Bisimulation Lattice of Chi Processes

Chi calculus was proposed as a process algebra that has a uniform treatment of names. The paper carries out a systematic study of bisimilarities for chi processes. The notion of L-bisimilarity is introduced to give a possible classification of bisimilarities on chi processes. It is shown that the set of L-bisimilarities forms a four element lattice and that well-known bisimilarities for chi processes fit into the lattice hierarchy. The four distinct L-bisimilarities give rise to four congruence relations. Complete axiomatization system is given for each of the four relations. The bisimulation lattice of asynchronous chi processes and that of asymmetric chi processes are also investigated. It turns out that the former consists of two elements while the latter twelve elements. Finally it is pointed out that the asynchronous asymmetric chi calculus has a bisimulation lattice of eight elements. The χ-calculus ([4]) was introduced with two motivations in mind. One is to remove the ad hoc nature of prefix operation in π-calculus ([11]) by having a uniform treatment of names ([3]), thus arriving at a conceptually simpler language. The second is to materialize a communication-as-cut-elimination viewpoint ([5]), therefore taking up a proof theoretical approach to concurrency theory, an approach that has been proved very fruitful in the functional world. Independently Parrow and Victor have come up with essentially the same language, Update Calculus as they term it ([13]). They share, we believe, the first motivation but have quite a different second one originated from concurrent constraint programming. The difference between π and χ lies mainly in the way communications happen. The former adopts the familiar value-passing mechanism whereas the latter takes an information exchange or information update viewpoint. The algebraic theory of the language has been investigated in the above mentioned papers. Parrow and Victor have looked into strong bisimilarity and axiomatization of it for Update Calculus, while Fu has examined an observational bisimilarity for χ-processes. More recently, Parrow and Victor have proposed Fusion Calculus ([14, 17]), which is a polyadic version of χ-calculus. The authors have also studied an observational equivalence called weak hyperbisimilarity. What we know about the language, albeit little, tells us that it can practically do everything π can do and its algebraic properties are just as satisfactory. The studies carried out so far are however preliminary. ? ASIAN ’98 , Lecture Notes in Computer Science 1538, 245-262, 1998. ?? Supported by the National Nature Science Foundation of China. The objective of this paper is to continue our examination of the algebraic theory of χ-calculus. Section 1 reviews the operational semantics of χ. Section 2 defines L-bisimilarities and investigates their relationship. Section 3 gives alternative characterizations of L-bisimilarities. Section 4 presents a complete axiomatization system for each of the congruence relations induced by the four L-bisimilarities. The next three sections look into the L-bisimilarities of asynchronous, asymmetric, asynchronous asymmetric χ-calculi respectively. 1 Operational Semantics In π-calculus there are two kinds of closed names, one has dummy names as x in m(x).P and local names as x in (x)mx.P . In simple words, the χ-calculus is obtained from π-calculus by unifying these names. This identification forces a unification of input and output prefix operations. The two π-processes just mentioned then turn into (x)m[x].P and (x)m[x].P respectively. In the resulting calculus communications are completely symmetric as exemplified by the following reductions: m[x].P |m[x].Q τ −→ P |Q (x)(R|(m[y].P |m[x].Q)) τ −→ R[y/x]|(P [y/x]|Q[y/x]), where y 6= x (x)m[x].P |(y)m[y].Q τ −→ (z)(P [z/x]|Q[z/y]), where z is fresh The reader is referred to [3–6, 13, 14, 17] for more explanations and examples. Let N be a set of names, ranged over by lower case letters. N , the set of conames, denotes {x | x ∈ N}. The following conventions will be used: α ranges over N ∪N , μ over {τ} ∪ {α[x], αx | x ∈ N}, and δ over {τ} ∪ {α[x], αx, [y/x] | x, y ∈ N}. The set C of χ-processes are defined by BNF as follows: P := 0 | α[x].P | P |P | (x)P | [x=y]P | P+P The process α[x].P is in prefix form. Here α or α is the subject name, and x the object name, of the prefix. The composition operator “|” is standard. In (x)P the name x is declared local; it cannot be seen from outside. The set of global names, or nonlocal names, in P is denoted by gn(P ). We will adopt the α-convention saying that a local name in a process can be replaced by a fresh name without changing the syntax of the process. The choice combinator ‘+’ is well-known. The process P+Q acts either as P or as Q exclusively. In this paper we leave out the replication operator. The result of this paper would not be affected had it been included. The operational semantics can be defined either by reduction semantics ([4]) or in terms of a labeled transition system ([3]). Here we opt for a pure transition semantics as it helps to present our results with clear-cut proofs. The labeled transition system given below defines an early semantics. The reason to use an early semantics is that the definition of weak bisimulation is more succinct in early semantics than in late semantics. In the following formulation, symmetric rules are systematically omitted: α[x].P α[x] −→ P Sqn P δ −→ P ′ [x=x]P δ −→ P ′Cnd P δ −→ P ′ P+Q δ −→ P ′ Sum P μ −→ P ′ P |Q μ −→ P ′|Q0 P [y/x] −→ P ′ P |Q [y/x] −→ P ′|Q[y/x]1 P αx −→ P ′ Q α[x] −→ Q′ P |Q τ −→ P ′|Q′ Cmm0 P αx −→ P ′ Q αx −→ Q′ x 6∈ gn(P |Q) P |Q τ −→ (x)(P ′|Q′) Cmm1 P α[x] −→ P ′ Q α[x] −→ Q′ P |Q τ −→ P ′|Q′ Cmm2 P α[x] −→ P ′ Q α[y] −→ Q′ x 6= y P |Q [y/x] −→ P ′[y/x]|Q′[y/x] Cmm3 P δ −→ P ′ x 6∈ n(δ) (x)P δ −→ (x)P ′ Loc0 P α[x] −→ P ′ x 6∈ {α, α} (x)P αy −→ P ′[y/x] Loc1 P [y/x] −→ P ′ (x)P τ −→ P ′ Loc2 Labeled transitions of the form [y/x] −→, called update transitions, are first introduced in [3, 13] to help define communications in a transition semantics. In applying Loc1 local names need be renamed if necessary to prevent y from being captured. In Loc0, n(δ) denotes the set of names appeared in δ. The notation [y/x] occurred in P [y/x] for example is an atomic substitution of y for x. A general substitution σ is the composition of atomic substitutions, whose effect is defined by P [y1/x1] . . . [yn/xn] def = (P [y1/x1] . . . [yn−1/xn−1])[yn/xn]. The composition of zero atomic substitution is an empty substitution [] whose effect is vacuous. The next lemma collects some technical results whose proofs are simple inductions on derivation. Lemma 1. (i) If P μ −→ P ′ then Pσ μσ −→ P ′σ. (ii) If P [y/x] −→ P ′ and xσ 6= yσ then Pσ [yσ/xσ] −→ P ′σ[yσ/xσ]. (iii) If P [y/x] −→ P ′ and xσ = yσ then Pσ τ −→ P ′σ. (iv) If P [y/x] =⇒ P ′ then P [x/y] =⇒ P ′[x/y]. (v) P αx −→ P ′ if and only if P αz −→ P1 for some fresh z such that P ′ ≡ P1[x/z]. (vi) Suppose a 6∈ gn(P ). If (x)(P |a[x]) τ −→ ay −→ P ′ then (x)(P |a[x]) ay −→ τ −→ P ′. (vii) Suppose a 6∈ gn(P ). If (x)(P |a[x]) τ −→ P ′|a[y] then P [y/x] −→ P ′. Let =⇒ be the reflexive and transitive closure of τ −→. We will write μ =⇒ ( δ =⇒) for =⇒ μ −→=⇒ (=⇒ δ −→=⇒). We will also write μ̂ =⇒ ( δ̂ =⇒) for μ =⇒ ( δ =⇒) if μ 6= τ (δ 6= τ) and for =⇒ otherwise. A sequence of names x1, . . . , xn will be abbreviated to x; and consequently (x1) . . . (xn)P will be abbreviated to (x)P . When the length of x is zero, (x)P is just P . 2 Bisimulation Lattice We introduce in this section L-bisimilarities, which are refinement of the local bisimilarity of [4]. The reason to study L-bisimilarities is that they provide a framework to understand bisimilarity relations of interest. In a symmetric calculus such as χ, it does not make much sense to say that an action with positive, respectively negative, subject name is an input, respectively output, action. An action is an input or output, depending on if the object name is being received or being sent out. Let o denote the set {a[x] | a, x ∈ N} of output actions, o the set {a[x] | a, x ∈ N} of co-output actions, i the set {ax | a, x ∈ N} of input actions, i the set {ax | a, x ∈ N} of co-input actions and u the set {[y/x] | x, y ∈ N} of updates. Let L stand for {∪S | S ⊆ {o, o, i, i, u}∧S 6= ∅}. Definition 2. Let R be a binary symmetric relation on C and let L be an element of L. The relation R is an L-bisimulation if whenever PRQ then for any process R and any sequence x of names it holds that if (x)(P |R) φ −→ P ′ for φ ∈ L ∪ {τ} then there exists some Q′ such that (x)(Q|R) φ̂ =⇒ Q′ and P ′RQ′. The L-bisimilarity ≈L is the largest L-bisimulation. This is a uniform definition of 31 L-bisimilarities. The intuition behind is that ≈L is what an observer recognizes if he/she is capable of observing actions in L and only in L. We will show that the L-bisimilarities collapse to four distinct relations. In the rest of this section let L be an arbitrarily fixed element of L. First we establish a few technical lemmas. The next one follows directly from definition. Lemma 3. If P =⇒ P1 ≈L Q and Q =⇒ Q1 ≈L P then P ≈L Q. For φ ∈ L, let 〈φ〉 be a process such that (i) 〈φ〉 φ −→ 0 and (ii) if 〈φ〉 φ −→ A then A ≡ 0. Lemma 4. Suppose a 6∈ gn(P |Q). Then (i) (x)(P |a[x]) ≈L (x)(Q|a[x]) implies P ≈L Q; and (ii) P |a[x] ≈L Q|a[x] implies P ≈L Q. Proof. (i) Suppose φ ∈ L and n(φ) ∩ gn(P |Q) = ∅. As (x)(P |a[x])|a[x].〈φ〉 φ =⇒ (P |0)|0,Q1 exists such that (x)(Q|a[x])|a[x].〈φ〉 φ =⇒ (Q1|0)|0 ≈L (P |0)|0, which implies (x)(Q|a[x]) ax =⇒ Q1|0 ≈L P |0, which in turn implies Q =⇒ Q1. Similarly P1 exists such that P =⇒ P1 ≈L Q. By Lemma 3, P ≈L Q. (ii) can be proved similarly. ut Lemma 5. If P ≈L Q then Pσ ≈L Qσ for an arbitrary substitution σ. Proof. Suppose P ≈L Q. We only have to show that for x ∈ gn(P |Q) and y 6= x one has that P [y/x] ≈L Q[y/x]. Let b be a distinct fresh name. Suppose φ ∈ L and n(φ) ∩ gn(P |Q) = ∅. By definition the actions (x)(P |(b[y]|b[x].〈φ〉)) φ =⇒ P [y/x]|(0|0) must be matched up by (x)(Q|(b[y]|b[x].〈φ〉)) φ =⇒ Q1|(0|0). (1) If (x)(Q|(b[y]|b[x].〈φ〉)) τ −→ (x)(Q2|(b[y]|b[x].〈φ〉)) then by symmetry and αconvention the reduction is the same as (x)(Q|(b[y]|b[x].〈φ〉)) τ −→ (x)(Q3|(b[y]|b[x].〈φ〉)) such that Q τ −→ Q3. It follows that (1) can be factorized as follows (x)(Q|(b[y]|b[x].〈φ〉)) =⇒ (x)(Q′|(b[y]|b[x].〈φ〉)) τ −→ Q′[y/x]|(0|〈φ〉) φ =⇒ Q1|(0|0) for some Q′ and Q1 such that Q =⇒ Q′ and Q′[y/x] =⇒ Q1 ≈L P [y/x]. By Lemma 1, Q[y/x] =⇒ Q′[y/x]. Similarly P1 exists such that P [y/x] =⇒ P1 ≈L Q[y/x]. By Lemma 3, P [y/x] ≈L Q[y/x]. ut By definition the L-bisimilarity is closed under localization and composition. Using Lemma 5 it can be easily seen that ≈L is closed under prefix operation. Theorem 6. If P ≈L Q and O ∈ C then (i) α[x].P ≈L α[x].Q; (ii) P |O ≈L Q|O; (iii) (x)P ≈L (x)Q; and (iv) [x=y]P ≈L [x=y]Q. We investigate next the order structure of L-bisimilarities. Theorem 7. The following properties hold of the L-bisimilarities: (i) ≈o 6⊆≈o; ≈o 6⊆≈o. (ii) ≈L⊆≈u. (iii) ≈L⊆≈i=≈i. Proof. (i) It is obvious that (x)a[x].(b)(b[x]|b[z]) 6≈o a[z]+(x)a[x].(b)(b[x]|b[z]). It takes a while to see that (x)a[x].(b)(b[x]|b[z]) ≈o a[z]+(x)a[x].(b)(b[x]|b[z]). (ii) To prove ≈L⊆≈u, one only has to show that if P ≈L Q and P [y/x] −→ P ′ then Q′ exists such that Q [y/x] =⇒ Q′ and P ′ ≈L Q′. Now P [y/x] −→ P ′ implies that (x)(P |a[x]) τ −→ P ′|a[y] for a fresh a. So (x)(Q|a[x]) =⇒ Q′|a[y] for some Q′ such that P ′|a[y] ≈L Q′|a[y]. It follows from Lemma 4 that P ′ ≈L Q′. Clearly (x)(Q|a[x]) =⇒ Q′|a[y] can be factorized as (x)(Q|a[x]) =⇒ (x)(Q1|a[x]) τ −→ Q2|a[y] =⇒ Q′|a[y], where Q =⇒ Q1. By (vii) of Lemma 1, (x)(Q1|a[x]) τ −→ Q2|a[y] implies Q1 [y/x] −→ Q2. Hence Q [y/x] =⇒ Q′. (iii) Assume P ≈L Q and P αx −→ P ′. Suppose φ ∈ L and n(φ)∩gn(P |Q) = ∅. Now P |(α[z]+〈φ〉) τ −→ P1|0 for some fresh z such that P ′ ≡ P1[x/z]. There has to be some Q1 such that Q|(α[z]+〈φ〉) =⇒ Q1|0 ≈L P1|0. So Q αz =⇒ Q1 ≈L P1. Therefore Q αx =⇒ Q1[x/z] ≈L P1[x/z] ≡ P ′ by Lemma 1 and Lemma 5. Hence ≈L⊆≈i=≈i. ut