Typed Kleene Algebra with Products and Iteration Theories

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1 Typed Kleene Algebra with Prodcts and Iteration Theories Dexter Kozen and Konstantinos Mamoras Compter Science Department Cornell University Ithaca, NY , USA Abstract We develop a typed eqational system that sbsmes both iteration theories and typed Kleene algebra in a common framework. Or approach is based on cartesian categories endowed with commtative strong monads to handle nondeterminism. I. INTRODUCTION In the realm of eqational systems for reasoning abot iteration, two chief complementary bodies of work stand ot. One of these is iteration theories (IT), the sbject of the extensive monograph of Bloom and Ésik [1] as well as many other athors (see the cited literatre). The primary motivation for iteration theories is to captre in abstract form the eqational properties of iteration on strctres that arise in domain theory and program semantics, sch as continos fnctions on ordered sets. Of central interest is the dagger operation, a kind of parameterized least fixpoint operator, that when applied to an object representing a simltaneos system of eqations gives an object representing the least soltion of those eqations. Mch of the work on iteration theories involves axiomatizing or otherwise characterizing the eqational theory of iteration as captred by. Complete axiomatizations have been provided [2 4] as well as other algebraic and categorical characterizations [5, 6]. Bloom and Ésik claim that... the notion of an iteration theory seems to axiomatize the eqational properties of all comptationally interesting strctres... [7]. This is tre to a certain extent, certainly if one is interested only in strctres that arise in domain theory and programming langage semantics. However, it is not the entire story. Another approach to eqational reasoning abot iteration that has met with some sccess over the years is the notion of Kleene algebra (KA), the algebra of reglar expressions. KA has a long history going back to the original paper of Kleene [8] and was frther developed by Conway, who coined the name Kleene algebra in his 1971 monograph [9]. It has since been stdied by many athors. KA relies on an iteration operator that characterizes iteration in a different way from. Its principal models are not those of domain theory, bt rather basic algebraic objects sch as sets of strings (in which gives the Kleene asterate operation), binary relations (in which gives reflexive transitive closre), and other strctres with applications in shortest path algorithms on graphs and geometry of convex sets. Complete axiomatizations and complexity analyses have been given; the reglar sets of strings over an alphabet A form the free KA on generators A in mch the same way that the rational Σ -trees form the free IT on a signatre Σ. Althogh the two systems flfill many of the same objectives and are related at some level, there are many technical and stylistic differences. Whereas iteration theories are based on Lawvere theories, a categorical concept, Kleene algebra operates primarily at a level of abstraction one click down. For this reason, KA may be somewhat more accessible. KA has been shown to be sefl in several nontrivial static analysis and verification tasks (see e.g. [10, 11]). Also, KA can model nondeterministic comptation, whereas IT is primarily deterministic. Nevertheless, both systems have claimed to captre the notion of iteration in a fndamental way, and it is interesting to ask whether they can somehow be reconciled. This is the investigation that we have ndertaken in this paper. We start with the observation that ITs se the objects of a category to represent types. Technically the objects of interest in ITs are morphisms f : n m in a category whose objects are natral nmbers, and the morphism f : n m is meant to model fnctions f : A m A n (the arrows are reversed for technical reasons). Ths ITs might be captred by a version of KA with types. Althogh the primary version of KA is ntyped, there is a notion of typed KA [12], althogh it only has types of the form A B, whereas to sbsme IT it wold need prodcts as well. The presence of prodcts allow ITs to captre parameterized fixpoints throgh the rle f : n n + m f : n m giving the parameterized least fixpoint f : A m A n of a parameterized fnction f : A m A n A n. This wold be possible to captre in KA if the typed version had prodcts, which it does not. On the other hand, KA allows the modeling of nondeterministic comptation, which IT does not, at least not in any obvios way. Ths to captre both systems, it wold seem that we need to extend the type system of typed KA, or extend the categorical framework of IT to handle nondeterminism, or both. The reslt of or investigation is a common categorical framework based on cartesian categories (categories with

2 prodcts) combined with a monadic treatment of nondeterminism. Types are represented by objects in the category, and we identify the appropriate axioms in the form of typed eqations that allow eqational reasoning on the morphisms. Or framework captres iteration as represented in ITs and KAs in a common langage. We show how to define the KA operations as enrichments on the morphisms and how to define in terms of. Or main contribtions are as follows. Commtative strong monads. To accommodate nondeterminism, we need to lift the comptation to the Kleisli category of a monad representing nondeterminate vales. However, the ordinary powerset monad does not sffice for this prpose, as it does not interact well with non-strictness. We axiomatize the relevant properties for an arbitrary commtative strong monad, where (i) the property of commtativity captres the idea that the comptation of a pair can be done in either order, and (ii) strength refers to tensorial strength, which axiomatizes the interaction of pairs with the nondeterminism monad. Lazy pairs. We need to model non-strict (lazy) evalation of programs in the presence of prodcts. Ordinarily, a pair x, wold be by strictness. This reqires the development of the concept of lazy pairs and its categorical axiomatization. Intitively, in the case of eager pairs, the comptation of a pair v,, where v is a vale and denotes a diverging comptation, wold also be diverging, i.e., v, =. This makes it impossible to recover the left component v of the pair: v, ; π 1 =. Simplified axiomatization of commtative strong monads and lazy pairs. We have given a simplified axiomatization of commtative monads with lazy pairs in terms of a certain operator that captres the interaction of these concepts in a very concise form, mch simpler than the axiomatizations of the two of them separately. This is an adaptation of a constrction that can be fond in the work of Kock in the 1970s [13, 14]. We se this extensively in or development to simplify argments. Deterministic arrows. Certain properties work only for deterministic comptations. We show how to captre the necessary properties of determinism in the Kleisli category. A separate syntactic arrow provides a convenient notation for reasoning abot deterministic comptation in the nderlying category when working in the Kleisli category, and we provide an axiomatization of the necessary properties. Lifting the cartesian strctre. We show how the cartesian strctre (pairing and projections) in the nderlying category can be lifted in a smooth way to corresponding operations in the Kleisli category. Captring nondeterminism. We give three eqivalent ways of captring (angelic) nondeterminism in the homsets of the Kleisli category of a monad. These characterizations make essential se of cartesian strctre of the base category. Captring IT and KA. We show how to enrich the homsets of the Kleisli category with the KA operations, inclding, to obtain typed KA with prodcts, and that all the axioms of KA (except the strictness axiom) are satisfied. Seqential composition is modeled by Kleisli composition. We also show that Park theories [4] are sbsmed by KA with prodcts. This is or main reslt. Model theory. Finally, we show that two particlar monads, the lowerset monad and the ideal completion monad, provide natral concrete models in that they are commtative strong monads with lazy pairs. The ideal completion monad involves ideal completion in ω-complete partial orders (ω-cpos) and models nondeterminism in those strctres. A detailed accont of related work is given in IX. II. CARTESIAN CATEGORY WITH BOTTOM ELEMENTS In order to model non-strict (lazy) evalation of programs and lazy pairs we consider or base category to be a cartesian category C with bottom elements. For an object X, we write the identity for X as id X : X X. The composition operation is written as ; and the operands are given in diagrammatic order. f : X Y g : Y Z. f; g : X Z For objects X and Y, we denote by X Y their prodct with corresponding left and right projections π1 XY : X Y X and π2 XY : X Y Y respectively. The typing rle for the pairing operation, is f : X Y g : X Z. f, g : X Y Z The terminal object is denoted by 1. We write X : X 1 for the niqe arrow from X to 1. The typed eqational axioms f : X Y g : X Z f, g ; π 1 = f : X Y h : X Y Z h; π 1, h; π 2 = h : X Y Z f : X Y g : X Z f, g ; π 2 = g : X Y f : X 1 f = X : X 1 say that the operations, π 1, π 2,,, 1, endow C with cartesian strctre. For every object X, the bottom element of X is written as 1X : 1 X. Define the bottom morphism XY from X to Y by XY := X ; 1Y. The bottom morphisms satisfy the axiom f : X Y f; Y Z = XZ : X Z. A morphism f : X Y satisfiying 1X ; f = 1Y is called strict. Remark 1: Let C be a category with binary prodcts given by π 1, π 2,,. We define the prodct fnctor from C C to C by (X, Y ) X Y and f 1 : X 1 Y 1 f 2 : X 2 Y 2 f 1 f 2 := π 1 ; f 1, π 2 ; f 2 : X 1 X 2 Y 1 Y 2. 2

3 t,y P τ X,Y P Y P ( Y ) P 2 (X Y ) τ X,P Y X,Y X Y P (X P Y ) P 2 (X Y ) P (X Y ) P t X,Y X Y Fig. 1. Commtativity axiom for the strong monad (P, η, ), t. The fact that is a fnctor, as well as the eqations f; g, h = f; g, f; h and f 1 ; g 1, f 2 ; g 2 = f 1, f 2 ; (g 1 g 2 ), can be easily proved with eqational reasoning. III. COMMUTATIVE STRONG MONADS WITH LAZY PAIRS For this section, we consider a cartesian category C, whose cartesian strctre is explicitly given by, π 1, π 2,,, 1,. A monad (P, η, ) over C consists of the following: An endofnctor P : C C, and natral transformations η X : X and X : P 2 X, called the nit and mltiplication of the monad respectively. Additionally, they satisfy the eqations: P X ; X = ; X, η ; X = id, and P η X ; X = id. A monad (P, η, ) is strong with tensorial strength t X,Y : X P Y P (X Y ) if t is a natral transformation satisfying the axioms given by the commtative diagrams of Figre 8 (in the Appendix). The arrow α X,Y,Z : (X Y ) Z X (Y Z) is the natral isomorphism defined by α := π 1 ; π 1, π 1 ; π 2, π 2. We define the dal tensorial strength τ X,Y : Y P (X Y ) as the composite of s,y t Y,X P s Y,X Y Y P (Y X) P (X Y ), where s X,Y : X Y Y X is the natral isomorphism given by s X,Y = π 2, π 1. The properties satisfied by t imply that τ is also a natral transformation and the diagrams of Figre 9 (in the Appendix) commte. The arrow β X,Y,Z : X (Y Z) (X Y ) Z is the natral isomorphism which is the inverse of α X,Y,Z. For all objects X, Y define the morphism X,Y : P Y P (X Y ) as the composite of t,y P τ X,Y X Y P Y P ( Y ) P 2 (X Y ) P (X Y ). Using the fact that t and τ are natral transformations, it can be shown that is also a natral transformation. A strong monad (P, η, ), t is commtative if the diagram of Figre 1 commtes. Intitively, the commtativity condition says that when compting a pair it does not matter in which order the components are compted. We are interested in strong monads that model lazy pairs. In the case of eager pairs, the comptation of a pair v,, where v is a vale and denotes a diverging comptation, wold also be diverging, i.e., v, =. Therefore, it is not possible to recover the left component v of the pair: v, ; π 1 =. So, for lazy pairs we need to stiplate an additional axiom, X P Y π 1 X t η P (X Y ) P π 1 Y π 2 Y τ η P (X Y ) P Y P π 2 Fig. 2. Lazy pairs axiom in terms of t and eqivalently in terms of τ. P Y P (X Y ) id P π 1, P π 2 P Y X Y η η P Y η P (X Y ) id ( P Y ) P Z P (X Y ) P Z P ((X Y ) Z) α P α (P Y P Z) P (Y Z) P (X (Y Z)) id P 2 X P 2 Y X Y P Y,P Y P X,Y P ( P Y ) P 2 (X Y ) X,Y X Y P (X Y ) P Y P Y P (Y X) P (X Y ) s P s Fig. 3. Commtative strong monad with lazy pairs (given in terms of ). which can be given eqivalently in terms of t or τ. The lazy pairs axiom says that the diagrams of Figre 2 commte. Definition 2: Let C be a category with explicitly given cartesian strctre. We say that (P, η, ), t is a commtative strong monad with lazy pairs if (P, η, ) is a monad with tensorial strength t so that the commtativity axiom of Figre 1 and the lazy pairs axiom of Figre 2 hold. Remark 3: Now, we will discss how a commtative monad with lazy pairs can be eqivalently given in terms of. In Figre 3 we give properties that satisfies, when it is defined in terms of t as before [13]. Conversely, we consider a monad (P, η, ) over C together with a natral transformation : P Y P (X Y ) satisfying the axioms of Figre 3. Then, we can define t as the composite of η X id P Y X,Y X P Y P Y P (X Y ) and recover all the axioms we had given for (P, η, ), t [14]. A. Kleisli Category, Unit Fnctor, Deterministic Arrows Let (P, η, ) be a monad over C. The Kleisli category C P has the same objects as C. For all objects X, Y the homset C P (X, Y ) is eqal to the homset C(X, P Y ). We se the notation for an arrow in C P (X, Y ), which is also an arrow f : X P Y in C(X, P Y ). The composition operation in C P is the Kleisli composition operation, denoted ;, which is defined as g : Y Z f; g := f; P g; Z : X Z. 3

4 For object X, the identity in C P is η X : X X. The eqations η X ; f = f; η Y = g : Y Z h : Z W (f; g); h = f; (g; h) : X W stating that C P is a category can be shown from the definitions and the monad axioms. Remark 4: For f : X Y and g : Y P Z, it holds that f; g = (f; η Y ); g. Indeed, we have that (f; η Y ); g = f; η Y ; P g; Z = f; g; η P Z ; Z = f; g, sing the fact that η is natral and one of the monad axioms. Definition 5 (nit fnctor): We define the map H = ( ; η) from the category C to the Kleisli category C P as follows: H : X X f : X Y Hf := f; η Y : X Y We verify that H is a fnctor. First, note that it sends the identity id X : X X of C to the identity Hid X = η X : X X of C P. Moreover, the rle is valid. f : X Y g : Y Z H(f; g) = Hf; Hg : X Z Hf; Hg = f; η Y ; P (g; η Z ); Z [def.] = f; η Y ; P g; P η Z ; Z [P fnctor] = f; η Y ; P g [P monad] = f; g; η Z [η natral] = H(f; g). [H def] So, H is a fnctor from C to C P. We call this the nit fnctor of the monad. Definition 6 (deterministic arrows): We say that an arrow of C P is a deterministic arrow if there exists an arrow f : X Y of C sch that f = f ; η Y = Hf : X P Y. So, the deterministic arrows of the Kleisli category are exactly the image of the arrows of C nder the nit fnctor H. We indicate that f is a deterministic arrow of C P by writing f : X Y. The Kleisli composite of two deterministic arrows is also a deterministic arrow: f : X Y g : Y Z. (3) f; g : X Z Sppose that f = Hf : X Y and g = Hg : Y Z are deterministic arrows. Then, f; g = Hf ; Hg = H(f ; g ) since H is a fnctor. So, f; g is a deterministic arrow. The identity η X : X X is a deterministic arrow becase η X = Hid X. B. Kleisli Pairing, Projections, Prodct Let (P, η, ), t be a commtative strong monad over C with lazy pairs. In this section we will define Kleisli versions of projections, the pairing operation, and the prodct fnctor. We will prove sefl properties that they satisfy. The notion of deterministic arrow trns ot to be relevant. (1) (2) Definition 7 (Kleisli pairing and projections): We define the Kleisli pairing operation, in C P by g : X Z f, g := f, g ; : X Y Z. We also define the Kleisli projections (left and right): ϖ 1 := π 1 ; η = Hπ 1 : X Y X ϖ 2 := π 2 ; η = Hπ 2 : X Y Y We note that the Kleisli projections are deterministic arrows. We claim that if f : X Y and g : X Z are deterministic arrows, then so is f, g : X Y Z: f : X Y g : X Z. (4) f, g : X Y Z This is an immediate conseqence of the following rle, which states that H commtes with the pairing operation: f : X Y g : X Z H f, g = Hf, Hg : X Y Z. This can be seen by: Hf, Hg = f; η, g; η ; = f, g ; (η η); = f, g ; η = H f, g. Theorem 8: The following typed eqations for Kleisli projections/pairing are valid: g : X Z f, g ; ϖ 1 = g : X Z f, g ; ϖ 2 = g : X Z h : X Y Z h; ϖ 1, h; ϖ 2 = h : X Y Z (5) (6) (7) f : X Y g i : Y Z i i = 1, 2 f; g 1, g 2 = f; g 1, f; g 2 : X Z 1 Z 2 (8) Proof. We will show now that the first rle is valid. The validity of the second rle is shown similarly. We have that f, g ; ϖ 1 = f, g ; P ϖ 1 ; [; def] = f, g ; ; P (π 1 ; η); [, & ϖ 1 defs] = f, g ; ; P π 1 ; P η; [P fnctor] = f, g ; ; P π 1 [P, η, monad] = f, g ; π 1 [ properties] = f. For the third rle, we have that h = h ; η = Hh : X P (Y Z) for some h : X Y Z. Since H is a fnctor and it also commtes with the pairing operation, we get that h; ϖ 1, h; ϖ 2 = Hh ; Hπ 1, Hh ; Hπ 2 = H h ; π 1, h ; π 2, 4

5 which is eqal to Hh = h. For the forth rle, we have that f = f ; η = Hf : X P Y for some f : X Y. Now, f; g 1, g 2 = f; P g 1, g 2 ; [; def] = f ; η; P g 1, g 2 ; [f deterministic] = f ; g 1, g 2 ; η; [η natral] = f ; g 1, g 2 [monad] = f ; g 1, g 2 ; [, def] = f ; g 1, f ; g 2 ; [C cartesian] = (f ; η); g 1, (f ; η); g 2 ; [Remark 4] = f; g 1, f; g 2, and the proof is complete. Definition 9: We define the operation on C P, which we call Kleisli prodct fnctor, as follows: f 1 : X 1 Y 1 f 2 : X 2 Y 2 f 1 f 2 := (f 1 f 2 ); : X 1 X 2 Y 1 Y 2. Eqivalently, we can define the Kleisli prodct as f 1 f 2 = ϖ 1 ; f 1, ϖ 2 ; f 2. Indeed, f 1 f 2 = (f 1 f 2 ); [ def] = π 1 ; f 1, π 2 ; f 2 ; [ def] = (π 1 ; η); f 1, (π 2 ; η); f 2 ; [Remark 4] = ϖ 1 ; f 1, ϖ 2 ; f 2 ; [ϖ def] = ϖ 1 ; f 1, ϖ 2 ; f 2. [, def] We observe the similarity in the definitions of the prodct fnctor in C and the Kleisli prodct fnctor in C P : f i : X i Y i i = 1, 2 f 1 f 2 := π 1 ; f 1, π 2 ; f 2 f i : X i Y i i = 1, 2 f 1 f 2 := ϖ 1 ; f 1, ϖ 2 ; f 2 We have to see that is indeed a fnctor. This is shown in Theorem 10 that follows. We also observe that H commtes with the prodct fnctor: H(f g) = H π 1 ; f, π 2 ; g = H(π 1 ; f), H(π 2 ; g) = ϖ 1 ; Hf, ϖ 2 ; Hg = Hf Hg. Theorem 10: The map is a fnctor from the prodct category C P C P to C P. Proof. We first verify that η X η Y = (η X η Y ); = η X Y : X Y X Y, sing the axioms. It remains to see that the rle f i : X i Y i g i : Y i Z i i = 1, 2 (9) (f 1 f 2 ); (g 1 g 2 ) = (f 1 ; g 1 ) (f 2 ; g 2 ) is valid. Indeed, we have that (f 1 f 2 ); (g 1 g 2 ) = (f 1 f 2 ); ; P [(g 1 g 2 ); ]; = [; & defs] (f 1 f 2 ); ; P (g 1 g 2 ); P ; = [P fnctor] (f 1 f 2 ); (P g 1 P g 2 ); ; P ; = [ natral] (f 1 f 2 ); (P g 1 P g 2 ); ( ); = [ axioms] (f 1 ; P g 1 ; f 2 ; P g 2 ; ); [C cartesian] which is eqal to (f 1 ; g 1 f 2 ; g 2 ); = f 1 ; g 1 f 2 ; g 2. IV. NONDETERMINISTIC MONADS In this section we look at three eqivalent ways of endowing with (angelic) nondeterministic strctre the homsets of the Kleisli category of a monad. See also [15]. The proofs of eqivalence we present make essential se of the cartesian strctre of the base category. Definition 11: Let (P, η, ) be a monad over the category C, XY : X P Y be a family of morphisms, and + be a binary operation on C(X, P Y ) f : X P Y g : X P Y, f + g : X P Y which we call (nondeterministic) choice. We say that the (P, η, ), +, is a nondeterministic monad if the axioms (f + g) + h = f + (g + h) f; (g 1 + g 2 ) = f; g 1 + f; g 2 f + g = g + f f + = f f + f = f (f 1 + f 2 ); g = f 1 ; g + f 2 ; g are satisfied. The above axioms state that every homset C(X, P Y ) is a commtative idempotent monoid w.r.t. + and, and that + distribtes over Kleisli composition. Assming the category C is cartesian, we will give an eqivalent definition of the nondeterministic monad in terms of a natral transformation X :, which we can intitively think of as binary nion. Then, we will also derive another eqivalent definition of the nondeterministic monad in terms of a natral transformation d X : X X, which we think of as an operation that forms nordered pairs. Theorem 12: Let C be a category with cartesian strctre given by, π 1, π 2,,, 1,. Sppose (P, η, ), +, is a nondeterministic monad. Define X := π 1 + π 2 :. Then, is a natral transformation and the diagrams of Figre 4 commte. Conversely, sppose that (P, η, ) is a monad, X : is a natral transformation, and 1X : 1 is a family of morphisms, so that the axioms of Figre 4 are satisfied. Define the operation + by f : X P Y g : X P Y f + g := f, g ; Y : X P Y. Also define the family of morphisms XY := X ; 1Y : X P Y. Then, (P, η, ), +, is a nondeterministic monad. Proof. First, we will show a few simple conseqences of (P, η, ), +, being a nondeterministic monad. f : X Y g i : Y P Z i = 1, 2 f; (g 1 + g 2 ) = f; g 1 + f; g 2 : X P Z (10) Using Remark 4 and the distribtivity property, we have that f; (g 1 + g 2 ) = (f; η); (g 1 + g 2 ) = (f; η); g 1 + (f; η); g 2 = f; g 1 + f; g 2. f i : X P Y i = 1, 2 P (f 1 + f 2 ); = P f 1 ; + P f 2 ; : P Y (11) 5

6 α id ( ) ( ) id id id 1 1 P 2 X P 2 X s π 2 π 1 id, id id P ( ) P P 2 X P π 1, P π 2 P 2 X P 2 X P 2 X X X P 2 X Fig. 4. Commtative diagrams for X :. Using the distribtivity property, we have that P (f 1 +f 2 ); = id; (f 1 + f 2 ) = id; f 1 + id; f 2 = P f 1 ; + P f 2 ;. f i : X P Y i = 1, 2 g : Y Z (f 1 + f 2 ); P g = f 1 ; P g + f 2 ; P g : X P Z (12) As before, we have that (f 1 + f 2 ); P g = (f 1 + f 2 ); (g; η Z ) = f 1 ; (g; η Z ) + f 2 ; (g; η Z ) = f 1 ; P g + f 2 ; P g. f i : X P 2 Y i = 1, 2 (f 1 + f 2 ); Y = f 1 ; Y + f 2 ; Y : X P Y (13) We have that (f 1 +f 2 ); = (f 1 +f 2 ); id P Y = f 1 ; id+f 2 ; id = f 1 ; + f 2 ;. Now, we will see that is a natral transformation. For f : X Y, we claim that X ; P f = (P f P f); : P Y. Indeed, sing rle (12) we see that ; P f = (π 1 + π 2 ); P f = π 1 ; P f + π 2 ; P f, and sing rle (10) that (P f P f); = π 1 ; P f, π 2 ; P f ; (π 1 + π 2 ) = π 1 ; P f + π 2 ; P f. We will now show that the first for diagrams of Figre 4 commte. These diagrams say that for every object X, the triple (X, X, 1X ) is a commtative idempotent monoid in the cartesian category C. α; (id ); = π 1 ; π 1, π 1 ; π 2, π 2 ; (id ); = π 1 ; π 1, π 1 ; π 2, π 2 ; ; = π 1 ; π 1, π 1 ; π 2, π 2 ; (π 1 + π 2 ) ; = π 1 ; π 1, π 1 ; π 2 + π 2 ; (π 1 + π 2 ) = π 1 ; π 1 + (π 1 ; π 2 + π 2 ) Also, ( id); = π 1 ;, π 2 ; (π 1 + π 2 ) = π 1 ; + π 2 = π 1 ; (π 1 +π 2 )+π 2 = (π 1 ; π 1 +π 1 ; π 2 )+π 2. From associativity of + we obtain that α; (id ); = ( id);. For the second diagram (left and right identity) we have that ( id); = π 1 ;, π 2 ; =, π 2 ; (π 1 + π 2 ) = + π 2 = π 2 and similarly (id ); = π 1. For commtativity and idempotence respectively we have that s; = π 2, π 1 ; (π 1 + π 2 ) = π 2 + π 1 = π 1 + π 2 = and id, id ; = id, id ; (π 1 + π 1 ) = id + id = id. For the last two diagrams of Figre 4, we have that ( ); = π 1 ;, π 2 ; ; (π 1 +π 1 ) = π 1 ; +π 2 ; = (π 1 +π 1 ); = ; sing rle (13), and also that P π 1, P π 2 ; ; = (P π 1 + P π 2 ); = P π 1 ; + P π 2 ; = P (π 1 + π 2 ); = P ; sing rle (13) and rle (11). For the converse, we will first verify the properties stating that C(X, P Y ) is a commtative idempotent monoid w.r.t. + and. For associativity, we have that (f + g) + h = f, g ;, h ; = f, g, h ; ( id); = f, g, h ; α; (id ); = f, g, h ; (id ); = f, g, h ; ; = f + (g + h). We also have that f + = f, ; = f, ; ; = f, ; (id ); = f, ; π 1 = f and similarly + f = f. For commtativity and idempotence we have: f + g = f, g ; = f, g ; s; s; = g, f ; = g + f and f + f = f, f ; = f; id, id ; = f. We show now the distribtivity properties f; (g 1 + g 2 ) = f; P ( g 1, g 2 ; ); = f; P g 1, g 2 ; P ; = f; P g 1, g 2 ; P π 1, P π 2 ; ; = f; P g 1, g 2 ; P π 1, P π 2 ; ( ); = f; P g 1, g 2 ; P π 1 ;, f; P g 1, g 2 ; P π 2 ; ; = f; P g 1 ;, f; P g 2 ; ; = f; g 1 + f; g 2 (f 1 + f 2 ); g = f 1, f 2 ; ; P g; = f 1, f 2 ; (P g P g); ; = f 1, f 2 ; (P g P g); ( ); = f 1 ; P g;, f 2 ; P g; ; = f 1 ; g + f 2 ; g and the proof is complete. Theorem 13: Let C be a category with cartesian strctre given by, π 1, π 2,,, 1,. Let (P, η, ) be a monad over C, and 1X : 1 be a family of morphisms. Sppose that X : is a natral transformation sch that the diagrams of Figre 4 commte. Define d X := (η X η X ); X : X X. Then, d is a natral transformation satisfying the axioms of Figre 5. Conversely, sppose that d X : X X is a natral transformation sch that the diagrams of Figre 5 commte. Define X := d ; X :. Then, is a natral transformation satisfying the axioms of Figre 4. Proof. First, we will see that d X : X X is a natral transformation. Using the facts that and η are natral 6

7 α η d d (X X) X X (X X) P 2 X d η d P 2 X id id 1 1 X X P 2 X P 2 X d P 2 X d d P 2 X π 2 s P 3 X P 2 X X X d P 2 X d X π 1 id, id η P (X X) X X d P d P π 1, P π 2 d P 2 X Fig. 5. Commtative diagrams for d X : X X. P 2 X transformations, we get that d; P f = (η η); ; P f = (η η); (P f P f); = (η; P f η; P f); = (f; η f; η); = (f f); (η η); = (f f); d. To verify that the first diagram commtes, we first notice that d ; = (η η ); ; = (η η ); ( ); = (η ; η ; ); =. Then, we have that α; (η d); d ; = α; (η (η η); ); = α; (η (η η)); (id ); = ((η η) η); α; (id );, and (d η); d ; = ((η η); η); = ((η η) η); ( id);. Since α; (id ); = ( id);, the diagram commtes. It is immediate that the second diagram commtes: ( id); d ; = ( id); = π 2 and similarly (id ); d ; = (id ); = π 1. For the next three diagrams we have: s; d = s; (η η); = (η η); s; = (η η); = d, id, id ; d = id, id ; (η η); = η; id, id ; = η, and d P 2 X; ; = ; = ( ); = ( ); d ; respectively. Finally, P π 1, P π 2 ; d ; = P π 1, P π 2 ; (η η ); ; = P π 1, P π 2 ; (η η ); ( ); = P π 1, P π 2 ; (P η; P η; ); = P π 1 ; P η, P π 2 ; P η ; ( ); = P ((η η); π 1 ), P ((η η); π 2 ) ; ( ); = P (η η); P π 1, P π 2 ; ;, which is eqal to P (η η); P ; = P d;. So, the last diagram commtes. For the converse, we will first show that : is a natral transformation. Since d and are natral transformations, we have that (P f P f); Y = (P f P f); d P Y ; Y = d ; P 2 f; Y = d ; X ; P f = X ; P f. For the first diagram we see that α; (id ); = α; (η; d; ); d; = α; (η d); ( ); d; = α; (η d); d; ; = (d η); d; ; = (d η); ( ); d; = (d; η; ); d; = ( id); and therefore it commtes. Showing that the rest of the diagrams commte is straightforward. V. NONDETERMINISTIC STRONG MONAD WITH ITERATION In Section IV we investigated the nondeterministic strctre of the Kleisli category of a monad in isolation from prodcts. This is not sfficient for or prposes, becase we also want to captre the interaction between nondeterminism and prodcts. One step towards this direction is made by considering an additional axiom that relates the tensorial strength with the nondeterministic strctre (Theorem 14). Theorem 14: Let C be a category with cartesian strctre given by, π 1, π 2,,, 1,. Let (P, η, ), t be a commtative strong monad over C with tensorial strength t X,Y : X P Y P (X Y ). Assme additionally that (P, η, ) is a nondeterministic monad together with + and, and also that the axiom f : X P Y g : X P Y id, f + g ; t = id, f ; t + id, g ; t : X P (X Y ). holds. Then, the rles f i : X P Y i = 1, 2 g : X P Z f 1 + f 2, g = f 1, g + f 2, g : X P (Y Z) f : X P Y g i : X P Z i = 1, 2 f, g 1 + g 2 = f, g 1 + f, g 2 : X P (Y Z) are valid. (14) (15) Proof. First, we will show that the axiom we assmed can be generalized into the rle f : X Y g i : X P Z i = 1, 2 f, g 1 + g 2 ; t = f, g 1 ; t + f, g 2 ; t : X P (Y Z). Since t is a natral transformation, we have that f, g 1 + g 2 ; t = id, g 1 + g 2 ; (f P id); t = id, g 1 + g 2 ; t; P (f id) = ( id, g 1 ; t + id, g 2 ; t); P (f id) = id, g 1 ; t; P (f id) + id, g 2 ; t; P (f id) = f, g 1 ; t + f, g 2 ; t. Now, it is easy to show that the dal rle f i : X P Y i = 1, 2 g : X Z f 1 + f 2, g ; τ = f 1, g ; τ + f 2, g ; τ : X P (Y Z) 7

8 holds for the dal tensorial strength τ X,Y := s; t; P s : Y P (X Y ). So, for rle (14) we have that f 1 + f 2, g = f 1 + f 2, g ; τ; P t; and, similarly, = ( f 1, g ; τ + f 2, g ; τ); P t; = f 1, g ; τ; P t; + f 2, g ; τ; P t; = f 1, g + f 2, g f, g 1 + g 2 = f, g 1 + g 2 ; t; P τ; for rle (15). = ( f, g 1 ; t + f, g 2 ; t); P τ; = f, g 1 ; t; P τ; + f, g 2 ; t; P τ; = f, g 1 + f, g 2. Definition 15: Let C be a category with cartesian strctre and bottom elements given by, π 1, π 2,,, 1,. We say that (P, η, ),,, is a nondeterministic strong monad with iteration is the following are satisfied: (i) P 1 = 1 (hence 1 is a terminal object of C P ) (ii) (P, η, ), is a commtative strong monad with lazy pairs. (iii) (P, η, ),, is a nondeterministic monad, where XY = XY ; η Y : X P Y. (iv) For all f, g : X P Y, the eqation id, f + g ; t = id, f ; t + id, g ; t : X P (X Y ) holds, where t is the tensorial strength indced by and + is the choice operation indced by. (v) The axiom id P (X Y ) P π 1, P π 2 ; X,Y holds, where is the partial order indced by +. (vi) The iteration operation sends a morphism f : X to f : X. The axioms η + f; f f, η + f ; f f, f; g g f ; g g, and g; f g g; f g are satisfied. Theorem 16: Let C be a category with cartesian strctre and bottom elements given by, π 1, π 2,,, 1,. Let (P, η, ),,, be a nondeterministic strong monad with iteration. Then, the Kleisli category C P with Kleisli composition ; and Kleisli identity η, together with the operations ϖ 1, ϖ 2,,,, +, (Kleisli projections, pairing, bottoms, nondeterministic choice, and iteration) satisfies the axioms of Table I. Proof. The axioms of the first grop follow from the definition of determinism and the fact that (P, η, ) is a monad. The axioms of the second grop follow from the fact that the strong monad (P, η, ), is commtative and satisfies the lazy pairs property (Theorem 8 and Theorem 10). The axioms of the third grop follow from the reqirement that P 1 = 1 and from the definition XY = H XY, where H is the nit fnctor of the monad (recall that 1X is a bottom global element). For the forth grop of axioms: Some of them are a restatement of the definition of (P, η, ), +, being a nondeterministic monad (see also Theorem 12). The last two of them follow from the reqirement that id, f + g ; t = Kleisli identity η X : X X, and Kleisli composition ;. g : Y Z f; g : X Z η X ; f = f : X Y g : Y Z f; g : X Z f; η Y = g : Y Z h : Z W (f; g); h = f; (g; h) : X W Kleisli projections ϖ 1 : X Y X, ϖ 2 : X Y Y, Kleisli pairing,, Kleisli prodct fnctor f g = f; ϖ 1, g; ϖ 2. f : X Z g : Y Z f, g : X Y Z g : X Z f, g ; ϖ 1 = f : X Z g : Y Z f, g : X Y Z g : X Z f, g ; ϖ 2 = g : X Z h : X Y Z h; ϖ 1, h; ϖ 2 = h : X Y Z f : X Y g i : Y Z i i = 1, 2 f; g 1, g 2 = f; g 1, f; g 2 : X Z 1 Z 2 f i : X i Y i g i : Y i Z i i = 1, 2 (f 1 f 2 ); (g 1 g 2 ) = (f 1 ; g 1 ) (f 2 ; g 2 ) Kleisli bottom morphisms XY : X Y. f : X 1 f = : X 1 f; = : X Z Nondeterministic choice operation +. Partial order defined as: f g iff f + g = g, for f, g : X Y. g : X Y f + g : X Y f, g, h : X Y (f + g) + h = f + (g + h) : X Y Iteration operation. f + = f, g : X Y f + g = g + g 1, g 2 : Y Z f; (g 1 + g 2 ) = f; g 1 + f; g 2 : X Z f 1, f 2 : X Y f + f = g : Y Z (f 1 + f 2 ); g = f 1 ; g + f 2 ; g : X Z h : X Y Z h h; ϖ 1, h; ϖ 2 = h : X Y Z f 1, f 2 : X Y g : X Z f 1 + f 2, g = f 1, g + f 2, g : X Y Z g 1, g 2 : X Z f, g 1 + g 2 = f, g 1 + f, g 2 : X Y Z f : X X η X + f; f f : X X f : X X η X + f ; f f : X X f : X X f : X X f : X X g : X Y f; g g f ; g g : X Y g : X Y f : Y Y g; f g g; f g : X Y TABLE I KLEISLI CATEGORY OF A NONDETERMINISTIC STRONG MONAD WITH ITERATION. 8

9 id, f ; t + id, g ; t as shown in Theorem 14. Finally, h h; ϖ 1, h; ϖ 2 follows from id P π 1, P π 2 ;, becase it holds that h; ϖ i = h; P (π i ; η); = h; P π i ; P η; = h; P π i and therefore h; ϖ 1, h; ϖ 2 = h; P π 1, h; P π 2 = h; P π 1, h; P π 2 ; = h; P π 1, P π 2 ; h. The fifth grop of axioms is an assmption we have made. A. The Lowerset Monad In the sal relational interpretation of programs, a (strict) nondeterministic program is interpreted as a morphism in the category Rel of sets and binary relations. The category Rel is isomorphic to the Kleisli category Set of the powerset monad over the category Set of sets and total fnctions. In order to deal explicitly with partiality we introdce a symbol that denotes divergence. We note that Set is isomorphic to the category FlatST of flat posets and strict total fnctions. By total we mean here that a morphism f : X Y of FlatST sends every x to f (x). The isomorphism is witnessed by the lifting fnctor ( ), defined as follows: X in Set X in FlatST f : X Y in Set f : X Y in FlatST where X := X { X } together with the flat order, f ( X ) = Y and f (x) = f(x) for x X. Define the monad over FlatST by (X ) = {non-empty lower sets of X } (f : X Y ) = λs (X ).{f (x) x S} The bottom element of (X ) is the singleton { X }. The nit of the monad is given by x X {x, X } and the mltiplication is nion. Under these definitions, the Kleisli category Set is isomorphic to FlatST. The isomorphism is witnessed by the (nondeterministic) lifting fnctor given by: X in Set X in FlatST f : X (Y ) in Set φ : X (Y ) in FlatST where φ( X ) = { Y } and φ(x) = f(x) { Y } for x X. Remark 17: We observe that in FlatST, the natral operation of forming eager pairs of elements does not give rise to a categorical prodct. Define the eager pair x, y, where x X and y Y, to be eqal to X Y if one of x, y is bottom and eqal to the pair (x, y) otherwise. The smash prodct is defined as X Y = (X Y ). Now, y ; π 2 = ; π 2 =. Therefore, we cannot recover the right component y of the pair. Ths the smash prodct is not categorical. Consider now the category Pposet of pointed posets (posets with a bottom element) and monotone fnctions, in which we can interpret non-strict deterministic programs that form lazy pairs. The morphisms in Pposet can be partial, in the sense that they are allowed to send non-bottom elements to bottom. For an object (X, ) of Pposet, we nderstand the partial order as follows: x y intitively means that x has more diverging components than y. Remark 18: Pposet is a cartesian category with bottom elements. The prodct X Y of two objects X, Y is the cartesian prodct together with the pointwise partial order. So, the bottom element of X Y is X Y = X, Y. The projections morphisms π 1 and π 2 are given by π i (x 1, x 2 ) = x i. The pairing operation, is defined as f : X Y g : X Z f, g = λx X. f(x), g(x). The terminal object is some singleton poset 1 = { 1 }. The bottom global element 1X : 1 X sends 1 to the bottom X of X. So, XY = X ; 1Y is the fnction that always diverges. We define the pointwise partial order on every homset Pposet(X, Y ). Then, XY is the bottom element of Pposet(X, Y ). Definition 19 (lowerset monad): Let (X, ) be a pointed poset, the bottom element of which is denoted X. For a sbset S X define S = {y X y x for some x S} to be the lowerset generated by S. For x X, we write x to mean {x}. Define X to be the set of all non-empty lowersets of X. We observe that X is a complete lattice w.r.t. set inclsion. The top element is X and the bottom is { X }. The join is set-theoretic nion and the meet is set-theoretic intersection. It follows that the homset Pposet(X, Y ) (w.r.t. the pointwise order indced by Y, ) is also a complete lattice. We extend into an endofnctor Pposet Pposet by ptting f : X Y in Pposet f := λs X. {f(x) x S}. Together with the families of maps η X : x X x X X : S 2 X S X it forms a monad over Pposet. We call (, η, ) the lowerset monad over Pposet. It can be easily shown that f : X Y g : Y Z (f; g)(x) = y f(x) g(y) gives Kleisli composition. Theorem 20: Define the family of morphisms X,Y X,Y : X Y (X Y ) (S 1, S 2 ) S 1 S 2 and the family of morphisms X as X : X X X (S 1, S 2 ) S 1 S 2 For a morphism f : X X, define f i : X X, i < ω, by indction: f 0 = η X and f i+1 = f i ; f. Now, define the iteration operation by f := i<ω f i = λx X. i<ω f i (x) where is the join of the complete lattice Pposet(X, X). Then, the lowerset monad (, η, ) over the category Pposet, as 9

10 f 1 f 2 (S 1, S 2) ( f 1(S 1), f 2(S 2)) (f 1 f 2) S 1 S 2 (f 1(S 1) f 2(S 2)) ) ( ({S i} i I, {T j} j J i I Si, ) j J Tj {(S i, T j)} (i,j) I J () {S i T j} (i,j) I J (i,j) I J Fig. 6. is natral, and ; (); = ( );. Si Tj together with the operations,,, is a nondeterministic strong monad with iteration. Proof. First, we see that X,Y is well-defined. This is the case, becase if S 1, S 2 are non-empty lowersets of X, Y respectively, then S 1 S 2 is a non-empty lowerset of X Y. That is a natral transformation is a conseqence of the fact that A B = (A B), as can be seen in Figre 6. It is straightforward to show that satisfies the properties of Figre 3. For example, we show in Figre 6 that the forth diagram of Figre 3 commtes. We are making se of the simple fact that ( i I A i) ( j J B j) = (i,j) I J A i B j for any collections of sets {A i } i I and {B j } j J. We have ths established that (, η, ), is a commtative strong monad with lazy pairs. Now, we see that X is well-defined, becase S 1 S 2 is a lowerset whenever S 1 and S 2 are lowersets. We also define XY = XY ; η Y. So, XY (x) = { Y } for all x X. That is a natral transformation is a conseqence of the fact that f(s) f(t ) = (f(s) f(t )) = f(s T ), as can be seen in Figre 7. The properties of Figre 4 hold. In Figre 7 we show that last two diagrams of Figre 4 commte. We are making se of the simple properties: ({A i } i {B j } j ) = ( i A i) ( j B j) and ( i A i) ( i B i) = i (A i B i ). So, (, η, ),, is a nondeterministic monad. The operation + indced by is given by f + g = λx X.f(x) g(x) for f, g : X Y. The tensorial strength t X,Y : X Y (X Y ) indced by is given by t X,Y = (η id); = λ(x, S). x S. We verify the axiom id, f + g ; t = id, f ; t + id, g ; t for f, g : X Y. This simply amonts to showing that x (f(x) g(x)) = x f(x) x g(x) for all x X, which is tre. Moreover, we notice that for a lowerset {(x i, y i )} i I in (X Y ) we have: {(x i, y i )} i I ({x i } i I, {y i } i I ) {(x i, y j )} (i,j) I I throgh π 1, π 2 and respectively. Since the set {(x i, y i )} i I is contained in {(x i, y j )} (i,j) I I we conclde that the property id π 1, π 2 ; : (X Y ) (X Y ) holds. An easy indction gives s that f i+1 = f; f i for every i < ω (making se of the associativity property of Kleisli (S, T ) S T f f ({S i} i, {T j} j) ( i Si, j Tj) f ( f(s), f(t )) f(s T ) {S i} i {T j} j ( i Si) ( j Tj) (π 1), (π 2) {(S i, T i)} i ({S i} i, {T i} i) {S i} i {T i} i () {S i T i} i ( i Si) ( i Ti) Fig. 7. is natral, ( ); = ;, and (π 1 ), (π 2 ) ; ; = ();. composition). We show the properties: g : X (Y ) f : Y (Y ) (g; f )(x) = i<ω (g; f i )(x) f : X (X) g : X (Y ) (f ; g)(x) = i<ω (f i ; g)(x) (g; f )(x) = y g(x) f (y) = y g(x) i<ω f i (y) = i<ω y g(x) f i (y) = i<ω (g; f i )(x) (f ; g)(x) = y f (x) g(y) = y i<ω f i (x) g(y) = i<ω y f i (x) g(y) = i<ω (f i ; g)(x) (16) (17) Now, sing Property (16), we have that (f; f )(x) = i<ω (f; f i )(x) = i<ω f i+1 (x) f (x). So, η + f; f f. Using Property (17), we have that (f ; f)(x) = i<ω (f i ; f)(x) = i<ω f i+1 (x) f (x). So, η + f ; f f. To show the implication f; g g f ; g g, we sppose that f; g g and by indction we see that f i ; g g for all i < ω. Therefore, sing Property (16), we conclde that (f ; g)(x) = i<ω (f i ; g)(x) g(x) and hence f ; g g. The implication g; f g g; f g is shown similarly sing Property (17) instead. It only remains to observe that 1 = {{ 1 }} = { 1 } = 1 and the proof is complete. B. The Ideal-Completion Monad An ω-complete partial order (ω-cpo) is a partially ordered set (X, ) that has a least element X and is ω-complete in the sense that every ω-chain (contable chain) x 0 x 1 has a spremm sp i x i. A fnction f : X Y between ω-cpos is called ω-continos if it preserves sprema of ω-chains. That is, for every ω-chain x 0 x 1 in X, f(sp i x i ) = sp i f(x i ). An ω-continos fnction is monotone. An ω-continos fnction is strict if f( ) =. If X, Y are ω-cpos, then so is their cartesian prodct X Y nder the componentwise order: (x 1, x 2 ) (y 1, y 2 ) x 1 y 1 x 2 y 2. The least element is X Y = ( X, Y ). For an ω-chain (x 0, y 0 ) (x 1, y 1 ) in X Y, sp i (x i, y i ) = 10

11 (sp i x i, sp i y i ). We denote by [X Y ] the ω-cpo of all ω-continos fnctions from X to Y ordered pointwise: f g x P. f(x) g(x). The bottom element is XY = λx X. Y. The spremm of a chain f 0 f 1 in [X Y ] is sp i f i = λx X. sp i f i (x). For a chain x 0 x 1 in X, we have that (sp i f i )(sp j x j ) = sp i (f i (sp j x j )) = sp i sp j f i (x j ) = sp j sp i f i (x j ) = sp j (sp i f i )(x j ), therefore sp i f i is ω-continos. The operations ; and, are monotone in all argments. The ω-continos fnctions on ω-cpos are closed nder well-typed composition and pairing and contain all identities and projections. Ths, ω-cpos and ω-continos fnctions form a cartesian category with bottom elements, which we denote by CPO. Let (X, ) be an ω-cpo. A sbset I X is called an ideal of X if it is a non-empty lower set and is closed nder sprema of ω-chains. The set of all ideals of X is denoted IX. We denote by cl X (S) the smallest ideal containing S X. This is an operation of type cl X : X IX. We also write cl(s) instead of cl X (S) when no confsion arises. We say that cl(s) is the ideal generated by S. The set x is an ideal and we call it the principal ideal generated by x. If x 1 x 2... is an ω-chain in X, then cl[{x 1, x 2,...}] = sp i x i. The set IX of all ideals of an ω-cpo X is a complete lattice w.r.t. set-theoretic inclsion. The meet is set-theoretic intersection and the join is the ideal generated by the set-theoretic nion. The bottom element is { X } and the top element is X. If I, J are ideals of X, then so is I J. In particlar, cl(s 1 S 2 ) = cl(s 1 ) cl(s 2 ). Let X, Y be ω-cpos. If I is an ideal of X and J is an ideal of Y, then I J is an ideal of the ω-cpo X Y. If K is an ideal of X Y, then the left and right projections of K are ideals of X and Y respectively. Lemma 21: Let X, Y be ω-cpos. Then, the following properties hold: (1) Let S be a non-empty collection of non-empty sbsets of X. Then, cl[ S S cl(s)] = cl[ S]. (2) Let S be a non-empty sbset of X that is bonded above by X. Then, cl(s) is also bonded above by. (3) Let S be an ideal of (I(X), ), i.e, S is in I 2 (X). Then, S is an ideal of (X, ). (4) Let S be a non-empty collection of ideals of X. Then, clix [S] = cl X [ S]. (5) If f : X Y is an ω-continos fnction, then cl[f(cl(s))] = cl[f(s)]. (6) cl(a B) = cl(a) cl(b). Proof. Omitted. Definition 22 (the ideals endofnctor): We extend I to an endofnctor CPO CPO. For f : X Y in CPO define If : IX IY by (If)(S) := cl Y [f(s)] = cl Y {f(x) x S}, for S IX. We need to verify that I is a fnctor. Indeed, I(id X )(S) = cl X (id X (S)) = cl X S = S becase S is an ideal, and (If; Ig)(S) = cl(g(cl(f(s)))) = cl(g(f(s))) = (I(f; g))(s) sing Lemma 21(5). Definition 23 (the ideal-completion monad): We define the family of fnctions η X : X IX by η X (x) := x. For a morphism f : X Y, we have by monotonicity of f that f( x) = f(x) is an ideal of IY. So, (η X ; If)(x) = (If)( x) = cl[f( x)] = cl[ f(x)] = f(x) = (f; η Y )(x). Ths, η is a natral transformation. We define the family of fnctions X : I 2 X IX by X (S) := S = cl[ S] = S, by Lemma 21(3), becase S is an ideal of IX. We arge that is a natral transformation: For a fnction f : X Y and S I 2 X, we have that: (I 2 f; Y )(S) = (IIf)(S) [ def] = cl IY [{(If)(S) S S}] [I def] = cl Y {(If)(S) S S} [L. 21(4)] = cl Y S S (If)(S) = cl Y S S cl Y [f(s)] [I def] = cl Y S S f(s) [L. 21(1)] = cl Y [f( S)] ( X ; If)(S) = (If)( S) [ def] = cl Y [f( S)] [I def] Now, we claim that (I, η, ) is a monad, which we call the ideal-completion monad of CPO. It remains to verify the eqations I X ; X = IX ;, η IX ; X = id IX, and Iη X ; X = id IX. These are not hard to prove, given that we have already shown that is arbitrary nion. We give Kleisli composition in simple set-theoretic terms: (f; g)(x) = (f; Ig; Z )(x) [; def] = (Ig)(f(x)) [ def] = cl IZ {g(y) y f(x)} [I def] = cl Z {g(y) y f(x)} [L. 21(4)] = y f(x) g(y). for f : X IY and g : Y IZ. Theorem 24: Define the family of morphisms X,Y X,Y : IX IY I(X Y ) (I, J) I J and the family of morphisms X as X : IX IX IX (I, J) I J Also define the iteration operation by f := i<ω f i = λx X. i<ω f i (x) as 11

12 Then, the ideal-completion monad (I, η, ) over the category CPO, together with the operations,,, is a nondeterministic strong monad with iteration. Proof. Arging as in the proof of Theorem 20 for the lowerset monad, we see that (I, η, ), is a commtative strong monad with lazy pairs, and (I, η, ),, is a nondeterministic monad. For the tensorial strength t indced by, the axiom id, f +g ; t = id, f ; t+ id, g ; t holds, as well as the axiom id Iπ 1, Iπ 2 ;. For the iteration operation, we first show the properties: (g; f )(x) = y g(x) g : X IY f : Y IY (g; f )(x) = i<ω (g; f i )(x) f : X IX g : X IY (f ; g)(x) = i<ω (f i ; g)(x) i<ω f i (y) = [; & defs] [ def] [ cl y g(x) cl ( i<ω f i (y) )] = [L. 21(1)] [ ] cl y g(x) i<ω f i (y) = [ ] cl i<ω y g(x) f i (y) = [L. 21(1)] [ ( )] cl i<ω cl y g(x) f i (y) = [ def] [ ] cl i<ω y g(x) f i (y) = [; def] cl [ i<ω (g; f i )(x) ] = i<ω (g; f i )(x) [ def] (f ; g)(x) = [;, defs] cl {g(y) y f (x)} = [L. 21(4)] [ cl {g(y) y f (x)} ] = [, defs] [{ ( cl g(y) y cl i<ω f i (x) )}] = [L. 21(5)] [{ cl g(y) y i<ω f i (x) }] = [L. 21(4)] cl [ { g(y) y i<ω f i (x) }] = [ ] cl i<ω y f i (x) g(y) = [L. 21(1)] [ [ ]] cl i<ω cl y f i(x) g(y) = [ def] [ ] cl i<ω y f i (x) g(y) = [; def] cl [ i<ω (f i ; g)(x) ] = i<ω (f i ; g)(x) [ def] Now, the iteration axioms can be shown exactly as in the case of the lowerset monad. VI. TYPED KLEENE ALGEBRA WITH PRODUCTS Let Ω be a set of atomic types. Let 1 / Ω be a special constant called the nit type. The set of types over Ω, denoted Types(Ω), is the set of terms freely generated by Ω and 1 nder the binary prodct type constrctor. The terms of the langage are typed. The types of terms are expressions of the form X Y, where X, Y Types(Ω). We indicate the type of a term by writing. Some of the terms of the langage will be designated as deterministic. We indicate the deterministic terms by writing f : X Y. We think of X Y as a sbtype of X Y and hence we inclde the typing rle f : X Y. (18) Let H be a set of atomic arrows, each endowed with a fixed type h : X Y. We write h : X Y for a deterministic atomic arrow of H. Let π 1 : X Y X π 2 : X Y Y (19) id : X X : X Y (20) be special deterministic polymorphic constants called left projections, right projections, identities, and bottoms, respectively. Where necessary, we se sbscripts or sperscripts to denote the specialization at a particlar type; e.g., id X : X X, π1 XY : X Y X, or XY : X Y. Let ; and, be polymorphic constrctors called composition and pairing, respectively, satisfying the typing rles g : Y Z f; g : X Z f : X Y g : Y Z f; g : X Z g : X Z f, g : X Y Z f : X Y g : X Z f, g : X Y Z (21) (22) (23) (24) Note that compositions f; g are written in diagrammatic order. Composition and pairing preserve determinism. We add to the langage the polymorphic constrctors + and, called (nondeterministic) choice and iteration respectively, with typing rles g : X Y f + g : X Y f : X X f : X X (25) Choice and iteration introdce nondeterminism. Definition 25: A typed Kleene algebra with prodcts is a mlti-sorted algebraic strctre K = ( K 0,, 1, K 1, K d 1, dom, cod, +, ;,,, id,,, π 1, π 2 ), where K 0 is the set of objects of K, K 1 is the set of arrows or elements of K, and K d 1 K 1 is the set of deterministic arrows/elements. The operations dom and cod are fnctions (called domain and codomain) that map arrows to objects. The type of an element f of K is the expression X Y, where X = domf and Y = codf. We write to denote this. If f K d 1, we write f : X Y. The distingished prodct operation is a fnction : K 0 K 0 K 0. The object 1 K 0 is the distingished terminal object of the strctre. 12

13 f + (g + h) = (f + g) + h f + g = g + f f + = f f + f = f f; = id + f; f f id + f ; f f f; g g f ; g g f; (g 1 + g 2 ) = f; g 1 + f; g 2 g; f g g; f g (f 1 + f 2 ); g = f 1 ; g + f 2 ; g f, g ; π 1 = f f = : X 1 f, g ; π 2 = g h h; π 1, h; π 2 det. h: h; π 1, h; π 2 = h det. f: f; g 1, g 2 = f; g 1, f; g 2 (f 1 f 2 ); (g 1 g 2 ) = f 1 + f 2, g = f 1, g + f 2, g (f 1 ; g 1 ) (f 2 ; g 2 ) f, g 1 + g 2 = f, g 1 + f, g 2 TABLE II AXIOMS FOR TYPED KLEENE ALGEBRAS WITH PRODUCTS. The polymorphic operations and constants +, ;,,, id,,, π 1, and π 2 satisfy the expected typing rles. Additionally, the strctre is a model of the well-typed instances of the axioms in Table II. The partial order is indced by +: f g iff f + g = g. The prodct is defined by: f g = π 1 ; f, π 2 ; g. The first grop of axioms in Table II are the axioms of Kleene algebra [16, 17] except for the strictness axiom ; f =. We denote by KA the qasi-eqational system of typed Kleene algebra with prodcts. Notice that (p to a slight change in notation to make it less cmbersome), the typing rles and axioms satisfied by a typed Kleene algebra with prodcts are exactly those given in Table I. This means that any small sbcategory of the Kleisli category of a nondeterministic strong monad with iteration that is closed nder the appropriate operations is a typed Kleene algebra with prodcts. VII. ITERATION THEORIES The langage of iteration theories consists of atomic typed actions h : n m, where n, m are natral nmbers, and polymorphic operation symbols ; (composition), id (identity), ι (injection), [,,, ] (cotpling), (bottom), (dagger). The typing rles for the langage are the following: id n : n n ι n i : 1 n nm : n m f : n m g : m p f; g : n p f i : 1 m i = 1,..., n [f 1, f 2,..., f n ] : n m f : n n + p f : n p The typed terms f : n m of the langage are bild from the atomic actions and the operation symbols according to the typing rles. Consider now the category CPO of ω-cpos and ω- continos maps, which will provide the standard interpretation for the langage of iteration theories. An interpretation in CPO consists of an ω-cpo A and a mapping of every atomic symbol h : n m to a morphism h : A m A n in CPO, where A k denotes the k-fold associative cartesian prodct of A. The identity symbol id n : n n is interpreted as the identity fnction id n : A n A n. The injection symbol ι n i : 1 n is interpreted as the i-th projection ι n i : An A. The bottom symbol nm : n m is interpreted as the least fnction nm : A m A n of CPO(A m, A n ). Now, ; and [,,, ] are interpreted as fnction composition and tpling respectively. f i : 1 m i = 1,..., n [f 1,..., f n ] := λ x A m. f 1 ( x),..., f n ( x). Every ω-continos fnction φ : X X in CPO has a least fixpoint, which is the spremm of the ω-chain φ( ) φ 2 ( ) φ n ( ), where φ 0 = id and φ n+1 = φ n ; φ. We denote the least fixpoint of φ by (φ) = sp i φ i ( ). For an ω-continos fnction φ : X Y X, we define the fnction φ : Y X by φ (y) := (φ y ) = sp i φ i y( ), where φ y = λx X.φ(x, y) : X X. The fnction φ : Y X is also ω-continos. We call the parametric fixpoint operation. The operation is monotone. So, we interpret the dagger symbol of the langage as parametric fixpoint : f : n n + p ) f = ( f. : A n A p A n : A p A n We have ths defined for every typed term f : n m its interpretation f : A m A n. Every homset CPO(X, Y ) is eqipped with the pointwise partial order. For terms s, t we write CPO = s t to mean that s t for every interpretation of the langage in CPO. Define Th(CPO) to be the set of all valid ineqalities over CPO, that is, Th(CPO) := {s t CPO = s t}, where s, t are terms of the langage of iteration theories. Th(CPO) is the (in)eqational theory of iteration, in the words of Ésik [4]. Theorem 26 (Ésik [4]): Let Park be the niversal Horn system (with eqality) that incldes the following: (1) Axioms for categories. f : n m g : m p h : p q (f; g); h = f; (g; h) : n q f : n m id; f = f : n m f : n m f; id = f : n m (2) Axioms asserting that ι, [,,, ], and give associative categorical coprodcts. ι 1 1 = id 1 : 1 1 f k : 1 m k = 1,..., n ι n i ; [f 1, f 2,..., f n ] = f i : 1 m h : n p [ι n 1 ; h, ι n 2 ; h,..., ι n n; h] = h : n p (3) Axioms stating that is a partial order. (4) Axioms stating that ; and [,,, ] are motonone in all argments w.r.t.. (5) Axiom stating that nm : n m is the least element of Hom(n, m). f : X Y XY f : X Y 13