What are Iteration Theories?

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1 What are Iteration Theories? Jiří Adámek and Stefan Milius Institute of Theoretical Computer Science Technical University of Braunschweig Germany Jiří Velebil Department of Mathematics Faculty of Electrical Engineering Czech Republic In the setting of algebraic theories enriched with an external fixed-point operation, the notion of an iteration theory seems to axiomatize the equational properties of all the computationally interesting structures of this kind. S. L. Bloom and Z. Ésik (996), see [3] Abstract We prove that iteration theories can be introduced as algebras for the monad on the category of signatures assigning to every signature the rational- -tree signature. This supports the claim that iteration theories axiomatize precisely the equational properties of least fixed points in domain theory: is the monad of free rational theories and every rational theory has a continuous completion.. Introduction In domain theory recursive equations have a clear semantics given by the least solution. The function assigning to every system of recursive equations the least solution has a number of equational properties. The short but informal answer to the question in the title of our paper is: iteration theories are those Lawvere theories in which recursive equations have solutions subject to all equational laws that the least-solution-map obeys in domain theory. The same question also has a long and precise answer given by a list of all the equational laws involved unfortunately, various lists are used (in the fundamental monograph of Stephen Bloom and Zoltan Ésik under Summary of the Axioms, see [], 6.8., three lists are presented, a slightly different list is used in [7], etc.). And none of the lists is easy to memorize. The aim of the present paper is to offer a short and precise answer: Support by the grant 0/06/664 of the Grant Agency of the Czech Republic. Iteration theories are precisely the algebras for the rational-tree monad on the category of signatures. To be more specific: let Sgn be the category of signatures, that is, the slice category Set. Every signature generates a free rational theory in the sense of the ADJ-group: it is the theory RT of all rational trees (which means trees having, up to isomorphism, only finitely many subtrees) on the signature!#", for a new nullary symbol!. This follows from results in [] and [8], and it yields a freerational-theory monad %$ & RT on the category Sgn of signatures. We are going to verify that the equational laws of ( satisfied by the least solution in domain theory are precisely those that all rational theories satisfy: in fact, every rational theory is proved to have an extension to a continuous theory preserving least solutions. The main result of our paper is the following Theorem. The category of iteration theories is isomorphic to the category of Eilenberg-Moore algebras for the rational-tree monad on Sgn. Let us compare this result with the classical case of algebraic theories without recursion. The canonical forgetful functor )+* Th Sgn from theories to signatures (assigning to a theory, the signature with,.-0/$03 as -ary operation symbols for 46 ) has a right adjoint 7 FT, where FT is the theory of finite terms (or of finite trees) on. This defines a monad 839;: on the category of all signatures. Jean Bénabou proved that ) is monadic, in other words, the category of algebras for the monad 839<: is isomorphic to Th, see []. We certainly do not advise the reader to start thinking about theories as algebras for 839<: but this presents a concise slogan what theories are, and by analyzing the slogan, you realize they are up to isomorphism precisely the categories with natural numbers as

2 4 objects where / /. Quite analogously, a concise slogan states that iteration theories are the algebras for the rational-tree monad. The analysis of what this actually means can be found in the monograph []. And, in fact, one half of the proof of the above theorem is also already in []: it is the fact that a free iteration theory on a signature is the rational-tree theory RT above.. Rational Theories The aim of the present section is to introduce the rational-tree monad without using the result of [] that is the monad of free iteration theories: instead we use the free rational theories of [8]. Thus, we first recall the concept of a rational theory of the ADJ group and show that every rational theory has a free completion to a continuous theory preserving solutions. It then follows that the axioms for solutions that all continuous theories satisfy are precisely those that all rational theories satisfy. As a consequence we also obtain a short proof of the fact (established in [8] but with a sketch of proof only) that free rational theories are theories RT.. Remark. Throughout the paper an (algebraic) theory is a category whose objects are natural numbers and every object is a coproduct (/ / / ( copies) with chosen coproduct injections. Theory morphisms are the functors which are identity maps on objects and which preserve finite coproducts (on the nose). The resulting category of theories is denoted by Th. Examples. () Every signature - defines the -tree theory CT as follows. By a -tree on a set of generators is meant a tree labelled so that every leaf is labelled in and every node with children is labelled in. The theory CT has as morphisms / all -trees on generators, thus: CT - $ all -tuples of -trees on generators. Composition is given by tree substitution. () Analogously, the finite- -tree theory FT is given by FT - $ all -tuples of finite -trees on generators. (3) The theory Trees are understood to be rooted, ordered, labelled trees that one considers up to isomorphism. which is the full subcategory of Set on natural numbers is initial: for every theory, we have a theory morphism, uniquely determined by assigning to the of the notation morphism - / $ the coproduct injections of, - / $. We use - $ 46, - $ for this functor, and we call the morphisms of the form basic..3 Notation. We denote by )+* Th Sgn the forgetful functor assigning to every theory, the signature ) -, whose -ary symbols are,.- /$ 3 for all 4..4 Remark. (i) ) has a left adjoint FT * Sgn Th assigning to every signature the finite-tree theory FT. This gives us a monad on the category of signatures: (ii) We denote by 839<: * ) - FT CPO the category whose objects are posets with joins of -chains (a least element is not required, if an object has it we speak about a strict CPO). Morphisms are the continuous functions; they are monotone functions preserving joins of - chains. Morphisms between strict CPO s preserving the least element are called strict continuous maps.. Definition (see [8]). () Theories enriched over Pos, the category of posets and monotone functions, are called ordered theories. These are the theories with ordered homsets such that both composition and tupling are monotone. () An ordered theory is called pointed provided that every hom-set, - $ has a least element (notation:! or just! ) and composition is left-strict, i.e.,! all 4, - $.! for (3) A continuous theory is a pointed theory enriched over the category CPO, which means that both composition and tupling preserve joins of -chains (but there is no condition on tupling concerning! )..6 Remark. () In an algebraic theory, equation morphisms are morphisms of the form *

3 !! +!! For example, if, FT then represents a recursive system of equations - $ $ $ $ $. - $ $ $ $ $ (.) where the right-hand sides are terms in for $ $ " and $ $ ". () A solution of an equation morphism 3* # is a morphism * such that the triangle commutes. In case of (.) this is a substitution of terms - for the given variables such that the formal equations of (.) become identities in FT. It is obvious that many systems (.) fail to have a solution in FT (because the obvious tree expansions are not finite). (3) In contrast, in continuous theories all equation morphisms have a solution. In fact, the least solution always exists because the endofunction $ 9! of, - $ is continuous. By Kleene s Fixed-Point Theorem, is the join of the following where! -chain of approximations: #" * and given then %$ is the morphism &)( & (.) (4) Observe that the left-hand coproduct injection in, - $ has the solution #! in every continuous theory..7 Example: the continuous theory CT. Given a signature we denote by the extension of by a nullary symbol! 4 (no ordering assumed apriori!). The theory CT of -trees, see., carries a very natural ordering: given trees and * in CT - / $ then,+-.* holds iff can be obtained from * by cutting away some subtrees and labelling the new leaves by!. And the ordering of CT - $ is componentwise..8 Example. We illustrate solutions of recursive equations in CT for the signature 0/ 7 #$ " The system of recursive equations 3 0! represents a morphism in CT -4 $6 given by the pair of trees The solution 4! CT -4 $ /% is given by the two expansions and For the left-hand tree the approximations! +!!! analogously for the right-hand tree..9 Notation. We denote by CTh are + the category of all continuous theories and strict, continuous theory morphisms. Its forgetful functor CTh Sgn is the domain restriction of ) from Notation.3..0 Theorem [8]. For every signature a free continuous theory on is CT. That is, the forgetful functor CTh Sgn has a left adjoint given by & CT.. Remark. Let, be a pointed ordered theory. For every equation morphism 3* we can form the morphisms * as in. and we clearly obtain an -chain + + / in, - $. We call these chains admissible and extend this to composites 7 for morphisms 7 *!8 : 3

4 . Definition [8]. In a pointed ordered theory, an - chain in, - 8 $ is called amissible if it has the form 7-4 for some morphisms 3* & and 7 *8. The theory, is called rational if it has joins of all admissible -chains and if tupling preserves these joins..3 Observation. () For every equation morphism 3* in a rational theory the join is a least solution of. This follows from Kleene s Fixed-Point Theorem applied to the endomorphism of, - $ given by $09. () For every * the -chains - 4 are admissible (where *. is arbitrary). In fact, let - 9 *, then. This is easy to verify by induction on. Consequently, all sequences 7 are admissible..4 Examples. () Every continuous theory is rational. () The free continuous theory CT has a nice rational subtheory: the theory RT of all rational -trees. Recall that a -tree is called rational if it has up to isomorphism only finitely many subtrees, see []. For instance all trees in Example.8 are rational. It is easy to see that RT is a pointed subtheory of CT : by substituting rational trees for (the finitely many!) variables of a rational tree one always creates a rational tree. The fact that RT is a rational theory follows from the next observation.. Observation. (a) A solution of a system (.) of recursive equations in CT where the right-hand sides $ $ are rational trees on is an -tuple of rational trees. For a precise proof see e.g. []. (b) Conversely, every rational tree 4 CT - /$ can be obtained by solving a system (.) of recursive equations in CT where the right-hand sides are finite (in fact: flat) -trees: let $ $ be all (isomorphism types of) subtrees of with recursive equations: the first one is. We obtain a canonical system of - ( $ $ ) and where the where 4 is the root label of ( children of the root from left to right are ( $ $.. Analogously, the second one is / * - ( $ $ where * 4 is the root label of / and ( $ $ are the children, etc. If the root label is one of the generators of or!, the right-hand side is simply that label. This way we obtain an 8 -tuple of finite trees on 8 generators: the generators of 4 CT -0/$ plus the new 8 $ $. In other words, we get #4 FT - 8 $8 generators such that is the first component of *!8 7/ /. Analogously, for every -tuple of rational trees 4 CT - $ one can find 4 FT - 8 $8 such that the solution * 8 fulfils. RT for some base morphism * 8 in.6 Corollary. The rational-tree theory RT is the closure of the theory FT under solutions of equation morphisms in CT. Proof. (a) RT is closed under in CT : given 4 RT - $0 then the morphism 4 CT - $ lies in RT - $ by.(a). (b) Every subtheory, of CT - $ closed under and containing all finite -trees contains all rational - trees. In fact,, - $ contains! as a solution of the lefthand coproduct injection 3*. The inclusion RT - $,.- $ now follows from.(b).!.7 Notation. We denote by RTh the category of rational theories and order-enriched strict theory morphisms preserving least solutions. That is, given rational theories, and ", a morphism is a theory morphism ##*, " which (i) is monotone and strict on $&%( -sets and (ii) fulfils #- )# - 0 for all 4, - $..8 Remark. Given a strict poset * and a set +,-*/. of -chains having joins in *, we can form a free completion of * conservative w.r.t. +. This a strict CPO *0 together with a strict order-embedding **.*0 such that (i) preserves joins of chains in + and (ii) for every strict CPO 3 and every strict monotone function *&* 43 preserving joins of chains in + there exists a unique continuous extension **60 3. In fact, * 0 can be described in the poset * 7 of all down-sets of * closed under joins of + -chains (ordered by inclusion) as the smallest sub-cpo of * 7 containing *. Given a strict poset * * and a collection + * of -chains having joins, then the product *098 -:* * ;0 is a free completion of *<8* * conservative w.r.t all -chains whose first projection lies in + and with the second one in + *. We leave the easy proof to the reader.. 4

5 .9 Free completion,-0. Let, be a rational theory. We denote by, 0 the category whose objects are natural numbers and whose $% -set, 0-8 $ is a free completion of,.- 8 $ conservative w.r.t. admissible joins. Composition is defined by extending the composition of, :,.- 8 $ 8, - $,.- 8 $ uniquely to a continuous map, 0-8+$ 8, 0 - $, 0-8 $ This is well-defined because the composition of, preserves admissible joins. (In fact, composition preserves these joins in each variable separately, thus, it preserves them jointly.) It is easy to see that,-0 is a well-defined category such that the embedding *3,, 0 is a well-defined functor. Moreover,,-0 is a continuous theory and is a morphism of rational theories. In fact,,-0 is pointed because each, 0- $ has the least element inherited from, - $ and the fact that! - *, - $, - $03 is the constant function const implies that its unique continuous extension, 0- $, 0-3$03 is also const ; from the above definition of composition in, 0 it follows that this extension is! - in, 0. To verify that, 0 has CPO-enriched binary coproducts, observe that the natural isomorphism, - $ 8, - 8 $ %,.- 8 $ expressing the universal property of 8 in, preserves admissible joins. (In fact, for a fixed morphism 4, - 8 $ the map $! *, - $, - 8+$ preserves admissible joins 7 7 because an easy computation shows that 7 $ 7 $!, since tupling is order-preserving. Analogously for $,.) We thus have a unique continuous extension, 0 - $ 8, 0-8+$ & 0, 0-8 $ (.3) It is easy to verify that that extension is a natural isomorphism and that it corresponds to the universal property of 8 from which it follows that, 0 has finite coproducts and *,, 0 is an order-enriched theory morphism..0 Example. - RT 0 CT.. Proposition. Continuous theories form a reflective subcategory of rational theories: the above theory, 0 is continuous and the embedding *3,, 0 has the universal property that for every morphism ##*,( " of rational theories, where " continuous extension is continuous, there exists a unique # *3, 0 ". Proof. (a) We already observed that, 0 is an orderenriched theory. Since the embedding, - $, 0- $ is strict, it follows that, 0 is pointed: in fact the composite! - *,-0- $, 0 - $; is the constant map of value! because it is the continuous extension of! - of,. By the definition of composition of,-0 it follows that composition is continuous, and by our argument about coproducts in,-0 above we conclude that,-0 is CPO-enriched. Therefore,, 0 is a continuous theory. (b) Let us verify the universal property of *,, 0. Given a continuous theory " and a morphism ##*, " of rational theories, each # *, - 8 $ "&- 8 $ preserves joins of admissible sequences 7 7 (for 4,.- $0 ). In fact, since #- #- #-, it is sufficient to verify that #- #- This follows by a trivial induction argument on since # preserves composition, finite coproducts and!. Consequently, we have the unique continuous extension # *, 0-8 $ " - 8 $. It is easy to verify that this # *,-0 " extending #. yields a well-defined functor This functor is CPO-enriched and strict (since # is strict). Since # preserves finite coproducts, so does # : the isomorphism (.3) yields a commutative square of natural transformations, 0 - $ 8, 0-8+$, 0-8+$ "&- $ 8 " - 8 $ "&- 8 $ which easily follows from the universal property of, 0- $ 8, 0-8 $ as a completion of, - $ 8, - 8 $. Consequently, # *, 0 " is the required extension of #. Uniqueness is obvious.!. Corollary. The equational laws for the solution function valid in all continuous theories are precisely those valid for in all rational theories..3 Proposition [8]. A free rational theory on a signature is the rational-tree theory RT. More precisely, the forgetful functor * RTh Sig (a domain restriction of ) in.3) has a left adjoint RT Proof. To give a signature morphism from to -0,&, where, is a rational theory, is equivalent to giving a strict

6 & signature morphism from to -0,, and this is equivalent to giving a strict theory morphism # * FT, It is our task to prove that # has a unique extension to a morphism # * RT, of rational theories. In fact, due to Theorem.0 we have a unique strict, continuous theory morphism # * CT, 0 such that the square FT, RT RT CT, 0 commutes. For every morphism 4 RT - $ use the presentation RT where 4 FT - 8+$8 in.(b) to conclude that # - #- 4, - $ Thus, # has a domain-codomain restriction # * RT,. This is a morphism of rational theories because solutions of morphisms 4 RT - 8 $8 are the same in RT and CT. Uniqueness: suppose * RT, is a morphism of rational theories extending #, then - - #-!.4 Corollary. The monad of free rational theories on the category Sgn is defined by & That is, to every signature - RT 3 it assigns the signature whose -ary operation symbols are the rational -trees on generators. We call the rational-tree monad. 3. Iteration Theories Here we first recall the definition of an iteration theory from [] and then prove the main result: iteration theories are algebras for the rational-tree monad. Our proof uses Beck s theorem characterizing categories of -algebras for a monad - $ $3 on a category. Recall that a -algebra is an object * of together with a morphism * * * such that 9 and. The category of -algebras and homomorphisms (defined via an obvious commutative square in ) is equipped with a forgetful functor * given by -:*$ *. This is a right adjoint, and it creates coequalizers of -split pairs. The latter means that for every parallel pairs $ 7 *. in the domain category of and every diagram in and satisfying!$# %!" 7 (i) 9 (ii) 9 (iii) 7 ( (iv) there exists a unique morphism * in the domain category with, and moreover is a coequalizer of ans 7. Beck s theorem states that monadic algebras are characterized by the above two properties of the forgetful functor. That is, whenever a functor *) + is a right adjoint creating -split coequalizers, then ) is isomorphic to for the monad given by the adjoint situation of. See [6] for a proof. 3. Definition. An iteration theory is a theory, together with a function assigning to every ( equation ) morphism 3* a morphism * in such a way that the following five axioms hold: () Fixed Point Identity. This states that is a solution of, i.e., a fixed point of $09 : (3.) () Parameter Identity. We use the following notation: given an equation morphism 3* 6, then every morphism * yields a new equation morphism $0 -, /. (3.) The parameter identity tells us how the solutions of and -, are related: the triangle 6

7 (3.3) commutes. (3) Simplified Composition Identiy. We use the following notation: given morphisms : ; and we obtain an equation morphism. 8 ; 8 $ % 8 (3.4) The simplified composition identity states that the triangle (3.) commutes. (4) Double Dagger Identity. This is a statement about morphisms of the form 3* A solution yields * which we can solve again and get - 0 * &. On the other hand, the codiagonal * &6 yields an equation morphism 3* 6. The double-dagger identity states - * (3.6) In other words, the square, - 8 $0 6,.- $0 $, - $0, - $ (3.7) commutes. () Commutative identity. This is in fact an infinite set of identities: one for every 8 -tuple of base endomorphisms of 8 : $ $ $8 and for every decomposition 8 such that the corresponding codiagonal * in fulfils for $ $8 /. The commutative identity concerns an arbitrary morphism * 8 in,. We can form two equation morphisms: (see (3.4)) and *!8 8 defined by the individual components 7 * 6( in * / 8 for $ $8< &/ as follows: 7 in. in $ / & The conclusion is that the triangle 8 (3.8) (3.9) commutes. (Remark: the notation in [] for 7 is $ $ and instead of a general surjective base morphism is assumed. The simplification working with was proved in [4].) 3. Definition. Let -0, $; and -"! $# be iteration theories. A theory morphism ##*,! is said to preserve solutions if for every morphism 4, - $ + we have #- %$ )#-. The category of iteration theories and solution-preserving morphisms is denoted by We denote by &* ITh ITh Sgn the canonical forgetful functor (a restriction of ) in.3). 3.3 Example. The rational-tree theory RT is an iterative theory (for the choice the least solution of ). In fact, as proved in [], Theorem 6.., this is a free iteration theory on. In other words: 3.4 Theorem []. The forgetful functor &6* ITh Sgn is a right adjoint and the corresponding monad is the rationaltree monad. 3. Theorem. Iteration theories are precisely the algebras for the rarional-tree monad on Sgn. That is, the forgetful functor &* ITh Sgn is monadic. Proof. We are going to use Beck s theorem; due to 3.4 it Sgn creates coequal- is sufficient to verify that &* ITh izers of & -split pairs. From the result of Bénabou mentioned in the Introduction we know that )+* Th Sgn is 7

8 " # " & " & % - monadic, thus, it creates coequalizers of ) -split pairs. Consequently, our task is the following: given a parallel pair of solution-preserving morphisms $ 7 * -0,$# -"! $ in ITh, and given a split coequalizer # %,! " in Sgn, where is a coequalizer of and 7 in Th and the above equations (i) (iv) hold, then there exists a unique function * "&- $ "&- $ (for all $ 4 ) such that (a) is solution preserving: 7-0 for all 4! - $0, (3.0) (b) the axioms of Definition 3. hold for, and (c) is a coequalizer of and 7 in ITh. In fact, (a) determines as follows: 0 - for all 4 "&- $. (3.) To see this, put, then due to (ii), thus Conversely, by using (3.) we get (3.0) for every morphism 4! - $0-0 - (3.) - 7 (iv) 7 - $ 7 in ITh - $ (i) -: in ITh (iii) We now prove the axioms of iteration theories for and then we will get immediately (c): Suppose that with,! " % * -! $; - " $ 7. We have a unique theory morphism * " " with in Th and we only need to prove that is a morphism of ITh for all 4 "&- $. (3.) In fact, since is a morphism of ITh we have by (3.) This follows from being and it remains to verify an epimorphism since from (iv) and (iii) we get ( 7 Consequently, the theorem will be proved by verifying the individual axioms of iteration theories for the function from (3.) above. Observe that since is a theory morphism, it preserves, see (3.4): (3.3) () Fixed Point Identity. Given 4 "&- $0, we have, since preserves coproducts: 0 - (3.) - $09 (3.) - $09 (ii) 0 $ 9 (3.) () Parameter Identity. Given 4 " - $0+ and 4 "&- $, then (ii) implies, since preserves finite coproducts, the equality Therefore -, , (3.4) -, 0 - -, (3.) and (3.4) ( - -, 0 (3.0) - -, 0 (ii) -, (3.0) - (3.3) - (ii) 0 (3.). (3) Simplified Composition Identity. Given morphisms 4 "&- 8 $0 and 4 "&- $8, we have (3.0) - - (ii) - (3.) (3.0) 7-0 (ii). (ii) and (3.3) 8

9 7 (4) Double Dagger Identity. Given 4 "&- $0 #, since - (recall that preserves finite coproducts), we have (3.) - 0 (3.0) - (3.6) - 0 (3.0) and (3.3) - 0 (ii) and. () Commutative Identity. Given 4-8 $8 and 4 "&- $ then first observe - 7 (3.) In fact, preserves coproducts and thus it maps base morphisms (in,, etc.) of! to the corresponding base morphisms of ". Thus (3.) follows from (3.8). Consequently: 0-0 (3.) - (3.0) - (3.9) (3.0) 7-0 (i) and (3.3) This completes the proof.! References [] J. Bénabou, Structures algébriques dans les catégories, Cah. Topol. Géom. Différ. Catég. 0 (968), 6. [] S. L. Bloom and Z. Ésik, Iteration Theories, Springer Verlag, 993. [3] S. L. Bloom and Z. Ésik, Fixed-point operations on ccc s, Part I, Theoret. Comput. Sci. (996), 38. [4] Z. Ésik, Axiomatizing iteration categories, Acta Cybernetica 4 (999), 6 8. [] S. Ginali, Regular trees and the free iterative theory, J. Comput. Syst. Sci. 8 (979), 8 4. [6] S. MacLane, Categories for the Working Mathematician, second ed., Springer Verlag, 998. [7] A. Simpson and G. Plotkin, Complete axioms for categorical fixed-point operators, IEEE Symposium Logic in Computer Science 998, [8] J. B. Wright, J. W. Thatcher, E. G. Wagner and J. A. Goguen, Rational algebraic theories and fixed-point solutions, Proc. 7th IEEE Symposium on Foundations of Computing, Houston, Texas, 976, Conclusions and Future Research The goal of our paper was to prove that iteration theories of Stephen Bloom and Zoltan Ésik are monadic over the category signatures. This provides the possibility of using the corresponding monad (of rational tree signatures) as a means for defining iteration theories. More important is the way our results supports the claim that iteration theories precisely sum up the equational properties that the dagger function, assigning to every equation morphism its least solution, satisfies in all continuous theories. In fact, since is the monad of free rational theories, see [8], and every rational theory has a solution-preserving completion to a continuous theory, it is obvious that all continuous theories and all rational theories satisfy precisely the same equational laws for. In the future we intend to study the analogous question where the base category is, in lieu of Sgn, the category of all finitary endofunctors of Set. We hope that the corresponding monadic algebras will turn out to be precisely the iteration theories that are parametrically uniform in the sense of Simpson and Plotkin [7]. 9

What are Iteration Theories?

What are Iteration Theories? What are Iteration Theories? Jiří Adámek 1, Stefan Milius 1 and Jiří Velebil 2 1 Institute of Theoretical Computer Science, TU Braunschweig, Germany {adamek,milius}@iti.cs.tu-bs.de 2 Department of Mathematics,

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