1. Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Marcello Bonsangue, Stefan Milius, Alexandra Silva Coalgebras and Generalized Regular Expressions

2. IFIP WG 1.3 2013 | Stefan Milius | March 16 | S. 2 Regular Expressions S. Kleene (1956): regular expressions are equivalent to deterministic automata D. Kozen (1994): Kleene-Algebras axiomatize the equivalence of regular expressions. Syntatic description of regular languages: S. C. Kleene: Representation of events in nerve nets and finite automata, Automata Studies 1956 D. Kozen: A completeness theorem for Kleene algebras and the algebra of regular events, I&C 1994 Problem:

3. IFIP WG 1.3 2013 | Stefan Milius | March 16 | S. 3 Does this work for other systems (i.e. coalgebras) in a generic way? What does „regular language“ mean? Calculus of „regular expressions“ for coalgebras? Syntax and semantics? Correct and complete axiomatization? Decidability?

4. IFIP WG 1.3 2013 | Stefan Milius | March 16 | S. 4 Does this work for other systems (i.e.coalgebras) in a generic way? What does „regular language“ mean? Calculus of „regular expressions“ for coalgebras? Syntax and semantics? Correct and complete axiomatization? Decidability?

5. IFIP WG 1.3 2013 | Stefan Milius | March 16 | S. 5 Rational Fixpoint = „Regular Languages“ for Coalgebras Given. Construction. Theorems. 1.+2.: J. Adamek, S. Milius, J. Velebil: Iterative Algebras at Work, MSCS 2006 3.: S. Milius, A sound and complete calculus for finite stream circuits, Proc. LICS 2010.

6. IFIP WG 1.3 2013 | Stefan Milius | March 16 | S. 6 Examples

7. IFIP WG 1.3 2013 | Stefan Milius | March 16 | S. 7 Does this work for coalgebras in a generic way? What does „regular language“ mean? Calculus of „regular expressions“ for coalgebras? Syntax and semantics? Correct and complete axiomatization? Decidability?

8. IFIP WG 1.3 2013 | Stefan Milius | March 16 | S. 8 Expression calculi for bisimilarity M. Bonsangue, J. Rutten, A. Silva et al.:  Regular expressions for coalgebras for set functors F from an inductively defined class  „Kleene Theorem“  Correctness and completeness for F-behavioral equivalence  Decidability  Applications:  regular expressions  Milner‘s calculus for finite state processes  Simple Segala Systems  New calculi for behavioral equivalence of: (1) weighted automata; (2) stratified systems; (3) Pnüeli-Zuck-Systems A. Silva, M. Bonsangue, J. Rutten: Non-deterministic Kleene Coalgebras, LMCS 2010. F. Bonchi, M. Bonsangue, J. Rutten, A. Silva: Quantitative Kleene Coalgebras, I&C 2011. M. Bonsangue, G. Caltais, E.-I. Goriac, D. Lucanu, J. Rutten, A. Silva: A decision procedure for bisimilarity of generalized regular expressions, SBMF 2010. ♥

9. IFIP WG 1.3 2013 | Stefan Milius | March 16 | S. 9 But what about language equivalence? Example. R. Milner‘s calculus for finite state processes R. Milner: A complete inference system for a class of regular behaviours, J. Comput. Syst. Sci., 1984.

10. IFIP WG 1.3 2013 | Stefan Milius | March 16 | S. 10 But what about language equivalence? Example. R. Milner‘s calculus for finite state processes Theorem. Axioms 1.-4. are sound and complete for bisimilarity. A. Rabinovich: Theorem. Axioms 1.-5. are sound and complete for trace-congruence. R. Milner: A complete inference system for a class of regular behaviours, J. Comput. Syst. Sci., 1984. A. Rabinovich: A complete axiomatization for trace congruence of finite state behaviors, Proc. MFPS, 1994.

11. IFIP WG 1.3 2013 | Stefan Milius | March 16 | S. 11 Does this work for coalgebras in general? What does „language equivalence“ mean for coalgebras?

12. IFIP WG 1.3 2013 | Stefan Milius | March 16 | S. 12 1. Coalgebraic Trace Semantics I. Hasuo, B. Jacobs, A. Sokolova: Generic trace semantics via coinduction, LMCS, 2007. Applications:  labelled transition systems  probabilistic transition systems  contextfree grammars Difficulties:  weighted systems  probabilistic automata

13. IFIP WG 1.3 2013 | Stefan Milius | March 16 | S. 13 Towards language equivalence of coalgebras Examples. (1) nondeterministic automata But The nondeterministic/weighted branching does not occur in the desired final coalgebra. Observation. (2) weighted automata M. P. Schützenberger: On the definition of a family of automata, I & C, 1961 semiring But

14. IFIP WG 1.3 2013 | Stefan Milius | March 16 | S. 14 Coalgebraic Language Equivalence Two ideas are combined: 1. Generalized powerset construction: Definition (language equivalence). Theorem. Silva, Bonchi, Bonsangue, Rutten: Generalizing the powerset construction, coalgebraically, Proc. FSTTCS 2010.

15. IFIP WG 1.3 2013 | Stefan Milius | March 16 | S. 15 Coalgebraic Language Equivalene Two ideas are combined: 1.Generalized powerset construction 2.Final coalgebras and ½ F in algebraic categories S. Milius, A sound and complete calculus for finite stream circuits, Proc. LICS 2010. Example.correct and complete calculus for linear circuits

16. IFIP WG 1.3 2013 | Stefan Milius | March 16 | S. 16 Relating final coalgebras and rational fixpoints M. Bonsangue, S. Milius, S. Silva: Sound and complete axiomatizations of coalgebraic language equivalence, ACM ToCL, 2013. Assumptions. Bisimilarity and language equivalence: finite

17. IFIP WG 1.3 2013 | Stefan Milius | March 16 | S. 17 Relating final coalgebras and rational fixpoints M. Bonsangue, S. Milius, A. Silva: Sound and complete axiomatizations of coalgebraic language equivalence, ACM ToCL, 2013. Assumptions. Bisimilarity and language equivalence: finite

18. IFIP WG 1.3 2013 | Stefan Milius | March 16 | S. 18 What is this good for? Adding axioms a la Rabinovich is possible for FT-coalgebras such that … ! Generic tools: abstract Kleene Theorem & Soundness + Completeness Theorems Application: Weighted Automata  Concrete expression syntax  Kleene Theorem  New correct+complete axiomatization of weighted language equivalence  Special cases: 1. Rabinovich‘s result (for nondeterministic automata) 2. Calculus for linear circuits Independent and almost at the same time: algebraic characterization of rational power series M. Bonsangue, S. Milius, A. Silva: Sound and complete axiomatizations of coalgebraic language equivalence, ACM ToCL, 2013. Z. Esik, W. Kuich: Free iterative and iteration K-semialgebras, Algebra Univ., 2012.

19. IFIP WG 1.3 2013 | Stefan Milius | March 16 | S. 19 Example Koalgebren und die Axiome von Iteration | Dr. S. Milius | 24. Januar 2013 19

20. IFIP WG 1.3 2013 | Stefan Milius | March 16 | S. 20 Conclusions + Future Work Rational fixpoints characterize finite state behavior of coalgebras Method+Framework for sound and complete generalized regular expression calculi for coalgebras Adding axioms to obtain sound+complete calculus for language equivalence is „always“ possible (concrete example: weighted automata) Future work  Decidability  Generic calculus: (1)deterministic system type (functor F) from an inductively defined class (2)generic branching type (monad T)  Relationship to presentations of functor (e.g. Rob Myers‘ PhD thesis)  Other concrete calculi (e.g. probabilistic systems)

21. IFIP WG 1.3 2013 | Stefan Milius | March 16 | S. 21 Proof obligations for extended calculi injective

22. IFIP WG 1.3 2013 | Stefan Milius | March 16 | S. 22 Application: Expression Calculus for Weighted Automata Syntax. Example. M. Bonsangue, S. Milius, A. Silva: Sound and complete axiomatizations of coalgebraic language equivalence, ACM ToCL, 2013.

23. IFIP WG 1.3 2013 | Stefan Milius | March 16 | S. 23 Axioms + Rules

24. IFIP WG 1.3 2013 | Stefan Milius | March 16 | S. 24 Graphically…

25. IFIP WG 1.3 2013 | Stefan Milius | March 16 | S. 25 Homework: algebraic proof of language equivalence

26. IFIP WG 1.3 2013 | Stefan Milius | March 16 | S. 26 Results (1) Theorem. closed syntactic expressions axioms + proof rules Kleene Theorem. (weighted automata  expressions) ! M. Bonsangue, S. Milius, S. Silva: Sound and complete axiomatizations of coalgebraic language equivalence, 2012

27. IFIP WG 1.3 2013 | Stefan Milius | March 16 | S. 27 Ergebnisse (2) Theorem (Soundness + Completeness). Proof. injective M. Bonsangue, S. Milius, A. Silva: Sound and complete axiomatizations of coalgebraic language equivalence, 2013