Sound and complete axiomatizations of coalgebraic language - - PowerPoint PPT Presentation

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Sound and complete axiomatizations of coalgebraic language - - PowerPoint PPT Presentation

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Platzhalter fr 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

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Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Marcello Bonsangue, Stefan Milius, Alexandra Silva

Sound and complete axiomatizations of coalgebraic language equivalence

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Seminar Erlangen | Stefan Milius | June 12, 2012 | 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
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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?

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

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?

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

Examples

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Seminar Erlangen | Stefan Milius | June 12, 2012 | 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?

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Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 8

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 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.

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Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 9

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.
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Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 10

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.
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Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 11

Does this work for coalgebras in general? What does „language equivalence“ mean for coalgebras?

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Seminar Erlangen | Stefan Milius | June 12, 2012 | 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

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Seminar Erlangen | Stefan Milius | June 12, 2012 | 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
  • f a family of automata, I & C, 1961

semiring But

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Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 14

Coalgebraic Determinization

Generalized powerset construction:

Definition (language equivalence). Theorem.

Silva, Bonchi, Bonsangue, Rutten: Generalizing the powerset construction, coalgebraically, Proc. FSTTCS 2010.

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Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 15

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

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Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 16

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

  • 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.
  • Z. Esik, W. Kuich: Free iterative and iteration K-semialgebras, Algebra Univ., 2012.
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Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 17

Proof obligations for extended calculi

injective

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Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 18

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.

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

Axioms + Rules

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

Graphically…

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Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 21

Homework: algebraic proof of language equivalence

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Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 22

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

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Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 23

Proof obligations for extended calculi

injective

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Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 24

Ad 3. Proving uniqueness

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Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 25

Ergebnisse (2)

Theorem (Soundness + Completeness). Proof.

injective

  • M. Bonsangue, S. Milius, S. Silva: Sound and complete

axiomatizations of coalgebraic language equivalence, 2012

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Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 26

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)

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Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 27

Spezialfall: Rabinovichs Kalkül

Syntax. Axiome+Regeln.

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Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 28

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Agenda Kapiteltrennung

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Seminar Erlangen | Stefan Milius | June 12, 2012 | S. 29

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