Algebraic Synchronization Trees and Processes Luca Aceto ICE-TCS, - PowerPoint PPT Presentation

The Research Question and its Context Results Algebraic Synchronization Trees and Processes Luca Aceto ICE-TCS, School of Computer Science, Reykjavik University IMT Lucca, 30 May 2012 an ´ Joint work with Arnaud Carayol (University Paris-Est), Zolt´ Esik (University of Szeged) and Anna Ing´ olfsd´ ottir (Reykjavik University) Luca Aceto (ICE-TCS and SCS) Algebraic Synchronization Trees and Processes

The Research Question and its Context Results What is this talk about? The research question What is the expressive power of ‘algebraic’ recursion schemes as a means for defining synchronization trees up to isomorphism, bisimilarity and language equivalence? Some of the questions to be answered before we start 1 What is an ‘algebraic’ recursion scheme? 2 What are synchronization trees? 3 What do you mean by ‘define’? Luca Aceto (ICE-TCS and SCS) Algebraic Synchronization Trees and Processes

The Research Question and its Context Results What is this talk about? The research question What is the expressive power of ‘algebraic’ recursion schemes as a means for defining synchronization trees up to isomorphism, bisimilarity and language equivalence? Some of the questions to be answered before we start 1 What is an ‘algebraic’ recursion scheme? 2 What are synchronization trees? 3 What do you mean by ‘define’? Luca Aceto (ICE-TCS and SCS) Algebraic Synchronization Trees and Processes

The Research Question and its Context Results Recursion schemes (by example) An algebraic scheme A regular scheme F ( n ) = ifzero ( n , 1 , mult ( 2 , F ( pred ( n )))) , X = f ( X , Y ) where the symbols ifzero, mult, pred, 1 Y = a and 2 denote given function symbols. What is the meaning of these recursion schemes? Answer of initial algebra semantics The semantics of a recursive program scheme is the infinite term tree that is the ‘least fixed point’ of the system of equations associated with the program scheme. Luca Aceto (ICE-TCS and SCS) Algebraic Synchronization Trees and Processes

The Research Question and its Context Results Recursion schemes (by example) An algebraic scheme A regular scheme F ( n ) = ifzero ( n , 1 , mult ( 2 , F ( pred ( n )))) , X = f ( X , Y ) where the symbols ifzero, mult, pred, 1 Y = a and 2 denote given function symbols. What is the meaning of these recursion schemes? Answer of initial algebra semantics The semantics of a recursive program scheme is the infinite term tree that is the ‘least fixed point’ of the system of equations associated with the program scheme. Luca Aceto (ICE-TCS and SCS) Algebraic Synchronization Trees and Processes

The Research Question and its Context Results Synchronization trees: A classic model of process behaviour Synchronization trees arise as unfoldings of labelled transition systems and have been used to give denotational semantics to process description languages. Example of a synchronization tree • A synchronization tree that describes a a a a process that can perform any finite • • • sequence of a ’s and terminate ex a a • • • successfully thereafter. ex a Legenda: ex = successful termination. • • We only consider countable trees. ex • Luca Aceto (ICE-TCS and SCS) Algebraic Synchronization Trees and Processes

The Research Question and its Context Results Operations on synchronization trees Assume an alphabet A of actions, with a , b , c , d ∈ A . Constants: 0 (one node tree), 1 (the tree r ex → r ′ ), a (the tree → r ′ ex a → r ′′ ) for each a ∈ A r Unary operations: action prefixing a . , for each a ∈ A Binary operations: sum: t + t ′ is obtained by glueing the two trees at the root and sequential composition: t · t ′ is obtained by replacing each edge of t labelled ex by a copy of t ′ . Two signatures for synchronization trees: Γ and ∆ Γ contains + , 0 , 1 and each letter a ∈ A as a unary symbol. (Cf. Milner’s Basic CCS) ∆ contains + , · , 0 , 1 and each letter a ∈ A as a constant symbol. (Cf. Bergstra and Klop’s Basic Process Algebra (BPA)) Luca Aceto (ICE-TCS and SCS) Algebraic Synchronization Trees and Processes

The Research Question and its Context Results Bisimilarity and language equivalence over synchronization trees We consider synchronization trees up to isomorphism, bisimilarity and language equivalence. Two synchronization trees t = ( V , v 0 , E , l ) and t ′ = ( V ′ , v ′ 0 , E ′ , l ′ ) are bisimilar if there is some symmetric relation R ⊆ ( V × V ′ ) ∪ ( V ′ × V ) that relates their roots, and such that if ( v 1 , v 2 ) ∈ R and there is some edge ( v 1 , v ′ 1 ) , then there is an equally-labelled edge ( v 2 , v ′ 2 ) with ( v ′ 1 , v ′ 2 ) ∈ R . The path language of a synchronization tree is composed of the words in A ∗ that label a path from the root to the source of an exit edge. Two trees are language equivalent if they have the same path language. Luca Aceto (ICE-TCS and SCS) Algebraic Synchronization Trees and Processes

The Research Question and its Context Results Bisimilarity and language equivalence at work Example • a a a Infinitely many subtrees up to • • • language equivalence, and ex a a therefore up to isomorphism • • • and bisimilarity ex a Path language: { a n | n ≥ 1 } • • ex • Questions: 1 How can we use recursion schemes to define synchronization trees? 2 What type of recursion schemes do we consider? Luca Aceto (ICE-TCS and SCS) Algebraic Synchronization Trees and Processes

The Research Question and its Context Results Examples of recursion schemes defining synchronization trees A Γ -algebraic recursion scheme A ∆ -regular recursion scheme = F 2 ( a . 1 ) F 1 X = ( X · a ) + a F 2 ( v ) = v + F 2 ( a . v ) . Both these schemes define the synchronization tree • a a a • • • ex a a • • • ex a • • ex • Luca Aceto (ICE-TCS and SCS) Algebraic Synchronization Trees and Processes

The Research Question and its Context Results Examples of recursion schemes defining synchronization trees A Γ -algebraic recursion scheme A ∆ -regular recursion scheme = F 2 ( a . 1 ) F 1 X = ( X · a ) + a F 2 ( v ) = v + F 2 ( a . v ) . Both these schemes define the synchronization tree • a a a • • • ex a a • • • ex a • • ex • Luca Aceto (ICE-TCS and SCS) Algebraic Synchronization Trees and Processes

� � � � The Research Question and its Context Results The tree defined by a scheme (defined) Motto: A Γ - or ∆ -algebraic recursion scheme defines the synchronization tree that is the initial solution of the scheme. Example: F = 1 + a · F · b ���� ���� � ���� ���� ���� � ���� � ���� ���� � · · · a a a a ex b � b � b � ���� ���� ���� ���� ���� ���� ���� ���� · · · ex b � b � ���� ���� ���� ���� ���� ���� · · · ex b � ���� ���� ���� ���� · · · ex ���� ���� · · · Luca Aceto (ICE-TCS and SCS) Algebraic Synchronization Trees and Processes

� � � � � � � � � � � � � � � The Research Question and its Context Results The tree defined by a scheme (example) The Γ -algebraic recursion scheme b + c + a . F 2 ( b 2 , c 2 ) F 1 = F 2 ( v 1 , v 2 ) = v 1 + v 2 + a . F 2 ( b . v 1 , c . v 2 ) defines the unfolding of ���� ���� ���� ���� ���� ���� ���� ���� · · · b b b b b b ���� � ���� ���� ���� ���� ���� a a a · · · c c c ���� ���� ���� ���� ���� ���� ���� ���� · · · c c c This LTS is not expressible in BPA modulo bisimilarity. Luca Aceto (ICE-TCS and SCS) Algebraic Synchronization Trees and Processes

The Research Question and its Context Summary of the results Results A taste of the results and of the proof techniques Pictorial summary of our results Question: What is the expressive power of Γ - or ∆ -regular and of Γ - or ∆ -algebraic recursion schemes as a means of defining synchronization trees? Tree 3 Graph 3 Graph 3 = ∆ -alg. ∆ -alg. ∆ -alg. Γ -alg. Γ -alg. Γ -alg. = = Tree 2 Tree 2 = ∆ -reg. Tree 2 ∆ -reg. ∆ -reg. Tree 1 Tree 1 Tree 1 = = = Graph 1 Graph 1 Graph 1 = = = Γ -reg. Γ -reg. Γ -reg. (Language equivalence) (Bisimilarity) (Isomorphism) Luca Aceto (ICE-TCS and SCS) Algebraic Synchronization Trees and Processes

The Research Question and its Context Summary of the results Results A taste of the results and of the proof techniques Pictorial summary of our results Question: What is the expressive power of Γ - or ∆ -regular and of Γ - or ∆ -algebraic recursion schemes as a means of defining synchronization trees? Tree 3 Graph 3 Graph 3 = ∆ -alg. ∆ -alg. ∆ -alg. Γ -alg. Γ -alg. Γ -alg. = = Tree 2 Tree 2 = ∆ -reg. Tree 2 ∆ -reg. ∆ -reg. Tree 1 Tree 1 Tree 1 = = = Graph 1 Graph 1 Graph 1 = = = Γ -reg. Γ -reg. Γ -reg. (Language equivalence) (Bisimilarity) (Isomorphism) Luca Aceto (ICE-TCS and SCS) Algebraic Synchronization Trees and Processes