Ex nihilo: a reflective higher-order process calculus

Ex nihilo: a reflective higher-order process calculus The -calculus L.G. Meredith1 & Matthias Radestock2 1Djinnisys Corporation 2LShift, Ltd Foundations of Interaction ETAPS `05

Agenda • Motivations • -calculus • Syntax • Structural equivalence • Operational semantics • A warm-up: replication • Encoding the -calculus • Conclusions and future work Foundations of Interaction ETAPS `05

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 Foundations of Interaction ETAPS `05

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 Foundations of Interaction ETAPS `05

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 Foundations of Interaction ETAPS `05

The -calculus syntax • Grammar P, Q ::=0 null process x(y).P input x^P_ lift P|Q 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: x[y] @ x^_y^_ Foundations of Interaction ETAPS `05

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 likemachine code! Foundations of Interaction ETAPS `05

Structural equivalence, -equivalence and name equivalence • Clearly, we want 0 7 0|0 7 0|0|0 7 … should ^0_7N^0|0_7N^0|0|0_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 7P7 0 | P P | Q7Q | P (P | Q) | R7P | (Q | R ) Foundations of Interaction ETAPS `05

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 Foundations of Interaction ETAPS `05

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) x{^Q_/^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_}^@ 0 (R|S) _{^Q_/^P_}^@ (R _{^Q_/^P_}^ ) | (S_{^Q_/^P_}^ ) (x(y).R) _{^Q_/^P_}^@ x{^Q_/^P_}(z). ((R _{z/y}^) _{^Q_/^P_}^ ) (x^R_) _{^Q_/^P_}^@x {^Q_/^P_}^R{^Q_/^P_}^_ (_x^) _{^Q_/^P_}^@ ^Q_ if x N^P_ , _x^ otherwise where z is chosen distinct from the names in R, ^P_ and ^Q_ Foundations of Interaction ETAPS `05

Substitution • Semantic substitution -- same as above except for drop where the process is instantiated at substitution time (_x^) _{^Q_/^P_}^@ Q if x N^P_ , _x^ otherwise • Examples w^y[z]_{u/z} = w^y[u]_w[^y[z]_]{u/z} = w[^y[z]_] w^_x^_{^Q_/x} = w^Q_ Foundations of Interaction ETAPS `05

Operational semantics The operational semantics is given by a reduction relation   PROC  PROC recursively specified by the following rules. comm: xsrc Nxtrgt xsrc^P_ | xtrgt(y).Q Q _{^P_/y}^ par: P  P P | Q  P | Q equiv: P  P, P  Q, Q  P P  Q Foundations of Interaction ETAPS `05

Replication • Replication is defined by the following equation D(x) = x(y).(_y^ | x[y] ) !xP=D(x)| x^P | D(x)_ x(y).(_y^ | x[y] ) |x^P | D(x)_  P | D(x) | x[_P | D(x)^] =P |D(x) | x^P | D(x)_ • Replication is defined by the following equation D(x) = x(y).(_y^ | x[y] ) !xP=D(x)| x^P | D(x)_ x(y).(_y^ | x[y] ) |x^P | D(x)_  P | D(x) | x[_P | D(x)^] =P |D(x) | x^P | D(x)_ • Replication is defined by the following equation D(x) = x(y).(_y^ | x[y] ) !xP=D(x)| x^P | D(x)_ x(y).(_y^ | x[y] ) |x^P | D(x)_  P | D(x) | x[_P | D(x)^] =P |D(x) | x^P | D(x)_ Foundations of Interaction ETAPS `05

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)m[^0_] _ [P]=[P](h) | h [^h[^0_] _] [(x)P](h)=h(x). (h^x[^0_]_ | [P](h)) [! x(y).P](h)=h(z).(h^z[^0_]_ | z^x(y).(D(z) | [P](h))_ | D(z)) where z fn(P) and D(z) as in replication Foundations of Interaction ETAPS `05

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 y[v] 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] Foundations of Interaction ETAPS `05

180 6x104 6x1010 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}^ Foundations of Interaction ETAPS `05

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 Foundations of Interaction ETAPS `05

Ex nihilo: a reflective higher-order process calculus