollowing result without a proo. Lemma 3. Let X be a set and let ξ 1 : X F X and ξ 2 : X F X be two coalgebraic structures. Finally, let : X A, g : X B be mappings such that is a homomorphism rom X, ξ 1 to A, α, g is a homomorphism rom X, ξ 2 to B, β and the mappings, g satisy: X ξ 1 g B sβ F X F g F B X ξ 2 A sα F X F F A then the set, g (X) {((x), g(x)) A B x X} is a weak bisimulation between A, α and B, β.

10 Theorem 3. Let F : Set Set weakly preserve pullbacks and let A, α, B, β and C, δ be F -coalgebras rom the class C. Let R 1 be a weak bisimulation between A, α and B, β and R 2 be a weak bisimulation between B, β and C, δ. Then R 1 R 2 {(a, c) b B s.t. (a, b) R 1 and (b, c) R 2 } is a weak bisimulation between A, α and C, δ. Corollary 1. I F : Set Set weakly preserves pullbacks then the greatest weak bisimulation on a coalgebra A, α is an equivalence relation. Deinition 5. A relation R A B is said to be a saturated weak bisimulation between A, α and B, β provided that there is a structure γ : R F R or which the ollowing diagram commutes: sα A R π 2 B F A γ sβ F R F B Remark 1. We see that a saturated weak bisimulation between A, α and B, β is deined as a standard bisimulation between saturated models A, sα and B, sβ. Hence, any property true or standard bisimulation is also true or a saturated weak bisimulation. Proposition 2. Let R A B be a standard bisimulation between A, α and B, β. Then R is also a saturated weak bisimulation between A, α and B, β. Theorem 4. Let F : Set Set weakly preserve kernel pairs and let R A A be an equivalence relation which is a weak bisimulation on A, α. Then R is a saturated weak bisimulation on A, α. Proo. Let γ 1 : R F R be a structure or which α γ 1 and γ 1 sα π 2. By properties o the saturator s it ollows that sα sγ 1 and sγ 1 sα π 2. In other words, sα π A 2 R A F A sγ 1 sα F R F A Let p : A A/R; a a/r. Since F : Set Set preserves kernel pairs the ollowing diagram is a weak pullback diagram: F R F A F A F p F p F (A/R)

11 Since F p F p we have F p sα F p sγ 1 F p sγ 1 F p sα π 2. Let k : R R; (a, b) (b, a). We see that F p sα k F p sα π 2 k. Since k π 2 and π 2 k it ollows that F p sα π 2 F p sα. Hence, F p sα F p sα π 2. In other words, the set R together with sα and sα π 2 is a cone over the diagram F A F p F (A/R) F p F A. Recall that F R with and is a weak pullback o F A F p F (A/R) F p F A. The act that F preserves weak pullbacks provides us with a mediating morphism γ : R F R satisying sα γ and sα π 2 γ. We say that two elements a, b A are weakly bisimilar, and write a w b i there is a weak bisimulation R A A on A, α or which (a, b) R. We say that a and b are saturated weakly bisimilar, and write a sw b, i there is a saturated weak bisimulation R on A, α containing (a, b). Corollary 2. Let F : Set Set be a unctor weakly preserving pullbacks. Then the relations w and sw are equivalence relations and w sw. Deinition 6. We say that a unctor F : Set Pos preserves downsets provided that or any : X Y and any x F X the ollowing equality holds: F (x ) F ({x F X x x}) F (x) {y F Y y F (x)}. It is easy to see that all unctors rom Example 2 preserve downsets. In the ollowing example we will present a unctor weakly preserving pullbacks and not preserving downsets or which the greatest weak bisimulation and the greatest saturated weak bisimulation do not always coincide. Example 9. Deine a unctor F : Set Set by F Id 2 +Id. Clearly, the unctor F weakly preserves pullbacks. For any set X let us introduce a partial order on F X as the smallest partial order satisying x (x, x) or any x X. The order is well deined and turns the unctor F into F : Set Pos. Now consider sets X {x 1, x 2 }, Y {y} and the unique mapping : X Y. Take (y, y) F Y and note that y (y, y) ((x 1 ), (x 2 )) F ((x 1, x 2 )). In other words, y F (x 1, x 2 ) and (x 1, x 2 ) {(x 1, x 2 )}. Hence, the unctor F does not preserve downsets. For any F -coalgebra A, α deine an operator sα : A F A by sα(a) : i α(a) b then (b, b) else α(a).

12 The operator s : Set F Set F is a coalgebraic saturator with respect to the class o all F -coalgebras. Now consider a set A {x, y} and deine a structure α : A F A by α(x) x and α(y) (x, y). Clearly, x sw y since sα(x) (x, x) and sα(y) (x, y) and i we put R {(x, y), (x, x)} then or γ : R F R deined by γ(x, y) ((x, x), (x, y)), γ(x, x) ((x, x), (x, x)) we have sα γ and sα π 2 γ. At the same time the states x and y are not weakly bisimilar. Indeed, i there was a weak bisimulation R containing (x, y) then it would imply existence o γ 1 : R F R satisying α γ 1 and γ 1 sα π 2. Since (x, y) {(x, y)} we would then have which is impossible. x α(x) α( (x, y)) (γ(x, y)), (x, y) sα(y) sα(π 2 (x, y)) (γ(x, y)) Theorem 5. Let F : Set Set weakly preserve kernel pairs and preserve downsets. Let R A A be an equivalence relation which is a saturated weak bisimulation on A, α. Then R is a weak bisimulation on A, α. Proo. Let γ : R F R be the structure or which sα γ and γ sα π 2. Let p : A A/R; a a/r. Since F : Set Set preserves kernel pairs the ollowing diagram is a weak pullback diagram: F R F A F A F p F p F (A/R) Consider the mappings F p α and F p sα π 2 and observe that F p α F p sα F p γ F p γ F p sα π 2. This means that or any pair (a, b) R we have F p(α(a)) F p(sα(b)). In other words, F p(α(a)) F p(sα(b)) Since F preserves downsets, there exists an element x F A such that x sα(b) or which F p(α(a)) F p(x). Because F R together with and is a weak pullback o the diagram F A F p F (A/R) F p F A, there is an element r (a,b) F R such that (r (a,b) ) α(a) and (r (a,b) ) x. Deine γ 1 : R F R; (a, b) r (a,b). The structure γ 1 satisies α γ 1 and γ 1 sα π 2. Similarily, we prove existence o γ 2 : R F R satisying sα γ 2 and γ 2 α π 2. Corollary 3. Let S : Set Set weakly preserve pullbacks and preserve downsets. Then or any S-coalgebra A, α the relations w and sw are equivalence relations and w sw.

13 5 Weak coinduction principle In this section we assume that the saturator s we work with is an operator deined on the whole category o F -coalgebras. In the category Set F or any mapping : A B we have : A, α B, β is a homomorphism i and only i the relation gr() {(a, (a)) A B a A} is a standard bisimulation (see [2],[11]). This motivates considering the ollowing category (F. Bonchi, personal communication). Let Set w F denote the category in which objects are standard F -coalgebras and in which a map : A B is a morphism between two objects A, α and B, β provided that the relation gr() {(a, (a)) a A} is a weak bisimulation. Lemma 4. Let A, α be an F -coalgebra and let a amily S i, σ i, where S i A, be a amily o subcoalgebras o A, α in Set F such that sσ i σ i. Then there is a structure σ : S i F ( S i ) making S i, σ a subcoalgebra o A, α and sσ σ. Corollary 4. Let A, α be an F -coalgebra. There is the greatest subcoalgebra S, σ o A, α such that sσ σ. Lemma 5. Let T, t be a terminal coalgebra in the category Set F. Then T, st is a weakly terminal object in Set w F. Let T s, t be the greatest subcoalgebra o the terminal coalgebra T, t in Set F such that st t. Lemma 6. I F admits a terminal coalgebra T, t in Set F and weakly preserves pullbacks then the greatest weak bisimulation on T s, t is the equality relation. Proo. Assume that or x, y T s we have x w y. This implies x sw y. Since, st t this means that the saturated weak bisimulation sw is a standard bisimulation on T s, t. Since, the coalgebra T s, t is a subcoalgebra o the terminal coalgebra T, t the bisimulation sw is an equality relation (see [2, 11] or details). Hence x y. The above lemma allows us to ormulate a weak coinduction principle or T s, t. For two states x, y T s we have x w y x y. Theorem 6. I F admits a terminal coalgebra T, t in Set F and weakly preserves pullbacks then the coalgebra T s, t is a terminal object in Set w F. Proo. To prove that T s, t is weakly terminal it is enough to construct a homomorphism in Set w F rom T, st to T s, t and apply Lemma 5. Since T, t is terminal in Set F there exists a unique homomorphism st : T, st T, t. Any homomorphic image is a subcoalgebra o the codomain (see [2], [11] or details). Thereore, we can consider st as an onto homomorphism between T, st and T st, t T st, where the object T st, t T st is a subcoalgebra o T, t in Set F. In other words, we have

14 T st st T st t T st F T F T st F st Hence, T st st T st st T st F T F T st F st Thereore, st T st t T st and T st, t T st is a subcoalgebra o Ts, t. For uniqueness consider two homomorphisms 1, 2 rom an F -coalgebra A, α to T s, t in Set w F. This means that the relations gr( 1 ) and gr( 2 ) are weak bisimulations between A, α and T s, t. By the properties o weak bisimulations the relation gr( 1 ) 1 gr( 2 ) {( 1 (a), 2 (a)) a A} is a weak bisimulation on T s, t. By Lemma 6 we get that 1 (a) 2 (a) or any a A. For an F -coalgebra A, α let w α denote the unique homomorphism rom A, α to T s, t in Set w F. We see in the proo o Theorem 6 that w α st α. Theorem 7. Let F weakly preserve pullbacks. For two elements a, b A we have a w b a w α b w α Proo. Assume a w b. Since w α is a homomorphism in Set w F the relation gr( w α ) is a weak bisimulation between A, α and T s, t. Since F weakly preserves pullbacks the relation gr( w α ) 1 w gr( w α ) is a weak bisimulation on T s, t such that (a w α, b w α ) gr( w α ) 1 w gr( w α ). By Lemma 6 we get that a w α b w α. Conversely, let a w α b w α. This means that the weak bisimulation gr( w α ) gr( w α ) 1 on A, α contains a pair (a, b). Hence, a w b. 6 Summary and uture work In this paper we introduced a coalgebraic setting in which we can deine weak bisimulation in two ways generalizing Deinition 1 and Deinition 2 and compared them. We showed that the deinitions coincide with the standard deinitions o weak bisimulation or labelled transition systems and simple Segala systems. The approach towards deining weak bisimulation presented in this paper has two main advantages. First o all, it is a very general and simple approach. In

15 particular it does not require an explicit speciication o the observable and unobservable part o the unctor. Second o all, it easily captures the computational aspects o weak bisimilarity. It is worth noting that it has some limitations. Part o the author s ongoing research is to establish the reason why it does not work or e.g. ully probabilistic processes introduced in [1] and studied rom the perspective o coalgebra in [9]. Moreover, it may seem that the setting presented in the paper is too general. To justiy the statement note that or instance or an LTS coalgebra A, α we may deine a saturator as ollows: sα(a) : {(τ, a)} {(σ, a ) a τ σ a }. The above deinition o a saturator would lead to a dierent deinition o weak bisimulation or LTS. Thereore, it is necessary to establish more concrete ways or deinining standard saturators o coalgebras that lead to standard deinitions o weak bisimulations. Acknowledgements. I would like to thank Jan Rutten or many interesting comments and hospitality in CWI, Amsterdam, where part o the work on this paper was completed. I am very grateul to F. Bonchi, M. Bonsangue, H. Hansen, B. Klin and A. Silva or various suggestions or improvement o my results and or uture work. I would also like to express my gratitude to anonymous reerees o the paper or valuable comments and remarks. Reerences 1. Baier, Ch., Hermanns H.: Weak bisimulation or ully probabilistic processes. In Proc. CAV 1997, LNCS, vol. 1254, pp (1997) 2. Gumm, H.P.: Elements o the general theory o coalgebras. LUATCS 99, Rand Arikaans University, Johannesburg (1999) 3. Hughes, J., Jacobs, B.: Simulations in coalgebra. In TCS, vol. 327 Issue 1-2, pp (2004) 4. Milner, R.: A Calculus o Communicating Systems. In LNCS, vol. 92 (1980) 5. Rothe, J.: A syntactical approach to weak (bi)-simulation or coalgebras. In: L. Moss (ed.) Proc. CMCS02. ENTCS, vol 65, pp (2002) 6. Rothe, J., Masulović D.: Towards weak bisimulation or coalgebras. In: Proc. Categorical Methods or Concurrency, Interaction and Mobility, ENTCS, vol. 68 issue 1, pp (2002) 7. Segala, R., Lynch, N.: Probabilistic simulations or probabilistic processes. In: B. Jonsson, J. Parrow (eds.) Proc. CONCUR94, LNCS, vol. 836, pp (1994) 8. Segala, R.: Modeling and veriication o randomized distributed real-time systems. Ph.D. thesis, MIT (1995) 9. Sokolova, A., de Vink, E., Woracek, H: Coalgebraic Weak Bisimulation or Action- Type Systems, In Sci. Ann. Comp. Sci., vol. 19, pp (2009) 10. Rutten, J. J. M. M.: A note on coinduction and weak bisimilarity or while programs, In Theoretical Inormatics and Applications (RAIRO), vol. 33, pp (1999) 11. Rutten, J. J. M. M.: Universal coalgebra: a theory o systems. In TCS, vol. 249 issue 1, pp (2000)

16 Appendix Proo (Lemma 2). Let s be a saturator and let A, α C. We see that or the identity mapping id A : A A we have F (id A ) sα sα sα sα id A. This implies that F (id A ) α sα id A. Hence, α sα. We see that by extensivity rule and the assumption s(c) C we get sα s(sα). Thereore, to prove idempotency it is enough to show that s(sα) sα. To do this consider the ollowing inequality F (id A ) sα sα sα sα id A. By order preservation it implies that F (id A ) s(sα) sα id A. Thus, s(sα) sα. Finally, consider two coalgebras A, α and B, β rom C and a mapping : A B such that F α β. Note that F α β sβ. Since F α sβ then by the act that s is a saturator it ollows that F sα sβ. Now in order to prove the converse consider a coalgebraic operator s : C C which is a closure operator (i.e. is extensive, idempotent and monotonic). Let A, α and B, β be coalgebras rom C and let : A B be a mapping. Assume that F α sβ. By idempotency and monotonicity we get F sα sβ. By extensivity we conclude that F α sβ. Proo (Lemma 4). Let σ : S i F ( S i ) be the unique structure making S i, σ a subcoalgebra o A, α (such a structure always exists [2],[10]). Let e i : S i S i denote the inclusions. Then σ e i F (e i ) σ i. Since s is a saturator this means that sσ e i F (e i ) sσ i F (e i ) σ i. Thereore, or any i we have sσ e i σ e i. Hence, sσ σ. Proo (Lemma 5). Let A, α be any F -coalgebra. Let α : A, α T, t be the unique homomorphism in Set F with T, t as codomain. Then α is also a homomorphism between A, sα and T, st. Since α is a standard homomorphim the relation gr( α ) {(a, a α ) a A} is a standard bisimulation between A, α and T, t. I.e. there is γ : gr( α ) F (gr( α )) such that π A 2 gr( α ) T α F A F gr( α ) Since t st and sst st this implies that γ π A 2 gr( α ) T α F A F gr( α ) γ t F T F T sstst

17 Moreover, by saturating the same diagram we get sα π A 2 gr( α ) T sγ F A F gr( α ) This means that gr( α ) together with γ and sγ is a weak bisimulation between A, α and T, st which concludes the proo. F T st