A chemical abstract machine for graph reduction Alan Jeffrey

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1 UNIVERSITY OF SUSSEX COMPUTER SCIENCE A chemical abstract machine for graph reduction Alan Jeffrey Report 3/92 August 1992 Computer Science School of Cognitive and Computing Sciences University of Sussex Brighton BN1 9QH ISSN

2 A chemical abstract machine for graph reduction ALAN JEFFREY ABSTRACT. Graph reduction is an implementation technique for the lay λ-calculus. It has been used to implement many non-strict functional languages, such as lay ML, Gofer and Miranda. Parallel graph reduction allows for concurrent evaluation. In this paper, we present parallel graph reduction as a Chemical Abstract Machine, and show that the resulting testing semantics is adequate wrt testing equivalence for the lay λ-calculus. We also present a π-calculus implementation of the graph reduction machine, and show that the resulting testing semantics is also adequate. 1 Introduction The lay reduction strategy for the λ-calculus investigated by ABRAMSKY (1989) has only two reduction rules: λxeµf EFx E E ¼ EF E ¼ F This can be compared with the full evaluation strategy of BARENDREGT (1984): λxeµf EFx E E ¼ EF E ¼ F F F ¼ EF EF ¼ E E ¼ λxe λxe ¼ If the full evaluation strategy can terminate, then the lay evaluation strategy will. For example, if we define: à λxyx Á λxx λx λyx yyµµ λyx yyµµµ then Á but ÃÁ Áµ, whereas ÃÁ Áµ. However, the lay evaluation strategy is very inefficient, since it may duplicate arguments when applying a function. For example, if we define: 0 Á i 1 λxxxµi Copyright c 1992 Alan Jeffrey. Author s address: COGS, University of Sussex, Brighton BN1 9QH, UK. alanje@cogs.sussex.ac.uk This work was supported by SERC project GR/H Miranda is a trademark of Research Software Limited.

3 2 Alan Jeffrey Then i 2i Á but i 2i 1 2 Á, that is the lay strategy can be exponentially worse than the full strategy. Thus, the early functional languages, such as LISP (MCCARTHY et al., 1962) used a strict reduction scheme rather than the lay reduction scheme. Graph reduction was introduced by WADSWORTH (1971) as a means of efficiently implementing the lay reduction strategy. Rather than reducing syntax trees, we reduce syntax graphs which allows a more efficient representation of sharing. For example, we can represent the reduction of i 1 x x i Á Á Graph reduction has been used to implement non-strict functional languages such as JOHNSSON s lay ML (1984), JONES s Gofer (1992) and TURNER s Miranda (1985). It is discussed in PEYTON JONES s textbook (1987). However, there has been little work in the formal semantics of graph reduction. BARENDREGT et al. (1987) have shown that graph reduction is sound and complete with respect to term reduction. LESTER (1989) has shown that the G-machine of AUGUSTSSON (1984) and JOHNSSON (1984) is adequate wrt a denotational model of the lay λ-calculus. In this paper, we provide an alternative presentation of graph reduction, as a Chemical Abstract Machine (CHAM), in the style of BERRY and BOUDOL (1990). The CHAM was introduced as a way of presenting the operational semantics of parallel languages in a clean fashion. It has been used to give a semantics for MILNER s CCS (1989) and MILNER, PARROW and WALKER s π-calculus (1989). Here, we shall give a semantics for parallel graph reduction with blocking, as described by PEYTON JONES (1987). We will show that this is an adequate semantics for the lay λ-calculus, and that it can be implemented in a variant of the π-calculus. 2 The lay lambda-calculus The λ-calculus, introduced by CHURCH (1941), has the following syntax: E :: x EE λxe where x ranges over an infinite set of variables. This can be given a number of operational semantics, but we shall only look at two of these. We shall call these

4 the lay semantics: and the full semantics: λxeµf EFx A CHAM for graph reduction 3 λxeµf EFx E E ¼ EF E ¼ F E E ¼ EF E ¼ F F F ¼ EF EF ¼ E E ¼ λxe λxe ¼ Here, EFx is E, with every free occurrence of x replaced by F, up to the usual renaming of bound variables. PROPOSITION If E E ¼ and E E ¼¼ then E ¼ E ¼¼. 2. If E and E E ¼ then E ¼. 3. If E µ then E. PROOF. Part 1 is a simple induction. Part 2 follows from Theorem of BARANDREGT (1984), since E E ¼ is the leftmost full reduction of E. Part 3 follows from Theorem of BARANDREGT (1984). ¾ We can define a variant of MORRIS s testing pre-order (BARENDREGT, 1984, Exercise ): E Ú F iff C CF µ CE We can also define a variant of the λ-calculus with recursive declarations and strictness annotations: Here: M :: x xy λxm Öx : D Ò M D ::?M!M Öx :?M ÒN declares x recursively to be M in the context N. For example, a fixed point of f is Öx :? f Ò Öy :?xy Ò y. Öx :!M Ò N is the same, except that x is strict in N, and so evaluation of M can be sparked off as a parallel computation. We shall let bound variables be α-converted. The free variables of M are fv M: fvx x fv xyµ xy fv λxmµ fvm Ò x fv Öx : D ÒMµ fvd fvmµ Ò x fv!mµ fvm fv?mµ fvm

5 4 Alan Jeffrey There is a translation from the λ-calculus to the λ-calculus with Ö: x x EF Öx :!E Ò Öy :?F Ò xy λxe λxe Note that in the translation of EF, we know that E will be used, and so it can be evaluated strictly. On the other hand, we do not know if F will be used or not, so it cannot be annotated. 3 The chemical abstract machine The Chemical Abstract Machine (CHAM) of BERRY and BOUDOL (1990) is a way of presenting the operational semantics of parallel systems. We shall use it to give a semantics for parallel graph reduction of the λ-calculus with Ö. A CHAM gives reductions between solutions, which are multisets (or bags) of molecules. The definition of molecules is specific to each CHAM, but a solution can always be regarded as a molecule. In such a molecule, m 1 m n, the multiset brackets are called a membrane. There are three types of reduction: Heating rules, of the form S S ¼. Cooling rules, of the form S S ¼. Reaction rules, of the form S S ¼. Heating and cooling rules are always given in pairs S µ S ¼, whereas reaction rules are irreversible. We shall write µ for the transitive, reflexive, symmetric closure of µ, write for µ µ, and let µ range over, and. All CHAMs have the following structural rules, where m is a molecule containing precisely one hole: S µ S ¼ S S ¼¼ µ S ¼ S ¼¼ S µ S ¼ ms µ ms ¼ In addition, the CHAMs we shall consider in this paper allow the outermost membrane of any solution to be ignored. This allows us to write m 1 m n µ m ¼ 1 m¼ n ¼ for m 1 m n µ m ¼ 1 m¼ n ¼ : S µ S The molecules and reduction rules are specific to each CHAM. In the case of the graph reduction CHAM, molecules are defined: The free variables of m are fvm: m :: x : D S νxs fv x : Dµ x fvd

6 The defined variables of m are dvm: A CHAM for graph reduction 5 fvm 1 m n fvm 1 fvm n fv νxmµ fvm Ò x dv x : Dµ x dvm 1 m n dvm 1 dvm n dv νxmµ dvm Ò x We shall only consider solutions which do not define any variables twice, so in any solution m 1 m n, the defined variables of each m i are distinct. For example, we do not allow solutions such as νxx :!λwmx :!λwny :!xw which could reduce nondeterministically to y :!M or to y :!N. If x x 1 x n then we can write νxm for νxm and ν xm for νx 1 νx n m. Define: a molecule is a positive ion with valency x iff it is x :?M or x :!λym. a molecule is a negative ion with valency y iff it is x :!y or x :!y. a molecule is ionic iff it is a positive or negative ion. a solution is plasmic iff it is νỹm 1 m n or m 1 m n where each m i is ionic. A plasma is positive (negative) iff it contains only positive (negative) ions Plasmas can be regarded as graphs, for example the graph reduction:!λx!x?x? i? ² º? i? ² º! i 7i? ² º!λx!x!λx?!x!λx!x?!λx!x!λx!x!λx!x!λx!x

7 6 Alan Jeffrey is represented by the CHAM reduction: In these diagrams: νxy :!xyy :? ix :!λwww νuvy :!uvy :? iv :?yu :!y νuvy :!uvy :! iv :?yu :!y 7i νuvy :!uvy :! Áv :?yu :!y νuvy :!uvy :! Áv :?yu :! Á νvy :!vy :! Áv :?y νvy :!vy :! Áv :!y νv :!vv :! Á :! Á Tagged nodes x :!M are labelled with a!. Untagged nodes x :?M are labelled with a?. Application nodes x : y are labelled with Indirection nodes x : y are labelled with a y, if y is free, and with otherwise. Function nodes x : λym are labelled with a λy, and have the graph for :!M drawn beneath them, for some fresh variable. The supercombinators of HUGHES (1984) can be regarded as a form of solution. PEYTON JONES (1987) presents supercombinators as a collection of definitions, plus a variable to be evaluated. These are drawn: Definitions Variable These can be regarded as solutions. For example, the supercombinator program: is equivalent to the solution: $X 0 x 0 E 0 $X 1 x 1 E 1.. $X n x n E n $X 0 νx 0 X n X 0 :!λ x 0 E 0 X 1 :?λ x 1 E 1 X n :?λ x n E n The most important heating rule allows recursive declarations to become part of a solution, whilst hiding the bound variable. This is only valid when it would

8 A CHAM for graph reduction 7 not cause the free variable x to become bound by y, which we can achieve by α-converting y first. x :! Öy : D ÒM µ νyx :!My : D x yµ The scope of a hidden variable can migrate, as long as this does not result in variable capture: mνxm ¼ µ νxmm ¼ x ¾ fvmµ Hidden variables may be α-converted, exchanged and evaporated: νxm µ νy myx µ νxym µ νyxm νx µ y ¾ fvmµ Finally, we can perform garbage collection on positive plasmas, since a hidden positive plasma can never make any reductions: ν x x : D µ x : D is a positive plamaµ We shall sometimes write µ γ for this thermal action, and µγ for any other thermal action. For example, the graph reduction:!á?á?á can be derived: νyxx :!Áy :?Á :!xy νyxx :!Áy :?Á :!y νyxx :!Áy :?Á :!y νyνxx :!Á y :?Á :!y γ νyy :?Á :!y νyy :?Á :!y A reaction can occur whenever one positive and one negative ion with the same valency exist in a solution. Since there are two kinds of positive ion and two kinds of negative ion, there are four reaction rules. The first two allow untagged molecules to become tagged: x :!yy :?M x :!yy :!M x :!yy :?M x :!yy :!M

9 8 Alan Jeffrey These can be drawn:?m!m º?M º!M This models the first phase of graph reduction we search along the spine of a graph, tagging nodes for evaluation. Note that strict reactions do not occur: x :!y :?M x :!y :!M If a tagged indirection node points to a function, we can just copy the function. The SKIM (STOYE et al., 1984) and G-machine (JOHNNSON, 1984) use this as a method of eliminating indirection nodes. It was shown by LESTER (1989) to be adequate: This can be drawn: x :!yy :!λwm x :!λwmy :!λwm!λwm!λwm!λwm If an application node points to a function, it can be β-reduced: This can be drawn: x :!yy :!λwm x :!Mw y :!λwm!λw N M w w M!λw ²º M N w w We shall sometimes write β for this reaction, and β for any other reaction. This CHAM is summaried in Table 1. This CHAM implements the algorithm for parallel graph reduction described by PEYTON JONES (1987). A process is assigned to evaluating a node, which is tagged. It then searches along the spine, tagging each node as it passes. If

10 A CHAM for graph reduction 9 S µ S x :! Ö y : D Ò M µ νyx :!My : D x yµ mνxm ¼ µ νxmm¼ x ¾ fvmµ νxm µ νy myx µ νxym µ νyxm νx µ ν x x : D µ x :!yy :?M x :!yy :!M x :!yy :?M x :!yy :!M y ¾ fvmµ x :!yy :!λwm x :!λwmy :!λwm x : D is a positive plasmaµ x :!yy :!λwm x :!Mw y :!λwm TABLE 1. Summary of the graph reduction CHAM it reaches a function node which can be β-reduced, it does so. If it reaches a function node which cannot be β-reduced, this is returned as the result. If it reaches a previously tagged application or indirection node, it is blocked until the tagged node is evaluated. For example, in the graph: x ²?M?N only one process will evaluate M. This is mirrored in the CHAM by the fact that M will only be reduced once. However, this algorithm produces some surprising results with cyclic graphs. The solution y :! Öx :!x Ò x heats to become the plasma νxy :!xx :!x and the graph: y This has no reductions, because it is negative. This is mirrored in the parallel graph reduction algorithm, since the process evaluating y will discover that the indirection node at x has already been tagged. Thus, it is possible for evaluations to deadlock, when a sequential algorithm would diverge. Our translation of the λ-calculus will not produce cyclic graphs, although it

11 10 Alan Jeffrey can still produce divergent terms. For example, the translation of λxxxµ λxxxµ has the reductions: λx λx x x x x λx x x λx x x Since E has no tight loops, we will be able to show that the CHAM semantics for the λ-calculus is adequate. To do this, we define the testing preorder on molecules: m Ú m ¼ iff C Cm ¼ µ Cm and show in the next section that the CHAM semantics is adequate, that is if x :?Eµ Ú x :?Fµ then E Ú F. PROPOSITION For any S there is a plasmic S ¼ such that S γ S ¼. 2. If S is plasmic and S S ¼ then S µ S ¼. 3. S S ¼ iff S γ µ S ¼ 4. S γ µ iff S 5. S β iff S βγ 6. S β 7. If S S ¼ and S S ¼¼ then S ¼ µ S ¼¼ or S ¼ S ¼¼¼ and S ¼¼ S ¼¼¼. 8. S iff S PROOF. 1. Prove by induction on S. 2. Prove by case analysis that if S is plasmic and S µ S ¼ then S µ S ¼. Then show by induction on µ that if S µ S ¼ that S µ S ¼. 3. If S S ¼ then by parts 1 and 2, S γ S¼¼ µ S ¼ for some plasmic S ¼¼. Thus S γ µ S ¼. 4. If S then define S i by induction: S 0 S. Assuming S i, S i S ¼, so by part 3, S i γ γ S i 1 µ S ¼, so S i 1.

12 A CHAM for graph reduction 11 Thus, S S 0 γ S 1 γ and so S implies S γ µ. The reverse implication is trivial. 5. Is proved similarly. 6. Define #m by: Then: #m 1 m n #m 1 #m n # νxmµ #m # x : Dµ #D #!Mµ #M #?Mµ #M 1 # λxmµ 0 #y 1 #y 1 # Öx : D Ò Mµ #D #M S µγ S ¼ µ #S #S ¼ S β S ¼ µ #S #S ¼ Thus, S βγ, so by part 5, S β. 7. Show by case analysis that if S is plasmic and S S ¼ and S S ¼¼ then S ¼ S ¼¼ or S ¼ S ¼¼¼ and S ¼¼ S ¼¼¼. Then for any S, if S S ¼ and S S ¼¼ then by parts 1 and 2 there is a plasmic S 1 such that S µ S 1 S1 ¼ µ S ¼ and S 1 S1 ¼¼ µ S ¼¼. Then either S1 ¼ S¼¼ 1, so µ S¼ S ¼¼, or S1 ¼ S¼¼¼ and S1 ¼¼ S¼¼¼, so S ¼ S ¼¼¼ and S ¼¼ S ¼¼¼. 8. If S and S then we must have: Then for 0 i j n 1, define S i j S S 0 0 S0 1 S0 n S0 n 1 S S 0 0 S1 1 Sn n If S i j 1 1 µ S i j 1 then let Si j Si j 1. Otherwise, S i 1 j 2 Si 1 j 1 such that S i 1 j 1 Si j and Si j 1 Si j. such that Si 1 j 1 Si j and Si j 1 Si j : and Si 1 j 2 Si j 1 so by part 7, we can find Si j Then Sn n Sn 1 n which contradicts our hypothesis. Thus, if S then S. ¾ Note that Proposition 2.7 shows that this CHAM is Church-Rosser.

13 12 Alan Jeffrey 4 The CHAM semantics is adequate MILNER, PARROW and WALKER (1989) have shown a very close relationship between the lay reduction strategy and their translation of the λ-calculus into the π-calculus, that E F iff E F. This is not be true of the CHAM semantics for graph reduction, where there is no one-to-one correspondence between CHAM reductions and λ-calculus reductions. For example, the CHAM reduction: x?@?e?g?f x?e?g?f corresponds to the null reduction GFE 0 GFE, and the CHAM reduction:!λx x!λx x corresponds to ÁÁ ÁÁµ ÁÁ. In general, each CHAM reduction S β S ¼ corresponds to a λ-calculus reduction E 0 E, and each CHAM reduction S β S ¼ corresponds to a λ-calculus reduction E E ¼. In order to formalie this, we need to know when a CHAM solution can implement a λ-term. For example, some possible implementations of ÁÁ ÁÁµ are: x! Á! Á! Á! Á x! Á! Á x! Á! Á! Á x! Á We will write S M» E iff M implements E in the solution S. For example, S x» ÁÁ ÁÁµ for any of the above solutions. S M» E is defined in Table 2. PROPOSITION 3.

14 A CHAM for graph reduction 13 xµ S x» x x ¾ dvs S x» E S y» F xyµ S xy» EF S M» E λxµ x ¾ fvs S λxm» λxe S M» E νxµ x ¾ fvme νxs M» E µ S M» E S ¼ M» E S¼ S Sx :!M N» E?µ Sx :?M N» E Sx :!M x» E : Eµ x ¾ fvems S M» E S M» E : Iµ x ¾ fvems Sx :!M x» E TABLE 2. The definition of S M» E. 1. If S M» E and S M» F then E F. 2. If Sx :!y x» E then S y» F, S» G and E FG. 3. If Sx :!λym x» E then S M» F and E λyf. 4. If νxs M» E and x ¾ fvme then S M» E 5. If Sx :!M x» E and Sx :!M N» F then S N» G and F GEx. 6. If Sx :!yy :!M N» E then Sx :!My :!M N» E. 7. If x ¾ fve then x :!E x» E. PROOF. 1. First, prove by induction on the proof of S x»e that if x ¾ dvs and S x»e then E x. Then prove by induction on the proof of S M» E that if S M»E and S M»F then E F. The only tricky case is if S M»E follows from xµ, in which case M E x, and since S x» F the above gives that F x. 2. Prove by induction on the proof that if ν w Sx :!yµ x» E or S y» E then S y» F and S» G and E FG. 3. Prove by induction on the proof that if ν w Sx :!λymµ x» E or S λym»e then for any, Sy M»F and E λyf. The only tricky case is if S λym»e follows from νyµ, in which case S νys ¼ and S ¼ λym»e. By induction, S ¼ uy M» F and E λyf for some fresh u, so by νuµ, νu S ¼ uy µ M»F so by µ, νys ¼ M»F, so S M»F, so since y ¾ fvs,

15 14 Alan Jeffrey Sy M» F. 4. Prove by induction on the proof of ν xxs M» E that if ν xxs M» E and x ¾ fvme that ν xs M» E. 5. Prove by induction on the proof of ν w Sx :!Mµ N»F that if ν w Sx :!Mµ x» E and ν w Sx :!Mµ N» F then S N» G and F GEx. 6. First, prove by induction on the proof of S M» E that if y ¾ fvems and S M» E then Sy :!N M» E. Then prove by induction on the proof of S M» E that if ν w Sx :!yy :!Mµ N» E then Sx :!My :!M N» E and if ν w Sy :!Mµ y» E then Sy :!M M» E. 7. Proved by induction on E. ¾ However, the definition of S M» E ignores whether declarations are tagged or untagged. In order to reason about the reductions of a solution, we shall need to take tagging into account. The sequence x 0 : D 0 x n : D n is an x-spine in S iff: S µ νỹ x : DS ¼ where x x 0 and each x i : D i is a negative ion with valency x i 1. The variable x is lay in a plasma S iff the negative ions of S form an x-spine. The variable x is always lay in S iff for any S S ¼, x is lay in S ¼. For example, the -spine of the plasma: which can be drawn: νỹ :! 0 0 :! 1 y 1 n 1 :! n y n n :?My 0 :?N 1 y n :?N n...?n 1 º?M?N n is :! 0 0 :! 1 y 1 n 1 :! n y n. Since these are the only negative ions, this solution is lay. PROPOSITION 4. x is always lay in x :!E.

16 A CHAM for graph reduction 15 PROOF. Define x is invariantly lay (or i-lay) in S by: x is i-lay in If x is i-lay in S and x y then x is i-lay in νys If y is i-lay in S then x is i-lay in x :!ys. If y is i-lay in S then x is i-lay in x :!ys. If x is i-lay in S and y is i-lay in y :!M then x is i-lay in y :?MS. If x is i-lay in S and for all, y is i-lay in y :!Mw then x is i-lay in y :!λwms. If x is i-lay in S and S µ S ¼ then x is i-lay in S ¼. Then we can prove: If x is i-lay in S then x is lay in S. If x is i-lay in S and S S ¼ then x is i-lay in S ¼. x is i-lay in x :!E. Thus, if x is i-lay in S then x is always lay in S, and so x is always lay in x :!E. ¾ Then we can show the important properties, that link CHAM reductions to λ- calculus reductions. PROPOSITION 5. If S x» E and x is always lay in S then: 1. If S β S ¼ then S ¼ x» E. 2. If S β S ¼ then E E ¼ and S ¼ x» E ¼. 3. If E E ¼ then S β β S ¼, E ¼ E ¼¼ and S ¼ x» E ¼¼. PROOF. 1. If: then S ¼ x» E. If: then S ¼ x» E. If: S µ νỹu :!vv :?M S ¼ µ νỹu :!vv :!M S µ νỹu :!vwv :?M S ¼ µ νỹu :!vwv :!M S µ νỹu :!vv :!λym S ¼ µ νỹu :!λymv :!λym then S ¼ x» E. These are the only three rules for S β S ¼, so S ¼ x» E 2. Since x is lay in S, and S β S ¼, we know that: S µ ν x n 1 :!λwms ¼¼

17 16 Alan Jeffrey S ¼ µ ν x n 1 :!My n w S ¼¼ S ¼¼ µ x 0 :!M 0 x n :!M n M i x i 1 or x i 1 y i 1 x 0 x We shall prove the case when each M i x i 1 y i 1, since indirection nodes make little difference. By Propositions 3.4, 3.2 and 3.3 we have: and by Proposition 3.5: So: Since λwfµf n FF n w : so: E λwfµf n F 0 x n 1 :!λwms ¼¼ x i» λwfµf n F i x n 1 :!λwms ¼¼ M» F x n 1 :!λwms ¼¼ y i» F i S ¼¼ x n 1» λwf S ¼¼ y i» G i S ¼¼ y n» F n S ¼¼ x i» λwfµf n G n 1 G i F i G i λwfµf n x n S ¼ M» F S ¼ My n w» FF n w S ¼ x n» FF n w S ¼ y n» F n S ¼ y i» H i S ¼ x i» FF n w H n 1 H i S ¼ x» E ¼ H i G i FF n w x n E ¼ FF n w H n 1 H 0 F i G i λwfµf n x n G i FF n w x n H i Thus, S ¼ x» E ¼ and E E ¼. E λwfµf n F 0 FF n w F n 1 F 0 FF n w H n 1 H 0 E ¼

18 3. Define S i by induction: S 0 S. A CHAM for graph reduction 17 Let S i ν x 0 : D 0 x m : D m and let x 0 x n be the x-spine of S i. If D n 1?M then define: S i 1 ν x 0 : D 0 x n : D n x n 1 :!Mx n 2 : D n 2 x m : D m so S i β S i 1. If D n!x n 1 and D n 1!λwM then define: S i 1 ν x 0 : D 0 x n 1 : D n 1 x n : D n 1 x n 1 : D n 1 x m : D m so S i β S i 1. If D n!x n 1 y and D n 1!λwM then define: so S i β S ¼. Thus, either S β S ¼ ν x 0 : D 0 x n 1 : D n 1 x n :!Myw x n 1 : D n 1 x m : D m (which is impossible, by Proposition 2.6) or S β β S ¼. By parts 1 and 2, E E ¼¼¼ E ¼¼ and S ¼ x» E ¼¼. Since E E ¼ and E E ¼¼¼, by Proposition 1.1 E ¼ E ¼¼¼, and so E E ¼ E ¼¼. ¾ From the above propositions, we can show that a λ-term diverges iff its CHAM implementation diverges. PROPOSITION 6. If x ¾ fve then E iff x :!E PROOF. µ Define S i and E i by induction: S 0 x : E and E 0 E. By induction, S i x» E i and E i. Then by Proposition 5.3, S i S i 1, E i E i 1 and S i 1 x»e i 1. By Proposition 1.2, since E i, we know E i 1. Thus x :!E S 0 S 1 S 2. Since x :!E, by Proposition 2.6: so by Propositions 5.1 and 5.2: x :!E S 0 β β S 1 β β E E 0 E 1 so E µ, so by Proposition 1.3, E. ¾

19 18 Alan Jeffrey Thus, we can show that the CHAM semantics is adequate. THEOREM 7 (ADEQUACY). If x :?Eµ Ú x :?Fµ then E Ú F. PROOF. For any context C: Thus, E Ú F. CF µ λwcw µf µ :! λwcw µf µ νxy :!yxy :!λwcw x :?F µ νxy :!yxy :!λwcw x :?E µ :! λwcw µe µ λwcw µe µ CE However, it is not fully abstract. THEOREM 8. E Ú F does not imply x :?Eµ Ú x :?Fµ. PROOF. The CHAM semantics is not fully abstract because we can define ABRAM- SKY s (1989) combinator in the λ-calculus with Ö: λy Öx :!y Ò Á cannot be defined in the λ-calculus, and provides extra testing power to the λ-calculus with Ö. Define: E λxx λyxyµ F λxx xµ λx õ Á C νxy y :! :!xy ONG (1988, Theorem ) has shown that E and F are applicative bisimilar (ABRAMSKY, 1989) and so E Ú F. However: Cx :?E :! Cx :?F νxyx :!y :!x :! Thus, by Proposition 2.8, Cx :?E, so x :?Eµ Ú x :?Fµ. It is an open problem as to whether the CHAM semantics is fully abstract wrt the λ-calculus with, and as to whether the canonical semantics for the lay λ-calculus D ³ D Dµ is adequate wrt the CHAM semantics. ¾ ¾

20 A CHAM for graph reduction 19 S µ S P Q µ PQ νxp µ νxp mνxm ¼ µ νxmm ¼ x ¾ fvmµ νxm µ νy myx µ νxym µ νyxm νxxm µ νxm x x P µ P x y P µ P x yµ y ¾ fvmµ A xµ µ P xỹ A ỹµ def Pµ xy x vwµp Pyvw 5 The asynchronous pi-calculus TABLE 3. CHAM for the π-calculus The π-calculus, introduced by MILNER, PARROW and WALKER (1989) is a process algebra in which scope is considered important. MILNER has shown that it can be used to model pointer-structures (1991) and the lay λ-calculus (1992), which has been further investigated by SANGIORNI (1991). Since the π-calculus was designed with pointer structures and the λ-calculus in mind, it seems natural to use it to encode a parallel graph reduction algorithm. We shall consider a variant of BOUDOL s asynchronous π-calculus (1992). This has the syntax: Here: P :: xy x yµp P P νxp x y P x y P A xµ xy is the process which outputs the pair yµ along channel x. x yµp is the process which inputs a pair y ¼ ¼ µ along channel x, then behaves like Px ¼ xy ¼ y. P Q places P and Q in parallel. νxp creates a new channel x for use in P. x y P acts like P whenever x y, and deadlocks otherwise. x y P acts like P whenever x y, and deadlocks otherwise. A xµ is a recursive definition, in the style of MILNER (1989). We shall assume an environment of definitions A xµ def P, where fv P x. The CHAM for this variant of the asynchronous π-calculus is given in Table 3, and is very similar to BOUDOL s CHAM for the asynchronous π-calculus (1992). The only new rules are: S µ S, which is missing from BOUDOL s paper. This rule is required to

21 20 Alan Jeffrey prove the result that for any solution S there is a process P such that S µ P. For example, we cannot show xy µ xy without this rule. x x P µ P and x y P µ P whenever x y, which gives semantics for the conditional operators missing from BOUDOL s paper. A xµ µ P xỹ whenever A ỹµ def P, which gives semantics for recursive definitions which were not used in BOUDOL s paper. We can define much of the same vocabulary for this CHAM as we did for the graph reduction CHAM. In Proposition 10 we shall see that this abuse of notation is justified. A molecule is a positive ion with valency x iff it is x yµp. A molecule is a negative ion with valency x iff it is xy. A molecule is ionic iff it is a positive or negative ion. A solution is plasmic iff it is ν xp 1 P n or P 1 P n and each P i is ionic. A plasma is positive (negative) iff it contains only positive (negative) ions. We can give a translation of each molecule of the graph reduction CHAM into the π-calculus. This uses a special variable, which we shall use to represent a function which is being evaluated, but which has not (yet) been given an argument. The semantics for terms is: x def x xy def xy λxm def! xyµ x λxmy x Myµ Öx : D ÒM def νx Dx Mµ x µ where MILNER s (1991) replication operator is defined:!p def!p P Note that the definition of λxm is recursive, which is why we are taking recursion to be primitive, rather than replication. It is not obvious whether one could define semantics using replication for which Proposition 11 would hold. Note also that Öx : D Ò M is defined only when x, but we can use α-conversion on x to assure this. The semantics for declarations is: The semantics for molecules is:!m def M?M def xyµ xy Mµ x : D def Dx

22 !λxxx A CHAM for graph reduction 21? i?! i! Á?! Á 7i?? i?! Á?! Á! Á! Á TABLE 4. A sample graph reduction in the π-calculus νxm def νxm m 1 m n def m 1 m n! Á This semantics can be drawn with process graphs. For example, if we draw: D for D «Å «for!x x Å? for?x x Å «x y for!xy «Å?@ x y for?xy Then the reduction of i given on page 5 can be drawn (with some extraneous processes removed to account for garbage collection) in Table 4. This is exactly the same reduction as given on page 5. In general, we can show that each CHAM reduction is matched by exactly one π-calculus reduction, and thus that a CHAM solution diverges iff its π-calculus semantics diverges. This will allow us to show that the π-calculus semantics is adequate wrt the CHAM semantics for graph reduction (and so wrt the λ-calculus). PROPOSITION For any S there is a plasmic S ¼ such that S S ¼.

23 22 Alan Jeffrey 2. If S is plasmic and S S ¼ then S µ S ¼. 3. S iff S µ PROOF. As for Proposition 2. ¾ 6 The pi-calculus semantics is adequate We will show that the π-calculus semantics models the CHAM semantics for graph reduction without garbage collection. To begin with, we can justify an abuse of notation: PROPOSITION m is a (positive) (negative) ion iff m is a (positive) (negative) ion. 2. S is a (positive) (negative) plasma iff S is a (positive) (negative) plasma. PROOF. A simple case analysis. Then we can show that each reduction of the graph reduction CHAM can be matched by a π-calculus reduction: PROPOSITION If S T then T S ¼ and S γ S ¼. 2. If S γ S ¼ then S S ¼. 3. If S T then T S ¼ and S S ¼. 4. If S S ¼ then S S ¼. 5. S iff S PROOF. Parts 1 4 are simple case analyses. From parts 1 4, S µ iff S β µ, so by Propositions 2.4 and 9.3, S iff S, which proves part 5. This means that the π-calculus semantics is adequate. THEOREM 12 (ADEQUACY). If SÚ S ¼ then S Ú S ¼. PROOF. For any C, if CS ¼ then CS ¼ so CS ¼ so CS so CS so CS. Thus, S Ú S ¼. ¾ However, it is not fully abstract. THEOREM 13. m Ú m ¼ does not imply mú m ¼. PROOF. DefineR to be a simulation up to µ, similar to (MILNER, 1989, Definition 4.4), iff whenever mr m ¼ and m ¼ m ¼¼¼ then m m ¼¼ and m ¼¼ µ R µ m ¼¼¼. Let be the largest simulation up to µ, and let C be the largest precongruence contained in : m C m ¼ iff C Cm Cm ¼ ¾ ¾

24 A CHAM for graph reduction 23 Then if m m ¼ and m ¼ then m, so if m C m ¼ then m Ú m ¼. Define: m x :!y :! m ¼ x :!yy :! R Cx :!y :! x :!y :!λw 1 M 1 :!λw 1 M 1 x :!y :!λw n M n :!λw n M n Cx :!yy :! x :!yy :!λw 1 M 1 :!λw 1 M 1 x :!yy :!λw n M n :!λw n M n µ C is an n 1-place context ThenR is a simulation up to µ, so for any context C, Cm Cm ¼, so m C m ¼, so m Ú m ¼. We will now show that mú m ¼. Define: where is a divergent process such as: Then: whereas: so mú m ¼. C y uvµ def νw www w uvµµ Cm µ x y y uvµ Cm ¼ µ y x y y uvµ y SANGIORNI (1991) has investigated λ-calculi for which MILNER s π-calculus translation is fully abstract. It is an open problem as to whether similar results can be shown for the CHAM for graph reduction. Acknowledgements Many thanks to Lennart Augustsson for getting the ball rolling, to Thomas Hallgren, Magnus Carlsson and Niklas Röjemo for conversations about the innards of LML, and to K. V. S. Prasad, Jenny Petersson, Matthew Hennessy, Edmund Robinson, Steve Schneider and Mark Jones for many discussions. This work was conceived at Chalmers University and written at Sussex University and Drive Cottage, Ampney Knowle, Cirencester. References ABRAMSKY, S. (1989). The lay lambda calculus. In TURNER, D., editor, Declarative Programming. Addison-Wesley. ¾

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