Ong,C-H.L., The Lazy Lambda Calculus: An Investigation into the Foundation of Functional Programming, Phd thesis in Imperial College, May 31, 1988. 7  Home/Search   Document Not in Database   Summary   Related Articles   Check

....In the family of sound theories, we can pick up head normal forms as generic observables, though still without fi equality. Let = be an sound reduction theory over terms. Then: M = N ) M has some head normal form ) N has some head normal form) Since termination (cf. Lazy calculus[2, 16]) is not regarded as fundamental here, this notion of generic observable should be regarded as natural. An important corollary of this is again the existence of the maximum consistent theory in the family of theories, which coincides with K . Further investigation in this line is left to future ....

....theoretical and pragmatic. In the strong setting, the theory corresponding to the maximum consistent sound theory coincides with early strong bisimilarity as formulated in [12] We say M is a PO1 term, if for any natural number n, there is M such that M x 1 : x n 0 :M with n n[10, 16]. 6 ....

Ong,C-H.L., The Lazy Lambda Calculus: An Investigation into the Foundation of Functional Programming, Phd thesis in Imperial College, May 31, 1988. 7

....is easy to see that AE f is Church Rosser and strong normalizing. Definition 4.2 Convergence. N # f M if N AE f M 6AE f M 6 Hence # f , because 0 AE f x[x : I] 6 0 . Also all computable functions can be represented by f calculus (compare this to the case of Lazy calculus[9]) The following small lemma is essential. Lemma 4.3 Mf = x g # f M 0 V N # f N M[x : N ] # f M [x : N j M We prove the main result based on definability a la Curry. However it would be easy to extend the proof to another definability found e.g. in Barendregt[2] We use ....

C-H.L. Ong. The lazy lambda calculus: An investigation into the foundation of functional programming. Phd thesis in Imperial College, May 31 1988.

....of evaluation contexts to re ect the order of evaluation. 2.2 Call by value Models A denotational model for a call by value language is based on a collection of sets, indexed by type, suitable for giving meaning to terms. The following de nition is taken from [35] and resembles de nitions from [1, 28]. De nition 1 A call by value type frame over a type signature is a tuple D = fD j 2 g; f j 2 g; fApp j ; 2 g) 2 Table 1: Typing and Evaluation Rules for Call by Value Functional Languages. Terms (var) x : const) hf ; 1 ; n ) i 2 C M i : i f(M1 ; ....

C.-H. L. Ong. The Lazy Lambda Calculus: An Investigation into the Foundations of Functional Programming. PhD thesis, Imperial College, University of London, 1988.

.... ( p calculus) 5 Untyped c models It is well known that a categorical model for the untyped calculus is a reflexive object D D = D in a cartesian closed category (see [Sco80, Bar82] In a c model there are two analogs for a reflexive object: V V T = V and N TN T = N (see [Ong88] for similar definitions in the context of partial cartesian closed categories) In the first case we have a model of call by value. In fact the elements of V correspond to functions from values to computations (as V V T stands for V TV ) and therefore an element can be applied to a ....

....) T ( F Theta id TN ) eval T TN;N ) N Remark 5. 1 In call by value (let x=e in e 0 ) is equivalent to (x:e 0 ) e) but in callby name there is no way of expressing (let x=e in e 0 ) in terms of application and abstraction only, because e is evaluated before binding its value to x (see [Ong88] for an analysis of call by name for partial computations) We think that it is desirable (and very natural) for a programming language to have a let, which forces evaluation of an expression. We conjecture that the fi calculus (i.e. Plotkin s call by name calculus) proves exactly those ....

C.-H.L. Ong. Lazy Lambda Calculus: An Investigation into the Foundations of Functional Programming. PhD thesis, University of London, October 1988.

....for fixed points, but the restrictive type system ensures that we only have to find fixed points of terms of computation type, and those always have least elements. We can show that the denotational semantics is fully abstract for the operational semantics using a variant of Abramsky (1989) and Ong s (1988) lazy lambda calculus and Abramsky s (1991) domain theory in logical form. This is similar to Ong s (1993) use of a program logic for the untyped l calculus, but is simplified by the fact that nondeterminism can only occur at computation type. The simplified proof of full abstraction is due to the ....

....Proposition 15 means that if S consists of equality sorts, then NMMLS has a fully abstract semantics. 4. 5 Program logic In order to show the relationship between the operational and denotational semantics of NMMLS, we shall use a program logic similar to that used by Abramsky (1989) and Ong (1988) in modelling the untyped l calculus, based on Abramsky s (1991) domain theory in logical form. The logic is presented in two ways: ffl it has an operational characterization, similar to the operational characterization for HML (Milner, 1989) or the modal calculus (Kozen, 1983) and ffl it has ....

ONG, C.-H. L. (1988). The Lazy Lambda Calculus: An Investigation into the Foundations of Functional Programming.

....coincide with the # undefined ones. It is well known that the unsolvable terms are exactly the terms without head normal form. This implies that unsolvability is equivalent to 001 active, treated above. Hence all axioms hold. 8.2. 9 Strongly unsolvable 101 active A term is strongly unsolvable [Ong88, Sec. 2.1.1 2] if it is a zero term and it is not convertible to a term of the form xs. That is, it has no weak head normal form, or equivalently it is a 101 active form, treated above. Hence all axioms hold. 8.2.10 Mute hypercollapsing root active 111 active A term is mute if it is a zero term ....

C.-H. L. Ong. The Lazy Lambda Calculus: An Investigation into the Foundations of Functional Programming. PhD thesis, Imperial College of Science and Technology, University of London, May 1988.

....can define ABRAMSKY s (1989) C combinator in the l calculus with rec: C = ly:recx : y in I C cannot be defined in the l calculus, and provides extra testing power to the l calculus with rec. Define: E = lx:x(ly:x y) F = lx:x(x ) lx: YK) YI C[ nxy:fj ; y : C;z : xyjg ONG (1988, Theorem 4.5.1.1) has shown that E and F are applicative bisimilar (ABRAMSKY, 1989) and so E v F . However: C[x : E] fjz : jg 6 C[x : F] nxy:fjx : y : x; z : jg Thus, by Proposition 2.8, C[x : E] 6 , so (x : E) 6v (x : F) 2 It is an open problem ....

ONG, C.-H. L. (1988). The Lazy Lambda Calculus: An Investigation into the Foundations of Functional Programming. PhD thesis, Imperial College, London University.

....the finest usable notion of process equivalence in this setting. If processes P and Q are bisimilar, written P Q, then at all future times, they have exactly the same set of nondeterministic choices available. Bisimulation is a meaningful notion in many settings beyond process algebra (e.g. [34,30]) It admits several powerful proof methods, and protocol5 verification environments based on bisimulation have been built [16] Furthermore, bisimulation of n state, m transition processes can be computed in O(m lg n) time [39] making verification of even relatively large protocols a viable ....

L. Ong. The lazy lambda calculus: an investigation into the foundations of functional programming. PhD thesis, Imperial College, University of London, 1988.

....consider appealing the introduction of parallel operators. Various enrichments of the lazy lambda calculus with operators not lambda definable have already appeared in the literature. However either the operators themselves as in the case of convergency test and parallel convergency in [1, 2, 12] or the semantics given as for the non deterministic choice and parallel operator in [4] and [5] are rather ad hoc, chosen to achieve full abstraction for some canonical domain. Or at least, from a programming language point of view, they do not seem to be justified by any common ....

....give exact operational and denotational characterizations for . The latter in terms of Longo Trees and free lazy Plotkin Scott Engeler models. They represent the specialization to the lazy regime of, respectively, the Bohm Trees and the class of lambda models called Plotkin Scott Engeler models ([9, 12]) An alternative use of constants has played a crucial role in the operational study of . The standard way to treat a constant of the lambda calculus is to introduce it together with some rules describing its operational behaviour. In the sequel we will call these operators. We keep the word ....

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L., Ong, The Lazy Lambda Calculus: an Investigation into the Foundations of Functional Programming, PhD thesis, University of London, 1988. Also Prize Fellowship Dissertation, Trinity College, Cambridge 256 pp.

....terms coincide with the # undefined ones. It is well known that the unsolvable terms are exactly the terms without head normal form. This implies that unsolvability is equivalent to 001 active, treated above. Hence all axioms hold. Strongly unsolvable 101 active A term is strongly unsolvable [Ong88, Sec. 2.1.1 2] if it is a zero term and it is not convertible to a term of the form xs. That is, it has no weak head normal form, or equivalently it is a 101 active form, treated above. Hence all axioms hold. Mute hypercollapsing root active 111 active A term is mute if it is a zero term which ....

C.-H. L. Ong. The Lazy Lambda Calculus: An Investigation into the Foundations of Functional Programming. PhD thesis, Imperial College of Science and Technology, University of London, May 1988.

.... example is Milner s Context Lemma [Mil77] which says that applicative contexts are powerful enough to distinguish all terms (of the language in question) It has been called operational extensionality by Bloom [Blo87] and Abramsky and Ong [AO93] Ong also uses a form of the Context Lemma in [Ong88], as does Abramsky in [Abr90] To state the lemma, we rst need to de ne the simpler preorder. De ne s , a binary relation on 0 8 , as the largest relation satisfying the following co inductive de nition: 8v 1 2 V 8 : e 1 v 1 = 9v 2 2 V 8 : e 2 v 2 v 1 b s v 2 ....

C.-H. L. Ong. The Lazy Lambda Calculus: An Investigation into the Foundations of Functional Programming. PhD thesis, Department of Computer Science, Imperial College, University of London, March 1988.

....not be identified. The same holds for recursively defined functions. Hence, at second sight, it is not natural to consider terms without a normal form as meaningless. As an alternative, Abramsky and Ong propose to take the terms with a weak head normal form as the meaningful ones (Abramsky 1990, Ong 1988). As an argument in favour of this proposal Abramsky and Ong mention that in lazy functional languages no evaluation takes place inside weak head normal forms. However, this is only true since values of function type are not acceptable as output values. If terms of function type would be ....

Ong, C.-H.L. (1988),The Lazy Lambda Calculus: an Investigation into the Foundations of Functional Programming , Ph.D. Thesis, Imperial College, London.

....We also examine the possible weakenings of the theory m , by adding new axioms. We show that no such weakening can be as weak as . Our main result is the characterization of the preorder m over terms. We show that it coincides with the lazy Plotkin Scott Engeler preorder introduced by Ong in [13]. This is an ordering on an intensional representation of terms, the so called L evy Longo trees. These are like Bohm trees, fitted in with the lazy regime where any divergent term as Omega = INRIA The discriminating power of multiplicities 5 (x xx) x xx) is different from x The lazy PSE ....

.... Xi = fx:ff) fx:ff ) The lazy PSE ordering was introduced by Ong to characterize the local structure (following Barendregt s terminology) of some models of the lazy calculus. An immediate consequence of our main result and of results by Ong and Abramsky (namely the Theorem 3.4.1. 3 of [13] and Proposition 7.2.10 of [2] is that finite multiplicities provide us with strictly more discriminating power than the convergence testing combinators, introduced by Abramsky and Ong [2] to make the lazy calculus complete in some sense. Regarding the parallel convergence testing combinator, ....

[Article contains additional citation context not shown here]

L. Ong. The Lazy Lambda Calculus: an Investigation into the Foundations of Functional Programming. PhD thesis, Imperial College of Science and Technology, University of London, 1988.

.... ( p calculus) 5 Untyped c models It is well known that a categorical model for the untyped calculus is a re exive object D D = D in a cartesian closed category (see [Sco80, Bar82] In a c model there are two analogs for a re exive object: V V T = V and N TN T = N (see [Ong88] for similar de nitions in the context of partial cartesian closed categories) In the rst case we have a model of call by value. In fact the elements of V correspond to functions from values to computations (as V V T stands for V TV ) and therefore an element can be applied to a computation ....

....TN;N ) T ( F id TN ) eval T TN;N ) N Remark 5. 1 In call by value (let x=e in e 0 ) is equivalent to ( x:e 0 ) e) but in callby name there is no way of expressing (let x=e in e 0 ) in terms of application and abstraction only, because e is evaluated before binding its value to x (see [Ong88] for an analysis of call by name for partial computations) We think that it is desirable (and very natural) for a programming language to have a let, which forces evaluation of an expression. We conjecture that the calculus (i.e. Plotkin s call by name calculus) proves exactly those ....

C.-H.L. Ong. Lazy Lambda Calculus: An Investigation into the Foundations of Functional Programming. PhD thesis, University of London, October 1988.

....in part, to develop techniques for proving properties about code. We give some preliminary results in defining a logic (based on LCF [6, 13] for a fragment of PCF with up . The logic is shown to be sound for the model A Y . 1 These results were obtained independently from Abramsky [1] and Ong [9, 10], who have proven similar adequacy and full abstraction results for an untyped calculus. Cosmadakis [4] has extended our results to a language with product, sum, and recursive types. 2 Every isolated element [14] in the model may be given a Godel number n; an arbitrary element d is computable ....

.... ] A V [ oe] c A V [ where D c E is the cpo of continuous functions from D to E ordered pointwise [11, 13] In A Y , we lift each function space once: A Y [ oe ] D oe = A Y [ oe] c A Y [ If D is a domain, D) is D with a new bottom element added [1, 9, 10]. Concretely, the elements of (D) are fhd; 0i : d 2 Dg [ f g, ordered with v d for all d, and hd; 0i v hd 0 ; 0i iff d v d 0 . The function : D (D) with d = hd; 0i is an injection; the function : D) D, with hd; 0i = d and = is the corresponding projection. Given ....

[Article contains additional citation context not shown here]

C. L. Ong. The Lazy Lambda Calculus: An Investigation into the Foundations of Functional Programming. Ph.D. thesis, Imperial College, University of London, 1988.

....evaluation contexts to reflect the order of evaluation. 2.2 Call by value Models A denotational model for a call by value language is based on a collection of sets, indexed by type, suitable for giving meaning to terms. The following definition is taken from [35] and resembles definitions from [1, 28]. Definition 1 A call by value type frame over a type signature Sigma is a tuple D = fD oe j oe 2 Sigmag; f oe j oe 2 Sigmag; fApp ; j ; 2 Sigmag) Table 1: Typing and Evaluation Rules for Call by Value Functional Languages. Terms (var) x oe : oe (const) hf ; oe 1 ; oe ....

C.-H. L. Ong. The Lazy Lambda Calculus: An Investigation into the Foundations of Functional Programming. PhD thesis, Imperial College, University of London, 1988.

....D[ Delta] such that D[M] 6 D[N] Consider the type derivation x : s D[x] num. By Lemmas 3.2 and 3.3, there is an equivalent PCF term D 0 [x] such that x : s D 0 [x] num. Thus, D 0 [M] 6 D 0 [N] so M 6j PCF N. Such proofs can also be carried out for lazy and call by value PCF [2, 9, 26, 27, 31], but the set of rewrite rules are more complicated than the ones used in the above proof. The second style of proof is semantic, and has been used before in [23, 25] The proof idea is simple. Consider extending the language L 1 to L 2 . Suppose a fully abstract model, M 1 , of L 1 can be fully ....

C.-H. L. Ong. The Lazy Lambda Calculus: An Investigation into the Foundations of Functional Programming. PhD thesis, Imperial College, University of London, 1988.

....variables for two reasons: Firstly, they play logically distinct roles in the proof of Theorem 4.2 below in (Sangiorgi 1992) Secondly, we think that constants have their own natural interpretation, as described in Section 1. 3 Applicative Bisimulation over P C In (Abramsky 1987, Abramsky and Ong 1989, Boudol 1990 and 1991) the study of terms is conducted in terms of simulations and preorders 1 ; indeed it is always the case that a bisimulation coincides with the equivalence induced by the corresponding simulation. However this is not true in general with non determinism, hence we prefer to work with ....

.... Delta N r Gamma1 . Since the derivation of pM 1 Delta Delta Delta M r Gamma1 ) 2 M 00 has depth less than or equal to n Gamma 1, by induction pN 1 Delta Delta Delta N r Gamma1 ) 2 N 00 and N 00 P M 00 . Using App we infer pN 1 Delta Delta Delta N r ) 2 N 00 N r and, by the congruence of P , M 00 M r P N 00 N P . For N 0 = N 00 N r in (7) this concludes the case. Rule Trans) The premises of the rule are pM 1 Delta Delta Delta M r ) 2 M 00 and M 00 ) 2 M 0 . This case simply requires the application of the inductive hypothesis of the lemma twice. Behavioural ....

Ong, L. (1988a) The Lazy Lambda Calculus: an Investigation into the Foundations of Functional Programming, PhD thesis, University of London. Also Prize Fellowship Dissertation, Trinity College, Cambridge.

....of hnf s. A whnf is a term of the form x. M or x f M . This second proposal forms the basis of the lazy theory. Its tree structures under this proposal are the L#vy Longo Trees. There are interesting mathematical models of the calculus whose local structure is precisely the LT equality, see [Ong88] As an example, the terms x. Omega and Omega are distinguished in the lazy theory because only the former has a whnf; but they are identi ed as meaningless in the sensible theory because neither has an hnf. Similarly, Xi and x. Omega , are meaningless in a sensible theory, but are meaningful ....

....and Park in concurrency [Par81, Mil89] Since Abramsky s work, the idea of applicative bisimulation has been applied to a variety of higherorder sequential languages; see [Gor95, Pit97] for surveys. Open applicative bisimulation coincides with the equivalence induced by Ong s lazy PSE ordering [Ong88] however, a conceptual dioeerence between the two is the emphasis that Ong s preorder places on j expansion. The operational and denotational theory of N (that they call the lazy calculus) has been extensively studied by Abramsky and Ong [Abr87, Abr89, Ong88, AO93] The canonical model of N ....

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L. Ong. The Lazy Lambda Calculus: an Investigation into the Foundations of Functional Programming. PhD thesis, University of London, 1988. Also Prize Fellowship Dissertation, Trinity College, Cambridge, 256 pp.

.... from types with regions to F # types, and ( Delta) from types without regions to F # types, as follows: s; r) s # r) int = int (s j Gamma t) s j (t ) These types are related to the interpretation of call by value in call by name (see [14]) The translation also involves the translation of sets of effects to F # types. Define (get(r) 8b: b # r) b) put(r) 8b: b (b # r) Table 4. Translation of the region calculus. Var] G x : G(x) 0 ) G (lp : 0 : lift x) 0 (G(x) Const] G ....

C.-H. L. Ong. The Lazy Lambda Calculus: An Investigation into the Foundations of Functional Programming. PhD thesis, Imperial College, University of London, 1988.

....work with an aim similar to ours has already appeared [21, 4] the equational framework in the present paper has several significant properties not found in those predecessors. First, the basic formal apparatus is an extension of the well studied method found in both strict and lazy theories [2, 1, 22], using the maximality condition among a certain family of congruences to derive canonical equality over agents. Specifically a fundamental element of the construction is reduction closure for equality, which generalises fi equality (or convertibility) into the regime of concurrent processes. The ....

.... without committing ourselves to a specific notion of observation we even do not employ any kind of convergence predicate [21, 4] which is considered to be minimised observability in [21] Rather, we use a semantic scheme analogous to the one used effectively in strict and lazy calculi [2, 1, 22], where, in each setting, the identification of meaningless terms in equational theories leads to a certain canonical equality which is more general than convertibility and, moreover, which coincides with a syntax free model of the calculus. It turns out that the mechanism works well in the ....

[Article contains additional citation context not shown here]

Ong, C-H.L., The Lazy Lambda Calculus: An Investigation into the Foundation of Functional Programming. Phd thesis, Imperial College, 1988.

.... done to develop methods for reasoning about operational approximation and equivalence: Abramsky (1990, 1991) Bloom (1990) Egidi, Honsell, and Ronchi della Rocca (1992) Howe (1989, 1996) Gordon (1995) Lassen (1995b) Mason (1986) Mason and Talcott (1991a) Jim and Meyer (1991) Milner (1977) Ong (1988); Pitts and Stark (1993, 1996) Ritter and Pitts (1995) Pitts (1996) Smith (1992) Sullivan (1996) Talcott Reasoning about Functions with Effects 5 (1985) Methods developed for reasoning about operational approximation and equivalence include: general schemes for establishing equivalence; ....

Ong, C.-H. (1988). The Lazy Lambda Calculus: An investigation into the Foundations of Functional Programming. Ph. D. thesis, Imperial College, University of London.

....of equivalence as black boxes. Treating programs as black boxes requires only observing what effects and values they produce, and not how they produce them. Our definition extends the extensional equivalence relations defined by [40] and [46] to computation over memory structures. As shown by [3, 4, 9, 11, 25, 28, 32, 26, 38, 42, 45, 50, 51] operational equivalence and approximation can be characterized in various ways. Definition ( Two expressions are operationally equivalent, written e 0 = e 1 , if for any closing context C, C[e 0 ] is defined iff C[e 1 ] is defined. e 0 = e 1 , 8C 2 C FV(C[e 0 ] FV(C[e 1 ] ....

C-H.L. Ong. The Lazy Lambda Calculus: An investigation into the Foundations of Functional Programming. PhD thesis, Imperial College, University of London, 1988.

....N then C[M ] PSE C[N ] for any context C[ 6 Conclusions and Related Work The relationship between the three lazy behavioural preorders are as follows: Theorem 6.1 (Comparison) LT ae PSE ae B . Proof For a proof of the inclusion, see Propositions 4.6.6.2 and 4.6.6. 4 in [Ong88b]; to see that the inclusions are strict: let U def = x:x(x Upsilon Theta) Upsilon and V def = x:x(y:x Upsilon Thetay) Upsilon for any Theta 2 PO 0 and Upsilon 2 PO1 . We have: U = LT V and V = LT U ; U PSE V and V = PSE U ; U B V and V B U (proof of ....

....2 PO1 . We have: U = LT V and V = LT U ; U PSE V and V = PSE U ; U B V and V B U (proof of the last pair of assertions is non trivial, see e.g. Ong88b, Theorem 4.5.1. 1] or [AO93] Two other directions of research in the Lazy Lambda Calculus are reported in [Ong88b]: ffl We investigate the full abstraction problem in the Lazy Lambda Calculus a la Abramsky, focussing on the Lazy Lambda Calculus with convergence testing C and Labelled Lazy Lambda Calculus in Ch. 4 ibid. A general method for constructing fully abstract models which are retracts of D for ....

C.-H. L. Ong. The Lazy Lambda Calculus: An Investigation into the Foundations of Functional Programming. PhD thesis, University of London, 1988.

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L. Ong. The Lazy Lambda Calculus: an Investigation into the Foundations of Functional Programming. PhD thesis, University of London, 1988. Also Prize Fellowship Dissertation, Trinity College, Cambridge, 256 pp.

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