Constraint Automata
David Costa
CWI
IPA Lentedagen 2007
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Motivation
Timed Data Streams
Constraint Automata
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Automata Operators
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Behaviour equivalence and containment
Related work
Conclusions
Ongoing work
◦ Data Constraints
◦ Models of Reo connectors
◦ Product
◦ Hiding
Outline
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Observable data flow of coordinating connectors
◦ data flow at input/output ports (source/sink nodes) of
a connector
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Composition operators
◦ facilitate the modelling of large systems
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We abstain from what:
◦ data flow direction
◦ topology of the connector
Motivation
What do we want to model?
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Non-empty set of data: Data
◦ domain of data that can flow through the connector
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The set of a data streams over the set Data, are
all the infinite sequences over Data denoted by:
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The set of timed streams over the set IR+, are all
the infinite sequences over IR+ denoted by:
Timed Data Stream (TDS)
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The set of timed data streams, TDS, over the
set Data, is given by:
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A set of Names to use for the input or output
ports of the connector
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Assigning a TDS to a connector port Ai defines
the data flow behaviour of a port Ai
Timed Data Stream (TDS)
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Channels
◦ assigning a binary relations R µ TDS£TDS defines
the data flow behaviour of a channel.
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Example
◦ the data flow behaviour of a synchronous channel is
formally described by the relation:
Reference:
F.Arbab and J.J.M.M.Rutten. A coinductive calculus of
component connectors. WADT 2002.
Connectors as TDS-tupples
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Automata
◦ as acceptors of relations on timed data streams, such
automaton observes the data occurring at certain
input/output ports and either fires a transition according
to the observed data or rejects it if there is no
corresponding transition in the automaton.
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State
◦ possible configurations (buffer contents)
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Transition
◦ one-step possible data flow satisfying some data constraints
and its effect on the present configuration
Constraint Automata
The idea/Informaly
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Symbolic representation of sets of data
assignments (subsets of Data)
 Built from the atoms: dA = d with the grammar:
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Common derived data constraints:
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DC(N, Data)
◦ N non-empty subset of Names
◦ denotes the set of data constraints dA = d, A 2 N
Data Constraints
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Constraint Automata
Formal definition
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buffer FIFO1 with ports A and B
buffer FIFO1
1-Bounded FIFO Channel
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Channels
sync
syncdrain/syncspout
asyncdrain/asyncspout
Constraint Automata
Models of Reo connectors (I)
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Merger
merger
Constraint Automata
Models of Reo connectors (II)
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lossy (synchronous) channel
lossy/lossysync
Constraint Automata
Models of Reo connectors (III)
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Given a TDS-tuple we inspect whether it
corresponds to an accepting run of the automaton.
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Accepting runs (accepting behaviour)
◦ is given by all infinite runs of the automaton starting
from an initial state
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Rejecting runs (rejecting behaviour)
◦ is given by all finite (possibly empty) run of the
automaton
Intuitive behaviour of a CA
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Now we know how to model small connectors.
 The question next is: what can we do with these
models?
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◦ Combine them to build models of larger systems
 Composition and abstraction operators
◦ Check for equivalence between two models
◦ Check for behaviour containment of one model into
another model.
◦ Adapt know model checking methods from reactive
systems and !-automata for our constraint automata
Operators and Analysis methods
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Automata Operators
Product
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We consider 2 FIFO1 over Data = f1g with
ports fA, Cg and fC, Bg respectively
Product of two FIFO1
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Hiding a port C in constraint automata
corresponds to make unobservable the data flow
at that port.
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Removes all the information about port C.
Hiding operation
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Automata Operators
Hiding
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9C [FIFO1 ./ FIFO1]
Hiding C on product of two FIFO1
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An alternative characterization of language
equivalence and inclusion can be given using
branching time relations
◦ they allow a simpler way to verify if two automata are
language equivalent, or if the language is contained in
the language of the other.
Bisimulation and Simulation
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Behaviour Equivalence
Bisimulation vs. Language Equiv.
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Behaviour Containment
Simulation vs. Language Inclusion
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Congruence result for bisimulation equivalence
and the simulation preorder for the operators
product and hiding
What do you mean: Compositionality?
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Briefly mention:
◦ other similar automata formalism:
 IO automata
◦ labels with action names (data independent)
◦ input enabledness
◦ strict notion of time
 timed port automata
◦ input enabledness
◦ strict notion of time
 interface automata
◦ based on game theory
◦ allow automatic checking of compatibility between interfaces
Related Work
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CA allows to build formal models of the data
flow behaviour of coordinating connectors
 provides composition and abstraction operators
to build larger models out of existing models
 provides analysis and verification methods
adapted from known methods for reactive
systems or formal languages
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Conclusions
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Extend the formalism to allow models for
context sensitive connectors
◦ two approaches
 capturing intentional behaviour
 embedding some notion of priority in the behaviour
domain
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Implementation of model checking algorithms
Ongoing work
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