There have been several approaches to semantics of lambda calculus with intersection types (Barendregt et al., 1983; Dezani-Ciancaglini and Margaria, 1984; Barbanera et al., 1995; Ong and Tsukada, 2012; de Carvalho, 2018) nevertheless none of them addresses combinatory logic. In Barendregt et al. (1983), the authors have introduced the filter models for lambda calculus and proved the soundness and completeness of the type assignment system with respect to the proposed semantics. Further, in Dezani-Ciancaglini and Margaria (1984) a modification of intersection type discipline has been considered, and it has been proved that this type system is sound and complete with respect to the F-semantics. The type assignment system with intersection and union types has been introduced in Barbanera et al. (1995), where authors proposed three semantics for the introduced system and proved soundness and completeness results. In de Carvalho (2018), the author introduced a non-uniform semantics of lambda calculus which allows to measure execution time and presented the intersection type system induced by these semantics, which can be seen as a reformulation of the system of Coppo et al. (1980). In Ong and Tsukada (2012), the two-level game semantics is used to model intersection types.

On the other hand, semantics for combinatory logic with intersection types has not been investigated as much as the semantics for lambda calculus with intersection types. An overview of models of combinatory logic is given in Bimbó (2012), however most of these models are models of untyped combinatory logic. Particularly, one class of models for combinatory logic with intersection types is presented based on theories of types which are in 1 − 1 correspondence with filters. Nevertheless, to the best of our knowledge filter models for combinatory logic with intersection types has not been published anywhere. The lack of study of models of combinatory logic has been our motivation for the present research.

In Section 2, we briefly review basic notions of the untyped combinatory logic and intersection type assignment system for combinatory logic. We then define the semantics in Section 3. Section 4 presents the results of the paper: in Section 4.1, we prove the soundness and completeness of the equational theory of the untyped combinatory logic with respect to the proposed semantics; in Section 4.2, we prove the soundness and completeness of the combinatory logic with intersection types with respect to the proposed semantics. Section 5 concludes and contains the discussion of the related work.

The language of combinatory logic is build up from a countable set of term variables V and a set of term constants using application, the only (binary) operation. Terms of combinatory logic, called CL-terms, are generated by the following grammar:

where x is a term variable and S, K, I are term constants, or constants for short. The set of all CL-terms is denoted by CL and is ranged over by capital letters M, N, …, M₁, …. The set of variables that occur in a term M is denoted by FV(M), and the result of substitution of the term N for variable x in the term M is denoted by M{N/x}. The use of parentheses is minimized by the convention that application associates to the left: M₁M₂…M_n means […(M₁M₂)…M_n].

The main objects of study in combinatory logic are the relations between terms. We will consider the equivalence relation generated by the equational theory EQ given in Figure 1.

If M = N can be derived from the set of axioms and rules of Figure 1 we say that terms M and N are equal, and we write M = N. The equational theory obtained by adding the rule

to the equational theory EQ will be denoted by EQη, and it represents the extensional equational theory. We write M =_η N, whenever M = N is provable in EQη. We use the notion of equivalence classes: [M]={N∣M=N is provable in EQη}.

Intersection types for combinatory logic were introduced in Dezani-Ciancaglini and Hindley (1992). The set of intersection types is generated by the following grammar:

where a belongs to a countable set of type variables V_Type and ω is a type constant. There are two type forming operations: the arrow, →, which generates the simple types and the intersection, ∩, which provides the extension of the simple types. The set of all intersection types is denoted by Types and is ranged over by σ, τ, …, σ₁, …. We can omit parentheses in arrow types by removing outer parentheses and using the convention that, unless otherwise bracketed, nested arrows associate to the right: σ₁ → … → σ_n−1 → σ_n means (σ₁ → … → (σ_n−1 → σ_n)). Intersection types are associative: (σ ∩ τ) ∩ ρ is the same as σ ∩ (τ ∩ ρ), and an intersection type takes precedence over an arrow type: σ ∩ τ → ρ means (σ ∩ τ) → ρ.

Definition 2.1 (Dezani-Ciancaglini and Hindley, 1992; Barendregt et al., 2013).

The subtyping (pre-order) relation on the set of intersection types is defined in the following way.

Definition 2.2 (Dezani-Ciancaglini and Hindley, 1992). The relation ≤ is the smallest binary relation satisfying:

The induced equivalence relation ~ is defined by

The type-assignment system for combinatory logic with intersection types, denoted by CL_∩, is presented in Figure 2. If the typing judgment Γ ⊢ M : σ can be derived by the rules in Figure 2, then it means that the statement M : σ is derivable from the basis Γ and that the term M is typable in the given basis Γ with type σ.

The rules (axiom S), (axiom K), and (axiom I) type the combinators S, K, and I, respectively. The rule (axiom ω) enables every combinatory term M to be typable. The rules (→ elim), (∩ elim-l), and (∩ elim-r) are the usual elimination rules for → and ∩, respectively. The rule (∩ intro) is the introduction rule for ∩. The rule (sub-type) is the subsumption rule induced by the subtyping relation on types and the rule (eq) is the rule which enables type preservation under equality.

Remark 2.3. The type-assignment system TACLB1 of Dezani-Ciancaglini and Hindley (1992) is obtained from CL_∩ by omitting the rule (eq). Type-assignment statements in TACLB1 are preserved by the equality generated by the equational theory EQ (Dezani-Ciancaglini and Hindley, 1992, Theorem 3.12). In order to obtain a type-assignment system in which types are preserved by the equality generated by the extensional equational theory EQη, the rule (eq) has to be added.

Lemma 2.4 (Weakening lemma). If Γ ⊢ M : σ and Γ ⊆ Γ′ then Γ′ ⊢ M : σ.

Proof. By induction on the length of derivation of Γ ⊢ M : σ.     □

In the previous section we have introduced models for CL_∩ such that the interpretation of every term is defined in a model. A fundamental question to ask is: if two terms are proven to be equal in the equational theory are their interpretations equal in a model?—this is referred to as the soundness of the equational theory with respect to the model. The other direction: if interpretations of two terms are equal in every model are the terms proven to be equal in the equational theory?—is referred to as the completeness.

In the sequel, we prove that the equational theory of untyped combinatory logic EQη is sound and complete with respect to the proposed semantics.

Theorem 4.1 (Soundness of EQη). If M =_η N, then ⟦M⟧_ρ = ⟦N⟧_ρ for every model Mρ.

Proof. The proof is by induction on the length of the proof of M = N in EQη.

If M = N is an instance of axiom (id), then terms M and N represent the same term L and it holds that ⟦M⟧_ρ = ⟦L⟧_ρ = ⟦N⟧_ρ for every model Mρ.

If M = N is obtained by axiom (S), then there exist terms P, Q and R such that terms M and N are terms SPQR and (PR)(QR), respectively. Then, from Definitions 3.2 and 3.6 we obtain

If M = N is obtained by axiom (K), then there exist terms P and Q, such that terms M and N are terms KPQ and P, respectively. Then, from Definitions 3.2 and 3.6 we obtain

If M = N falls under axiom (I), then terms M and N are of the form IP and P, respectively, for some term P. In a similar way as in the previous two cases, we obtain

If M = N is obtained from N = M by rule (sym), then by induction hypothesis ⟦N⟧_ρ = ⟦M⟧_ρ and the statement holds.

If M = N is obtained from M = L and L = N by rule (trans), then ⟦M⟧_ρ = ⟦L⟧_ρ and ⟦L⟧_ρ = ⟦N⟧_ρ hold by induction hypothesis. Thus, ⟦M⟧_ρ = ⟦N⟧_ρ also holds.

Next, let us suppose that M = N is obtained from L = Q by rule (app-l), then terms M and N are of the form LP and QP, respectively, for some term P. By induction hypothesis ⟦L⟧_ρ = ⟦Q⟧_ρ. From the latter we obtain

Similarly to the previous case, if M = N is obtained by rule (app-r), then terms M and N are of the form PL and PQ, respectively, for some terms P, L, Q such that L = Q is provable in EQη. By induction hypothesis we obtain ⟦L⟧_ρ = ⟦Q⟧_ρ, and thus

Finally, we consider the case where M = N is obtained by applying rule (ext) to Mx = Nx, with x being a variable which appears neither in M nor in N. Let Mρ be a model for CL_∩. In order to prove that ⟦M⟧_ρ = ⟦N⟧_ρ holds, it is enough to prove that for every d ∈ D it holds that App(⟦M⟧_ρ, d) = App(⟦N⟧_ρ, d), because of the extensionality of the applicative structure M. Let d ∈ D. Since by induction hypothesis ⟦Mx⟧_ρ = ⟦Nx⟧_ρ holds in any model, it also holds in model Mρ(x:=d), that is App(⟦M⟧_{ρ(x: = d)}, ⟦x⟧_{ρ(x: = d)}) = App(⟦N⟧_{ρ(x: = d)}, ⟦x⟧_{ρ(x: = d)}). From the assumption that x does not appear in M and Lemma 3.8, we obtain ⟦M⟧_ρ = ⟦M⟧_{ρ(x: = d)}. Similarly, ⟦N⟧_ρ = ⟦N⟧_{ρ(x: = d)}. Now, we have

Thus, ⟦M⟧_ρ = ⟦N⟧_ρ due to the extensionality of the applicative structure.

This concludes the proof.     □

Recall that by [M] we denote the equivalence class of term M with respect to the equivalence relation generated by the equational theory EQη, [M]={N∣M=Nis provable inEQη}.

In order to prove the completeness, we first prove the following property.

Lemma 4.2. Let M=〈D,{Aσ},App〉 be an extensional applicative structure with combinators, such that D = {[M]∣M ∈ CL}, App([M], [N]) = [MN], s = [S], k = [K] and i = [I]. If an environment ρ^⋆ is defined by ρ^⋆(x) = [x], then ⟦M⟧ρ⋆=[M].

Proof. The proof is by induction on the structure of M.

     □

Theorem 4.3 (Completeness of EQη). If ⟦M⟧_ρ = ⟦N⟧_ρ holds in every model Mρ, then M =_η N.

Proof. Suppose that ⟦M⟧_ρ = ⟦N⟧_ρ holds for any model Mρ. We define a model Mρ⋆′=〈M0,ρ⋆〉 where M0 is an applicative structure 〈D, {A^σ}, App〉 with the following components:

The environment ρ^⋆ is defined by ρ^⋆(x) = [x].

First we prove that the structure M0 is an applicative structure, meaning that it satisfies conditions of Definition 3.1. The set D is a non-empty set. For every σ ∈ Types, the set A^σ = {[M] ∣ M ∈ CL and ⊢ M : σ} is a subset of the set D = {[M] ∣ M ∈ CL}. Since for every term M it holds that ⊢ M : ω, we have that A^ω = {[M] ∣ M ∈ CL and ⊢ M : ω} = {[M] ∣ M ∈ CL} = D. Next, we look at the sets A^σ and A^τ. By the definition of the applicative structure M0 we have A^σ = {[M] ∣ M ∈ CL and ⊢ M : σ} and A^τ = {[M] ∣ M ∈ CL and ⊢ M : τ}. If [M] ∈ A^σ ∩ A^τ, then ⊢ M : σ and ⊢ M : τ. By rule (∩ intro) of Figure 2 we obtain ⊢ M : σ ∩ τ, which is equivalent to [M] ∈ A^σ∩τ, and we conclude A^σ ∩ A^τ ⊆ A^σ∩τ. Similarly, we prove that A^σ∩τ ⊆ A^σ ∩ A^τ and we conclude A^σ∩τ = A^σ ∩ A^τ. Let σ, τ ∈ Types such that σ ≤ τ. If [M] ∈ A^σ, then ⊢ M : σ holds, and since σ ≤ τ we obtain ⊢ M : τ by rule (sub-type). It follows that [M] ∈ A^τ and we can conclude that σ ≤ τ implies A^σ ⊆ A^τ. The function App defined by App([M], [N]) = [MN] maps D × D to D. If [M] ∈ A^{σ → τ} and [N] ∈ A^σ, then by the definition of the applicative structure M0 we have ⊢ M : σ → τ and ⊢ N : σ. By rule (→ elim) of Figure 2 we obtain ⊢ MN : τ, thus we conclude [MN] ∈ A^τ and App ↾ (A^{σ → τ} × A^σ):A^{σ → τ} × A^σ → A^τ.

Next, we prove that the applicative structure M0 has combinators. We consider the equivalence classes [S], [K], [I] ∈ D and denote them by s, k, i, respectively. Since ⊢ S:(σ → (τ → ρ)) → ((σ → τ) → (σ → ρ)) holds by (axiom S) of Figure 2, we have that s = [S] ∈ A^{(σ → (τ → ρ)) → ((σ → τ) → (σ → ρ))} for any σ, τ, ρ ∈ Types. Similarly, we obtain k = [K] ∈ A^{σ → (τ → σ)} and i = [I] ∈ A^{σ → σ}. We also have to prove that the elements s, k, and i satisfy the Equations (1)–(3). By the rules of the equational theory EQη we obtain [SMNL] = [(ML)(NL)], [KMN] = [M] and [IM] = [M] and it follows that:

The extensionality of the applicative structure follows from the extensionality of combinatory logic. Let [M], [N] ∈ D such that for every [L] ∈ D, App([M], [L]) = App([N], [L]), i.e., [ML] = [NL]. Then, for a variable x, that does not appear in terms M and N, it holds that [Mx] = [Nx], that is Mx = Nx is provable in EQη. By rule (ext) we obtain M = N is provable in EQη, so [M] = [N].

We have proved that Mρ⋆′ is a model for CL_∩. If ⟦M⟧_ρ = ⟦N⟧_ρ in any model, then it also holds in the model Mρ⋆′. Hence, we have that ⟦M⟧ρ⋆=⟦N⟧ρ⋆. From Lemma 4.2 we get ⟦M⟧ρ⋆=[M] and ⟦N⟧ρ⋆=[N]. Now we can conclude that [M] = [N] holds, that is M =_η N.     □

Theorems 4.1 and 4.3 prove that the equational theory of untyped combinatory logic is both sound and complete with respect to the proposed semantics, hence the proposed semantics is a semantics of the untyped combinatory logic.

In the sequel we consider another fundamental question: if M : σ is (syntactically) derivable in the type system CL_∩, ⊢ M : σ, is it valid in all models, ⊧ M : σ?—this is referred to as the soundness of the type-assignment system with respect to the model. More general, if Γ ⊢ M : σ is (syntactically) derivable in the type system CL_∩, is it semantically derivable, Γ ⊧ M : σ? The other direction: if M : σ is valid in all models of CL_∩, is it derivable in CL_∩?—is referred to as the completeness. More general, does semantical derivability Γ ⊧ M : σ imply syntactical derivability Γ ⊢ M : σ?

We prove herein the type-assignment system CL_∩ to be sound and complete with respect to the proposed semantics.

Theorem 4.4 (Soundness of CL_∩). If Γ ⊢ M : σ, then Γ ⊧ M : σ.

Proof. We prove the statement by induction on the derivation of Γ ⊢ M : σ, considering the last rule used.

     □

Remark 4.5. As pointed out in Section 3, we have defined an environment as a total mapping, whereas in Kašterović and Ghilezan (2020) an environment was defined as a partial mapping. Every combinatory term is typable with the type ω, i.e., Γ ⊢ M : ω for any basis Γ, and in order to obtain the soundness of the type-assignment system we need to prove that M : ω holds in every model. Thus, we had to define environments so that the meaning of every term M is defined in all models. This is specific to intersection type systems and is due to the rule (axiom ω) which ensures typability of all terms.

We define a class of canonical models, following the approach used in Kašterović and Ghilezan (2020), in order to prove the completeness of the type-assignment system CL_∩ with respect to the models presented in Section 3. Each canonical model is defined with respect to some basis, so that it satisfies only statements that can be derived from that basis. More precisely, we will define a model Mρ⋆Γ such that

Definition 4.6. Let Γ be a basis. A canonical model is a pair Mρ⋆Γ=〈MΓ,ρ⋆〉, where

such that

and ρ^⋆(x) = [x].

Lemma 4.7. The canonical model Mρ⋆Γ is a model for CL_∩.

Proof. We need to prove that the tuple Mρ⋆Γ introduced in Definition 4.6 satisfies the conditions of Definition 3.5. First, we prove that MΓ=〈D,{Aσ},App〉 is an applicative structure. The set D is a non-empty set. The set A^σ = {[M] ∣ M ∈ CL and Γ ⊢ M : σ} is a subset of D = {[M] ∣ M ∈ CL}. For every CL-term M we have Γ ⊢ M : ω, thus A^ω = {[M] ∣ M ∈ CL and Γ ⊢ M : ω} = {[M] ∣ M ∈ CL} = D. The proofs that the family {A^σ} and the application function App satisfy the conditions of Definition 3.1 are obtained similarly as in the proof of Theorem 4.3.

We have proved that the tuple MΓ=〈D,{Aσ},App〉 is an applicative structure. Next, we prove that it has combinators and that it is extensional. Similarly as in the proof of Theorem 4.3 we denote equivalence classes [S], [K], [I] by s, k, i, respectively. From the type-assignment system (Figure 2) we know that for the basis Γ, Γ ⊢ S:(σ → (τ → ρ)) → ((σ → τ) → (σ → ρ)). From the latter it follows that s = [S] ∈ A^{(σ → (τ → ρ)) → ((σ → τ) → (σ → ρ))}. Similarly, we obtain k = [K] ∈ A^{σ → (τ → σ)} and i = [I] ∈ A^{σ → σ}. The proof that elements [S], [K], [I] satisfy Equations (1)–(3) is given in the proof of Theorem 4.3.

We have proved that the applicative structure introduced in Definition 4.6 has combinators. The proof that the applicative structure is extensional is the same as in the proof of Theorem 4.3.

The mapping ρ^⋆ defined by ρ^⋆(x) = [x] is an environment for the applicative structure MΓ.

This completes the proof that Mρ⋆Γ is a model for CL_∩.     □

Lemma 4.8. Let Mρ⋆Γ be the canonical model. For every CL-term M, it holds ⟦M⟧ρ⋆=[M].

Proof. This is a straightforward consequence of Lemma 4.2.     □

Theorem 4.9. Let Mρ⋆Γ be a canonical model. It holds that

Proof. By Definition 4.6 and Lemma 4.8 we obtain

     □

Theorem 4.10 (Completeness of CL_∩). If Γ ⊧ M : σ, then Γ ⊢ M : σ.

Proof. Suppose that Γ ⊧ M : σ. Let Mρ⋆Γ be a canonical model (Definition 4.6). First, we need to prove that Mρ⋆Γ⊧Γ. Let x : σ be a declaration from the basis Γ. By Lemma 4.8 and Definition 4.6 we have

It follows that Mρ⋆Γ⊧Γ. From the latter and the assumption Γ ⊧ M : σ we obtain Mρ⋆Γ⊧M:σ. By Theorem 4.9 we obtain Γ ⊢ M : σ.     □

Theorems 4.4 and 4.10 prove that the type-assignment system CL_∩ is both sound and complete with respect to the proposed semantics. Hence the proposed semantics is a semantics of the combinatory logic with intersection types and of the untyped combinatory logic, as proven in Section 4.1.