Process Algebra C alculus of C ommunicating S ystems

Process AlgebraCalculus of Communicating Systems Daniel Choi Provable Software Lab. KAIST

Content • Introduction • Calculus of Communicating Systems • Equivalence for CCS • Discussions Provable Software Laboratory Seminar

Why are we going to study Process Algebra? • Need • Mathematical models for concurrent communicating processes? • How • How can we define a mathematical models for concurrent communicating process? Provable Software Laboratory Seminar

The reason why we study Process Algebra • Need • Mathematical models for concurrent communicating processes? • Process Algebra, Petri Net, etc. • How • How can we define a mathematical models for concurrent communicating process? • By defining structured operational semantics Provable Software Laboratory Seminar

Families of Algebraic Approaches • Milner’s Calculus of Communicating Systems (CCS) • Hoare’s theory of Communicating Sequential Processes • The Algebra of Communicating Processes (ACP) of Bergstra & Klop Provable Software Laboratory Seminar

Content • Introduction • Why are we going to study Process Algebra? • Calculus of Communicating Systems • Definitions • Operational Semantic of CCS Terms • Examples • Equivalence for CCS • Discussions Provable Software Laboratory Seminar

DefinitionsTheoretical definitions • Assume a non-empty set S of states, together with a finite, non-empty set of transition labels A and a finite set of predicate symbols • Signature • Consist of a finite set of function symbols f, g, … where each function symbol f has an arity ar(f), being its number of arguments. • Symbol of arity zero : constant (a, b, c, …) • Countably infinite set of variables (x, y, z, …) • Finite non-empty set A of (atomic) actions • Each atomic action a is a constant that can execute itself, after which it terminates successfully. • Term • Set T(∑) of open terms, s, t, u, … over ∑ is defined as the least set satisfying • Each variable is T(∑); • If f ∈ ∑ and t1, …, tar(f) ∈ T(∑), then f(t1, …, tar(f)) ∈ T(∑) • A term is closed if it does not contain variables. The set of closed terms is denoted by T(∑) • Labeled transition system • A transition is a triple (s, a, s’) with a ∈ A, or a pair (s, P) with P a predicate, where s, s’ ∈ S. A labeled transition system (LTS) is a possibly infinite set of transitions. An LTS is finitely branching if each of its states has only finitely many outgoing transitions • The states of an LTS are always the closed terms over a signature ∑. • In view of the syntactic structure of closed terms over a signature, such transitions can be derived by means of inductive proof rules, where the validity of a number of transitions (the premises) may imply the validity of some other transition (the conclusion) • Process graph • A Process (graph) p is an LTS in which one state s is elected to be the root. If the LTS contains a transition s – a -> s’, then p – a -> p’ where p’ has root state s’. Moreover, if the LTS contains a transition sP, then pP. Provable Software Laboratory Seminar

DefinitionsTheoretical definitions • Assume a non-empty set S of states, together with a finite, non-empty set of transition labels A and a finite set of predicate symbols • Signature • Consist of a finite set of function symbols f, g, … where each function symbol f has an arity ar(f), being its number of arguments. • Symbol of arity zero : constant (a, b, c, …) • Countably infinite set of variables (x, y, z, …) • Finite non-empty set A of (atomic) actions • Each atomic action a is a constant that can execute itself, after which it terminates succefully. • Term • Set T(∑) of open terms, s, t, u, … over ∑ is defined as the least set satisfying • Each variable is T(∑); • If f ∈ ∑ and t1, …, tar(f) ∈ T(∑), then f(t1, …, tar(f)) ∈ T(∑) • A term is closed if it does not contain variables. The set of closed terms is denoted by T(∑) • Labeled transition system • A transition is a triple (s, a, s’) with a ∈ A, or a pair (s, P) with P a predicate, where s, s’ ∈ S. A labeled transition system (LTS) is a possibly infinite set of transitions. An LTS is finitely branching if each of its states has only finitely many outgoing transitions • The states of an LTS are always the closed terms over a signature ∑. • In view of the syntactic structure of closed terms over a signature, such transitions can be derived by means of inductive proof rules, where the validity of a number of transitions (the premises) may imply the validity of some other transition (the conclusion) • Process graph • A Process (graph) p is an LTS in which one state s is elected to be the root. If the LTS contains a transition s – a -> s’, then p – a -> p’ where p’ has root state s’. Moreover, if the LTS contains a transition sP, then pP. Provable Software Laboratory Seminar

DefinitionsDefinitions in CCS • Actions • Atomic • uninterruptible execution steps(with some other internal computation steps(τ)) • Representing potential interactions with its environment (inputs/outputs on ports) Provable Software Laboratory Seminar

DefinitionsOperator of CCS • nil • terminated process that has finished execution • a.p • Capable first of a and then behaves like p • + • Choice construct • p1 + p2 offers the potential of behaving like either p1 or p2, depending on the interactions offered by the environment • | • parallel composition • p1 | p2 offers interleaves the execution of p1 and p2 • Permitting complementary actions of p1 and p2 to synchronize (τ) • Restriction operator • Permits actions to be localized within a system • [f] • Actions in a process to be renamed • P[f] behaves exactly like p except that f is applied to each action that p wishes to engage in • Defining equation • C represents a valid system Provable Software Laboratory Seminar

DefinitionsLabeled transition Systems • Labeled transition system (LTS) • Triple <Q, A,→> • Q : a set of states • A : a set of actions • → : transition relation →⊆Qⅹ A ⅹ Q • B = ((a.(b.B + c.0) + b.0)|a’.0 )\a b (B|0)\a … ((b.B + c.0)|0)\a τ c (0|0)\a B b (0|a’.0)\a Temporal Structure Provable Software Laboratory Seminar

Operation Semantics of CCS Terms Referenced from lecture note of Prof. Kim Provable Software Laboratory Seminar

ExamplesLovers • Assume that there is a man and a woman in the society • Man and Woman can manifest their emotion independently (concurrently) • M = ‘man.(acc.M‘+ rej.M) • W = man.(‘acc.W’ + ‘rej.W) • M’ = lov.M’ + ‘lov.M’ + ‘neg_man.M • W’ = lov.W’ + ‘lov.W’ + neg_man.W • Does L = (M|W) is a model of happy lovers? Provable Software Laboratory Seminar

ExamplesLTS of Happy lovers M|W τ (acc.M‘+ rej.M)|(’acc.W’ + ‘rej.W) τ M‘|W’ HL = (M|W) \{man, lov, acc, rej} τ Provable Software Laboratory Seminar

ExamplesLTS of Happy lovers M|W τ (acc.M‘+ rej.M)|(’acc.W’ + ‘rej.W) proc HL = (M|W)\{manifest,love,neg_manifest,accept,reject} proc UHL = (M|W) proc M = 'manifest.(accept.M1 + reject.M) proc W = manifest.('accept.W1 + 'reject.W) proc M1 = love.M1 + 'love.M1 + 'neg_manifest.M proc W1 = 'love.W1 + love.W1 + neg_manifest.W τ M‘|W’ HL = (M|W) \{man, lov, acc, rej} τ Provable Software Laboratory Seminar

ExamplesProof • Proof of (M|W)\{man, lov, acc, rej} => (M|W)\{man, lov, acc, rej} (M|W)\{man, lov, acc, rej} -τ->((acc.M‘+ rej.M)|(’acc.W’ + ‘rej.W)) \{man, lov, acc, rej} Provable Software Laboratory Seminar

ExamplesProof • Proof of (M|W)\{man, lov, acc, rej} => (M|W)\{man, lov, acc, rej} Res (M|W)\{man, lov, acc, rej} -τ->((acc.M‘+ rej.M)|(’acc.W’ + ‘rej.W)) \{man, lov, acc, rej} Provable Software Laboratory Seminar

ExamplesProof • Proof of (M|W)\{man, lov, acc, rej} => (M|W)\{man, lov, acc, rej} Res ‘man.(acc.M‘+ rej.M) | man.(‘acc.W’ + ‘rej.W) -τ->(acc.M‘+ rej.M)|(’acc.W’ + ‘rej.W) (M|W)\{man, lov, acc, rej} -τ->((acc.M‘+ rej.M)|(’acc.W’ + ‘rej.W)) \{man, lov, acc, rej} Provable Software Laboratory Seminar

ExamplesProof • Proof of (M|W)\{man, lov, acc, rej} => (M|W)\{man, lov, acc, rej} Parτ Res ‘man.(acc.M‘+ rej.M) | man.(‘acc.W’ + ‘rej.W) -τ->(acc.M‘+ rej.M)|(’acc.W’ + ‘rej.W) (M|W)\{man, lov, acc, rej} -τ->((acc.M‘+ rej.M)|(’acc.W’ + ‘rej.W)) \{man, lov, acc, rej} Provable Software Laboratory Seminar

ExamplesProof • Proof of (M|W)\{man, lov, acc, rej} => (M|W)\{man, lov, acc, rej} Act Act ‘man.(acc.M‘+ rej.M) – ‘man-> (acc.M‘+ rej.M) man.(‘acc.W’ + ‘rej.W) – man-> (‘acc.W’ + ‘rej.W) Parτ Res ‘man.(acc.M‘+ rej.M) | man.(‘acc.W’ + ‘rej.W) -τ->(acc.M‘+ rej.M)|(’acc.W’ + ‘rej.W) (M|W)\{man, lov, acc, rej} -τ->((acc.M‘+ rej.M)|(’acc.W’ + ‘rej.W)) \{man, lov, acc, rej} Provable Software Laboratory Seminar

ExamplesProof • Proof of (M|W)\{man, lov, acc, rej} => (M|W)\{man, lov, acc, rej} Act Act ‘acc.W‘ – ‘acc -> W’ acc.M‘ – acc -> M’ ChoiceL ChoiceL (acc.M‘+ rej.M) - acc-> M’ (‘acc.W’ + ‘rej.W) – ‘acc -> W’ Parτ (acc.M‘+ rej.M) | (‘acc.W’ + ‘rej.W) -τ-> (M’|W’) Res ((acc.M‘+ rej.M)|(’acc.W’ + ‘rej.W)) \{man, lov, acc, rej} -τ-> (M’|W’) \{man, lov, acc, rej} Provable Software Laboratory Seminar

Content • Introduction • Why are we going to study Process Algebra? • Calculus of Communicating Systems • Definitions • Operational Semantic of CCS Terms • Examples • Equivalence for CCS • Trace Equivalence • Strong Bisimulation Equivalence • Weak Bisimulation Equivalence • Discussions Provable Software Laboratory Seminar

Trace EquivalenceDefinition • Language Equivalence • Two machines are equivalent if they accept the same sequences of symbol • Can we directly apply language equivalence to rooted LTS? No • Identify every state in a rooted LTS as being accepting • Definition Let <Q, A,→> be a labeled transition system • Let A* consists of the set of finite sequences of elements of A • Let s = a1 … an∈A* be a sequence of actions. Then q – s-> q’ if there are states q0, ..., qnsuch thatq = q0, qi –ai-> qi+1 and q’ = qn • s is a strong trace of q if there exists q’ such that q – s -> q’. We use S(q) to represent the set of all strong traces of q • p ≈s q exactly when S(p) = S(q) (strong traces do not distinguish between internal and external actions) • Can we use trace equivalence to decide whether two system are behavioral congruent? No Provable Software Laboratory Seminar

Trace Equivalence Definition • Language Equivalence • Two machines are equivalent if they accept the same sequences of symbol • Can we directly apply language equivalence to rooted LTS? No • Identify every state in a rooted LTS as being accepting • Definition Let <Q, A,→> be a labeled transition system • Let A* consists of the set of finite sequences of elements of A • Let s = a1 … an∈A* be a sequence of actions. Then q – s-> q’ if there are states q0, ..., qnsuch thatq = q0, qi –ai-> qi+1 and q’ = qn • s is a strong trace of q if there exists q’ such that q – s -> q’. We use S(q) to represent the set of all strong traces of q • p ≈s q exactly when S(p) = S(q) (strong traces do not distinguish between internal and external actions) • Can we use trace equivalence to decide whether two system are behavioral congruent? No Provable Software Laboratory Seminar

Trace EquivalenceDefinition • Language Equivalence • Two machines are equivalent if they accept the same sequences of symbol • Can we directly apply language equivalence to rooted LTS? No • Identify every state in a rooted LTS as being accepting • Definition Let <Q, A,→> be a labeled transition system • Let A* consists of the set of finite sequences of elements of A • Let s = a1 … an∈A* be a sequence of actions. Then q – s-> q’ if there are states q0, ..., qnsuch thatq = q0, qi –ai-> qi+1 and q’ = qn • s is a strong trace of q if there exists q’ such that q – s -> q’. We use S(q) to represent the smallest set of all strong traces of q (prefix-closed) • p ≈s q exactly when S(p) = S(q) (strong traces do not distinguish between internal and external actions) • Can we use trace equivalence to decide whether two system are behavioral congruent? No Provable Software Laboratory Seminar

Trace Equivalence Definition • Language Equivalence • Two machines are equivalent if they accept the same sequences of symbol • Can we directly apply language equivalence to rooted LTS? No • Identify every state in a rooted LTS as being accepting • Definition Let <Q, A,→> be a labeled transition system • Let A* consists of the set of finite sequences of elements of A • Let s = a1 … an∈A* be a sequence of actions. Then q – s-> q’ if there are states q0, ..., qnsuch thatq = q0, qi –ai-> qi+1 and q’ = qn • s is a strong trace of q if there exists q’ such that q – s -> q’. We use S(q) to represent the smallest set of all strong traces of q (prefix-closed) • p ≈s q exactly when S(p) = S(q) (strong traces do not distinguish between internal and external actions) • Can we use trace equivalence to decide whether two system are behavioral congruent? No Provable Software Laboratory Seminar

ExampleTrace Equivalence p0 q0 a a a q1 q1’ p1 b c b c p2 p3 q2 q3 P = a.(b.nil + c.nil)S(P) = {ε,a,ab,ac} Q = a.b.nil + a.c.nilS(Q) = {ε,a,ab,ac} Provable Software Laboratory Seminar

ExampleTrace Equivalence p0 q0 a a a S(P) = S(Q) q1 q1’ p1 b c b c p2 p3 q2 q3 P = a.(b.nil + c.nil)S(P) = {ε,a,ab,ac} Q = a.b.nil + a.c.nilS(Q) = {ε,a,ab,ac} Provable Software Laboratory Seminar

ExampleTrace Equivalence p0 q0 a a a S(P) = S(Q) q1 q1’ p1 b c b c p2 p3 q2 q3 Trace Equivalent P = a.(b.nil + c.nil)S(P) = {ε,a,ab,ac} Q = a.b.nil + a.c.nilS(Q) = {ε,a,ab,ac} Provable Software Laboratory Seminar

ExampleTrace Equivalence p0 q0 a a a S(P) = S(Q) q1 q1’ p1 b c b c p2 p3 q2 q3 Trace Equivalent P = a.(b.nil + c.nil)S(P) = {ε,a,ab,ac} Q = a.b.nil + a.c.nilS(Q) = {ε,a,ab,ac} It is not behavioral congruent Provable Software Laboratory Seminar

Strong Bisimulation EquivalenceDefinition • Execution sequences for equivalent systems ought to pass through equivalent states • Definition Let <Q, A,→> be an LTS. A relation R ⊆ Q x Q is a bisimulation if whenever <p, q> ∈R, then the following conditions hold for any a, p’ and q’ • If p –a-> p’ then q – a -> q’ for some q’ such that <p’, q’> ∈R • If q –a-> q’ then p – a -> p’ for some p’ such that <p’, q’> ∈R • Definition System p and q are bisimulation equivalent, or bisimilar, if there exists a bisimulation R containing <p, q>. We write p ~ q whenever p and q are bisimilar Provable Software Laboratory Seminar

Strong Bisimulation EquivalenceHow to find out P and Q are bisimular? • Strong Simulation • Let <Q, A,→> be an LTS, and let S be a binary relation over Q. Then S is called a strong simulation over <Q, A,→> if, whenever pSq, if p – a -> p’ then there exists q’ ∈ Q such that q – a -> q’ and p’ S q’ • q strongly simulates p if there exists a strong simulation S such that pSq Provable Software Laboratory Seminar

Strong Bisimulation EquivalenceHow to find out P and Q are bisimular? : Example p0 q0 a a a q1 q1’ p1 b c b c p2 p3 q2 q3 Suppose, (p0, q0)∈ S Provable Software Laboratory Seminar

Strong Bisimulation EquivalenceHow to find out P and Q are bisimular? : Example Suppose p0 strongly simulates q0, (q0, p0)∈ S or q0Sp0 q0 S p0 a a q1p1 Provable Software Laboratory Seminar

Strong Bisimulation EquivalenceHow to find out P and Q are bisimular? : Example Suppose p0 strongly simulates q0, (q0, p0)∈ S or q0Sp0 q0 S p0 q0 S p0 a a a a q1p1 q1'p1 q1 S p1 q1' S p1 Provable Software Laboratory Seminar

Strong Bisimulation EquivalenceHow to find out P and Q are bisimular? : Example Suppose p0 strongly simulates q0, (q0, p0)∈ S or q0Sp0 q0 S p0 q0 S p0 q1 S p1 a a b a a b q1p1 q1'p1 q2p2 q1 S p1 q1' S p1 q2 S p2 Provable Software Laboratory Seminar

Strong Bisimulation EquivalenceHow to find out P and Q are bisimular? : Example Suppose p0 strongly simulates q0, (q0, p0)∈ S or q0Sp0 q0 S p0 q0 S p0 q1 S p1 q1' S p1 a a b c a a b c q1p1 q1'p1 q2p2 q3p3 q1 S p1 q1' S p1 q2 S p2 q3 S p3 Provable Software Laboratory Seminar

Strong Bisimulation EquivalenceHow to find out P and Q are bisimular? : Example Suppose p0 strongly simulates q0, (q0, p0)∈ S or q0Sp0 q0 S p0 q0 S p0 q1 S p1 q1' S p1 a a b c a a b c q1p1 q1'p1 q2p2 q3p3 q1 S p1 q1' S p1 q2 S p2 q3 S p3 Therefore S = {(q0, p0), (q1, p1), (q1’, p1), (q2, p2), (q3, p3)} Provable Software Laboratory Seminar

Strong Bisimulation EquivalenceHow to find out P and Q are bisimular? : Example Suppose q0 strongly simulates p0, (p0, q0)∈ S or p0Sq0 p0 S q0 p1 S q1 p1 S q1 a b c a b p1q1 p2q2 p3 p1 S q1 q1' S p1 Provable Software Laboratory Seminar

Strong Bisimulation EquivalenceHow to find out P and Q are bisimular? : Example Suppose q0 strongly simulates p0, (p0, q0)∈ S or p0Sq0 p0 S q0 p1 S q1’ p1 S q1’ a c b a c p1q1’ p3q3 p2 p1 S q1’ p3 S q3 Provable Software Laboratory Seminar

Strong Bisimulation EquivalenceHow to find out P and Q are bisimular? • Strong Simulation • Let <Q, A,→> be an LTS, and let S be a binary relation over Q. Then S is called a strong simulation over <Q, A,→> if, whenever pSq, if p – a -> p’ then there exists q’ ∈ Q such that q – a -> q’ and p’ S q’ • q strongly simulates p if there exists a strong simulation S such that pSq • S-1 is the set of pairs (y, x) such that (x, y) ∈ S • Strong bisimulation • A binary relation S over Q is said to be a strong bisimulation over the LTS if both S and its converse are simulations Provable Software Laboratory Seminar

Strong Bisimulation EquivalenceHow to find out P and Q are bisimular? : Example b a p0 p1 q0 a q1 a a a b b p2 a a q2 S = {(p0, q0), (p1, q1), (p2, q1), (p0, q2)} S’ = {(q0, p0), (q1, p1), (q1, p2), (q2, p0)} Provable Software Laboratory Seminar

Strong Bisimulation EquivalenceHow to find out P and Q are bisimular? : Example b a p0 p1 q0 a q1 a a a b b p2 a a q2 Strong Bisimulation S = {(p0, q0), (p1, q1), (p2, q1), (p0, q2)} S’ = {(q0, p0), (q1, p1), (q1, p2), (q2, p0)} Provable Software Laboratory Seminar

Strong Bisimulation EquivalenceHow to find out P and Q are bisimular? : Example a p0 p1 q0 a q1 a b q2 b p2 p3 It is not Strong Bisimulation P strongly simulates Q S = {(q0, p0), (q1, p2), (q2, p3)} Q strongly simulates P S’ = {(p0, q0), (p1, q1), (p2, q1), (p3, q2)} Provable Software Laboratory Seminar

Strong Bisimulation EquivalenceHow to find out P and Q are bisimular? • Strong Simulation • Let <Q, A,→> be an LTS, and let S be a binary relation over Q. Then S is called a strong simulation over <Q, A,→> if, whenever pSq, if p – a -> p’ then there exists q’ ∈ Q such that q – a -> q’ and p’ S q’ • q strongly simulates p if there exists a strong simulation S such that pSq • S-1 is the set of pairs (y, x) such that (x, y) ∈ S • Strong bisimulation (P ~ Q) • A binary relation S over Q is said to be a strong bisimulation over the LTS if both S and its converse are simulations • Strong bisimulation equivalence : reflexive, symmetric, transitive • P ~ Q implies P ≈s Q • What about internal computation τ? • Weak bisimulation Provable Software Laboratory Seminar

Process Algebra C alculus of C ommunicating S ystems