Work It, Wrap It, Fix It, Fold It

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1 1 Work It, Wrap It, Fix It, Fold It NEIL SCULTHORPE University o Kansas, USA GRAHAM HUTTON University o Nottinham, UK Abstract The worker/wrapper transormation is a eneral-purpose technique or reactorin recursive prorams to improve their perormance. The two previous approaches to ormalisin the technique were based upon dierent recursion operators and dierent correctness conditions. In this article we show how these two approaches can be eneralised in a uniorm manner by combinin their correctness conditions, extend the theory with new conditions that are both necessary and suicient to ensure the correctness o the worker/wrapper technique, and explore the beneits that result. All the proos have been mechanically veriied usin the Ada system. 1 Introduction A undamental objective in computer science is the development o prorams that are clear, eicient and correct. However, these aims are oten in conlict. In particular, prorams that are written or clarity may not be eicient, while prorams that are written or eiciency may be diicult to comprehend and contain subtle bus. One approach to resolvin these tensions is to use proram transormation techniques to systematically rewrite prorams to improve their eiciency, without compromisin their correctness. The ocus o this article is the worker/wrapper transormation, a transormation technique or improvin the perormance o recursive prorams by usin more eicient intermediate data structures. The basic idea is simple and eneral: iven a recursive proram o some type A, we aim to actorise it into a more eicient worker proram o some other type B, toether with a wrapper unction o type B A that allows the new worker to be used in the same context as the oriinal recursive proram. Special cases o the worker/wrapper transormation have been used or many years. For example, the technique has played a key role in the Glasow Haskell Compiler since its inception more than twenty years ao, to lace the use o boxed data structures by more eicient unboxed data structures (Peyton Jones & Launchbury, 1991). However, it is only recently in two articles that lay the oundations or the present work (Gill & Hutton, 2009; Hutton et al., 2010) that the worker/wrapper transormation has been ormalised, and considered as a eneral approach to proram optimisation. The oriinal ormalisation (2009) was based upon a least-ixed-point semantics o recursive prorams. Within this settin the worker/wrapper transormation was explained and ormalised, proved correct, and a rane o prorammin applications presented. Usin

2 2 N. Sculthorpe and G. Hutton ixed points allowed the worker/wrapper transormation to be ormalised, but did not take advantae o the additional structure that is present in many recursive prorams. To this end, a more structured approach (2010) was then developed based upon initial-alebra semantics, a cateorical approach to recursion that is widely used in proram optimisation (Bird & de Moor, 1997). More speciically, a worker/wrapper theory was developed or prorams deined usin old operators, which encapsulate a common pattern o recursive prorammin. In practice, usin old operators results in simpler transormations than the approach based upon ixed points. Moreover, it also admitted the irst ormal proo o correctness o a new approach (Voitländer, 2008) to optimisin monadic prorams. While the two previous articles were nominally about the same technique, they were quite dierent in their cateorical oundations and correctness conditions. The irst was ounded upon least ixed points in the cateory CPO o complete partial orders and continuous unctions, and identiied a hierarchy o conditions on the conversion unctions between the oriinal and worker types that are suicient to ensure correctness. In contrast, the second was ounded upon initial alebras in an arbitrary cateory C, and identiied a lattice o suicient correctness conditions on the oriinal and worker alebras. This raises the question o whether it is possible to combine or uniy the two dierent approaches. The purpose o this new article is to show how this can be achieved, and to explore the beneits that result. More precisely, the article makes the ollowin contributions: We show how the least-ixed-point and initial-alebra approaches to the worker/ wrapper transormation can be eneralised in a uniorm manner by combinin their dierent sets o correctness conditions (sections 3 and 5). We identiy necessary conditions or the correctness o the worker/wrapper technique, in addition to the existin suicient conditions, thereby ensurin that the theory is as widely applicable as possible 1 (sections 3 and 5). We use our new theory to develop a specialised worker/wrapper theory or olds in CPO that eliminates all unnecessary strictness conditions (section 6). The article is aimed at readers who are amiliar with the basics o least-ixed-point semantics (Schmidt, 1986), initial-alebra semantics (Bird & de Moor, 1997), and the worker/wrapper transormation (Gill & Hutton, 2009; Hutton et al., 2010), but all necessary concepts and results are reviewed. A mechanical veriication o the proos in Ada is available as supplementary material on the JFP website, alon with an extended version o this article that includes a series o worked examples and all the proos. 2 Least-Fixed-Point Semantics The oriinal ormalisation o the worker/wrapper transormation was based on a leastixed-point semantics o recursion, in a domain-theoretic settin in which prorams are continuous unctions on complete partial orders. In this section we review some o the basic deinitions and properties rom this approach to proram semantics, and introduce our notation. For urther details, see or example Schmidt (1986). 1 Speciically, we identiy conditions that are necessary and suicient to ensure that the worker/wrapper actorisation and usion properties are both valid.

3 Work It, Wrap It, Fix It, Fold It 3 A complete partial order (cpo) is a set with a partial-orderin, a least element, and limits (least upper bounds) o all non-empty chains. A unction between cpos is continuous i it is monotonic and preserves the limit structure. I it also preserves the least element, i.e. =, the unction is strict. A ixed point o a unction is a value x or which x = x. Kleene s well-known ixed-point theorem (Schmidt, 1986) states that any continuous unction on a cpo has a least ixed point, denoted by ix. The basic proo technique or least ixed points is ixed-point induction (Winskel, 1993). Suppose that is a continuous unction on a cpo and that P is a chain-complete predicate on the same cpo, i.e. whenever the predicate holds or all elements in a non-empty chain then it also holds or the limit o the chain. Then ixed-point induction states that i the predicate holds or the least element o the cpo (the base case) and is preserved by the unction (the inductive case), then it also holds or ix : Lemma 2.1 (Fixed-Point Induction) I P is chain-complete, then: P ( x. P x P( x)) P(ix ) Fixed-point induction can be used to veriy the well-known ixed-point usion property (Meijer et al., 1991), which states that the application o a unction to a ix can be reexpressed as a sinle ix, provided that the unction is strict and satisies a simple commutativity condition with respect to the ix aruments: Lemma 2.2 (Fixed-Point Fusion) = h strict (ix ) = ix h Finally, a key property o ix that we will use is the rollin rule (Backhouse, 2002), which allows the irst arument o a composition to be pulled outside a ix, resultin in the composition swappin the order o its aruments, or rollin over : Lemma 2.3 (Rollin Rule) ix( ) = (ix( )) 3 Worker/Wrapper or Least Fixed Points Within the domain-theoretic settin o the previous section, consider a recursive proram deined as the least ixed point o a unction : A A on some type A. Now consider a more eicient proram that perorms the same task, deined by irst takin the least ixed point o a unction : B Bon some other type B, and then convertin the resultin value back to the oriinal type by applyin a unction : B A. The equivalence between these two prorams is captured by the ollowin equation: ix = (ix ) We call ix the oriinal proram, ix the worker proram, the wrapper unction, and the equation itsel the worker/wrapper actorisation or least ixed points. We now turn our attention to identiyin conditions to ensure that it holds.

4 4 N. Sculthorpe and G. Hutton 3.1 Assumptions and Conditions First, we require an additional conversion unction : A B rom the oriinal type to the new type. This unction is not required to be an inverse o, but we do require one o the ollowin worker/wrapper assumptions to hold: (A) = id A (B) = (C) ix( ) = ix These assumptions orm a hierarchy, with (A) (B) (C). Assumption (A) is the stronest and usually the easiest to veriy, and states that is a let inverse o, which in the terminoloy o data reinement means that the tract type A can be aithully resented by the concrete type B. For some applications, however, assumption (A) may not be true in eneral, but only or values produced by the body unction o the oriinal proram, as captured by the weaker assumption (B), or we may also need to take the recursive context into account, as captured by (C). Additionally, we require one o the ollowin worker/wrapper conditions 2 that relate the body unctions and o the oriinal and worker prorams: (1) = (1β) ix = ix( ) (2) = strict (2β) ix = (ix ) (3) = In eneral, there is no relationship between the conditions in the irst column, i.e. none implies any o the others, while the β conditions in the second column arise as weaker versions o the correspondin conditions in the irst. The implications (1) (1β) and (2) (2β) ollow immediately usin extensionality and ixed-point usion respectively, which in the latter case accounts or the strictness side condition in (2). We will return to the issue o strictness in Section 3.2. Furthermore, iven assumption (C), it is straihtorward to show that conditions (1β) and (2β) are in act equivalent. Nonetheless, it is still useul to consider both conditions, as in some situations one may be simpler to use than the other. Note that attemptin to weaken condition (3) in a similar manner ives ix = (ix ), which there is no merit in considerin as this is precisely the worker/wrapper actorisation result that we wish to establish. In terms o how the worker/wrapper conditions are used in practice, or some applications the worker proram ix will already be iven, and our aim then is to veriy that one o the conditions is satisied. In such cases, we use the condition that admits the simplest veriication, which is oten one o the stroner conditions (1), (2) or (3) that do not involve the use o ix. For other applications, our aim will be to construct the worker proram. In such cases, conditions (1), (1β) or (2β) provide explicit but ineicient deinitions or the worker proram in terms o the body unction o the oriinal proram, which we then attempt to make more eicient usin proram-usion techniques. This was the approach 2 The assumptions and conditions are both sets o equational properties; we use the dierin terminoloy or consistency with Gill & Hutton (2009) and Hutton et al. (2010).

5 Work It, Wrap It, Fix It, Fold It 5 that was taken by Gill & Hutton (2009). However, as shown by Hutton et al. (2010), in some cases it is preerable to use conditions (2) or (3), which provide an indirect speciication or the body unction o the worker, rather than a direct deinition. 3.2 Worker/Wrapper Factorisation We can now state the main result o this section: provided that any o the worker/wrapper assumptions hold, and any o the worker/wrapper conditions hold, then worker/wrapper actorisation is valid, as summarised in Fiure 1. To prove this result it suices to consider assumption (C) and conditions (1β) and (3) in turn, as (A), (B), (1) and (2) are already covered by their weaker versions, and (2β) is equivalent to (1β) in the presence o (C). For condition (1β), actorisation is veriied by the ollowin simple calculation: ix = { (C)} ix( ) = { rollin rule} (ix( )) = { (1β)} (ix ) For condition (3), at irst lance it may appear that we don t need assumption (C) at all, as condition (3) on its own is suicient to veriy the result by usion. But the use o usion requires that is strict. However, usin assumption (C) and ixed-point induction, we can prove the actorisation result without this extra strictness condition (see the extended version o this article or the details). But perhaps bein strict is implied by the assumptions and conditions? In act, iven assumption (A), this is indeed the case. However, or the weaker assumption (B), is not necessarily strict. A simple counterexample is shown in the ollowin diaram, in which bullets on the let and riht sides are elements o A and B respectively, dotted lines are orderins (x y) that are directed upwards, and solid arrows are mappins (x y): In particular, this example satisies assumption (B), condition (3), and worker/wrapper actorisation, but is non-strict. Because (B) implies (C), the same counterexample also shows that the strictness o is not implied by (C) and (3). It is interestin to note that in the past condition (3) was rearded as bein uninterestin because it just corresponds to the use o usion (Hutton et al., 2010). But in the context o ix this requires that is strict. However, as we have now seen, in the case o (B) and (C) this requirement can be dropped. Hence, worker/wrapper actorisation or condition (3) is applicable in some situations where usion is not, i.e. when is non-strict.

6 6 N. Sculthorpe and G. Hutton Given unctions : A A : B B or some types A and B, and conversion unctions : A B : B A then we have a set o worker/wrapper assumptions (A) = id A (B) = (C) ix( ) = ix and a set o worker/wrapper conditions (1) = (1β) ix = ix( ) (2) = strict (2β) ix = (ix ) (3) = Provided that any o the assumptions hold and any o the conditions hold, then worker/wrapper actorisation is valid: ix = (ix ) Furthermore, i any o the assumptions hold, and any o the conditions except (3) hold, then worker/wrapper usion is valid: ((ix )) = ix Fiure 1: Worker/wrapper transormation or least ixed points. Recall that showin (2) (2β) usin ixed-point usion requires that is strict. It is natural to ask i we can drop strictness rom (2) by provin worker/wrapper actorisation in another way, as we did above with condition (3). The answer is no, and we veriy this by exhibitin a non-strict that satisies = and assumption (A), but or which worker/wrapper actorisation does not hold, as ollows: Because (A) (B) (C), the same example shows that = on its own is also insuicient or assumptions (B) and (C). However, while the addition o strictness is suicient to ensure worker/wrapper actorisation, it is not necessary, which can be

7 Work It, Wrap It, Fix It, Fold It 7 veriied by exhibitin a non-strict that satisies =, assumption (A), and worker/wrapper actorisation, shown in the example below. As beore, this example also veriies that strictness is not necessary or (B) and (C). 3.3 Worker/Wrapper Fusion When applyin worker/wrapper actorisation, it is oten desirable to use toether instances o the conversion unctions and to eliminate the overhead o eatedly convertin between the new and oriinal types (Gill & Hutton, 2009). In eneral, it is not the case that can be used to ive id B. However, provided that any o the assumptions (A), (B) or (C) hold, and any o the conditions except (3) hold, then the ollowin worker/wrapperusion property is valid, as summarised in Fiure 1: ((ix )) = ix In a similar manner to Section 3.2, or the purposes o provin this result it suices to consider assumption (C) and condition (2β): ((ix )) = { worker/wrapper actorisation, (C) and (2β)} (ix ) = { (2β)} ix As with worker/wrapper actorisation, we conirm that strictness o is suicient but not necessary in the case o condition (2), by exhibitin a non-strict that satisies =, assumption (A), and worker/wrapper usion:

8 8 N. Sculthorpe and G. Hutton Finally, in the case o condition (3), the ollowin example shows that (3) and (A) are not suicient to ensure worker/wrapper usion: Furthermore, even i we were also to require that be strict, be strict, or both conversion unctions be strict, it is still possible to construct correspondin examples that demonstrate that worker/wrapper usion does not hold in eneral or condition (3). 3.4 Relationship to Previous Work The worker/wrapper results or ix presented in this section eneralise those in Gill & Hutton (2009). The key dierence is that the oriinal article only considered worker/wrapper actorisation or condition (1β), althouh it wasn t identiied as an explicit condition but rather inlined in the statement o the theorem itsel, whereas we have shown that the result is also valid or (1), (2), (2β) and (3). Moreover, worker/wrapper usion was only established or assumption (A) and condition (1β), whereas we have shown that any o the assumptions (A), (B) or (C) and any o the conditions (1), (1β), (2) or (2β) are suicient. We also exhibited a counterexample to show that (3) is not a suicient condition or worker/wrapper usion under any o the assumptions. We conclude by notin that in the context o assumption (C), the equivalent conditions (1β) and (2β) are not just suicient to ensure that worker/wrapper actorisation and usion hold, but are in act necessary too. In particular, iven these two properties, we can then veriy that condition (2β) holds by the ollowin simple calculation: ix = { worker/wrapper usion} ((ix )) = { worker/wrapper actorisation} (ix ) Hence, while previous work identiied conditions that are suicient to ensure actorisation and usion are valid, we now have conditions that are both necessary and suicient. 4 Initial-Alebra Semantics We now turn our attention to the other previous ormalisation o the worker/wrapper transormation, which was based upon an initial-alebra semantics o recursion in a cateorical settin in which prorams are deined usin old operators. In this section we review the basic deinitions and properties rom this approach to proram semantics, and introduce our notation. For urther details, see or example Bird & de Moor (1997). Suppose that we ix a cateory C and a unctor F : C C on this cateory. Then an F-alebra is a pair (A,) comprisin an object A and an arrow : F A A. An F- homomorphism rom one such alebra (A,) to another (B,) is an arrow h : A B such

9 Work It, Wrap It, Fix It, Fold It 9 that h = F h. Alebras and homomorphisms themselves orm a cateory, with composition and identities inherited rom the oriinal cateory C. An initial alebra is an initial object in this new cateory, and we write (µf,in) or an initial F-alebra, and old or the unique homomorphism rom this initial alebra to any other alebra (A,). Moreover, the arrow in : F µf µf has an inverse out : µf F µf, which establishes an isomorphism F µf = µf. The above deinition or old can also be expressed as the ollowin equivalence, known as the universal property o old: Lemma 4.1 (Universal Property o Fold) h = old h in = F h The universal property orms the basic proo technique or the old operator. For example, it can be used to veriy the correspondin versions o ixed-point usion (Lemma 2.2) and the rollin rule (Lemma 2.3) or initial alebras: Lemma 4.2 (Fold Fusion) h = F h h old = old Lemma 4.3 (Rollin Rule) old( ) = old( F ) 5 Worker/Wrapper or Initial Alebras Within the cateory-theoretic settin o the previous section, consider a recursive proram deined as the old o an alebra : F A A or some object A. Now consider a more eicient proram that perorms the same task, deined by irst oldin an alebra : F B B on some other object B, and then convertin the resultin value back to the oriinal object type by composin with an arrow : B A. The equivalence between these two prorams is captured by the ollowin equation: old = old In a similar manner to least ixed points, we call old the oriinal proram, old the worker proram, the wrapper arrow, and the equation itsel the worker/wrapper actorisation or initial alebras. The properties that we use to validate the actorisation equation are similar to those that we identiied or least ixed points, and are summarised in Fiure 2. As previously, the assumptions orm a hierarchy (A) (B) (C), the conditions (1β) and (2β) are weaker versions o (1) and (2) and are equivalent iven assumption (C), and in eneral there is no relationship between conditions (1), (2) and (3). As we are workin in an arbitrary cateory the notion o strictness is not deined, and hence there is no requirement that be strict or (2); we will return to this point in Section 6. Worker/wrapper usion can also be ormulated or initial alebras, as shown in Fiure 2. Moreover, the example rom Section 3.3 that shows that usion is not in eneral valid or condition (3) or least ixed points can readily be adapted to the case o initial alebras. Speciically, i we deine a constant unctor F : SET SET on the cateory o sets and

10 10 N. Sculthorpe and G. Hutton Given alebras : F A A : F B B or some unctor F, and conversion arrows : A B : B A then we have a set o worker/wrapper assumptions (A) = id A (B) = (C) old( ) = old and a set o worker/wrapper conditions (1) = F (1β) old = old( F ) (2) = F (2β) old = old (3) = F Provided that any o the assumptions hold and any o the conditions hold, then worker/wrapper actorisation is valid: old = old Furthermore, i any o the assumptions hold, and any o the conditions except (3) hold, then worker/wrapper usion is valid: old = old Fiure 2: Worker/wrapper transormation or initial alebras. total unctions by F X = 1 and F = id 1, where 1 is any sinleton set, then the ollowin deinitions satisy (3) and (A) but not worker/wrapper usion: F A A B F B The worker/wrapper results or old presented in this section eneralise those in Hutton et al. (2010). The key dierence is that the oriinal article only considered worker/wrapper actorisation or assumption (A) and conditions (1), (2) and (3) (in which context (1) is stroner than the other two conditions), whereas we have shown that the result is also valid or the weaker assumptions (B) and (C) (in which context (1), (2) and (3) are in eneral unrelated) and the weaker conditions (1β) and (2β). Moreover, worker/wrapper usion was essentially only established or assumption (A) and condition (1), whereas we have shown that any o the assumptions (A), (B) or (C) and any o the conditions (1), (1β), (2) or (2β)

11 Work It, Wrap It, Fix It, Fold It 11 are suicient. We also showed that (3) is not suicient or worker/wrapper usion under any o the assumptions. Finally, we note that as with least ixed points, in the context o assumption (C) the equivalent conditions (1β) and (2β) are both necessary and suicient to ensure worker/wrapper actorisation and usion or initial alebras. 6 From Least Fixed Points to Initial Alebras In Section 5 we developed the worker/wrapper theory or initial alebras. Given that the results were ormulated or an arbitrary cateory C, we would expect them to hold in the cateory CPO o cpos and continuous unctions used in the least-ixed-point approach. This is indeed the case, with one complicatin actor: when CPO is the base cateory, the universal property has a strictness side condition, which weakens our results by addin many strictness requirements. In this section, we show that all but one o these strictness conditions is unnecessary, by instantiatin our theory or least ixed points. 6.1 Strictness Recall that the basic proo technique or the old operator is its universal property. In the cateory CPO, this property has a strictness side condition (Meijer et al., 1991): Lemma 6.1 (Universal Property o Fold in CPO) I h is strict, then: h = old h in = F h The universal property o old, toether with derived properties such as usion and the rollin rule, orm the basis o our proos o worker/wrapper actorisation and usion or initial alebras in Section 5. Trackin the impact o the extra strictness condition above on these results is straihtorward but tedious, so we omit the details here (they are provided in the supplementary Ada proos) and just present the results: or conditions (1), (1β), (2) and (2β), both actorisation and usion require that, and are strict, while or (3), actorisation requires that and are strict. In summary, instantiatin the worker/wrapper results or initial alebras to the cateory CPO is straihtorward, but derivin the results in this manner introduces many strictness side conditions that may limit their applicability. Some o these conditions could be avoided by usin more liberal versions o derived properties such as old usion and the rollin rule that are proved rom irst principles rather than bein derived rom the universal property. However, it turns out that most o the strictness conditions can be avoided usin our worker/wrapper theory or least ixed points. 6.2 From Fix to Fold As noted earlier, the eneralised worker/wrapper results or initial alebras are very similar to those or least ixed points. Indeed, uniyin the results in this manner is one o the primary contributions o this article. In this section we show how the initial-alebra results in CPO can in act be derived rom those or least ixed points, by exploitin the act that in this context old can be deined in terms o ix (Meijer et al., 1991):

12 12 N. Sculthorpe and G. Hutton Lemma 6.2 (Deinition o Fold usin Fix in CPO) old = ix(λ h F h out) Suppose we are iven alebras : F A A and : F B B, and conversion unctions : A Band : B A. Our aim is to use the worker/wrapper results or ix to derive assumptions and conditions that imply the actorisation result or old, that is: old = old First, we deine unctions and such that old = ix and old =ix : : (µf A) (µf A) : (µf B) (µf B) = λ h F h out = λ h F h out Then we deine conversion unctions between the types or old and old : : (µf A) (µf B) : (µf B) (µf A) h = h h = h Usin these deinitions, the worker/wrapper equation old = old in terms o old is equivalent to the ollowin equation in terms o ix: ix = (ix ) This equation has the orm o worker/wrapper actorisation or ix, and is hence valid provided one o the assumptions and one o the conditions rom Fiure 1 are satisied or,, and. By expandin deinitions, it is now straihtorward to simpliy each o these assumptions and conditions in terms o the oriinal unctions,, and (see the extended version o this article or the details). A similar procedure can be applied to worker/wrapper usion. The end result is a worker/wrapper theory or initial alebras in CPO that has the same orm as Fiure 2, except that condition (2) requires that is strict. Compared to the derivation in Section 6.1, this new approach eliminates all but one strictness requirement, and hence the resultin theory is more enerally applicable. One miht ask i we can also drop strictness rom condition (2), but the answer is no. In order to veriy this, let us take Id : CPO CPO as the identity unctor, or which it can be shown by ixed-point induction that old = ix. Now consider the example rom Section 3.2 that shows that strictness cannot be dropped rom (2) in the theory or ix. This example satisies (A) and = Id, but not worker/wrapper actorisation old = old. In particular, i we assume actorisation is valid we could apply both sides to to obtain old = (old ), which by the above result is equivalent to ix = (ix ), which does not hold or this example as shown in Section 3.2. Hence, by contradiction, old = old is invalid. 7 Related Work A historical review o the worker/wrapper transormation and related work was iven in Gill & Hutton (2009), so we direct the reader to that article rather than eatin the details here. The transormation can also be viewed as a orm o data reinement (Hoare, 1972; Moran & Gardiner, 1990), a eneral-purpose approach to lacin a data structure by a

13 Work It, Wrap It, Fix It, Fold It 13 more eicient version. Speciically, the worker/wrapper transormation is a data reinement technique or unctional prorams deined usin the recursion operators ix or old. Recently, Gammie (2011) observed that the manner in which the worker/wrapper-usion rule was used in Gill & Hutton (2009) may lead to the introduction o non-termination. However, this is a well-known consequence o the old/unold approach to proram transormation (Burstall & Darlinton, 1977; Tullsen, 2002), which in eneral only preserves partial correctness, rather than bein a problem with the usion rule itsel, which is correct as it stands. Alternative, but less expressive, transormation rameworks that uarantee total correctness have been proposed, such as the use o expression procedures (Scherlis, 1980). Gammie s solution was to add the requirement that be strict to the worker/wrapperusion rule, which holds or the relevant examples in the oriinal article. However, we have not added this requirement in the present article, as this would unnecessarily weaken the usion rule without overcomin the underlyin issue with old/unold transormation. Gammie also pointed out that the stream memoisation example in Gill & Hutton (2009) incorrectly claims that assumption (A) holds, but we note that the example as a whole is still correct as the weaker assumption (B) does hold. In this article we have ocused on developin the theory o the worker/wrapper transormation, with the aim o makin it as widely applicable as possible. Meanwhile, a team in Kansas is puttin the technique into mechanised practice as part o the HERMIT project (Farmer et al., 2012). In particular, they are developin a eneral purpose system or optimisin Haskell prorams that allows prorammers to write suicient annotations to permit the Glasow Haskell Compiler to apply custom transormations automatically. The worker/wrapper transormation was the irst hih-level technique encoded in the system, and it then proved relatively straihtorward to mechanise a selection o new and existin worker/wrapper examples (Sculthorpe et al., 2013). Workin with the automated system has also revealed that other, more specialised, transormation techniques can be cast as instances o worker/wrapper, and consequently that usin the worker/wrapper inrastructure can simpliy mechanisin those transormations (Sculthorpe et al., 2013). 8 Conclusions and Future Work The oriinal worker/wrapper article (Gill & Hutton, 2009) ormalised the basic technique usin least ixed points, while the ollow-up article (Hutton et al., 2010) developed a worker/wrapper theory or initial alebras. In this article we showed how the two approaches can be eneralised in a uniorm manner by combinin their dierent sets o correctness conditions. Moreover, we showed how the new theories can be urther eneralised with conditions that are both necessary and suicient to ensure the correctness o the transormations. All the proos have been mechanically checked usin the Ada proo assistant, and are available as supplementary material on the JFP website. It is interestin to recount how the conditions (1β) and (2β) were developed. Initially we ocused on combinin assumptions (A), (B) and (C) rom the irst article with conditions (1), (2) and (3) rom the second. However, the resultin theory was still not powerul enouh to handle some examples that we intuitively elt should it within the ramework. It was only when we looked aain at the proos or worker/wrapper actorisation and

14 14 N. Sculthorpe and G. Hutton usion that we realised that conditions (1) and (2) could be urther weakened, resultin in conditions (1β) and (2β), and proos that they are equivalent and maximally eneral. In terms o urther work, practical applications o the worker/wrapper technique are bein driven orward by the HERMIT project in Kansas, as described in Section 7. On the oundational side, it would be interestin to exploit additional orms o structure to urther extend the enerality and applicability o the technique, or example by usin other recursion operators such as unolds and hylomorphisms, ramin the technique usin more eneral cateorical constructions such as limits and colimits, and considerin more sophisticated notions o computation such as monadic, comonadic and applicative prorams. Acknowledements The irst author was supported by NSF award number We would like to thank Jennier Hackett or the counterexample in Section 6, and the anonymous reerees or their detailed and helpul reviews. Reerences Backhouse, Roland. (2002). Galois Connections and Fixed Point Calculus. Paes o: Alebraic and Coalebraic Methods in the Mathematics o Proram Construction. Spriner. Bird, Richard, & de Moor, Oee. (1997). Alebra o Prorammin. Prentice Hall. Burstall, Rod. M., & Darlinton, John. (1977). A Transormation System or Developin Recursive Prorams. Journal o the ACM, 24(1), Farmer, Andrew, Gill, Andy, Komp, Ed, & Sculthorpe, Neil. (2012). The HERMIT in the Machine: A Pluin or the Interactive Transormation o GHC Core Lanuae Prorams. Paes 1 12 o: Haskell Symposium. ACM. Gammie, Peter. (2011). Strict Unwraps Make Worker/Wrapper Fusion Totally Correct. Journal o Functional Prorammin, 21(2), Gill, Andy, & Hutton, Graham. (2009). The Worker/Wrapper Transormation. Journal o Functional Prorammin, 19(2), Hoare, Tony. (1972). Proo o Correctness o Data Representations. Acta Inormatica, 1(4), Hutton, Graham, Jaskelio, Mauro, & Gill, Andy. (2010). Factorisin Folds or Faster Functions. Journal o Functional Prorammin, 20(3&4), Meijer, Erik, Fokkina, Maarten M., & Paterson, Ross. (1991). Functional Prorammin with Bananas, Lenses, Envelopes and Barbed Wire. Paes o: Functional Prorammin Lanuaes and Computer Architecture. Spriner. Moran, Carroll, & Gardiner, P. H. B. (1990). Data Reinement by Calculation. Acta Inormatica, 27(6), Peyton Jones, Simon, & Launchbury, John. (1991). Unboxed Values as First Class Citizens in a Non-Strict Functional Lanuae. Paes o: Functional Prorammin Lanuaes and Computer Architecture. Spriner. Scherlis, William Louis. (1980). Expression Procedures and Proram Derivation. Ph.D. thesis, Stanord University.

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