3.2 Equivalence, Evaluation and Reduction Strategies

Transcription

1 3.2 Equivalence, Evaluation and Reduction Strategies The λ-calculus can be seen as an equational theory. More precisely, we have rules i.e., α and reductions, for proving that two terms are intensionally equivalent 12 which implies that the two terms have the same meaning. In particular, if we take the reduction relation as the embodiment of the behavioural meaning of terms, intensional equivalence should then imply that two equivalent terms have the same behaviour. In the context of the λ-calculus, one intuitive interpretation of such behaviour would then be that two equivalent λ-terms denote the same function. A similar discussion was already held earlier for α-reductions. In fact we have already argued that the two terms λx.x and λy.y denote the same function, i.e., the identity function Id over some domain and co-domain, because the specific name of the bound variable is not important. Correspondingly, Def. 64(α-equivalence) allowed us to equate these two terms as the relation M α M. We now extend this discussion to computation in general, so as to be able to equate terms such as λx.x and (λx.x)(λy.y) on the basis that they both behave like the identity function Id. In fact, if we apply these two terms to any arbitrary argument M, they both return M (after some reductions). Multi-Step reduction (Evaluation) Definition 68 (Evaluation). The least relation satisfying M N λx.m = λx.n M = M M α M M = N N α N M = M N = N MN = M N M = L L = N Slide 93 We start off by formalising computations (evaluations) over λ-terms, =, which relates terms that are reachable using an arbitrary number of -reductions (up-to α-equivalence); this can be seen as the reflexive transitive closure of, i.e.,. It is defined as the least relation satisfying the rules on Slide 93. We here point out a number of characteristics of the relation and of -reduction in particular: -reduction is asymmetric (as opposed to α-reduction, which was symmetric). Thus if M N,it is generally not the case that N M. From Def. 68, it should be clear that multi-step reduction, =, is also asymmetric. In fact, the last two rules admit only reflexivity (because M 0 M and transitivity (because M n L and L m N implies M n+m N). This gives us a preorder over λ-terms. 12 Two terms are intentionally equivalent when they denote the same meaning; on the other hand, they are syntactically equivalent when they have an identic syntactic form. 74

2 Multi-step reduction abstracts over α-equivalence. This is characterised by the second rule on Slide 68. If a term M has a normal form, then it must be the case that for some N which is in normal form Confluence Before we consider equivalence up-to -reduction, we have to understand a fundamental property relating to multi-step -reductions (evaluations) in the λ-calculus. This property is often referred to as the diamond property, and is stated on Slide 94. As it turns out, this property justifies the use of -reductions as rules for determining equivalence. More formally, Theorem 69, called the Church-Rosser Theorem, states that evaluation in the λ-calculus is confluent, meaning that, when starting from the same λ-term, if two sequences of reductions reach different terms, we can always eventually evaluate back to a common term from the resulting distinct terms. The Church Rosser (Diamond) Property Theorem 69 (Church-Rosser). For any term M, whenever M = M 1 and M = M 2 then there exists some term M 3 such that M 1 = M 3 and M 2 = M 3. M implies M 1 M 2 M 1 M M 3 M 2 Slide 94 This theorem has particular significance with respect to normal forms i.e., terms with no further reductions (see Def. 67). In fact, since terms in normal form can only evaluate up to themselves, i.e., N = N, the Church-Rosser Theorem implies that if two normal forms N 1 and N 2 can be reached when evaluating some common term M, then these terms must be equivalent (up-to α-equivalence) - otherwise they would violate Theorem 69. Stated otherwise, we are justified in abstracting away from the actual reduction sequence used to reach a normal form i.e., = of Slide 93, as this is independent of the order in which the individual -reductions are performed on a term, i.e., of Slide 91. Example 70 (Evaluations and normal forms). Consider the term (λx.(λz.z)x) λy.y. There are two evaluations that lead to the same normal form, namely: (λx.(λz.z)x) λy.y λz.z(λy.y) λy.y (λx.(λz.z)x) λy.y (λx.x) λy.y λy.y Note that the diamond property only holds for evaluations and does not hold for single-step -reductions. 75

3 In other words, we can find a counter example that violates the diagram below: M implies M 1 M 2 M 1 M M 2 M 3 For instance, consider the term (λx.x x) ((λy.y)(λz.z)) in Example 71. It can admit two different reductions as shown on Slide 95. However, we have no way how to then complete the diamond diagram. The problem is caused by the term λx.x x which replicates it arguments. Thus, if the argument in this case (( λy.y) (λz.z)) is not fully evaluated before being passed to λx.x x, then the resulting term also duplicated any evaluations in this argument. This, in turn, prohibits us from completing the diagram using a single -reduction. Church-Rosser does not hold for single-step reductions Example 71. The term (λx.x x) ((λy.y)(λz.z)) can have the two reductions: ((λy.y)(λz.z)) ((λy.y)(λz.z)) (λx.x x) ((λy.y)(λz.z)) (λx.x x) (λz.z) But we cannot close the diamond using a single -reduction! Slide 95 As an aside, from a computational point of view, this fact has an important implication in terms of the efficiency of term evaluation; in fact it seems to imply that it is generally more efficient to evaluate arguments before they are used in an application, because they may be used more than once in the body of an abstraction, thereby duplicating the effort through multiple reductions of the same argument. This discourse is however not as clearcut as this example seems to indicate. Consider the term (λx.y)((λz.z)(λw.w)) as a second example. Evaluating the argument first yeilds two reductions: (λx.y) ((λz.z) (λw.w)) λx.y (λw.w) y whereas not evaluating arguments beforehand leads to just one reduction till we hit a term in normal-form. We revisit this argument later on in Section Equivalence (λx.y) ((λz.z) (λw.w)) y Theorem 69 also justifies the use of evaluations as a mechanism for determining whether two terms M and N are equivalent: we evaluate the two terms down to their normal form and if the respective normal-form terms are α-equivalent, then M and N are equivalent. More precisely, by using = as the basis for our 76

4 proof system, we obtain an equational theory that is consistent. 13 More precisely, in order to show that two terms, say M and N, have the same semantic meaning, we can just show, using the rules on Slide 93, that M and N have the same normal form. Thanks to the Church-Rosser property, we know that there is no way we could reach two different normal forms by following different reduction strategies. All this is formalised as Def. 72 on Slide 96. When two terms, M and N, are equivalent in this sense, we say that they are -equivalent, and denote it as M N. Example 73 shows two terms that are -equivalence. -equivalence Definition 72 (-equivalence). Two terms M and N are - equivalent, denoted as M N iff whenever M = L where L is in normal, then N = L, and viceversa. Example 73. We have (λx.(λz.z)x) λy.y (λx.λz.x) (λy.y) w Slide Non-termination of Evaluation Although different reduction strategies cannot reach distinct normal forms (up to α-equivalence), they can still yield different outcomes. In particular, one evaluation may terminate whereas the other may diverge and evaluate forever. Typically this happens when a term has a normal form but contains a subterm having no normal form. Church-Rosser and non-termination Example 74. Recall the terms Cnst = λy.z Ω =(λx.x x)(λx.x x) The composite term Cnst Ω can reach a normal form : (λy.z)ω z which implies (λy.z)ω = z But it can also reduce forever Cnst(λx.x x)(λx.x x) Cnst(λx.x x)(λx.x x)... Slide 97 Example 74 on Slide 97 shows one such term that can both yield a term in normal form and also evaluate forever. In particular, the infinite reduction sequence is caused by the subterm called Ω (encountered earlier at the end of Sec. 3.1.) 13 An equational theory is inconsistent if all equations are provable; or alternately, if an equation can be proved true and false within the same theory. 77

5 Conflicting aims complicate the search for an efficient reduction strategy for evaluating a term to its normal-form (when it has one): on Slide 95 we saw how it is desirable to evaluate the argument before performing a -reduction in order not to duplicate reductions unnecessarily; at the same time, Slide 97 illustrates the danger with this strategy, as there are cases where this will not lead to a normal-form but instead result in an infinite evaluation even when such normal form exists. Normal Order Reduction Definition 75 (Normal Order Reduction). The least relation satisfying (λx.m) N no M{x := N} M = λx.m M no L MN no LN M no λx.m no N λx.n M = λx.m M no N no L MN no ML Slide 98 If we are merely interested in finding the normal form of a term (whenever it exists) then there is a reduction strategy that guarantees to find it. It is called normal-order reduction and it always evaluates the leftmost-outermost redex in a term; by leftmost we mean that in a term of the form MN we reduce M before N; by outermost we mean that in a term of the form (λx.m) N we first substitute N for x in M before reducing either M or N. The Normalisation Property Theorem 76 (Normalisation). If M has a normal form N, then there exist a finite number n of normal-order reductions such that M n no L such that L α N Example 77. Recall the terms Cnst = λy.z Ω =(λx.x x)(λx.x x) The composite term Cnst Ω can reach a normal form using 1 normal-order reduction: (λy.z) Ω = z which implies M.(λy.z)Ω no M then M α z Slide 99 This reduction strategy is formalised by the relation no defined by the rules of Def. 75 on Slide 98. It is important to note that the normal-order reduction strategy is deterministic: when there is more than one 78

6 possible redex to reduce in a term, it always picks the same one, namely the leftmost-outermost one. More importantly though, normal order reduction avoids running into infinite computations when alternative reductions exist, as in the case of Example 74, by not evaluating arguments before substituting them. Although this is (in general) inefficient (it may replicate reductions), it guarantees to evaluate down to a normal form term if one exists. We state (without proving) this property in Theorem Equivalence and Undecidability Back to our notion of equivalence between λ-terms, we note that Def. 72 with respect to divergence i.e., non-terminating terms is somewhat too weak as it does not distinguish between any of these divergent terms (i.e., we always equate them). In some sense though, we would like to, at least, distinguish amongst some of these divergent terms based on the intuition that these terms may yield a normal form when applied to certain arguments. One such example of terms would be (λx.x x)(λx.x x) and λw.(w (λx.x x)(λx.x x)) The left hand term will always yield an infinite computation irrespective of what argument it is applied to. However, the right term may yeild a normal form, when applied to an argument such as λy.z. Thus,inthis sense, they denote qualitatively different functions. Naive attempts at addressing this problem may however introduce complications. For instance, it turns out that any system that equates the λ-terms (λy.y (λu.λv.u))(λx.x x)(λx.x x) and (λy.y (λu.λv.λw.u w (vw)))(λx.x x)(λx.x x) will lead to an inconsistent system whereby every term can be equated to any other term. The implication of this fact is also that general λ-term equivalence is an undecidable problem and bears a direct correspondence to the Halting problem and Turing Machines. There exist (partial) solutions for this problem in terms of a strictly stronger notion of equivalence based on -reduction, whereby -reductions are only allowed to occur in certain positions within a λ-term. More specifically, this involves identifying what is called a head redex and then limiting -reduction up to head normal forms. This allows us to differentiate between more terms, more precisely, between different classes of divergent terms. In this course we shall not delve further into this aspect of the topic but interested readers are encouraged to consult the reference literature stated at the beginning of the course Reduction Strategies As we already saw, -reductions as defined in Def. 66 are agnostic as to which redex to choose to evaluate when multiple redex instances occur in a term. This leads to a non-deterministic specification of reduction which is problematic for implementations. Functional Programming languages (which are directly inspired by the λ-calculus) often limit themselves to a reduction strategy, thereby making reductions (and the behaviour of their programs) deterministic. Broadly speaking, these reduction strategies are usually partitioned into two classes, often refered to as eager strategies and lazy strategies. Implementations based on eager reduction strategies such as Standard-ML and OCaml usually prioritise efficiency. More specifically, in function applications arguments are eagerly evaluated before being substituted in function bodies. One such strategy is the Call-by-Value strategy. It is based on the partitioning of λ-terms into values and non-value terms and requires that arguments are evaluated down to values before they are substituted in function bodies. 14 Call-by-Value reduction is formalised through Def. 78 (whereby values consist of either variables or abstractions) and the rules in Def. 79 on Slide 100. Implementations based on lazy reduction strategies such as Haskell prioritise the clean theoretical programming model of the λ-calculus with its associated properties such as normalisation (see Thm. 76). In such strategies arguments are not evaluated before being substituted in function bodies. Perhaps the simplest lazy strategy from a theoretical standpoint is the Call-by-Name strategy. In fact, it does not rely on the 14 In standard programming languages, values are assumed to be in normal form. 79

7 Call-by-Value Reduction Definition 78 (Call-by-Value Terms and Values). M,N ::= x MN λx.m V,U ::= x λx.m Definition 79 (Call-by-Value Reduction). M cbv L N cbv L (λx.m) V cbv M{x := V } MN cbv LN VN cbv VL Slide 100 definition of values in the calculus and can be specified by the two rules on Slide 101. As in normal-order reduction, the Call-by-Name strategy is inefficient because it replicates the reductions in unevaluated arguments substituted in abstraction bodies. 15 There exist more complex lazy evaluation strategies that are not inefficient. One such example is the Call-by-Need reduction strategy which reduces function arguments at most once; this way it attains the best of both Call-by-Value and Call-by-Name strategies. Languages such as Haskell adopt a Call-by-Need reduction strategy. Call-by-Name Reduction Definition 80 (Call-by-Name Reduction). (λx.m) N cbn M{x := N} M cbn MN cbn L LN Slide 101 It is worth highlighting the fact that different reduction strategies yields different equivalence relations. In particular for every reductions strategy we can seen so far, we can define three variants of the evaluation reduction, namely = no,= cbv and = cbn, by adapting the relation defined in Definition 68 on Slide 93 and replacing the first rule by the following rules respectively (the rest of the rules carry over unchanged 15 The two reduction strategies more or less correspond. In fact they are both optimal with respect to term normalisation. The only difference is that normal-order reduction reduces under abstractions whereas Call-by-Name does not. Thus, for example if we evaluate λy.ω under normal-order reduction we reduce forever whereas Call-by-Name stops reducing at λy.ω. 80

8 with the exception of the congurence rules 16 ): M no N M = no N M cbv N M = cbv N M cbn N M = cbn N These three evaluations can then be used to define 3 equivalence relations, i.e., no, cbv and cbn respectively. Notice that the notion of a normal form under = no is different from the one for both = cbv and = cbn, since the latter two reduction strategies do not evaluate under λ-abstractions Exercises 1. Argue why the diamond property does not hold for single-step -reductions, using a similar argument to the one used in Example 71, but instead using the term (λx.y)((λz.z)(λw.w)). 2. Show that (λx.x)m M and (λx.x)(λy.y)m M using the rules on Slide Show that (λx.x)m = M and (λx.x)(λy.y)m = M using the rules on Slide Derive (λz.λx.zx) λy.x = λw.x using the rules on Slide Prove that, for all n, M n N implies. (Hint: use induction on n, the number of -reductions used to reach N from M.) 6. Prove that (λx.(λz.z)x) λy.y (λx.λz.x) (λy.y) w. 7. Evaluate the term (λx.λy.x x) ((λz.z)(λz.z)) using normal-order reduction, Call-by-Value and Callby-Name strategies. 16 In this case, the congurence rules are M = M N = N λx.m = λx.n MN = M N 81