Foundations of Interaction

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Ex nihilo a reflective higher-order process
calculus

• The ?-calculus

L.G. Meredith1 Matthias Radestock2
1Djinnisys Corporation 2LShift, Ltd
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Agenda

• Motivations
• ?-calculus
• Syntax
• Structural equivalence
• Operational semantics
• A warm-up replication
• Encoding the ?-calculus
• Conclusions and future work

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Motivations

• ?-calculus is not a closed theory
• dependent upon a theory of names
• such a theory will at least dictate computation
  of name-equality
• Name-equality is a computation
• nowhere is there an infinite set of atomic
  elements available to the computer scientist
• all countably infinite sets available to the
  computer scientist are generated from a finite
  presentation
• perforce the elements of these sets have
  structure -- and this structure is used to
  compute equality

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Motivations

• If interaction is to provide a foundational
  theory of computation, then this computation must
  be accounted for, too!
• All realizations (e.g., implementations) of
  mobile process calculi face this fact
• Would our theory better serve our practitioners
  therefore if it accounted for name structure as
  well?
• Synchronization and Substitution play very
  different roles in ?-like mobile process calculi
  
• requiring different computations

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Motivations potential applications

• Biology sites in molecular biology are decidedly
  not atomic locations
• Ligand-binding receptors, phosphorylation sites,
  etc, have extension and behavior
• modeling these as atomic names may miss important
  behavior
• Security concrete realizations of a naming
  scheme will have names with structure,
• subject to guessing attacks
• theory of interaction with a structural account
  of names can facilitate reasoning about this

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The ?-calculus syntax

• Grammar
• P, Q 0 null process
• x(y).P input
• xP_ lift
• PQ parallel composition
• _x drop
• x,y P_ quote
• PROC denotes the set of processes generated by
  this grammar
• PROC_ denotes the set of names generated by
  this grammar
• Syntactic sugar xy _at_ x _y _

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The ?-calculus syntax - examples
the ur-process, everything literally comes ex
nihilo, out of nothing! the first name the first
output process the first input process some new
names

• 0
• 0_
• 0_0_
• 0_(0_).0
• 0_0_ _ , 0_(0_).0 _

Looks remarkably like machine code!
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Structural equivalence, ?-equivalence and name
equivalence

• Clearly, we want 0 7 00 7 000 7
• should 0_7N 00_ 7N 000_ 7N ?
• Name equivalence, ?N ? PROC_ ? PROC_ , is the
  smallest equivalence relation respecting
• x ?N _x_ P 7 Q ? P_7N Q_
• Structural equivalence, ? ? PROC ? PROC, is the
  smallest equivalence relation, containing
  ?-equivalence, respecting
• P 0 7 P 7 0 P
• P Q 7 Q P
• (P Q) R 7 P (Q R )

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Structural equivalence, ?-equivalence and name
equivalence

• First subtlety -- a cycle in Structural
  equivalence
• structural equivalence depends on ?-equivalence
• ?-equivalence depends on name equality
• name equality depends on structural equivalence!
• Each recursive call is one level of quotes
  fewer
• Quote Depth
• (P_) 1(P)
• (P) max( (Q_) Q_ ? N(P))
• Grammar enforces strict alternation of quoting
  and process constructor
• Calculation of structural equivalence terminates
  by easy induction on quote depth

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Substitution

• Syntactic substitution
• A substitution is a partial map, ? PROC_ ?
  PROC_
• Q_/P_ denotes the map which sends P_ to
  Q_ we write x? for ?(x)
• xQ_/P_ Q_ if x ?N P_, x otherwise.
• A substitution, ?, is uniquely extended to a
  map, _? PROC ? PROC
• by the following recursive definition
• 0 _Q_/P_ _at_ 0
• (RS) _Q_/P_ _at_ (R _Q_/P_ ) (S
  _Q_/P_ )
• (x(y).R) _Q_/P_ _at_ xQ_/P_ (z). ((R
  _z/y ) _Q_/P_ )
• (xR_) _Q_/P_ _at_ x Q_/P_RQ_/P_
  _
• (_x) _Q_/P_ _at_ Q_ if x ?N P_ , _x
  otherwise
• where z is chosen distinct from the names in R,
  P_ and Q_

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Substitution

• Semantic substitution -- same as above except for
  drop where the process is instantiated at
  substitution time
• (_x) _Q_/P_ _at_ Q if x ?N P_ , _x
  otherwise
• Examples
• wyz_ u/z wyu_ wyz_ u/z
  wyz_
• w_x_Q_ /x wQ_

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Operational semantics

• The operational semantics is given by a reduction
  relation
• ? ? PROC ? PROC
• recursively specified by the following rules.
• comm xsrc ?N xtrgt
• xsrcP_ xtrgt(y).Q ? Q _P_ /y
• par P ? P?
• P Q ? P? Q
• equiv P ? P?, P? ? Q?, Q? ? P
• P ? Q

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Replication

• Replication is defined by the following equation
• D(x) x(y).( _y xy )
• !xP D(x) xP D(x)_
• x(y).( _y xy ) xP D(x)_
• ? P D(x) x_P D(x)
• P D(x) xP D(x)_

• Replication is defined by the following equation
• D(x) x(y).( _y xy )
• !xP D(x) xP D(x)_
• x(y).( _y xy ) xP D(x)_
• ? P D(x) x_P D(x)
• P D(x) xP D(x)_

• Replication is defined by the following equation
• D(x) x(y).( _y xy )
• !xP D(x) xP D(x)_
• x(y).( _y xy ) xP D(x)_
• ? P D(x) x_P D(x)
• P D(x) xP D(x)_

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Encoding the ?-calculus

• Paper presents a distributed encoding in which
  par-ands are mapped to separate namespaces
• Below we present a centralized encoding (due to
  Radestock) in which there is a single resource
  against which all ?-requests are synchronized
• Both encodings use a trick for free names build
  a ?-calculus with the name set PROC_
• Let h be a name not in fn(P), e.g. h ?m ?
  fn(P) m0_ _
• P P(h) h h0_ _
• (? x)P(h) h(x). (hx0__ P(h))
• ! x(y).P(h) h(z).(hz0__ zx(y).(D(z)
  P(h))_ D(z))
• where z ? fn(P) and D(z) as in replication

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Correctness of the encoding

• names are global in the ?-calculus
• ?-calculus contexts can make observations that
  ?-calculus contexts cannot
• to prove correctness of the encoding one must
  restrict to name-sets visible in ?-calculus
  contexts
• an observation relation, ?N, parameterized in a
  set of names, N, is given by
• x ?N y P ?N x or Q ?N x
• yv ?N x P Q x
• an P ?N x if there is a Q s.t. P?Q and Q ?N x
• an N-barbed bisimulation, SN, is a symmetric
  relation s.t.
• P ? P? implies Q ? Q? , P? SN Q?
• P ?N x implies Q ?N x
• P 3N Q if there is an N-barbed bisimulation, SN
  , P SN Q
• THM P 1? Q iff P 3FN(P)?FN(Q)Q

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Operational semantics revisited

• An alternative operational semantics may be given
  by
• commannihil ?R.(Pchan Pcochan ? R)?R ?
  0
• Pchan_P_ Pcochan_(y).Q ? Q _P_ /y

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Conclusions and future work

• Presented a higher-order asynchronous
  message-passing calculus built on a notion of
  quoting
• Provides an account of structured names
• Eliminates ? and replication
• Work underway on
• Abstract data types
• Destructuring on input
• Hennessy-Milner style logic
• Silent ?-calculus
• Fully abstract encoding of Ambient calculus