Sound and complete axiomatizations of coalgebraic language - PowerPoint PPT Presentation

Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Sound and complete axiomatizations of coalgebraic language equivalence Marcello Bonsangue, Stefan Milius, Alexandra Silva

Regular Expressions Syntatic description of regular languages: S. Kleene (1956): regular expressions are equivalent to deterministic automata D. Kozen (1994): Kleene-Algebras axiomatize the equivalence of regular expressions. 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 Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 2

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? Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 3

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? Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 4

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. Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 5

Examples Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 6

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? Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 7

Expression calculi for bisimilarity M. Bonsangue, J. Rutten, A. Silva: � Regular expressions for coalgebras for set functors F from an inductively defined class � „Kleene Theorem“ ! � Correct and complete axiomatization of behavioral equivalence for F: � Decidability M. Bonsangue, G. Caltais, E.-I. Goriac, D. Lucanu, J. Rutten, A. Silva: A decision procedure for bisimilarity of generalized regular expressions, SBMF 2010. � 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. Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 8

But what about language equivalence? Example. Milner‘s calculus for finite state processes R. Milner: A complete inference system for a class of regular behaviours, J. Comput. Syst. Sci., 1984. Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 9

But what about language equivalence? Example. Milner‘s calculus for finite state processes Theorem. Axioms 1.-4. are sound and complete for bisimilarity. 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. Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 10

Does this work for coalgebras in general? What does „language equivalence“ mean for coalgebras? Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 11

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 Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 12

Towards language equivalence of coalgebras Examples. (1) nondeterministic automata But (2) weighted automata M. P. Schützenberger: On the definition of a family of automata, I & C, 1961 But semiring Observation. The nondeterministic/weighted branching does not occur in the desired final coalgebra. Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 13

Coalgebraic Determinization Generalized powerset construction: Definition (language equivalence). Theorem. Silva, Bonchi, Bonsangue, Rutten: Generalizing the powerset construction, coalgebraically, Proc. FSTTCS 2010. Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 14

Relating final coalgebras and rational fixpoints M. Bonsangue, S. Milius, S. Silva: Sound and complete axiomatizations of coalgebraic language equivalence, ACM ToCL, 2012. Assumptions. Bisimilarity and language equivalence: finite Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 15

What is this good for? Adding axioms a la Rabinovich is always possible! Abstract Kleene Theorem Abstract Soundness + Completeness Theorems Application: Weighted Automata − Concrete expression syntax − Kleene Theorem − Correct+complete axiomatization of weighted language equivalence: 1. Rabinovich‘s result (for nondeterministic automata) 2. New calculus for linear circuits Z. Esik, W. Kuich: Free iterative and iteration K-semialgebras, Algebra Univ., 2012. M. Bonsangue, S. Milius, S. Silva: Sound and complete axiomatizations of coalgebraic language equivalence, ACM ToCL, 2012. S. Milius, A sound and complete calculus for finite stream circuits, Proc. LICS 2010. Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 16

Proof obligations for extended calculi injective Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 17

Application: Expression Calculus for Weighted Automata Syntax. Example. M. Bonsangue, S. Milius, S. Silva: Sound and complete axiomatizations of coalgebraic language equivalence, ACM ToCL, 2012. Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 18

Axioms + Rules Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 19

Graphically… Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 20

Homework: algebraic proof of language equivalence Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 21

Results (1) Kleene Theorem. (weighted automata �� expressions) Theorem. ! axioms + proof rules closed syntactic expressions M. Bonsangue, S. Milius, S. Silva: Sound and complete axiomatizations of coalgebraic language equivalence, 2012 Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 22

Proof obligations for extended calculi injective Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 23

Ad 3. Proving uniqueness Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 24

Ergebnisse (2) Theorem (Soundness + Completeness). Proof. injective M. Bonsangue, S. Milius, S. Silva: Sound and complete axiomatizations of coalgebraic language equivalence, 2012 Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 25

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 � Relationship of our proof method to Esik‘s & Kuich‘s work � Decidability � Generic calculus: (1) deterministic system type (functor F ) from an inductively defined class (2) generic branching type (monad T ) � Relationship to Rob Myers‘ PhD thesis � Other concrete calculi (e.g. probabilistic systems) Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 26

Spezialfall: Rabinovichs Kalkül Syntax. Axiome+Regeln. Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 27

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Hier steht eine einzeilige Headline � Nullam pulvinar lorem sed enim placerat vel malesuada purus laoreet. Nulla ultrices urna sapien, venenatis ultrices risus. Pellentesque congue, magna laoreet congue semper, eros dui commodo metus, sed fringilla nisi massa a justo. Vivamus gravida accumsan nibh, at semper enim egestas in. � Proin eget blandit quam. Duis semper scelerisque semper Lorem ipsum dolor sit amet, consectetur adipiscing elit. Proin eget blandit quam. Duis semper scelerisque semper. � Mauris cursus orci iaculis libero Imperdiet ac egestas odio fermentum. Orci iaculis libero imperdiet ac egestas odio fermentum. sin � � � ��� Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 29