The π-calculus Semantics. Equivalence and Value-Passing. Infinite Sums 4/12/2004

Transcription

1 The π-calculus Semantics Overview ate and early semantics Bisimulation and congruence Variations of the calculus eferences obin Milner, Communicating and Mobil Systems Davide Sangiorgi and David Walker, The π-calculus: A Theory of Mobile rocesses obin Milner, Joachim arrow, David Walker, A Calculus of Mobile rocesses, art I+II 22 Equivalence and Value-assing Consider the following two processes: = a(x).(if x = 3 then else Q) + a(x). S = a(x).(if x = 3 then ) + a(x).(if x 3 then Q) Do and S exhibit the same behavior? 23 Infinite Sums rocesses defined in a calculus with value-passing can be translated into processes of a value-free calculus using infinite sums: = Σ n ωa n. n + Σ n ωa n. where 3 =, and n = Q for n 3. Similarly, S = Σ n ωa n. n + Σ n ωa n.q n where 3 = and Q 3 =, while n = and Q n = Q for n 3. The sums are equivalent, that is, is equivalent to S. 24 1

2 Indexed rocess Families rocesses are not capable to perform infinitely many actions a n. The number of actions is given by the prefixes: a(x) if x = 3 then else Q a(x) S a(x) if x = 3 then S a(x) if x 3 then Q The action a(x) yields a family of processes indexed by the variable x. Two processes and S are equivalent, if both yield under a(x) indexed process families, which are elementwise (that is, for each value of x) equivalent. In this view, and S are not equivalent. 25 π-calculus Actions Kind fn() bn() Comment silent free output {x, y} bound output {x} {y} local name y exported along channel x free input {x, y} early instantiation bound input {x} {y} any name can be received along x, (y) designates the place 26 A ate abeled Transitions EFIX:. : SUM Q + Q ES: x n( ) ( ν x) ( ν x)' ' SUM : + Q MATCH: π. [x = x] π. E:!! OEN: ' x y ( ν y) ' : A bn( ) fn(q) = Q Q x(z) ' Q Q ' {y/z} Q : A bn( ) fn(q) = Q x(z) ' Q Q {y/z}' COSE : ' Q ' Q COSE : Q ( ν y)(' ) Q ( ν y)(' ) 27 2

3 An Early abeled Transitions EFIX:. INUT: a(x). υ {u/x} ES: x n( ) ( ν x) ( ν x)' : SUM Q + Q ' SUM : + Q MATCH: π. [x = x] π. E:!! OEN: ' x y ( ν y) ' : A bn( ) fn(q) = Q Q ' Q Q ' Q : A bn( ) fn(q) = Q ' Q Q ' COSE : ' Q ' Q COSE : Q ( ν y)(' ) Q ( ν y)(' ) 28 ate vs. Early Consider the interaction between. and.q. Both semantics yield the same -transitions. ate semantics: : COM EFIX:. EFIX:.Q..Q {z/y} Q Q Early semantics: : COM INUT:. {z/y} EFIX:...Q {z/y} Q 29 The A-ule The A / has an extra condition that Q does not contain a name bound in. Note, bound names are just references to occurrences. So, in the conclusion Q Q the action should not refer to any occurrences in Q. Consider the inference A :. Q Q Combined with an output. we get. Q Q. (. Q). {z/y}( Q) which is only correct if y fn(q) that can be guaranteed if y is a fresh name (that is, y n(, Q)). 21 3

4 Bisimilarity A generic definition of bisimulation is that it is a symmetric binary relation on processes satisfying the property, whenever Q ' : Q ' The intuition is that if can do an action then Q can do the same action and the derivates lie in the same relation. 211 π-calculus Bisimilarity For the π-calculus extra care has to be taken for actions involving bound names. Consider = a(u), Q = a(x).( ν v)vu Both processes represent the same behavior: they can do an input along a and then nothing more. However, Q has the name u free, whereas has not. Therefore, x in Q cannot be alphaconverted to u, and the transition a(u) cannot be simulated by Q. Such a difference between and Q is not important since, if has an action a(u), then by alpha-conversion it also has a similarly derived action a(w) for infinitely many w. Hence, it is sufficient for Q to simulate only the bound actions where the bound object is not free in Q. 212 Strong ate Bisimulation A strong bisimulation is a symmetric binary relation S on processes satisfying the following: Whenever SQ and where bn() is fresh implies that 1. If = a(x) then there exists Q such that Q a(x) and u: {u/x} S{u/x}Q. 2. If is not an input then there exists Q such that Q and SQ. and Q are strongly bisimular, written ~ Q, if they are related by a strong bisimulation

5 Examples Consider 1 = a(x). + a(x). 2 = a(x). + a(x).[x=u] Assume that is not bisimular to. Then /~ 1 2 a(x) since the transition cannot be simulated 1 by 2. For example, a(x) [x = u] does not suffice 2 since, for the substitution {u/x}, the derivates are not bisimular. This also holds for 2 = a(x). + a(x). + a(x).[x=u] 214 Input An input a(x) Q can be thought of as a λx.q, clause (1.) says that the derivates (which are functions from names to processes) must be pointwise bisimular. It holds that a b ~ a.b + b.a a a ~ a.a + a.a + However, this illustrates that ~ is not in general closed under substitution, that is, from ~ Q we cannot conclude that σ ~ σq. In fact, ~ is not preserved by the input prefix. 215 Strong Early Bisimulation A strong early bisimulation is a symmetric binary relation S on processes satisfying the following: Whenever SQ and where bn() is fresh implies that 1. If = ax then u: there exists Q such that Q ax and {u/x} S{u/x}Q. 2. If is not an input then there exists Q such that Q and SQ. and Q are strongly early bisimular, written ~ E Q, if they are related by a strong early bisimulation. Note, ~ E is also not preserved by the input prefix

6 ate Name Instantiation 1 = a(x). + a(x). 2 = a(x). + a(x).[x=u] a λx. u w {u/x} {w/x} a u λx. w {u/x} {w/x} 1 a u 2 a u [u=u]{u/x} λx. w λx.[x=u] w [w=u]{w/x} 217 Early Name Instantiation 1 = a(x). + a(x). 2 = a(x). + a(x).[x=u] au {u/x} au {u/x} aw {w/x} aw {w/x} 1 au aw 2 au aw [u=u]{u/x} [w=u]{w/x} 218 Asynchronous Communication In contrast to synchronous communication, in asynchronous communication there is an unpredictable delay between output and input, while the message is in transit. In fact, we are not able to observe the consumption of an output. We can model this behavior by inserting a process representing an asynchronous communication medium between sender and receiver. For example, an unbound medium not preserving the order of messages is: M = i(x).(ox M) When M receives u along I it evolves to ou M, and can at any time deliver u along o, and also continue to accept more messages

7 The Asynchronous π-calculus We can define an asynchronous subcalculus that does not require an explicit medium. The subcalculus constists of processes satisfying the following requirements: Only can follow an output prefix. An output prefix may not occur as an unguarded operand of +. The first requirement disallows agents such as ax.by, where a process other than follows ax. The second requirement disallows processes like o x + b(y), but allows processes like.o x + b(y). 22 Design ationale An unguarded output prefix ax occurring in a process term represents a message that has been sent but not yet been received. The action of sending a message is placing it in an unguarded position, as in the following. ( ax ) ax After this transition, ax can interact with a receiver, and the sender proceeds concurrently as. Due to the first requirement the fact that the message has been received cannot be detected by unless the receiver explicitly send an acknowledgement. Due to the second requirement a message cannot disappear unless it is received. 221 Asynchronous ate Bisimulation An asynchronous strong bisimulation is a symmetric binary relation S on processes satisfying the following: Whenever SQ and where bn() is fresh implies that 1. If = a x, a(x), or then there exists Q such that and SQ. 2. If a(x) ' then there exists Q such that either Q a(x) and u: {u/x} S{u/x}Q or Q and u :{u/x}' S({u/x} au). and Q are strongly bisimular, written ~ Q, if they are related by a strong asynchronous bisimulation. Q 222 7

8 Expressive ower The asynchronous π-calculus is not as expressive as the full π-calculus. The asynchronous π-calculus is incapable to break certain symmetries possibly present in the initial communication graph the leader election problem cannot be solved. roof: Catuscia alamidessi, Comparing the Expressive ower of the Synchronous and Asynchronous π-calculi