2.1 Basics on rewriting

We recall here some standard definitions and notations in rewriting that we shall use in this paper (see for instance Terese 2003 or Baader and Nipkow Reference Baader and Nipkow1998 for details).

Rewriting System. An abstract rewriting system (ARS) is a pair consisting of a set A and a binary relation on A whose pairs are written and called steps. A -sequence from is a sequence of steps, where or for some , for all and (in particular, the sequence is empty for , i.e. ). We denote by (resp. ; ) the transitive-reflexive (resp. reflexive; transitive) closure of , and stands for the transpose of , that is, if . We write for a -sequence of steps. If are binary relations on then denotes their composition, i.e. if there exists such that . We often set .

A relation is deterministic if for each there is at most one such that . It is confluent if .

We say that is -normal (or a -normal form, noted ) if for all , that is, there is no such that ; and we say that has a normal form u if with u -normal. Confluence implies that each has unique normal form, if any exists.

Normalization. Let be an ARS. In general, a term may or may not reduce to a normal form. And if it does, not all reduction sequences necessarily lead to a normal form. A term is weakly or strongly normalizing, depending on if it may or must reduce to normal form. If a term is strongly normalizing, any choice of steps will eventually lead to a normal form. However, if is weakly normalizing, how do we compute a normal form? This is the problem tackled by normalization and normalizing strategies: by repeatedly performing only specific steps, a normal form will be computed, provided that can reduce to any. We recall two important notions of normalization.

Note that in Definition 2.2, need not be deterministic, and and need not be confluent.

Factorization. In this paper, we will extensively use factorization results.

Relation postpones after , written , if

It is an easy result that -factorization is equivalent to postponement, which is a more convenient way to express it.

(1) Postponement: .

(2) Factorization: .

Hindley (Reference Hindley1964) first noted that a local property implies factorization. Let . We say that strongly postpones after , if

Observe that the following are special cases of strong postponement. The first one is linear in ; we refer to it as linear postponement. In the second one, recall that .

(1) .

(2) .

Linear variants of postponement can easily be adapted to quantitative variants, which allow us to “count the steps” and are useful to establish termination properties. We do this in Section 6.3.

Diamonds. We recall also another quantitative result, which we will use.

Fact 2.6 (Newman 1842). In an ARS , if is quasi-diamond, then it has random descent, where quasi-diamond and random descent are defined below.

(1) Quasi-Diamond: For all , if , then or for some .

(2) Random Descent: For all , all maximal -sequences from have the same number of steps, and all end in the same normal form, if any exists.

Clearly, if is quasi-diamond then it is confluent and uniformly normalizing.

Postponement, Confluence and Commutation. Both postponement and confluence are commutation properties. Two relations and on commute if

So, a relation is confluent if and only if it commutes with itself. Postponement and commutation can be defined in terms of each other, simply taking for and for ( postpones after if and only if commutes with ). As propounded in Reference van Oostromvan Oostrom (2020b), this fact allows for proving postponement by means of decreasing diagrams (van Oostrom Reference van Oostrom1994, Reference van Oostrom2008). This is a powerful and general technique to prove commutation properties: it reduces the problem of showing commutation to a local test; in exchange for localization, diagrams need to be decreasing with respect to a labeling.

where for any , and .

Modularizing Confluence. A classic tool to modularize a proof of confluence is Hindley–Rosen lemma: the union of confluent reductions is itself confluent if they all commute with each other.

Like for postponement, strong commutation implies commutation.

2.2 Basics on the -calculus

We recall the syntax and some relevant notions of the -calculus, taking Plotkin’s call-by-value (CbV, for short) -calculus (Plotkin Reference Plotkin1975) as a concrete example.

Terms and values are mutually generated by the grammars below.

where x ranges over a countably infinite set of variables. Terms of shape TS and are called applications and abstractions, respectively. In , binds the occurrences of x in T. The set of free ( i.e. non-bound) variables of a term T is denoted by . Terms are identified up to (clash-avoiding) renaming of their bound variables ( -congruence).

Reduction.

• Contexts (with exactly one hole ) are generated by the grammar

stands for the term obtained from by replacing the hole with the term T (possibly capturing some free variables of T).

• A rule is a binary relation on , also noted , writing ; R is called a -redex.

• A reduction step is the closure of under contexts . Explicitly, if then if and , for some context and some .

The Call-by-Value -calculus. The CbV -calculus is the rewrite system , the set of terms equipped with -reduction , that is, the contextual closure of the rule :

where is the term obtained from T by capture-avoiding substitution of V for the free occurrences of x in T. Notice that here -redexes can be fired only when the argument is a value.

Weak evaluation (which does not reduce in the body of a function) evaluates closed terms to values. In the literature of CbV, there are three main weak schemes: reducing from left to right, as defined by Plotkin (Reference Plotkin1975), from right to left (Leroy Reference Leroy1990), or in an arbitrary order (Lago and Martini Reference Lago and Martini2008). Left contexts , right contexts , and (arbitrary order) weak contexts are, respectively, defined by

Given a rule on , weak reduction is the closure of under weak contexts ; non-weak reduction is the closure of under contexts that are not weak. Left and non-left reductions ( and ), right and non-right reductions ( and ) are defined analogously.

Note that and are deterministic, whereas is not.

CbV Weak Factorization. Factorization of allows for a characterization of the terms which reduce to a value. Convergence below is a remarkable consequence of factorization.

3.1 The computational core vs. computational calculi with let-notation

It is natural to compare the computational core with other untyped computational calculi, and wonder if the analysis of the rewriting theory of we present in this paper applies to them. There is indeed a rich literature on computational calculi refining Moggi’s (Moggi 1988 Reference Moggi1989, Reference Moggi1991), most of them use the let -constructor. A standard reference is Sabry and Wadler’s (Sabry and Wadler Reference Sabry and Wadler1997, Section 5), which we display in Figure 1.

Figure 1.

: Syntax and reduction.

has a two sorted syntax that separates values (i.e., variables and abstractions) and computations. The latter are either let-expressions (aka explicit substitutions, capturing monadic binding), or applications (of values to values), or coercions [V] of values V into computations ([V] is the notation for in Sabry and Wadler (Reference Sabry and Wadler1997), so it corresponds to in ).

• The reduction rules in are the usual and from Plotkin’s call-by-value -calculus (Plotkin Reference Plotkin1975), plus the oriented version of three monad laws: , , (see Figure 1).

• Reduction is the contextual closure of the union of these rules.

To state a correspondence between and , consider the translations in Figure 2: translation from to (resp. from to ) is defined via the auxiliary encoding (resp. ) for values. The translations induce an equational correspondence by adding -equality to . More precisely, let be the closure of the rule (below left) under contexts (below right).

Figure 2.

Translations between and .

Let be the reflexive transitive and symmetric closure of the reduction , and similarly for with respect to .

Proposition 3.9 establishes a precise correspondence between the equational theories of (including -conversion) and . We had to consider since

(7)

(where is the reflexive transitive and symmetric closure of ) and so condition (1) in Proposition 3.9 would not hold if we replace with .

• the condition corresponding to Point 1, namely , fails since when for any value V, see (7) above;

• the condition corresponding to Point 3, namely implies , fails because but .

Surface vs. Weak Reduction. In this paper, we study not only weak but also surface reduction, as the latter has better rewriting properties than the former. Surface reduction in can be seen as a natural counterpart to Plotkin’s weak reduction in calculi with the let-constructor, such as . Intuitively, a term of the form can be interpreted as syntactic sugar for , however, in the expression , it is not obvious that weak reduction should avoid firing redexes in N. The distinction between and let allows for a clean interpretation of weak reduction: it forbids reduction under , but not under let, which is compatible with surface reduction in .

Technically, when we embed in the computational core via the translation in Figure 2, weak reduction in (defined as the restriction of that does not fire under ) corresponds to surface reduction in : if then but not necessarily . Indeed, consider in , with ; then , which is not a weak step, in .

5.1 Weak reduction: the impact of associativity and identity

Weak (left) reduction (Section 2.2) is one of the most common and studied way to implement evaluation in CbV, and more generally in calculi with effects.

Weak reduction , that is, the closure of under weak contexts is a deterministic relation. However, when including the rules induced by the monadic equation of associativity and identity, the reduction is nondeterministic, nonconfluent, and normal forms are not unique.

This is somehow surprising, given the prominent role of such a reduction in the literature of calculi with effects. Notice that the issues only come from and , not from . To resume:

(1) Reductions and are nondeterministic, but are both confluent.

(2) are nondeterministic, nonconfluent, and their normal forms are not unique, i.e., adding , weak reductions lose confluence and uniqueness of normal forms.

Because of the rule, weak reductions and are not confluent and their normal forms are not unique (Point 2 above). Indeed, consider where and . Then,

where and are different -normal forms (clearly, ).

Reduction (like , Theorem 2.11.1) admits weak factorization (Fact F.2 in Appendix F)

This is not the case for . The following counterexample is due to van Oostrom (Reference van Oostrom2020a).

M is -normal and cannot reduce to by only performing steps (note that is -normal), hence it is impossible to factorize the sequence from M to as .

Let: Different Notation, Same Issues. We stress that the issues are inherent to the associativity and identity rules, not to the specific syntax of . Exactly the same issues appear in Sabry and Wadler’s (Sabry and Wadler Reference Sabry and Wadler1997) (see our Figure 1), as we show in the example below.

We write for the closure of the rules (in Figure 1 ) under contexts . We observe two problems, the first one due to the rule , the second one to the rule .

(1) Non-confluence . Because of the associative rule , reduction is nondeterministic, nonconfluent, and normal forms are not unique. Consider the following term

There are two weak redexes in T, the overlined and the underlined one. Therefore,

where T’ are T” different -normal forms.

(2) Non-factorization . Because of the -rule, factorization w.r.t. sequencing does not hold. That is, a reduction sequence cannot be reorganized as weak steps followed by non-weak steps. Consider the following variation on van Oostrom’s Example 5.2:

M is -normal and cannot reduce to z z by only performing steps (note that is -normal), so it is impossible to factorize the sequence form M to zz as .

5.2 Surface reduction

In , surface reduction is nondeterministic, but confluent, and well-behaving.

Fact 5.4 (Nondeterminism). For , is nondeterministic (because in general more than one surface redex can be fired).

We now analyze confluence of surface reduction. We will use confluence of (Point 2 below) in Section 8 (Theorem 8.13).

Proof. We rely on confluence of (by Theorem 4.2), and on Hindley–Rosen Lemma (Lemma 2.9). We prove commutation via strong commutation (Lemma 2.10). The only delicate point is the commutation of with (Points 3 and 4).

(1) is locally confluent and terminating, and so confluent; is quasi-diamond (in the sense of Fact 2.6), and hence also confluent.

(2) It is easily verified that and strongly commute and similarly for and . The claim then follows by Hindley–Rosen Lemma.

(3) strongly commutes with . This point is delicate because to close a diagram of the shape may require a step, see Example 5.7. The claim then follows by Hindley–Rosen Lemma.

(4) A counterexample is provided by the same diagram mentioned in the previous point (Example 5.7), requiring a step to close.

Example 5.6. Let us give an example for nondeterminism and confluence of surface reduction.

(1) Non-determinism: Consider the term where R and R’ are any redexes.

(2) Confluence: Consider the same term as in Example 5.1: . Then,

Now we can close the diagram:

Note that , and so , since is the term obtained from by renaming its bound variable y to x.

This is also a counterexample to confluence of and of .

In Section 6, we prove that surface reduction does factorize , similarly to what happens for Simpson’s calculus (Theorem 4.1). Surface reduction also has a drawback: it does not allows us to separate and steps. This fact makes it difficult to reason about returning a value.

Here it is not possible to postpone a step after a step .

5.3 Confluence properties of , and

Finally, we briefly revisit the confluence of , already established in de' Liguoro and Treglia (2020), in order to analyze the confluence properties of the different subsystems too. This completes the analysis given in Proposition 5.5. In Section 8, we will use confluence of (Theorem 8.14).

6.1 Surface factorizations in , modularly

We prove surface factorization in . We already know that surface factorization holds for (Fact 4.3), so we can rely on it, and work modularly, following the approach proposed in Accattoli et al. (Reference Accattoli, Faggian and Guerrieri2021). The tests for call-by-name head factorization and call-by-value weak factorization in Accattoli et al. (Reference Accattoli, Faggian and Guerrieri2021) easily adapt to surface factorization in , yielding the following convenient test. It modularly establishes surface factorization of a reduction , where is a new reduction added to . Details of the proof are in Appendices B.2 and B.3.

(1) Surface factorization of : .

(2) is substitutive:

(3) Root linear swap: .

We will use the following easy property (an instance of Lemma B.4 in the Appendix).

Root Lemmas. Lemmas 6.7 and 6.8 below provide everything we need to verify the conditions of the modular test in Proposition 6.5, and so to establish surface factorization in .

Lemma 6.7 ( -Roots). Let . The following holds:

Lemma 6.8 ( -Roots). Let . The following holds:

Let us make explicit the content of the two lemmas above. By instantiating in Lemmas 6.7 and 6.8, and combining them with Lemma 6.6, we obtain the following facts:

Fact 6.9. implies and so (Lemma 6.6) implies ( i.e. strong postponement holds).

(2) implies and so (Lemma 6.6) implies .

(3) implies .

Fact 6.10. (1) implies , and so (Lemma 6.6) implies ( i.e. strong postponement holds).

(2) implies , and so (Lemma 6.6) implies .

(3) implies .

Surface Factorization of . We can now combine the facts above concerning and steps, using the modular approach proposed in Accattoli et al. (Reference Accattoli, Faggian and Guerrieri2021) (see Theorem B.3 in the Appendix), to prove surface factorization of .

Surface Factorization of , Modularly. We are now ready to use the modular test for surface factorization with (Proposition 6.5) to prove Theorem 6.1.

Interestingly, the same machinery can also be used to prove another surface factorization result, which says that surface factorization, when applied to only, does not create steps.

Proposition 6.12 (Surface factorization of ). Reduction admits surface factorization:

The fact that two similar factorization results (Theorem 6.1 and Proposition 6.12) can be proven by means of the same modular test (Proposition 6.5) fed on similar lemmas shows one of the benefits of our modular approach: passing from one result to the other is smooth and effortless.

6.2 A closer look at steps, via postponement

We show a postponement result for steps on which evaluation (Section 7) and normalization (Section 8) rely. Note that overlaps with . We define as a step that is not .

Clearly, . In the proofs, it is convenient to split into steps that are also steps, and those that are not.

Notice that .

We prove that steps can be postponed after both and steps, by using van Oostrom’s decreasing diagrams technique (van Oostrom Reference van Oostrom1994, Reference van Oostrom2008) (Theorem 2.8). The proof closely follows van Oostrom’s proof of the postponement of after β (Reference van Oostromvan Oostrom 2020b).

We prove that the pair of relations is decreasing by checking the following three local commutations hold (the technical details are in the Appendix):

(1) (see Lemma C.3);

(2) (see Lemma C.4);

(3) (see Lemma C.5).

Hence, by Theorem 2.8, the relations and commute. That is, postpones after :

Or equivalently (Lemma 2.4), .

6.3 Weak factorization

Thanks to surface factorization plus -postponement (Corollary 6.14), every -sequence can be rearranged so that it starts with an -sequence. We show now that such an initial -sequence can in turn be factorized into weak steps followed by non-weak steps. This weak factorization result will be used in Section 7 to obtain evaluation via weak steps (Theorem 7.6).

Remarkably, weak factorization of an -sequence preserves the number of steps. This property has no role with respect to evaluation, but it will be crucial when we investigate normalizing strategies in Section 8. For this reason, we include it in the statement of Theorem 6.16.

Quantitative Linear Postponement. Let us take an abstract point of view. The condition in Lemma 2.5 – Hindley’s strong postponement – can be refined into quantitative (linear) variants, which allow us to “count the steps” and are useful to establish termination properties.

(8)

Then, implies and the two sequences have the same number of steps, for each .

Observe that in (8), the last step is , not necessarily .

Weak Factorization. We show two kinds of weak factorization: a surface sequence can be reorganized, so that non-weak steps are postponed after the weak ones; and a weak sequence can in turn be rearranged so that weak steps are before weak steps.

(9)

Assume : M and L have the same shape, which is not , otherwise no weak or surface step from M is possible. We examine the cases.

- The step has empty context:

- Then, , and with . Therefore, .

- . Then, , and where exactly one among reduces to V, P, Q, respectively, the other two are unchanged. So, .

- The step has nonempty context. Necessarily, we have , with :

- Case with and , then .

- Case with . We conclude by i.h. .

Observe that we have proved more than (9), namely we proved

So, we conclude that the two sequences have the same number of steps, by Lemma 6.15.

(2) We prove similarly to Point 1 and conclude by Lemma 6.15.

Combining Points 1 and 2 in Theorem 6.16, we deduce that

and the two sequences from M to have the same number of steps.

7.1 Values via weak steps

Thanks to factorization, we can prove that steps suffice to return a value. This is an immediate consequence of surface factorization plus postponement (Corollary 6.14), and weak factorization (Theorem 6.16), and the fact that non-weak steps, steps, and steps cannot produce -terms.

If then by surface factorization plus postponement (Corollary 6.14). By weak factorization (Theorem 6.16.1-2), we have

By iterating Lemma 7.5 from backward (and since ), we have that all terms in the sequence from M’ to are -terms. So in particular, M’ has shape for some value W.

7.2 Values via root steps

We also show an alternative way to evaluate a term in . Let us call root steps the rules and . The first two suffice to return a value, without the need for any contextual closure.

Note that this property holds only because terms are restricted to computations (for example, in Plotkin’s CbV -calculus, (II)(II) can be reduced, but it is not itself a redex, so ).

Looking closer at the proof of Corollary 7.3, we observe that any closed ( i.e. without free variables) computation has the following property: it is either a returned value (when ), or a -redex (when ) or a -redex (when ). More generally, the same holds for any (possibly open) computation that returns a value (Corollary 7.9 below).

Thus, , where and the ’s are abstractions, and if then .

Corollary 7.9 states a progression results: a -sequence from M may only end in a -term. We still need to verify that such a sequence terminates.

(10)

• If M is -redex, then by determinism of (Fact 7.1), and the claim is proved.

• If M is a -redex, observe that by Lemma 7.8,

- , and

- .

We apply all possible steps starting from M, obtaining

which is a -redex, so (note that we used the hypothesis on free variables of ). We observe that . We conclude, by using (10) and the fact that is deterministic (Fact 7.1).

The converse of Proposition 7.10 is also true and immediate. We can finally prove that root steps and suffice to return a value, without the need for any contextual closure.

Proof. Trivially (2) (1). Conversely, (1) (2) Proposition 7.10 and Theorem 7.6.

7.3 Observational equivalence

We now adapt the notion of observational equivalence, introduced in Plotkin (Reference Plotkin1975) for the CbV -calculus, to . Informally, two terms are observationally equivalent if they can be substituted for each other in all contexts without observing any difference in their behavior. For a computation M in , the “behavior” of interest is returning a value: for some value V, also noted .

A consequence of Theorem 7.6 is that the behavior of interest in Definition 7.12 can be equivalently defined as for some value V, instead of : the resulting notion of observational equivalence would be exactly the same. The definition using instead of is more in the spirit of Plotkin’s original one for the CbV -calculus (Plotkin Reference Plotkin1975). Reduction is deterministic, and for closed terms it terminates if and only if it ends in a returned value (Corollary 7.3). Hence, for closed terms, returning a value amounts to say that their evaluation halts.

The advantage of our Definition 7.12 is that it allows us to prove an important property of observational equivalence – the fact that it contains the equational theory of (Corollary 7.15)—in a very easy way, thanks to the following obvious lemma and adequacy (Theorem 7.14).

An easy argument, similar to that in Crary (Reference Crary2009) (which in turn simplifies the one in 1975) gives:

Conversely, suppose , that is, for some value V. By confluence of (Proposition 3.5), since , there is such that and . Since is a returned value, so is L by value persistence (Lemma 7.13). Therefore, .

The converse of Corollary 7.15 fails. Indeed, but .

8.1 Irrelevance of steps for normalization

We show that postponement of steps (Theorem 6.13) implies that steps have no impact on normalization, i.e., whether a term M has or not a -normal form. Indeed, saying that M has a -normal form is equivalent to say that M has a -normal form.

On the one hand, if and N is -normal, to reach a -normal form it suffices to extend the reduction with steps to a -normal form (since is terminating, Proposition 8.6). Notice that here we use Lemma 8.1. On the other hand, the proof that -normalization implies -normalization is trickier, because -normal forms are not preserved by performing a step backward (the converse of Lemma 8.1.2 is false). Here is a counterexample.

Consequently, the fact that M has a -normal form N means (by postponement of ) that for some P that Lemma 8.1 guarantees to be -normal only, not -normal. To prove that M has a -normal form is not even enough to take the -normal form of P, because a step can create a -redex. To solve the problem, we need the following technical lemma.

We also use that and are strongly normalizing (Proposition 8.6). Instead of proving that and are – separately – so, we state a more general result (its proof is in Appendix D).

Now we have all the elements to prove the following.

(11)

By Lemma 8.1.1, P is -normal in (11).

For any sequence of the form (11), let , where and are the lengths of the maximal -sequence and of the maximal -sequence from P, respectively; they are well-defined because and are strongly normalizing (Proposition 8.6).

We proceed by induction on w(P) ordered lexicographically to prove that for some P’ -normal (and so M is -normalizing).

- If then , so P is -normal and hence -normal.

- If , then P is -normal and hence -normal.

- Otherwise with . By Lemma 8.5, for some P’ with : indeed, or . By i.h. , we can conclude.

: As M is -normalizing, for some -normal N. As is strongly normalizing (Proposition 8.6), for some P -normal. By Lemma 8.1.1-2, P is also -normal and -normal. Summing up, with P -normal, i.e., M is -normalizing.

8.2 The essential role of σ steps for normalization

In λ_©, for normalization, σ steps play a crucial role, unlike ι steps. Indeed, σ steps can unveil “hidden” β_c-redexes in a term. Let us see this with an example, where we consider a term that is β_c-normal but diverging in λ_© and this divergence is “unblocked” by a σ step.

M_z is β_c-normal, but not ©-normal. In fact, M_z is diverging in λ_© (that is, it is not ©-normalizing):

Note that the σ step is weak and that N^z is normal for but not for .

The fact that a σ step can unblock a hidden β_c-redex is not limited to open terms. Indeed, is closed and β_c-normal, but divergent in λ_©:

Example 8.8 shows that, contrary to ι steps, σ steps are essential to determine whether a term has or not a normal form in λ_©. This fact is in accordance with the semantics. First, it can be shown that the term M_z above and Δ!Δ are observational equivalent. Second, the denotational models and type systems studied in Ehrhard (Reference Ehrhard2012), de' Liguoro and Treglia (2020) (which are compatible with λ_©) interpret M_z in the same way as Δ!Δ, which is a β_c-divergent term. It is then reasonable to expect that the two terms have the same operational behavior in λ_©. Adding σ steps to β_c-reduction is a way to obtain this: both M_z and Δ!Δ are divergent in λ_©. Said differently, σ-reduction restricts the set of ©-normal forms, so as to exclude some β_c-normal (but not β_cσ-normal) forms that are semantically meaningless.

Actually, σ-reduction can only restrict the set of terms having a normal form: it may turn a β_c-normal form into a term that diverges in λ_©, but it cannot turn a β_c-diverging term into a λ_©-normalizing one. To prove this (Proposition 8.10), we rely on the following lemma.

Roughly, Lemma 8.9 says that a σ step on a term that is not β_c-normal cannot erase a β_c-redex, and hence it can be postponed. Lemma 8.9 does not contradict Example 8.8: the former talks about a σ step on a term that is not β_c-normal, whereas the start terms in Example 8.8 are β_c-normal.

We prove now the part of the statement about normalization. If M is β_cσ-normalizing, there exists a reduction sequence with N β_cσ-normal. Let be the number of steps in , and let be the number of β_c steps after the last σ step in (when is just the length of ). We prove by induction on ordered lexicographically that M is β_c-normalizing. There are three cases.

(1) If contains only β_c steps ( ), then and we are done.

(2) If ( ends with a nonempty sequence of σ steps), then L is β_c-normal by Lemma 8.9, as N is β_c-normal; by i.h. applied to the sequence , M is β_c-normalizing.

(3) Otherwise is the last σ step in , followed by a β_c step). By Lemma 8.9, either there is a sequence , then and then . In both cases , so by i.h. M is β_c-normalizing.

8.3 Normalizing strategies

Irrelevance of ι steps (Theorem 8.7) implies that to define a normalizing strategy for λ_©, it suffices to define a normalizing strategy for β_cσ. We do so by iterating either surface or weak reduction. Our definition of β_cσ-normalizing strategy and the proof of normalization (Theorem 8.14) is parametric on either.

The difficulty here is that both weak and surface reduction are nondeterministic. The key property we need in the proof is that the reduction we iterate is uniformly normalizing (see Definition 2.1). We first establish that this holds for weak and surface reduction. While uniform normalization is easy to prove for the former, it is nontrivial for the latter, its proof is rather sophisticated. Here we reap the fruits of the careful analysis of the number of β_c steps in Section 6.3. Finally, we formalize the strategies and tackle normalization.

Notation. Since we are now only concerned with β_cσ steps, for the sake of readability in the rest of the section, we often write and for and , respectively.

Understanding Uniform Normalization. The fact that and are uniformly normalizing is key in the definition of normalizing strategy and deserves some discussion.

The heart of the normalization proof is that if M has a -normal form N, we can perform surface steps and reach a surface normal form. Note that surface factorization only guarantees that there exists a -sequence such that if then , where L is -normal. The existential quantification is crucial here because is not a deterministic reduction. Uniform normalization of transforms the existential into a universal quantification: if M has a -normal form (and so a fortiori a surface normal form), then every sequence of steps will terminate. The normalizing strategy then iterates this process, performing surface reduction on the subterms of a surface normal form, until we obtain a -normal form.

Uniform Normalization of Weak and Surface Reduction

We prove that both weak and surface reduction are uniformly normalizing, i.e., for , if a term M is -normalizing, then it is strongly -normalizing. In both cases, the proof relies on the fact that all maximal -sequences from a given term M have the same number of β_c steps.

Fact 8.11 (Number of β_c steps). Given a -sequence , the number of its β_c steps is finite if and only if is finite.

Proof. The right-to-left implication is obvious. The left-to-right is an immediate consequence of the fact that is strongly normalizing (Proposition 8.6).

A maximal -sequence from M is either infinite or ends in a -normal form. Theorem 8.13 states that for , all maximal -sequences from the same term M have the same behavior, also quantitatively (with respect to the number of β_c steps). The proof relies on the following lemma. Recall that weak reduction is not confluent (Example 5.1); however, is deterministic.

Lemma 8.12. (Invariant). Given , every sequence where N is -normal has the same number k of β_c steps. Moreover,

(1) the unique maximal -sequence from M has length k, and

(2) there exists a sequence for some .

Proof. The argument is illustrated in Figure 3. Let k be the number of β_c steps in a sequence where N is -normal. By weak factorization (Theorem 6.16.2) there is a sequence with the same number k of β_c steps. As N is -normal, so is L (Lemma 8.2). Thus, is a maximal -sequence from M, and it is unique because is deterministic.

Figure 3:

Weak reduction.

Theorem 8.13. (Uniform normalization).

(1) Reduction is uniformly normalizing.

(2) Reduction is uniformly normalizing.

Moreover, all maximal -sequences (resp. all maximal -sequences) from the same term M have the same number of β_c steps.

Proof. We write (resp. ) for (resp. ).

Claim 1. Let where N is -normal, and so, in particular -normal. By Lemma 8.12, where is the (unique) maximal -sequence from M. We prove that no -sequence from M may have more than k β_c steps. Indeed, every sequence can be factorized (Theorem 6.16.2) as with the same number of β_c steps as , and is a prefix of the maximal -sequence from M (since is deterministic).

We deduce that no infinite -sequence from M is possible (by Fact 8.11).

Claim 2. Assume that with N -normal. Recall that is confluent (Proposition 5.5.2), so N is the unique -normal form of M.

First, by induction on N, we prove that given a term M,

(#) all sequences have the same number of β_c steps.

Let be two such sequences. Figure 4 illustrates the argument. By weak factorization (Theorem 6.16.1), there is a sequence (resp. ) with the same number of β_c steps as (resp. ), and whose steps are all surface. Note that and are -normal (by Lemma 8.3, because N is in particular -normal), and so in particular -normal. By Lemma 8.12, , have the same number k of β_c steps, and so do the sequences and .

Figure 4:

Surface reduction.

To prove (#), we show that the sequences and have the same number of β_c steps.

By confluence of (Proposition 5.5.1), , for some , and (by confluence of , Proposition 5.5.2) there is a sequence . By Lemma 8.2, since are -normal, terms in these sequences are -normal, and so all steps are not only surface, but also steps. That is, , and . Hence, have the same shape by Fact 6.2.

We examine the shape of N, and prove claim (#) by showing that and have the same number of β_c steps as (note that the sequences and have no β_c steps).

• . In this case, , and the claim (#) is immediate.

• , and . We have and . Since P and Q are -normal, by i.h. we have:

- the two sequences and have the same number of β_c steps, and similarly and . Hence, and have the same number of β_c steps.

- Similarly, and have the same number of β_c steps.

This completes the proof of (#). We now can conclude that is uniformly normalizing. If the term M has a sequence where N is -normal, then no -sequence can have more β_c steps than , because given any sequence then (by confluence of ) , and (by #) has the same number of β_c steps as . Hence, all -sequences from M are finite.

Normalizing strategies

We are ready to define and deal with normalizing strategies for λ_©. Our definition is inspired, and generalizes, the stratified strategy proposed in Guerrieri (Reference Guerrieri2015), Guerrieri et al. (Reference Guerrieri, Paolini and Ronchi Della Rocca2017), which iterates weak reduction (there called head reduction) according to a more strict discipline.

Iterated -Reduction. We define a family of normalizing strategies, parametrically on the reduction to iterate, which can be surface or weak. Let . Reduction is defined as follows, by iterating in the left-to-right order (Theorem 8.14 then shows that is a normalizing strategy).

(1) If M is not -normal:

(2) If M is -normal (below, “ -normal” means or with L β_cσ-normal):

Theorem 8.14 (Normalization for β_cσ). Assume where N is -normal. Let . Then, every maximal -sequence from M ends in N.

Proof. By induction on the term N. Let be a maximal -sequence from M.

We write

,

,

and

for

, respectively. We observe that

Indeed, from , by -factorization, we have that . Since N is -normal, so is L (by Lemma 8.3) and (**) follows by uniform normalization of (Theorem 8.13).

Let be the maximal prefix of such that . Since it is finite, is , where is -normal. Let be the sequence such that .

Note that all terms in are -normal (by repeatedly using Lemma 8.3 from ), hence , and (by shape preservation, Fact 6.2) all terms in have the same shape as .

By confluence of (Proposition 5.9.2), . Again, all terms in this sequence are -normal, by repeatedly using Lemma 8.3 from . So, , and (by shape preservation, Fact 6.2) and N have the same shape.

We have established that and all terms in have the same shape as N. Now we examine the possible cases for N.

• , and . Trivially .

• and with . Since is β_cσ-normal, by i.h. every maximal -sequence from P terminates in , and so every maximal -sequence from terminates in . Since the sequence is a maximal -sequence, we have that is as follows

• and , with and . Since and are both β_cσ-normal, by i.h. :

- every maximal -sequence from P ends in . So every -sequence from eventually reaches ;

- every maximal -sequence from Q ends in . So every -sequence from eventually reaches .

Therefore is as follows

• and . Similar to the previous one.

From a normalizing strategy for (Theorem 8.14), we derive a normalizing strategy in λ_©.

Corollary 8.15 (Normalization for λ_©) Let . If M is ©-normalizing, then any maximal -sequence from M followed by any maximal -sequence ends in the ©-normal form of M.

Proof. By Theorem 8.14, every maximal -sequence from M ends in a -normal form L. Since is strongly normalizing (Proposition 8.6), every maximal -sequence from L ends in a -normal form N, which is also -normal by Lemma 8.1. Therefore, N is ©-normal and this is the unique ©-normal form of M since is confluent (Proposition 5.9.3).

9.1 Discussion: reduction and evaluation

In computational calculi, it is standard practice to define evaluation as weak reduction, aka sequencing (Dal Lago et al. Reference Dal Lago, Gavazzo and Levy2017; Filinski Reference Filinski1996; Jones et al. Reference Jones, Shields, Launchbury, Tolmach, MacQueen and Cardelli1998; Levy et al. Reference Levy, Power and Thielecke2003). Despite the prominent role that weak reduction has in the literature, in particular for calculi with effects, what one discovers when analyzing the rewriting properties is somehow unexpected. As we observe in Section 5, where we consider both the computational core λ_© (de' Liguoro and Treglia 2020), and a widely recognized reference such as the calculus by Sabry and Wadler (Reference Sabry and Wadler1997) (in turn inspired by Moggi Reference Moggi1988, Reference Moggi1989, Reference Moggi1991), while full reduction is confluent, the closure of the rules under evaluation contexts turns out to be non-deterministic, non-confluent, and its normal forms are not unique. The issues come from the monadic rules of identity and associativity, hence they are common to all computational calculi.

A Bridge between Evaluation and Reduction. On the one hand, computational λ-calculi have an unrestricted non-deterministic reduction that generates the equational theory of the calculus, studied for foundational and semantic purposes. On the other hand, weak reduction models evaluation in an ideal programming language. It is then natural to wonder what is the relation between reduction and evaluation. This is the first contribution of this paper. We establish a bridge between evaluation and reduction via a factorization theorem stating that every reduction can be rearranged so as to bring forward weak reduction steps.

We focused on the rewriting theory of a specific computational calculus, namely the computational core λ_© (de' Liguoro and Treglia 2020). We expect that our results and approach can be adapted also to other computational calculi such as . This demands further investigations. Transferring the results is not immediate because the correspondence between the two calculi is not direct with respect to the rewriting (see Remark 3.10).

9.2 Technical contributions

We studied the rewriting theory of the computational core λ_© introduced in de' Liguoro and Treglia (2020), a variant of Moggi’s λ_c-calculus Moggi (Reference Moggi1988), focusing on two questions:

• how to reach values?

• how to reach normal forms?

For the first point, we show that weak β_c-reduction is enough (Section 7). For the second question, we define a family of normalizing strategies (Section 8).

We have faced the issues caused by identity and associativity rules (which internalize the monadic rules in the syntax) and dealt with them by means of factorization techniques.

We have investigated in depth the structure of normalizing reductions, and we assessed the role of the σ-rule (aka associativity) as computational and not merely structural. We found out that it plays at least three distinct, independent roles in λ_©:

• σ unblocks “premature” β_c-normal forms so as to guarantee that there are not ©-normalizing terms whose semantics is the same as diverging terms, as we have seen in Section 8.2;

• it internalizes the associativity of Kleisli composition into the calculus, as a syntactic reduction rule, as explained in Section 1 after Equation (3);

• it “simulates” the contextual closure of the β_c-rule for terms that reduce to a value, as we have seen in Theorem 7.4.

9.3 Related work

Relation with Moggi’s Calculus. Since our focus is on operational properties and reduction theory, we chose the computational core λ_© (de' Liguoro and Treglia 2020) among the different variants of computational calculi in the literature inspired by Moggi’s seminal work (Moggi 1988 Reference Moggi1989, Reference Moggi1991). Indeed, the computational core λ_© has a “minimal” syntax that internalizes Moggi’s original idea of deriving a calculus from the categorical model consisting of the Kleisli category of a (strong) monad. For instance, λ_© does not have to consider both a pure and a (potentially) effectful functional application. So, λ_© has less syntactic constructors and less reductions rules with respect to other computational calculi, and this simplifies our operational study.

Let us discuss the difference between λ_© and Moggi’s λ_c. As observed in Sections 1 and 3, the first formulation of λ_c and of its reduction relation was introduced in Moggi (Reference Moggi1988), where it is formalized by using let-constructor. Indeed, this operator is not just a syntactical sugar for the application of λ-abstraction. In fact, it represents the extension to computations of functions from values to computations, therefore interpreting Kleisli composition. Combining let with ordinary abstraction and application is at the origin of the complexity of the reduction rules in Moggi (Reference Moggi1988). On the other hand, this allows extensionality to be internalized. Adding the η-rule to λ_© breaks confluence, as shown in de' Liguoro and Treglia (2020).

Besides using let or not, a major difference of λ_© with respect to λ_c is the neat distinction among the two syntactical sorts of terms, restricting the combination of values and non-values since the very definition of the grammar of the language. In spite of these differences, in de' Liguoro and Treglia (2019, Section 9) it has been proved that there exists an interpretation of λ_c into λ_© that preserves the reduction, while there is a reverse translation that preserves convertibility, only.

Other Related Work. Sabry and Wadler (Reference Sabry and Wadler1997) is the first work on the computational calculus to put on center stage the reduction. Still the focus of the paper are the properties of the translation between that and the monadic metalanguage – the reduction theory itself is not investigated.

In Herbelin and Zimmermann (Reference Herbelin, Zimmermann and Curien2009) a different refinement of λ_c has been proposed. Its reduction rules are divided into a purely operational, a structural and an observational system. It is proved that the purely operational system suffices to reduce any closed term to a value. This result is similar to our Theorem 7.6, with weak β_c steps corresponding to head reduction in Herbelin and Zimmermann (Reference Herbelin, Zimmermann and Curien2009). Interestingly, the analogous of our rule σ is part of the structural system, while the rule corresponding to our is generalized and considered as an observational rule. Unlike our work, normalization is not studied in Herbelin and Zimmermann (Reference Herbelin, Zimmermann and Curien2009).

Surface reduction is a generalization of weak reduction that comes from linear logic. We inherit surface factorization from the linear λ-calculus in Simpson (Reference Simpson and Giesl2005). Such a reduction has been recently studied in several variants of the λ-calculus, especially for semantic purposes (Accattoli and Guerrieri Reference Accattoli and Guerrieri2016; Accattoli and Paolini Reference Accattoli and Paolini2012; Bucciarelli et al. Reference Bucciarelli, Kesner, Ros, Viso, Nakano and Sagonas2020; Carraro and Guerrieri Reference Carraro and Guerrieri2014; Ehrhard and Guerrieri Reference Ehrhard and Guerrieri2016; Guerrieri and Manzonetto Reference Guerrieri and Manzonetto2019; Guerrieri and Olimpieri Reference Guerrieri and Olimpieri2021; Guerrieri Reference Guerrieri2019).

Regarding the σ-rule, in Carraro and Guerrieri (Reference Carraro and Guerrieri2014) two commutation rules are added to Plotkin’s CbV λ-calculus in order to remove meaningless normal forms – the resulting calculus is called shuffling. The commutative rule there called is literally the same as σ here. In the setting of the shuffling calculus, properties such as the fact that all maximal surface -reduction sequences from the same term M have the same number of steps, and so such a reduction is uniformly normalizing, were known via semantic tools (Carraro and Guerrieri Reference Carraro and Guerrieri2014; Guerrieri Reference Guerrieri2019), namely non-idempotent intersection types. In this paper, we give the first syntactic proof of such a result.

A relation between the computational calculus, Simpson (Reference Simpson and Giesl2005) and other linear calculi are well known in the literature, see for example Egger et al. (Reference Egger, Møgelberg, Simpson, Grädel and Kahle2009), Sabry and Wadler (Reference Sabry and Wadler1997), Maraist et al. (Reference Maraist, Odersky, Turner and Wadler1999).

In de' Liguoro and Treglia (2020), Theorem 8.4 states that any closed term returns a value if and only if it is convergent according to a big-step operational semantics. That proof is incomplete and needs a more complex argument via factorization, as we do here to prove Theorem 7.6 (from which that statement in de’ Liguoro and Treglia Reference de’ Liguoro and Treglia2020 easily follows).

Acknowledgements. We are in debt with Vincent van Oostrom, for the technical issues he pinpointed, and for his many insightful technical suggestions. We also thank the referees for their valuable comments. This work was partially supported by the ANR project PPS: ANR-19-CE48-0014.