RESEARCH STATEMENT My current research interests lie in ...

RESEARCH STATEMENTSTEFAN MILIUSMy current research interests lie in category theory and its applications to theoretical computer science.In particular, the theory of coalgebras and its application to the semantics of recursion have been a majorfocus of my work. I will first give some background and then describe those of my recent results on whichmy doctoral thesis is based. In the third part I will formulate goals for future research.1. BackgroundRecursion arises everywhere in science. It allows for a clear, concise and finite description of complicatedobjects and phenomena. One example is the usual recursive definition of the factorial function; the goldenratio ϕ is the unique positive real number satisfying the recursive equationϕ = 1 + 1 ϕ ;and the famous Cantor space is the unique non-empty closed subset of the interval [0, 1] such that theequationc = 1 3 c + ( 23 + 1 3 c )holds, where we write 1 3 c to mean { 1 3 x | x ∈ c }.In computer science, recursion is a major ingredient of every programming language or formalism describingthe work of an information system. For example, a programmer might write a statement likefac(n) := if n = 0 then 1 else n ∗ fac(n − 1) (1)to program the factorial function. In process algebra one uses recursion to specify repeated or potentiallyinfinite system behaviour by finite means. A very basic example is a process termP := a.P (2)describing a system which can perform the action a repeatedly without ever terminating.Every recursive definition immediately raises the question about its semantics, i. e., what its meaningshould be. Mathematics provided structures and results to answer such questions.In the literature there are several approaches to the semantics of recursion. The starting point of ourresearch is the classical work of Elgot and his collaborators [19, 16, 20]. Their aim was to study the semanticsof recursion at a purely algebraic level. In [19] Elgot introduced iterative theories, which are algebraic theoriesin the sense of Lawvere [29] with the property that certain finite systems of mutually recursive equationshave unique solution. Bloom and Elgot [16] proved that for every signature Σ of operation symbols a freeiterative theory exists, and in [20] Elgot et al. showed that free iterative theories are formed by all rationalΣ-trees, i. e., Σ-trees with finitely many different subtrees; this description was provided by Ginali [22].The original proof of Elgot et al. is quite involved and it is spread out over more than a hundred pagesin the three papers [19, 16, 20]. So it was a great step forward when Nelson [41] and Tiuryn [43] introducediterative algebras for a signature to obtain a more easy approach to iterative theories. In fact, Nelson provedthat for every set X of generators a free iterative algebra on X is given by rational Σ-trees over X, andmoreover, the algebraic theory of free iterative algebras yields the free iterative theory of Elgot.Similar to iterative theories there are completely iterative theories in which unique solutions are requiredto exists for non-finite recursive equations as well. A free completely iterative theory on a signature Σ is thetheory of all (finite and infinite) Σ-trees, see [20].(Completely) iterative theories do not capture recursive function definitions such as the definition of thefactorial function. This is the topic of algebraic semantics, see e. g. the book of Guessarian [23]. Againinfinite trees for a signature play a major rôle in the theory of algebraic semantics.1

2 STEFAN MILIUS2. Recent Research2.1. Completely Iterative Algebras. My paper [34] concerns a generalization and extension of the classicalwork of Elgot and Nelson using ideas from the theory of coalgebras, see e. g. [2, 24, 27, 42] for introductorytexts. The basic observation is that infinite Σ-trees form the terminal coalgebra for the polynomial endofunctorassociated to the signature Σ. Then one can prove the classical results using category-theoreticmethods, i. e., the universal property of a terminal coalgebra is sufficient to obtain those result.More precisely, in lieu of a signature one works with an endofunctor H of Set (or a more general categorysatisfying some rather mild side conditions). Then a (co-)algebra is an underlying set A together with a mapfrom HA to A (or from A to HA, respectively). I introduce the concept of completely iterative algebras(cia). For a signature Σ the algebra T Σ X of all Σ-trees over the set X of generators is completely iterative.Other important examples stem from the realm of algebras on complete metric spaces. We already saw thegolden ratio and the Cantor space as unique solutions of recursive equations. The respective cias are carriedby the interval [1, 2] and by the set of non-empty closed subsets of [0, 1] with the Hausdorff metric. A basicopening result of the theory of cias is theTheorem 2.1. [34] A free cia on a set X is precisely a terminal coalgebra for the functor H( ) + X.The importance of this result, bringing algebraic and coalgebraic methods under one roof, lies in applications:terminal coalgebras often serve as domains for semantics of recursive definition; infinite trees, streams,formal languages accepted by sequential automata form terminal coalgebras. The characterization of terminalcoalgebras as free algebras of some sort opens the door for coalgebraic tools in algebraic semantics.And so we work with endofunctors that have enough terminal coalgebras, i. e., for every set X there exists aterminal coalgebra T X for H( ) + X. This is a mild assumption. For example, every accessible endofunctorof Makkai and Paré [30] has enough terminal coalgebras.Theorem 2.2. [1, 34] Let H have enough terminal coalgebras T X. Then T is the object assignment of amonad, and this monad is the free completely iterative monad on H.This result is an analogue of Nelson’s result on iterative algebras; the above theorem states that free ciasyield a free completely iterative theory. At the same time our result is more general and we feel that ourproof is clearer using category-theoretic language and tools. For the category-theoretic semantics of recursivefunction definitions Theorem 2.2 is an important prerequisite. In the special case of a polynomial endofunctorarising from a signature, the universal property of the monad T captures second-order substitution, i. e.,substitution of infinite trees for operation symbols. This notion is necessary to formulate the notion of asolution of a recursive function definition in the appropriate generality. We believe that this is the first timesecond-order substitution is understood by a categorical universal property.2.2. Iterative Algebras. In joint work with Adámek and Velebil [4, 5, 6] we deal with iterative algebrasand iterative theories. Recall that here we are concerned with unique solutions of finite systems of recursiveequations. Hence, the suitable category-theoretic environment for this study are locally finitely presentablecategories of Gabriel and Ulmer [21], e. g., sets, posets, graphs, finitary varieties of algebras, etc. In this settingwe introduce iterative algebras for an endofunctor. All completely iterative algebras for an endofunctorare, of course, iterative. Rational trees for a signature Σ form an iterative algebra which is not completelyiterative (for the polynomial endofunctor arising from Σ).Theorem 2.3. [6] For a finitary endofunctor H a free iterative algebra RX on an object X is obtained asthe colimit of all coalgebras for H( ) + X with a finitely presentable carrier.The existence of free iterative algebras is easy to prove from general category-theoretic results. However,the coalgebraic construction is a non-trivial and crucial result. All our subsequent results on iterativity relyon it. Furthermore, it yields explicit descriptions of free iterative algebras. For example, the constructionexplains that for a signature Σ free iterative algebras are rational Σ-trees as expected. The main resultof [4, 6] isTheorem 2.4. For every finitary endofunctor of a locally finitely presentable category the monad of freeiterative algebras is a free iterative monad.

RESEARCH STATEMENT 3Again, in the special case of a polynomial endofunctor of Set arising from a signature the above Theoremyields Nelson’s classical result. Our proof is much shorter then Elgot’s original proof and, moreover, and isconceptually much clearer than any of the previous proofs.2.3. Elgot Algebras. In another joint paper [7] with Adámek and Velebil we characterize the Eilenberg-Moore algebras of the free iterative monads and of the free completely iterative monads. For an endofunctorof a category with finite coproducts we introduce complete Elgot algebras: they are algebras for the functortogether with a (non-unique) choice of solutions of every recursive equation system of a certain very simpleform. This choice of solutions has to satisfy two simple and well-motivated axioms.Theorem 2.5. Let H be an endofunctor with enough terminal coalgebras. Then the category of completeElgot algebras is isomorphic to the category of Eilenberg-Moore algebras for the free completely iterativemonad on H.Similarly, we introduce (non-complete) Elgot algebras; they come with a choice of solutions of finitarysystems of equations. The choice of solutions is required to satisfy the same two axioms.Theorem 2.6. Let H be a finitary endofunctor of a locally finitely presentable category. Then the categoryof Elgot algebras for H is isomorphic to the category of Eilenberg-Moore algebras for the free iterative monadon H.These two results are an important tool in applications. It turns out that algebras used in classical algebraicapproaches to the semantics of recursion are complete Elgot algebras. For example, continuous algebrasfor a signature, which are carried by a complete partial order and where operations are continuous maps,are complete Elgot algebras. And so are algebras on complete metric spaces with contracting operations.In fact, all (completely) iterative algebras are (complete) Elgot algebras. So on the one hand, the notionof (complete) Elgot algebra builds a common roof over various notions of algebras used in semantics. Onthe other hand, complete Elgot algebras have enough structure to provide semantics of recursive functiondefinitions. This is the topic of the next section.2.4. Recursive Program Schemes. In the joint paper [38] with Moss we turn more to applications of ourmathematical results in semantics. We have started to rework the classical theory of algebraic semantics,see e. g. [23], using our category-theoretic machinery. In lieu of sets, signatures and infinite trees we workwith abstract categories and endofunctors having enough terminal coalgebras. In this setting we show howto formulate recursive program schemes and their uninterpreted and interpreted solutions. Our results mainresults state for every guarded 1 recursive program scheme:(a) A unique uninterpreted solution exists.(b) In every complete Elgot algebra there exists a canonical interpreted solution.(c) In every completely iterative algebra a unique interpreted solution exists.(d) The uninterpreted solution of (a) and the interpreted ones of (b) and (c) are consistent.These results generalize all previous work on the topic. Moreover, we can provide semantics of recursivefunction definitions which are not captured by the classical theory. For example, recursive program schemesdefining operations satisfying equations, e. g. commutativity. From our results on interpreted schemes onecan recover the usual denotational semantics using continuous algebras. Working with algebras on completemetric spaces yields new applications which had so far not been recognized as solutions of recursive programschemes, e. g., fractals like the Cantor space. These applications show that we have obtained a unified viewof solutions principles for a large class of recursive definitions including the usual algebraic semantics ofrecursive program schemes. Of course, our more abstract theory takes some effort to build. But we feel thatthe gain in conceptual clarity more than outweighs this effort. Our work unveils the basic mechanisms atwork in the semantics of recursion.3. Future ResearchThere are several topics that I consider very fruitful for future research. Some of them have already beenpartially attended to by our research group.1 Guardedness is an easy syntactic condition of a recursive program scheme inspired by a similar condition in the classicalwork called Greibach normal form.

4 STEFAN MILIUS3.1. Iterative Algebras for a Base. One line of our present research has been inspired by the workof Uustalu [44]. He proposed to study complete iterativity with respect to a so-called base in lieu of anendofunctor. A base is a functor from the category A to the category of (finitary) monads on A. Bases allowan interesting extension of our theory which captures algebras satisfying certain equations and where theiterativity can be restricted. In fact, we have taken up the task to extend the results on iterative algebrasand iterative monads of [4, 6] to bases with the series of papers [8, 9, 10, 11, 12] most of which are still inpreparation.Research Problem 3.1. Investigate completely iterative algebras for a base and extend our results of [1, 34]to this setting.3.2. Algebraic Trees. An important application of the theory of iterativity for a base is presented in [13]:a category theoretic description of the monad of algebraic trees. Classically, algebraic trees over a signatureΣ are those trees arising as the uninterpreted solutions of all recursive program schemes with the givens fromΣ, see [18]. We have shown how to obtain the monad of algebraic trees from the free iterative monad for abase on the category of finitary endofunctors of Set.Research Problem 3.2.(i) Formulate and establish the following properties of algebraic trees in the category-theoretic setting:(a) algebraic trees form an iterative theory;(b) algebraic trees are closed under second-order substitution;(c) algebraic trees are rational trees over a signature with additional explicit substitution operations.(ii) Find a universal property that characterizes the monad of algebraic trees up to isomorphism.In contrast to all Σ-trees and all rational Σ-trees, respectively, a universal property is not known foralgebraic trees in the classical setting. So it would be a great success of the category-theoretic approach toprovide such a result.3.3. Iterative Reflection of a Monad. Our previous work shows how to obtain a free iterative monad ona finitary endofunctor of a locally finitely presentable category. How about starting with a finitary monad?Research Problem 3.3. Does there exist an iterative reflection of a given finitary monad of a locally finitelypresentable category?This would yield another important new result in the classical setting; an iterative theory free on (thetheory given by) a signature and a finite set of equations involving operation symbols of that signature.I have a partial result [36] showing that every ideal finitary monad of Set has an iterative reflection. Theconcept of an ideal theory was introduced by Elgot in order to study iterativity, see [19]. Ideal monads forma subcategory of all finitary monads of Set. Although the proof of the partial result provides a coalgebraicconstruction one does not obtain an explicit description of the iterative reflection from that. But sucha description would be highly desirable in order to obtain further examples of iterative theories and tounderstand whether the partial result can be extended beyond Set.Research Problem 3.4. Give an explicit description of an iterative reflection of an ideal finitary monadof Set.For free iterative monads on endofunctors of Set an explicit description is provided by Adámek and myselfin [3].Similar problems as above should also be addressed in the completely iterative case. Among expectedapplications of the theory developed to answer those problems are a semantics of recursive program schemesusing operations satisfying equations.3.4. More on Recursive Program Schemes. Our work on solutions of recursive program schemes wasjust the beginning of a long term project to rework algebraic semantics with category-theoretic methods.The next steps include:(a) Find other applications. For example, working with the power-set functor we obtain applicationsin the theory of hypersets of Barwise and Moss [15]. Other possible applications lie in the realm ofprocess calculi, e. g. Milner’s CCS [39].

RESEARCH STATEMENT 5(b) Investigate higher-order recursive program schemes defining functions on function spaces. Matthesand Uustalu [31] have addressed the extremely useful concept of variable binding and this may wellbe relevant in this respect.(c) Study the equivalence of interpreted program schemes. One of the main goals of the original theorywas to provide foundations for the study of program equivalence. Since on the natural numbers theequivalence of recursive program schemes is undecidable one works with classes of interpretations.These classes are defined on ordered algebras, see [23]. Do complete Elgot algebras suggest tractableclasses of interpretations?3.5. Logical Calculi. For a variety of algebras defined by a signature and a finite set of equations there isthe calculus of equational logic, which is sound and complete.Research Problem 3.5. Investigate the equational logic of (completely) iterative algebras and (complete)Elgot algebras!Work on the formal language of recursion by Hurkens et al. [26] and on the equational logic of terminalcoalgebras for polynomial endofunctors of Set by Moss [40] are the starting point in this direction.Mersch [32] has studied a sound and complete logical calculus for uninterpreted recursive program schemesw. r. t. polynomial endofunctors of Set.Research Problem 3.6. Develop a similar calculus for arbitrary endofunctors (of other base categoriesthan Set)!3.6. Relation to Other Structures. I will soon investigate the connection of (completely) iterative monadsto other structures typically used in semantics.Research Problems 3.7.(i) Investigate equational properties of unique solutions in iterative monads and prove that the Kleislicategory of an iterative monad is a traced monoidal category of Joyal, Street and Verity [28], seealso [25]. Does this property of the Kleisli category yield an equivalent description of iterativemonads?(ii) For a locally finitely presentable category A the assignment of a free iterative monad to a finitaryendofunctor yields itself a monad on the category of finitary endofunctors of A. Characterize theEilenberg-Moore algebras of this monad!My preliminary work [37] indicates that the Eilenberg-Moore algebras in (ii) are closely related to iterationtheories of Bloom and Ésik [17].In a first serious step [14] towards the solution of the above problems we have shown that in (completely)iterative theories all recursive equations can be given a canonical solution.3.7. Other Topics. The intended future research presented so far is concentrated on results of my doctoralthesis and other recent previous work. But I would also like to follow different topics in future research.One such topic could arise by revisiting results from my Master’s thesis [33]. There first steps into theinvestigation of factorization systems in 2-categories were made. I hope to find the opportunity to comeback to that work and develop that theory further.References[1] Peter Aczel, Jiří Adámek, Stefan Milius and Jiří Velebil, Infinite Trees and Completely Iterative Theories: A CoalgebraicView, Theoret. Comput. Sci. 300 (2003), 1–45.[2] Jiří Adámek, Introduction the Coalgebra, Theory Appl. Categ. 14 (2005), 157–199.[3] Jiří Adámek and Stefan Milius, Terminal Coalgebras and Free Iterative Theories, accepted for publication in Inform. andComput., available via http://www.iti.cs.tu-bs.de/~milius.[4] Jiří Adámek, Stefan Milius and Jiří Velebil, Free Iterative Theories: a coalgebraic view, Math. Structures Comput. Sci. 13(2003), no. 2, 259–320.[5] Jiří Adámek, Stefan Milius and Jiří Velebil, On Rational Monads and Free Iterative Theories, Electron. Notes Theor. Comput.Sci. 69 (2002), 24 pp.[6] Jiří Adámek, Stefan Milius and Jiří Velebil, From Iterative Algebras to Iterative Theories, submitted, available viahttp://www.iti.cs.tu-bs.de/~milius, extended abstract appeared in Electron. Notes Theor. Comput. Sci. 106 (2004),3–24.

6 STEFAN MILIUS[7] Jiří Adámek, Stefan Milius and Jiří Velebil, Elgot Algebras, preprint (2005), available viahttp://www.iti.cs.tu-bs.de/~milius, extended abstract to appear in Electron. Notes Theor. Comput. Sci.[8] Jiří Adámek, Stefan Milius and Jiří Velebil, Algebras with Parameterized Iterativity, submitted, available viahttp://www.iti.cs.tu-bs.de/~milius.[9] Jiří Adámek, Stefan Milius and Jiří Velebil, Bases for Parameterized Iterativity, preprint (2005), available viahttp://www.iti.cs.tu-bs.de/~milius.[10] Jiří Adámek, Stefan Milius and Jiří Velebil, Iterative Algebras for a Base, Electron. Notes Theor. Comput. Sci. 122 (2005),147–170.[11] Jiří Adámek, Stefan Milius and Jiří Velebil, Base Modules for Parameterized Iterativity, preprint (2005).[12] Jiří Adámek, Stefan Milius and Jiří Velebil, Description of Bases for Parameterized Iterativity, preprint (2005).[13] Jiří Adámek, Stefan Milius and Jiří Velebil, Algebraic Trees Coalgebraically, manuscript (2005).[14] Jiří Adámek, Stefan Milius and Jiří Velebil, How iterative are Iterative Algebras?, submitted.[15] Jon Barwise and Lawrence S. Moss, Vicious Circles, CSLI Publications, Stanford, 1996.[16] Stephen L. Bloom and Calvin C. Elgot, The Existence and Construction of Free Iterative Theories, J. Comput. SystemSci. 12 (1974), 305–318.[17] Stephen L. Bloom and Zoltán Ésik, Iteration Theories: The equational logic of iterative processes, EATCS Monographson Theoretical Computer Science, Berlin: Springer-Verlag (1993).[18] Bruno Courcelle, Fundamental properties of infinite trees, Theoret. Comput. Sci. 25 (1983), no. 2, 95–169.[19] Calvin C. Elgot, Monadic Computation and Iterative Algebraic Theories, in: Logic Colloquium ’73 (eds: H. E. Rose andJ. C. Shepherdson), North-Holland Publishers, Amsterdam, 1975.[20] Calvin C. Elgot, Stephen L. Bloom and Ralph Tindell, On the Algebraic Structure of Rooted Trees, J. Comput. SystemSci. 16 (1978), 361–399.[21] Peter Gabriel and Friedrich Ulmer, Lokal präsentierbare Kategorien, Lecture N. Math. 221, Springer-Verlag, Berlin 1971.[22] Susanna Ginali, Regular trees and the free iterative theory, J. Comput. System Sci. 18 (1979), 228–242.[23] Irène Guessarian, Algebraic Semantics. Lecture Notes in Comput. Sci. 99, Springer, 1981.[24] Hans Peter Gumm, Elements of the General Theory of Coalgebras, LUATCS’99, Rand Africaans University, Johannesburg,South Africa, 1999.[25] Masahito Hasegawa, Recursion from Cyclic Sharing: Traced Monoidal Categories and Models of Cyclic Lambda Calculi,Proc. 3rd International Conference on Typed Lambda Calculi and Applications, Lecture Notes in Comput. Sci. 1210,196–213 Springer-Verlag, Berlin, 1997.[26] A. J. C. Hurkens, Monica McArthur, Yannis N. Moschovakis, Lawrence S. Moss and Glen T. Whitney, The logic of recursiveequations, J. Symbolic Logic 63 (1998), no. 2, 451–478.[27] Bart Jacobs and Jan J. M. M. Rutten, A Tutorial on (Co)Algebras and (Co)Induction, Bull. Eur. Assoc. Theor. Comput.Sci. EATCS 62 (1997), 222–259.[28] André Joyal, Ross Street and Dominic Verity, Traced Monoidal Categories, Math. Proc. Cambridge Philos. Soc. 119 (1996),no. 3, 447–468.[29] F. William Lawvere, Functorial Semantics of Algebraic Theories, PhD thesis, Columbia University, 1963.[30] Michael Makkai and Robert Paré, Accessible categories: the foundation of categorical model theory, Contemp. Math. 104(1989), Amer. Math. Soc., Providence, RI.[31] Ralph Matthes and Tarmo Uustalu, Substitution in Non-Wellfounded Syntax with Variable Binding, Electron.Notes Theor. Comput. Sci. 82 (2003), no. 1, 15 pp.[32] Jay Mersch, Equational Logic of Recursive Program Schemes, in Proceedings of the 1st International Conference on Algebraand Coalgebra in Computer Science (CALCO 2005), Lecture Notes in Comput. Sci. 3629 (2005), 293–312.[33] Stefan Milius, Relations in Categories, master’s thesis, York University, 2000.[34] Stefan Milius, Completely Iterative Algebras and Completely Iterative Monads, Inform. and Comput. 196 (2005), 1–41.[35] Stefan Milius, On Iteratable Endofunctors, Electron. Notes Theor. Comput. Sci. 69 (2002), 18 pp.[36] Stefan Milius, The Iterative Reflection of an Ideal Monad, manuscript (2005).[37] Stefan Milius, Elgot monads, manuscript (2005).[38] Stefan Milius and Lawrence S. Moss, The Category Theoretic Solution of Recursive Program Schemes, extendedabstract, in Proceedings of the 1st International Conference on Algebra and Coalgebra in Computer Science(CALCO 2005), Lecture Notes in Comput. Sci. 3629 (2005), 293–312, full version submitted and available viahttp://www.iti.cs.tu-bs.de/~milius.[39] Robin Milner, Communication and Concurrency, Prentice Hall, 1989.[40] Lawrence S. Moss, Recursion and corecursion have the same equational logic, Theoret. Comput. Sci. 294 (2003), 233–267.[41] Evelyn Nelson, Iterative Algebras, Theoret. Comput. Sci. 25 (1983), 67–94.[42] Jan J. M. M. Rutten, Universal coalgebra, a theory of systems, Theoret. Comput. Sci. 249 (2000), no. 1, 3–80.[43] Jerzy Tiuryn, Unique Fixed Points vs. Least Fixed Points, Theoret. Comput. Sci. 12 (1980), 229–254.[44] Tarmo Uustalu, Generalizing substitution, Theor. Inform. Appl. 37 (2003), no. 4, 315–336.