Complete Axioms for Stateless Connectors joint work with Roberto Bruni and Ugo Montanari Dipartimento di Informatica Università di Pisa Ivan Lanese Dipartimento di Informatica Università di Pisa CALCO 2005, Swansea, Wales, UK, 3-6 September 2005

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Roadmap Why connectors? The tile model Stateless connectors Axiomatization of synch-connectors Adding mutual exclusion Concluding remarks

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Roadmap Why connectors? The tile model Stateless connectors Axiomatization of synch-connectors Adding mutual exclusion Concluding remarks

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Interaction and connectors Modern systems are huge composed by different entities that collaborate to reach a common goal interactions are performed at some well- specified interfaces… …and are managed by connectors Connectors allow separation between computation and coordination

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Coordination via connectors Connectors useful to ensure compatibility among independently developed components allow to reuse them allow run-time reconfiguration Connectors exist at different levels of abstraction (architecture, applications, …)

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Which connectors? We follow the algebraic approach system as term in an algebra We propose an algebra of simple stateless connectors for synchronization and mutual exclusion expressive enough to model the architectural connectors of CommUnity [IFIP TCS 04] build on symmetric monoidal categories and P- monoidal categories related to Stefanescu’s flow algebras and REO connectors

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Roadmap Why connectors? The tile model Stateless connectors Axiomatization of synch-connectors Adding mutual exclusion Concluding remarks

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari The tile model Operational and observational semantics of open concurrent systems compositional in space and time Category based

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari parallel composition Configurations input interface output interface sequential composition

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Configurations input interface output interface parallel composition sequential composition functoriality

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Configurations input interface output interface parallel composition sequential composition functoriality + symmetries = symmetric monoidal cat

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Observations initial interface final interface concurrent computation

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Tiles Combine horizontal and vertical structures through interfaces initial configuration final configuration trigger effect

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Tiles Compose tiles horizontally

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Tiles Compose tiles horizontally (also vertically and in parallel) symmetric monoidal double cat

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Tiles as LTS Structural equivalence axioms on configurations (e.g. symmetries) LTS states = configurations transitions = tiles labels = (trigger,effect) pairs Observational semantics tile trace equivalence/bisimilarity congruence results for suitable formats

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Roadmap Why connectors? The tile model Stateless connectors Axiomatization of synch-connectors Adding mutual exclusion Concluding remarks

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Connectors Connectors to express synchronization and mutual exclusion constraints on local choices Possible outcomes: tick (1, action performed) or untick (0, action forbidden) Operational semantics via tiles and observational semantics via tile bisimilarity Denotational semantics via tick-tables (boolean matrices) Complete axiomatization of connectors and reduction to normal form

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Basic connectors !!00 Symmetry Duplicator Bang Mex Zero

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Notation Only two kinds of allowed observations Initial and final states always coincide (since connectors are stateless) Thus we can use a “flat” notation for tiles 1 0

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Operational semantics Tiles specify the behaviours of basic connectors When composed, connectors must agree on the observation at the interfaces

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Basic tiles (I) Dual connectors have dual tiles

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Basic tiles (II) ! ! 0

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Connectors can be seen as black boxes input interface output interface admissible observations on interfaces Denotations are just matrixes n inputs  2 n rows m outputs  2 m columns dual is transposition sequential composition is matrix multiplication parallel composition is matrix expansion cells are filled with empty/copies of matrices …  0101  0010 … …111001… Denotational semantics domain is {input 3, outputs 1,2,3}

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Denotational semantics  1   1  0 10  11  10  01   1  0. !  1   0. 0

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Semantic correspondance Tile bisimilarity coincides with tile trace equivalence (stateless property) Two connectors are tile bisimilar iff they have the same associated tick-tables Tile bisimilarity is a congruence

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Roadmap Why connectors? The tile model Stateless connectors Axiomatization of synch-connectors Adding mutual exclusion Concluding remarks

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Axiomatization We want to find a complete axiomatization for the bisimilarity of connectors Synch-connectors (without mex and zero) symmetries, duplicators and bangs form a gs- monoidal category adding dual connectors we get a P-monoidal category No simple known axiomatization works for mex, but we show an axiomatization for the full class of connectors

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Gs-monoidal axioms = = = = !

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Additional P-monoidal axioms == = !!.

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Synch-tables Entry with empty domain is enabled Entries are closed under (domains) union intersection difference complementation Base: set of minimal (non empty) entries w.r.t. domain intersection Each synch-table is determined by its base

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Normal form Sort connectors ! ! …… … … Central points (correspond to cells of the base) Hiding connectors directly connected to central points Central points are connected to at least one external interface

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Properties All the axioms bisimulate (correctness) Each connector can be transformed in normal form using the axioms Bijective correspondance between synch- tables and connectors in normal form

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Roadmap Why connectors? The tile model Stateless connectors Axiomatization of synch-connectors Adding mutual exclusion Concluding remarks

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Adding mex and zero Synch-connectors are not expressive enough (only synchronization) Adding mex and zero to express mutual exclusion constraints and enforce inactivity Just mex has to be inserted: zero and dual connectors can be derived Mex and zero form a gs-monoidal category

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Obtaining zero connector = 1  0 10  1   01   10 = x = ! def 1  0. 00

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Obtaining comex connector = ! ! ! Hiding and synchronization allow to flip wires

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Looking for axiomatization of mex = 

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Looking for axiomatization of mex = 

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Looking for axiomatization of mex =  

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Looking for axiomatization of mex =  

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Looking for axiomatization of mex =  

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Key axioms 

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Key axioms = ! ! = ! ! !

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Some axioms about mex-dup = =

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Some axioms about zero = 0 0 == 0 0 = 0 0 = 0 !0 !0 =.= 0

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari A sample proof 00 = ! 0 = ! 0 0 =. ! = 0

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Additional axioms = !

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari An axiom scheme ! ! ! ……

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari An axiom scheme ! … !

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Entry with empty domain is enabled All the tables with that property can be expressed Generalized sorted and normal form Full tables

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Full tables Zeros directly connected to free variables

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Full tables Hiding connected to roots of mex or to central points

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Full tables Each hidden variable is connected to at most two central points

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Full tables At most one path between a central point and a variable

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Full tables No hidden variables are connected to the same central points

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Full tables No two central points have the same set of variables

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Full tables Each central point is connected to at least a free variable

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Full tables Each pair of central points share at least a variable

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Full tables Hidden variables attached to roots of mex are on the left

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Properties Full extension of the properties of synch- connectors all the axioms bisimulate each connector can be transformed in normal form using the axioms bijective correspondance between tables and connectors in normal form More complex axiomatization and normalization

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Roadmap Why connectors? The tile model Stateless connectors Axiomatization of synch-connectors Adding mutual exclusion Concluding remarks

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Conclusions Full correspondences between observational semantics denotational semantics equivalence classes modulo axioms Normalization allows to find a standard representative

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Axiomatization and colimits In [IFIP TCS 04] connectors used to model CommUnity Translation of a diagram is isomorphic to the translation of the colimit Now: translation of a diagram is equal up to the axioms to the translation of the colimit Furthermore normalization allows to algebraically compute the colimit

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Comparison with REO connectors REO connectors add directionality and data flow For synchronization purposes the two kinds of connectors are almost equivalent REO connectors allow some state (buffers) and some priority among configurations (LossySync) Algebraic theory of REO connectors less developed (as far as we know)

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari Future work Open problem: does a finite axiomatization exist? maybe Wan Fokkink techniques Extend the results to larger classes of connectors actions ruled by a synchronization algebra (instead of just 0 and 1) REO connectors probabilistic connectors

CALCO 2005, 3-6 September, Swansea, Wales, UK Bruni, Lanese, Montanari