Third-Order Matching via Explicit Substitutions

Transcription

1 Third-Order Matching via Explicit Substitutions Flávio L. C. de Moura 1 and Mauricio Ayala-Rincón 1 and Fairouz Kamareddine 2 1 Departamento de Matemática, Universidade de Brasília, Brasília D.F., Brasil. flavio@mat.unb.br,ayala@mat.unb.br 2 School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh, Scotland. fairouz@macs.hw.ac.uk Abstract. We show how Dowek s third-order matching decision procedure can be adapted to the simply typed λσ-calculus of explicit substitutions. The advantages of this approach include being closer to low-level implementations based on the λ-calculus, where substitution has to be made explicit. The work is done by, firstly, reducing matching problems to interpolation problems in the language of the λσ-calculus and, secondly, showing that if an interpolation problem has a solution then it is possible to find a solution which depends only on a measure of the structure of the initial matching problem, where this measure is based on the notion of λσ-böhm trees. 1 Introduction Matching is a basic operation extensively used in implementations of automated deduction environments and programming languages. Calculi of explicit substitutions are refinements of the λ-calculus by internalizing the substitution operation. By studying matching using explicit substitutions, we can ensure a more accurate description of implementations based on the λ-calculus. This is useful when we consider low-level implementations in which matching algorithms are to be implemented in the level of the language itself. In this paper we prove the decidability of a special class of third-order matching problems in the λσ-calculus[accl91]. This class consists of those matching problems originated in the simply typed λ-calculus. Our result is achieved by extending the techniques in [Dow94] to the language of the simply typed λσcalculus. In the next subsection we present the necessary background. In section 2, we define λσ-böhm trees for λσ-terms and present relevant results related to this notion. In section 3, we show how a matching problem in the language of λσ is translated to an interpolation problem. In section 4, we show how to decide third-order interpolation problems. Corresponding author. Supported by Brazilian CAPES Foundation. Partially supported by Brazilian CNPq Council.

2 1.1 The λσ-calculus The λσ-calculus of explicit substitutions extends the λ-calculus with explicit operators to simulate the substitution (meta-)operation of the λ-calculus. The λσ-calculus uses indexes to represent variables, the well-known de Bruijn notation [db72], and its rewriting system is given in Table 1. (Beta) (λa b) a [b id] (App) (a b)[s] (a [s]) (b [s]) (Abs) (λa)[s] λ(a [1 (s )]) (Clos) (a [s])[t] a [s t] (VarCons) 1 [a s] a (Id) a[id] a (Assoc) (s t) u s (t u) (Map) (a s) t a [t] (s t) (IdL) id s s (IdR) s id s (ShiftCons) (a s) s (VarShift) 1 id (SCons) 1[s] ( s) s (Eta) λ(a 1) b if a = σ b[ ] Table 1. The λσ-rewriting System The syntax of the typed λσ-calculus is given by: Types A ::= K A B Contexts ::= nil A. Terms a ::= 1 X (a b) λ A.a a[s] where X X Substitutions s ::= id a.s s s The set of λσ-terms is written as Λ λσ (X ). The λσ-typing rules are given by: (var) A. 1 : A (lambda) A. a : B λ A.a : A B (app) a : A B b : A s a : A (clos) (a b) : B a[s] : A (id) id (shift) A. a : A s s s (cons) a.s A. (comp) s s In addition, to each meta-variable X we associate a unique type T X and a unique context X. We assume that for each pair (, A) there is an infinite set of meta-variables X such that X = and T X = A. We add the following typing rule for meta-variables: (Metavar) X X : T X. We write a A to say that the λσ-term a has type A under context. The simply typed λσ-calculus is weakly terminating and, hence every typed term has a normal form. The λσ-nf (λσ-nf for short) of a λσ-term is essential in this work and is given by the following definition taken from [Río93]. Proposition 1. Any λσ-term in λσ-nf has one of the following forms: 1. λ.a, where a is in λσ-nf. 2. (a b 1... b q ), where a and b i are in λσ-nf and a is either 1, 1[ n ], X or X[s] where s is a substitution term in λσ-nf and different from id. 3. a 1... a p n, where a 1,..., a p are in λσ-nf and a p n. 2

3 2 λσ-böhm Trees We introduce λσ-böhm trees for simply typed λσ-terms and for λσ-terms in η-long β-normal form and set the ground for our main result. Definition 2 (Occurrence). An occurrence is a list of strictly positive integers α = s 1,..., s n. The number n is called the length of the occurrence α. Definition 3 (Occurrence Path). Let T be a tree and α = s 1,..., s n be an occurrence in T. The path of α is the set of occurrences { s 1,..., s k k n}. We now define the so-called λσ-böhm trees for simply typed λσ-terms. Definition 4 (λσ-böhm Trees). A λσ-böhm tree is a tree whose nodes are labeled with pairs l, va such that l is a positive integer and v A is a λσ-term of type A under context. From now on, we use λσ-nf to denote the η-long β-normal form of a λσ-term. Below, we define the λσ-böhm tree for λσ-terms in η-long β-normal form. Notice that a similar definition can be done requiring just a kind of head normal form because each normal form is a head normal form. Definition 5 (λσ-böhm tree of a λσ-nf). Let a A = λ A1 λ Ak.(h B 1... B m B b 1 B1 b m Bm ) be a term in λσ-nf, where = A 1... A k. The Böhm tree of a A is recursively defined as the tree whose root is labeled with the pair k, h B 1... B m B and whose sons are the λσ-böhm trees of: 1. b 1B1,...,b mbm, if h B 1... B m B is a de Bruijn index; 2. a 1A1,..., a pap, b 1B1,..., b mbm, if h B 1... B m B is a meta-variable of the form XA [a 1 A 1... a pap n ], where a 1A1... a pap n is a substitution in λσ-nf. As in [Dow94], we write a A to denote the depth of the λσ-böhm tree of the λσ-nf of the term a A. Example 6. The λσ-böhm tree of the term λ A λ A λ A.(4 A A A X A 1 A), where = A A A A A A nil is given by: 3, 4 A A A 0, XA 0, 1 A 3

4 The λσ-böhm tree of the term λ A λ A λ A.(4 A A A (X A A [(λ A.1 A A ) 1 A 2 2 ] 2 A) 1 A), where = A A A A A A nil and = A A A 2 is given by: 0, X A [(λ A.1 A 1, 1 A A 3, 4 A A A A ) 1 A 2 2 ] 0, 1 A 0, 1 A 0, 2 A Definition 7 (Free occurrence). A free occurrence of a de Bruijn index i in the λσ-term a is defined by: 1. If a = X[s] then i does not occur free in a, where X is a meta-variable and s is any substitution. 2. If a = i then i occurs free in a. 3. If a = λ A.b and i occurs free in b then i + 1 occurs free in a. 4. If a = (b c) and i occurs free in b or c (or in both) then i occurs free in a. Definition 8 (Relevant term). A λσ-term a = λ A1... λ Ak.b is relevant in its i-th (1 i k) argument if the index k i + 1 occurs free in b. Lemma 9. Let a A be a λσ-nf and a 1 A 1,..., a pap be λσ-terms of atomic type, where = A 1... for some context. For all 1 j p, if the de Bruijn index j occurs free in a A then the λσ-nf of a A[a 1 A1... a p n ] contains a j Aj as a sub-term, and hence a A a A [a 1 A 1... a p Ap n ]. Proof. Note that all possible free occurrences of j in a are in non functional positions because they have atomic types; i.e., all these free occurrences are leaves. Therefore, the normalisation of a A [a 1 A 1... a p Ap n ] does not include any simulation of β-reduction. Thus, if β is a free occurrence of j in a A, then after a σ-normalisation, every occurrence α in the λσ-böhm tree of the λσnf of a j Aj will become the occurrence βα in the λσ-böhm tree of the λσ-nf of a A [a 1 A 1... a p n ]. This means that the λσ-böhm tree of a j A j is a sub-tree of the λσ-böhm tree of the λσ-nf of a A [a 1 A 1... a pap n ]. Lemma 10. Let a be a term in λσ-nf. Then a A a A n ] where k, n 0, i k n, =... and n is a well-typed substitution. 4

5 Proof. By structural induction on a A : If a A = r A and r k then k A n ] = i r I r = 0. Otherwise, r A n ] = r k + n B = 0 If a A = XΠ A [a 1 A 1... a p m Π ], then XΠ A [a 1 A 1... a p m def. Π ] = 1 + max( a 1A1,..., a pap ). In addition, a A n ] = { XA Π[a 1 A 1... a pap m Π ] n ] t1, if m k σ t 2, if m > k where t 1 = XA Π[a 1 A 1 n ]... a pap n ] i m I m n ] and t 2 = XA Π[a 1 A 1 n ]... a pap... i k n ] m k+n Π ]. In the first case, i.e., m k, by IH, for all 1 j p, a j A j a j A j n ] and by definition of λσ-böhm trees, we have that X Π A [a 1 A 1 n ]... a p Ap... i k n ] i m I m... i k n ] = 1+max( a 1 A1... i k n ],..., a p Ap max( a 1 A1,..., a p ) = X Π A [a 1 A 1... a p m Π ] In the second case, i.e., m > k, by IH, for all 1 j p, a j A j a j A j n ] n ], i m I m,..., i k ) 1 + and then XA Π[a 1 A 1 n ]... a pap n ] m k+n Π ] 1 + max( a 1 A 1 n ],..., a p Ap n ] ) 1 + max( a 1 A1,..., a p ) = X Π A [a 1 A 1... a p If a A = λ B.b B C i k B n+1b m Π ] then by IH we have that b B C bb C [1B B B i 1... ] for an adequate context. In addition, by the definition of λσ-böhm trees, we have that b B C [1B B λ B.b B C [1B B i 1 B B n+1b i 1 B ] = (λ B.b B C B n+1b ] = )... i n 1 ] and the lemma holds. If a A = (b B A c B ) then, by IH, we have that b B A b B A... i k n ] and cb = c B n ]. Since a A = (b B A a B ) is in λσ-nf, we have that a A = 1 + max( b B A, c B ) 1 + max( b B A... i k n ], c B n ] ) = a B n ]. Proposition 11. Let a A be a λσ-normal term and a 1 A 1,..., a p be λσ-normal terms of at most second order, where = A 1... for some context. For all 1 j p, if the de Bruijn index j occurs free in a A then a j A j a A[a 1 A1... a p n ] 5

6 for some n 0 such that n is a well-typed substitution. Proof. The proof is by structural induction on a A : If a A = j A then j A [a 1 A 1... a p Ap n ] = a j Aj. If a A = λ B.b B C then a A [a 1 A 1... a pap n ] = (λ B.b B C )[a 1 A 1... a pap n ] σ λ B.b B C [1B B a 1A1 [ B ]... a p [ B ] n+1b ] and by IH we have that, for all 1 j p + 1, if the de Bruijn index j occurs free in b B C then b j A j [ B b B C [1B B ] b B C [1 B B a 1A1 [ B ]... a p [ B { 1 B where b j A j [ B ] = B, if j = 1 a j 1Aj 1 [ B ], if 1 < j p. Therefore, using Lemma 10 and equation (1), we have that: j > 1: a j 1Aj 1 a j 1Aj 1 [ B ] a j Aj [ B ]... a p [ B ] n+1b ] = λ B.b B C [1B B (λ B.b B C j = 1: a 1A1 [ B )[a 1 A 1... a pap n ] 1 B B λ B.b B C [1B B b B C [1B B a 1 A1 [ B ]... a p [ B ] n+1b ] = a j Aj [ B ]... a p [ B ] n+1b ] = (λ B.b B C )[a 1 A 1... a p n ] If a A = (b B A c B ) then a A [a 1 A 1... a pap n ] = (b B A c B )[a 1 A 1... a p Ap n ] σ ] n+1b ] (1) ]... a p [ B ] n+1b ] = (b B A [a 1 A 1... a pap n ] c B [a 1 A 1... a pap n ]) and now we have the following cases according to the existence of a free occurrence of a de Bruijn index j with 1 j p in b B A and/or in c B : 1. Assume j with 1 j p occurs free in b B A. If this occurrence is in the head of the term then we have (we omit type information for clarity): (b[a 1... a p n ] c[a 1... a p n ]) = ((j b 1... b k )[a 1... a p n ] c[a 1... a p n ]) σ (a j b 1 [a 1... a p n ]... b k [a 1... a p n ] c[a 1... a p n ]) = (λ B1... λ Bk+1.d b 1 [a 1... a p n ]... b k [a 1... a p n ] c[a 1... a p n ]) λσ d[c[a 1... a p n ] b k [a 1... a p n ]... b 1 [a 1... a p n ] id]. Using Lemma 9, we conclude that a j = λ B1... λ Bk+1.d = d d[c[a 1... a p n ] b k [a 1... a p n ]... b 1 [a 1... a p n ] id] = (b[a 1... a p n ] c[a 1... a p n ]). If the occurrence of j is not in the head of b B A then the λσ-nf of (b B A [a 1 A 1... a p n ] c B [a 1 A 1... a p n ]) is of the form (b B A c B ), where b B A (resp. c B ) is the λσ-nf of b B A [a 1 A 1... a p Ap 6

7 n ] (resp. c B [a 1 A 1... a p Ap n ]). Therefore, b B A [a 1 A 1... a p Ap n ] (b B A [a 1 A 1... a p Ap n ] c B [a 1 A 1... a p Ap n ]) and by IH, we have that a j Aj b B A [a 1 A 1... a p Ap n ] because j occurs free in b B A. 2. Suppose that j with 1 j p occurs free in c B. By IH, we have that a j A j c B [a 1 A 1... a p n ] (b B A [a 1 A 1... a pap n ] c B [a 1 A 1... a pap n ]). Proposition 12. Let a A be a λσ-normal term and, a 1 A 1,..., a p be λσ-terms of at most second order, and = A 1... for some context. Let α = s 1,..., s n be an occurrence in the λσ-böhm tree of a A and β 1 = s 1,..., s k1,..., β l = s 1,..., s kl be all the occurrences of free de Bruijn indexes j with 1 j p that are in the path of α. For each 1 i l, if the term a j Aj is relevant in its r-th argument, where r is the position of the son of β in the path of α, i.e., r = s ki +1 then there exists an occurrence α in the λσ-böhm tree of the λσ-nf of a A [a 1 A 1... a p n ], for some n 0 where n is well typed and, such that all the labels occurring in the path of α, except j, occur in the path of α and the number of times they occur in the path of α is greater than or equal to the number of times they occur in the path of α. Moreover, if the occurrence α is labeled with a symbol different from j then the occurrence α is labeled with the same symbol and, if a j Aj = 0 then the length of α is greater than or equal to the length of α. Proof. By induction on the number of occurrences of free de Bruijn indexes j with 1 j p in the path of α. We consider occurrences from the lowest to the highest in the λσ-böhm tree of a A. In addition, whenever it is possible, we omit for clarity, the decoration of terms. Let α = β l β l 1... β 2 β 1 γ (2) be an occurrence in the λσ-böhm tree of a A. If there is no free occurrence of j (1 j p) in the path of α, i.e., l = 0 in (2), then α is also an occurrence in the λσ-böhm tree of the λσ-nf of a A [a 1 A 1... a pap n ] and then we take α = α = γ. If there exists exactly one free occurrence of j (1 j p) in the path of α, then α = β 1 r γ, where γ = r γ and suppose that the term a A at the occurrence β 1 has the form λ B1... λ Bv.(j e 1... e u ), where j corresponds to a free occurrence of j. By hypothesis a j Aj = λ C1... λ Cu.d is relevant in the r-th argument (that corresponds to the son of β 1 that is in the path of α). Let ξ be a free occurrence of u r + 1 in the λσ-böhm tree of d. After the substitution, every occurrence β 1 θ have been removed, where θ is an occurrence in the λσ- Böhm tree of e i ; and the following occurrences have been added: β 1 δ, where δ is an occurrence in the λσ-böhm tree of d not labeled with a free de Bruijn index less than or equal to u. 7

8 β 1 δγ, where δ is a free occurrence of a de Bruijn index less than or equal to u and γ is an occurrence in the λσ-böhm tree of e i. In this case we take α as follows: α = β 1 ξγ β 1 q + 1 ξγ, if the label at the occurrence β 1 in the λσ-böhm tree of the λσ-nf of a A [a 1 A 1... a p n ] has the form k, X A [b 1 B 1... b q B q m ]., if the label at the occurrence β 1 in the λσ-böhm tree of the λσ-nf of a A [a 1 A 1... a p n ] has the form k, k A. Now, suppose that the result holds for an occurrence α = β l 1... β 2 β 1 γ in a A containing l 1 occurrences of free de Bruijn indexes less than or equal to p. Hence, there exists an α satisfying the proposition. For α = β l β l 1... β 2 β 1 γ = β l α the induction step is built as above when there exists exactly one free occurrence of j in the path of α. If a j A j = 0 then the length of α can be less than the length of α because α = β 1 r γ and the above construction will give α = β 1 γ since the occurrence of u r + 1 is in the head of a j Aj. Therefore, when a j Aj = 0, the occurrence ξ above cannot be empty and then the length of α is greater than or equal to the length of α. Corollary 13. Let a A be a λσ-nf, a 1 A 1,..., a pap be λσ-normal terms of at most second order, and = A 1... for some context. If for all 1 i p, the terms a iai are relevant in all its arguments and a iai = 0 then a A a A[a 1 A1... a p n ] Proof. Let α be the longest occurrence in the λσ-böhm tree of a A. By Proposition 12, there exists an occurrence α in the λσ-böhm tree of the λσ-nf of a A [a 1 A 1... a pap n ] that is, at least, as long as α. For a given matching (or interpolation) problem Ψ, the way to find substitutions that are solutions to Ψ is by using the unification method developed in [DHK00]. Instead of higher-order substitution, this method uses grafting (firstorder substitution) and to do so, the given matching (or interpolation) problem needs to be precooked. The precooking translation protects the meta-variables occurring in the problem avoiding capture and is defined as follows.: Definition 14 ([DHK00]). Let a Λ db (X ) such that a : T. To every meta-variable X of type U in the term a, we associate the type U and the context in λσ-calculus. The precooking of a from Λ db (X ) to the set of λσterms, written Λ λσ (X ), is defined by a F = F (a, 0) where F (a, n) is defined by: (a) F ((λ B.a), n) = λ B (F (a, n + 1)) (b) F (k, n) = 1[ k 1 ] (c) F ((a b), n) = F (a, n) F (b, n) (d) F (X, n) = X[ n ] 8

9 3 From Matching Problems to Interpolation Problems We describe how matching problems can be converted into equivalent interpolation problems. Third-order interpolation problems in the language of the λσ-calculus are solved using the matching rules presented in Table 2, that are adapted from [dmkar05]. σ, P {λ A.a <<? λσ λ A.b} Dec m -λ σ, P {a <<? λσ b} Dec σ, P {(n a 1... a p ) <<? λσ (n b 1... b p )} m-app σ, P {a 1 <<? λσ b1,..., ap <<? λσ bp} σ, P {(n a 1... a p ) <<? λσ (m b 1... b q )} Dec m-fail, if m n. F ail σ, P Exp m -λ Y : (A 1..A k. Y : B), σ, P {X λ A1... λ Ak.Y } if ( X : A 1 A k B) T Var(P ), Y T Var(P ), and X is not a solved variable. where σ = σ{x λ A1... λ Ak.Y } Imit Proj σ, P {X[a 1..a p. n ] <<? λσ (m b 1... b q )} σ, P σ {(m n+p H 1... H q )[a 1 σ..a p σ. n ] <<? λσ (m b 1... b q )} if X has atomic type and m > n. where σ = σ{x (m n+p H 1... H q )}, H 1,..., H q are metavariables with appropriate type and with contexts Hi = X ( 1 i q). σ, P {X[a 1..a p. n ] <<? λσ (m b 1... b q )} σ, P σ {(j H 1... H q )[a 1 σ..a p σ. n ] <<? λσ (m b 1... b q )} if X has atomic type and the j-th element (1 j p) of the list a 1..a p has the same type of X, where σ = σ{x (j H 1... H q)}, H 1,..., H q are metavariables with appropriate type and with contexts Hi = X( 1 i q). σ, P {a <<? λσ b} Normalise m if a or b is not in Eta-long form. σ, P {a <<? λσ b } where a (resp. b ) is the Eta-long form of a (resp. b), and σ is obtained from σ by normalising all its terms. if a (resp. b) is not a solved variable and a (resp. b) otherwise. Table 2. The Matching Rules Definition 15. Let a? λσ b be a matching equation and σ a ground solution to this equation, i.e., the λσ-normal form of aσ is equal to b. We define the interpolation problem Φ(a? λσ b, σ) inductively over the number of occurrences of a as follows: If a = λ A.c then b is also an abstraction of the form λ A.d and then σ is also a solution of c? λσ d and we let Φ(a? λσ b, σ) = Φ(c? λσ d, σ). 9

10 If a = (k c 1... c m ) then b is also of the form (k d 1... d m ) because a? λσ b is solvable and we let Φ(a? λσ b, σ) = Φ(c i? λσ d i, σ). i If a = (X[a 1... a p n ] c 1... c m ) then we let Φ(a? λσ b, σ) = {(X[a 1... a p n ] c 1 σ... c m σ)? λσ b} i H i, where Φ(c i? λσ c iσ, σ), if the dummy symbol occurs in the λσ-normal form of H i = (Xσ[a 1 σ... a p σ n ] c 1 σ... c i 1 σ c i+1 σ... c m σ)., otherwise. Proposition 16. Let a? λσ b be a matching equation and σ a ground solution to this equation. Then the substitution σ is a solution to Φ(a? λσ b, σ) and, conversely, if σ is a solution to Φ(a? λσ b, σ) then σ is also a solution to the matching equation a? λσ b. Proof. By induction on the structure of the terms on the left-hand side of all the equations occurring in the problem. Let σ be a ground solution of a? λσ b. If a is an abstraction of the form λ A.c then b is also an abstraction of the form λ A.d and σ a ground solution of c? λσ d. By IH σ is a solution to Φ(c? λσ d, σ) and hence also a solution to all the equations of Φ(a? λσ b, σ). If a is an application of the form (k c 1... c m ) then b is also an application of the form (k d 1... d m ) and σ is also a ground solution of the equations c 1? λσ d 1,...,c m? λσ d m. By IH, we have that σ is a solution to Φ(c i? λσ d i, σ) i and hence σ is also a solution to Φ(a? λσ b, σ). If a is of the form (X[a 1... a p n ] c 1... c m ) then σ is a solution to X[a 1... a p n ] c 1 σ... c m σ? λσ b and c 1? λσ c 1σ,..., c m? λσ c mσ. By IH, σ is also a ground solution to m Φ(c i? λσ c i σ, σ) i=1 and therefore, σ is a solution to {X[a 1... a p n ] c 1 σ... c m σ? λσ b} where H i is as in Def. 15. m H i = Φ(a? λσ b, σ), i=1 Conversely, let σ be a ground solution to Φ(a? λσ b, σ): If a is an abstraction of the form λ A.c then b is also of the form λ A.d and Φ(c? λσ d, σ) Φ(a? λσ b, σ) and hence σ is also a solution to c? λσ d and therefore, a solution to a? λσ b. 10

11 If a is an application of the form (k c 1... c m ) then b is also of the form (k d 1... d m ) and, since Φ(c i? λσ d i, σ) Φ(a? λσ b, σ), we have that i σ is also a solution to all the equations c i? λσ d i for i = 1,..., m and therefore, σ is also a solution to a? λσ b. If a is of the form (X[a 1... a p n ] c 1... c m ) then σ is also a solution to (X[a 1... a p n ] c 1 σ... c m σ)? λσ b and to Φ(c i? λσ c i σ, σ), i.e., i c i σ = λσ c i σ for all i = 1,..., m. Therefore, (X[a 1... a p n ]σ c 1 σ... c m σ) = λσ b (X[a 1... a p n ]σ c 1 σ... c m σ ) = λσ b (X[a 1... a p n ] c 1... c m )σ = λσ b aσ = λσ b Definition 17. Let Ψ be a third-order matching problem and σ be a solution to Ψ. We let Φ(Ψ, σ) be the following third-order interpolation problem: Φ(Ψ, σ) = Φ(a? λσ b, σ). a? λσ b Ψ Proposition 18. Let Φ be an interpolation problem generated from a precooked matching problem Ψ and σ be a ground solution to Φ. Then all the equations of Φ are of the form (X[a 1... a p n ] c 1... c q )? λσ b, where all the terms a 1,..., a p, c 1,..., c q and b are ground. Proof. Def. 15 implies that any interpolation equation is flexible-rigid. Flexiblerigid equations are of the form (X[a 1... a p n ] c 1... c q )? λσ b. The right-hand side of a matching equation is a ground term and, since b comes from terms in the same side of matching equations b should be ground. The proof is by inspection on the matching rules. Let Ψ be a matching problem which is in the image of the precooking translation. Then all meta-variables occurring in Ψ have the form X[ n ], where n 0 means that X is under the scope of n abstractors. Therefore, after converting the matching problem Ψ into the interpolation problem Φ, we get that all interpolation equations are of the form (X[ n ] c 1... c q )? λσ b because no λσ-rule is applied in this conversion. Moreover, all the terms c 1,..., c q are ground. The matching rules that can be applied for this equation are Exp m -λ, Imit or Proj. In the first case, X is replaced by λ C1... λ Cq.Y, where C i is the type of c i and the equation must be normalised. During the normalisation, the λσ-rules that insert terms inside substitutions are (Beta) or (Abs). In the case of Imit or Proj, the introduced meta-variables are always in the head of the new equations. Applications of Dec m -λ, Dec m -App and Dec m -Fail do not change the structure of the explicit substitutions. 4 Third-Order Interpolation Problems We show how interpolation problems are decided concluding the decidibility of third-order matching. 11

12 Definition 19 (Occurrence accessible w.r.t. an equation). Consider an equation of the form (X[a 1... a p n ]c 1... c q ) = b and the term t = λ C1... λ Cq.u with the same type of X. The set of occurrences in the λσ-böhm tree of t accessible with respect to the equation (X[a 1... a p n ] c 1... c q ) = b is inductively defined as: the root of the λσ-böhm tree of t is accessible. if α is an accessible occurrence labeled with a de Bruijn index j with 1 j p + q and d j is relevant in its r-th argument then the occurrence α r is accessible, where: d j = { aj if q < j p + q, c q i+1 if 1 j q. if α is an accessible occurrence labeled with a de Bruijn index greater than p + q or with a meta-variable then all the sons of α are accessible. if α is an accessible occurrence labeled with a meta-variable then each son of α is accessible. Definition 20 (Occurrence accessible w.r.t. an interpolation problem [Dow94]). An occurrence is accessible with respect to an interpolation problem if it is accessible with respect to one of the equations of this problem. Definition 21 (λσ-term accessible w.r.t. to an interpolation problem). A λσ-term is accessible with respect to an interpolation problem if all occurrences of its λσ-böhm tree which are not leaves are accessible with respect to this problem. Definition 22 (Accessible solution built from a solution). Let Φ be an interpolation problem and let σ be a ground solution to this problem. For each meta-variable X occurring in the equations of Φ consider the λσ-term t such that {X t} σ. In the λσ-böhm tree of t, we prune all occurrences non accessible (that are not leaves) with respect to the equations of Φ in which X has an occurrence and put λσ-böhm trees of ground terms of depth 0 of the expected type as leaves. Call t the term whose λσ-böhm is obtained this way and σ the resulting substitution. Proposition 23. Let Φ be an interpolation problem generated from a precooked matching problem and let σ be a ground solution to Φ. Then the accessible solution σ, built from σ, is a solution to Φ. Proof. Let (X[a 1... a p n ] c 1... c q )? λσ b be an equation of Φ,σ be a ground solution to Φ, t = Xσ = λ C1... λ Cq.d and t = X σ = λ C1... λ Cq.d. It is enough to prove that d[c q... c 1 a 1... a p n ] = λσ d [c q... c 1 a 1... a p n ] because the left-hand side of the above equation reduces to b and the right-hand side is λσ-equivalent to (X[a 1... a p n ] c 1... c q ) σ. By construction, the λσ-böhm trees of d and d differ only on the non accessible occurrences. So, let β r be a non accessible occurrence in a path in the 12

13 λσ-böhm tree of t such that the occurrence β is accessible w.r.t the equation (X[a 1... a p n ] c 1... c q ). From the definition of accessible occurrence, we conclude that β is labeled with a de Bruijn index j (1 j p + q) and d j is { aj q, if j > q not relevant in its r-th argument, where d j =. During the c q j+1, if j q normalisation step, j will be replaced by d j and since d j is not relevant in its r-th argument, the label of β r will disappear during the normalisation. Proposition 24. Let Φ be an interpolation problem, σ a solution to Φ and σ the accessible solution built from σ. If h is the maximum depth in the λσ-böhm tree of the right-hand side of the equations of Φ and t = X σ = λ C1... λ Cq.u (u atomic) then there are at most h + 1 occurrences of de Bruijn indexes k with k > q on a path in the λσ-böhm tree of t. Proof. Notice that the occurrences in the λσ-böhm tree of t and those in the λσ- Böhm tree of u are the same, except that the label at the root differ. Suppose that there exists an occurrence α in the λσ-böhm tree of u such that there exists more than h + 1 occurrences of de Bruijn indexes k with k > q in the path of α. Let β be the h+1-th occurrence of such index. The occurrence β is not a leaf and hence it is accessible w.r.t. an equation of Φ, say (X[a 1... a p n ] c 1... c q )? λσ b. For each occurrence γ = s 1,..., s l, in the path of β, labeled with a de Bruijn index j with 1 j q, we have that c q j+1 is relevant in its r-th argument, where r = s l+1. Using Proposition 12 we conclude that there exists an occurrence β in the λσ-böhm tree of u[c q... c 1 n ] = λσ b such that β contains at least h + 1 occurrences and thus the length of β is at least h. Moreover, the labels of β and β are the same and hence β is not a leaf. In this case, the depth of b is greater than h which contradicts the hypothesis. Definition 25 (Compact λσ-term). A λσ-term t = λ C1... λ Cq.u (u atomic) is compact w.r.t. an interpolation problem Φ if no de Bruijn index j with 1 j q appears free in a path of the λσ-böhm tree of u more than h + 1 times, where h is the maximum depth in the λσ-böhm tree of the right-hand side of the equations of Φ. Proposition 26. Let Φ be an interpolation problem, σ be an accessible solution to Φ, h be the maximum depth in the λσ-böhm tree of the right-hand side of the equations of Φ and t = X σ = λ C1... λ Cq.u (u atomic). If α is an occurrence in the λσ-böhm tree of u that contains more than h + 1 free occurrences of the de Bruijn index j (1 j q) in its path, then: for all equations (X[a 1... a p n ] c 1... c q )? λσ b of Φ where the (h + 2)- th occurrence labeled with j is accessible w.r.t. this equation, the term c q j+1 is a projection, i.e., there exists an integer 1 r p such that c q j+1 = λ B1... λ Bp.r. Proof. Let β be the first occurrence with label j (1 j q) in the λσ-böhm tree of u that is in the path of α and suppose that the (h + 2)-th free occurrence of j in the path of α is accessible w.r.t. the equation (X[a 1... a p n ] c 1... c q ). If 13

14 α = β r β for some β then we have that c q j+1 is relevant in its r-th argument. If the head of c q j+1 is not first order then c q j+1 = 0 and, by Proposition 12, we have that there exists an occurrence α as long as the occurrence α in the λσ-böhm tree of u[c q... c 1 a 1... a p n ], where a 1,..., a p, c 1,..., c q are λσ-terms of at most second order and hence the depth of the λσ-böhm tree of u[c q... c 1 a 1... a p n ], which reduces to b, is at least h + 1 which is contradictory. Therefore, c q j+1 must be a λσ-term with depth equals to 0 and hence c q j+1 = λ B1... λ Bp.r. Definition 27. [Compact Accessible Solution built from an Accessible Solution] Let Φ be an interpolation problem, σ be an accessible solution to this problem and h be the maximum depth in the λσ-böhm tree of the right-hand side of the equations of Φ. The grafting σ is a compact accessible solution built from an accessible solution to Φ if, for all meta-variable X occurring in Φ, the term t = X σ = λ C1... λ Cq.u (u atomic) is such that there is no path in the λσ-böhm tree of u containing more than h+1 occurrences labeled with the de Bruijn index j (1 j q). If there exists a path in the λσ-böhm tree of u that has more than h + 1 free occurrences of the de Bruijn index j (1 j q) then the compact accessible solution is built as follows: By Proposition 26, if these occurrences are accessible w.r.t. the equation (X[a 1... a p n ] c 1... c q ), we have that c q j+1 is of the form λ B1... λ Bp.r. In this case, we replace all these occurrences of j by λ B1... λ Bp.r. The compact accessible solution built from the accessible solution σ is denoted by σ. Proposition 28. Let Φ be an interpolation problem, σ a solution to Φ, σ be the accessible solution built from σ and σ be the compact accessible solution built from σ. Then σ is also a solution to Φ. Proof. The proof is similar to that of Proposition 23. Notice that, in the construction of the compact accessible solution, the replacement of j for the term c q j+1 occurs during the normalisation step of the term ((λ C1... λ Cq.u)[a 1... a p n ] c 1... c q ), where j occurs free in u. Proposition 29. Let Φ be an interpolation problem, σ be a solution to Φ, σ be the accessible solution built from σ and σ be the compact accessible solution built from σ. If h is the maximum depth in the λσ-böhm tree of the right-hand side of the equations of Φ, then for every meta-variable X of arity q, the depth of the λσ-böhm tree of Xσ = λ C1... λ Cq.u is less than or equal to (n + 1)(h + 1) 1. Proof. By Def. 27 and Proposition 24, we know that a path in the λσ-böhm tree of u may have at most h + 1 occurrences of a free de Bruijn index j. Therefore, since X has arity q, a path in the λσ-böhm tree of u may have at most (q + 1)(h + 1) symbols, i.e., its depth is at most (q + 1)(h + 1) 1. Corollary 30. Let Φ be a third-order interpolation problem, σ be a solution to Φ, σ be the accessible solution built from σ and σ be the compact accessible solution built from σ. If h is the maximum depth in the λσ-böhm tree of the right-hand side of the equations of Φ, then for every meta-variable X of arity q, 14

15 the depth of the λσ-böhm tree of Xσ = λ C1... λ Cq.u is less than or equal to (n + 1)(h + 1) 1. Proposition 16 allows us to use the boundary of the solution of the interpolation problem in Corollary 30 for third-order matching problems. In this way we can build a decision procedure as follows: Theorem 31 (Decision procedure). The class of third-order λσ-matching problems that come from the simply typed λ-calculus is decidable. Proof. Let Ψ be a third-order matching problem in the λσ-calculus. Enumerate all ground substitutions for the meta-variables occurring in the equations of the form (X[a 1... a p n ] c 1... c q )? λσ b of Ψ, such that the terms to be substituted for X have depth less than or equal to (q + 1)(h + 1) 1, where h is the depth of the λσ-böhm tree of b. If none of these substitutions is a solution Φ then Φ is not solvable. Otherwise, it is solvable. 5 Conclusion We studied third-order matching problems in the λσ-calculus of explicit substitutions. Both matching and explicit substitutions are important tools in the implementation of higher order programming languages and proof assistants. Studies such as the one carried out in this paper, help to improve the design and implementation of languages based on the λ-calculus. Our main contributions can be summarized as follows: The notion of λσ-böhm trees for the λσ-calculus which extends the usual notion of Böhm trees. The extension of the decision algorithm of [Dow94] to the language of the λσ-calculus. We believe that this extension might be useful for developing implementations based on the simply typed λσ-calculus. Future work includes the analysis and comparison of third-order matching in different styles of explicit substitutions. References [ACCL91] M. Abadi, L. Cardelli, P.-L. Curien, and J.-J. Lévy. Explicit Substitutions. J. of Func. Programming, 1(4): , [db72] N.G. de Bruijn. Lambda-Calculus Notation with Nameless Dummies, a Tool for Automatic Formula Manipulation, with Application to the Church-Rosser Theorem. Indag. Mat., 34(5): , [DHK00] G. Dowek, T. Hardin, and C. Kirchner. Higher-order unification via explicit substitutions. Information and Computation, 157: , [dmkar05] F. L. C. de Moura, F. Kamareddine, and M. Ayala-Rincón. Second order matching via explicit substitutions. In F. Baader and A. Voronkov, editors, LPAR , volume Springer-Verlag, [Dow94] G. Dowek. Third-order matching is decidable. APAL 69: , [Río93] A. Ríos. Contributions à l étude de λ-calculs avec des substitutions explicites. Thèse de doctorat, Université Paris VII,