Introduction to the calculus

1
Introduction to the ?-calculus

• Stefan Kahrs
• source Joachim Parrows chapter in the Handbook
  of Process Algebra

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What is it?

• In short a process algebra, invented by Robin
  Milner around 1990. Process algebra are like toy
  programming languages for parallel programming
• Special features
• Name creation is internal to the calculus
• Orthogonality names serve both as data and as
  channels (mobility)

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Syntax

• P P1 P2 parallel
• P1 P2 n.d. choice
• (?x)P new name
• x1(x2). P input
• x1 x2 . P output
• 0 stop
• ? . P silent action
• A(x1,,xn) defined process
• if x1x2 then P match
• if x1?x2 then P mismatch

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Slight Variations

• the original paper by Milner et.al. had match but
  not mismatch some papers omit both
• instead of (recursively) defined agents, some
  papers have replication (!P) instead of recursion
• at least one paper omits non-det. choice
• syntactic variants xyP, (x)P, ?x.P

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Standard Process Algebra Stuff

• communication on a channel a
• notice this is synchronous, but...
• aside substitution avoids name capture

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...and also

• process descriptions of A(x1,...,xn)
• note the xi are just name parameters
• can be (mutually) recursive
• no renaming operator

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Names

• The distinguishing feature of the ?-calculus!
• originated in Standard ML semantics
  (implementations)
• type checking (for datatypes or abstract types)
• exception constructors
• not fundamentally different from object creation
  in OO-programming

8
Eh?

• the ? operator generates a new name
• or if you prefer hides a free name
• however these names are mobile communication
  can move them into other areas that seemingly
  were outside the scope
• this is achieved by the congruence relation which
  equates (some) equivalent processes

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Example of moving scope

• on the left ? inside parallel composition
• on the right it has moved outside

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Congruence ? (note Parrow)

• ? is the smallest congruence such that
• both and form commutative monoids with
  neutral element 0
• it includes ?-conversion
• ...and the unfolding of definitions
• ...and most importantly, the scope extension laws

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Scope Extension Laws
garbage collection
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Consequences and non-Consequences

• all binders can be moved to the top only
  recursive descriptions are in the way
• we do not have
• it should be clear why however, for we do not
  have this either and there it is less clear

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Operational Semantics

• defines a labelled transition relation
• when we have
• ...then t can evolve to u, and the outside world
  can observe the action ?
• three actions
• input a(x)
• output
• silent ?
• one of many presentations...

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Rules I
15
Comments

• because of the input prefix (also ?x labels, but
  they play only a minor rule) there are bound
  names in labels
• non-deterministic choice commits by making a
  transition (which can be silent), and this is the
  only thing it can do

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Rules II
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Comments

• because of the side-condition on the RES rule,
  any attempt to immediately send or receive on a
  fresh name will deadlock
• the CLOSE rule is not really necessary, because
  its effect can be achieved through the other
  rules, but without it the OPEN rule would fire
  blanks

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Bisimulation

• ...cannot quite be defined as usual, because we
  have two problems
• bound names in labels
• even if we take care of them, we do not get a
  congruence relation

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Reminder Standard Definition

• a (strong) bisimulation is a symmetric binary
  relation ? such that if and P?Q
  then ?Q. and P ? Q.
• modification for input prefix (?a(x))
• assume x is fresh and
• write P?Q if it holds for some bisimulation

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Input Prefix

• may not preserve strong bisimilarity
• because ? is not preserved by (arbitrary)
  substitutions an aliasing problem
• ...and input prefix can introduce aliasing

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Example

• is bisimilar to
• but the substitution b/a would break the
  equivalence since the left could silently evolve
  to 0, the right cannot
• therefore, the input prefix c(a) breaks
  bisimilarity as well

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Getting out of jail

• two agents P and Q are (strongly) congruent if
  and only if P? ? Q? for all substitutions ?.
• note strong congruence is the largest congruence
  relation in bisimilarity
• other bisimulations barbed, early, open,...
• also weak bisimilation, i.e. making ?
  unobservable

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Asynchronous ?-calculus

• As most process algebras, the ?-calculus is
  essentially synchronous.
• However, when we constrain the agents we can
  write in a certain way, we end up with an
  asynchronous calculus.
• What are the constraints?

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Constraints for asynchronous calculus

• Only 0 can follow an output prefix
• an output prefix may not occur as an unguarded
  operand of
• Rationale the agent ?.(ax.0 P) is essentially
  asynchronously outputting on a and continuing as P

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Recover...

• it is possible to recover from these constraints
  by using acknowledgements, e.g.
• the binder protects the name b and thus protects
  the acknowledgement

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Lots of other stuff

• if others are willing to continue that theme...
• PICT, a programming language based on the
  asynchronous ?-calculus
• proving (various) bisimilarities
• applications
• higher-order ?-calculus
• action semantics