J. Roger Hindley and Jonathan P. Seldin. Introduction to Combinators and #-Calculus. Cambridge University Press, 1986. 17 Ravara, Matos et al.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of the partial recursive functions over W. It is easy to find interpretations of the constants of so that all the axioms of PT are true in PRO . A further important model of PT is the open term model M(##) This model is based on the usual ## reduction of the untyped lambda calculus (cf. [1, 16]) and exploits the well known equivalence between combinatory logic with extensionality and ##. In order to deal with the constants di#erent from k and s, one extends ## reduction by the obvious reduction clauses for these new constants and checks that the so obtained new reduction relation enjoys ....

Hindley, J. R., and Seldin, J. P. Introduction to Combinators and #-Calculus. Cambridge University Press, 1986.

.... decidability of fragments of context uni cation [Com98,SS94,SS99b,SSS00,Lev96] or variants of context uni cation [CP97] In this paper we generalize the result on decidability of bounded second order uni cation to higher order uni cation in the simply typed lambda calculus with and rules [Bar84,HS86]. We show that solvability of uni cation problems is decidable if for any variable a bound on the number of lambda binders and occurrences of bound variables in the image of the variable under a uni er is given. Here each image is assumed to be in expanded normal form. The result implies ....

....degree = order 1. 6 2.4 Instantiation and Substitutions An instantiation of a variable x in s by the term t , written s[t=x] replaces free occurrences of the variable x in s by t, where before replacement, the bound variables in s have to be renamed to avoid variable capture. See, e.g. [HS86] for a precise de nition. After the replacement, it may be necessary to rename bound variables in the di erent copies of t, since we use the disjoint variable convention. The notation s[t=x] is only used if t and x have the same type. A closed substitution is a mapping from terms to closed ....

J.Roger Hindley and Jonathan P. Seldin. Introduction to combinators and -calculus. Cambridge University Press, 1986.

....Thus, we have assumed that the reader has a certain familiarity with logic and the lambda calculus. If the reader does not feel suciently comfortable with these topics, we suggest consulting Girard, Lafont, Taylor [9] or Gallier [6] for background on logic, and Barendregt [2] Hindley and Seldin [15], or Krivine [19] for background on the lambda calculus. For an in depth study of constructivism in mathematics, we highly recommend Troelstra and van Dalen [32] 2 Natural Deduction, Simply Typed Calculus We rst consider a syntactic variant of the natural deduction system for implicational ....

....every proof (term) reduces to a normal form, which is unique up to renaming. This result was rst proved by Prawitz [24] for i . speci ed in Definition 3.3) is con uent. Equivalently, conversion in is Church Rosser. parallel reduction (see also Barendregt [2] Hindley and Seldin [15], or Stenlund [27] 13 (as in Definition 3.3) is strongly normalizing. 1971) 11] 1972) see also Gallier [7] If one looks at the rules of the systems N (or ) one notices a number of unpleasant features: 1) There is an asymmetry between the lefthand side and the righthand side ....

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J.R. Hindley and J.P. Seldin. Introduction to Combinators and -Calculus, volume 1 of London Mathematical Society Student texts. Cambridge University Press, 1986.

....But here we are speaking Logicese, whence we say theory where the cited papers say filter. By # we mean the type free #K # calculus, invented by Church in the birth year of one of us, and exhaustively studied by Barendregt in [5] By CL we mean Curry s (weak) combinatory logic, as summarised in [15]. As CL is already definable in # in well known ways , it will suffice here to recount the [6] proof that CT = CT , is a model of #. First, we define an equivalence in ITD on Form by . A B = df A B and B [6] which uses # where we here use # ) rightly suggests that ITD may be ....

J. R. Hindley and J. P. Seldin. Introduction to combinators and #-calculus. Cambridge university press, Cambridge, 1986.

.... if one drops condition (i) For example, in the theory B it is well known that one can represent all recursive functions in the sense of (ii) The proof of this fact runs completely analogous to the argument in the untyped # calculus showing that all recursive function are representable there (cf. [1, 38]) The crucial ingredient in the proof is of course the recursion or fixed point lemma (Lemma 3) Hence, for example, it is possible to find a closed term exp representing a suitable form of exponentiation on W in the sense of condition (ii) above, but indeed none of the theories introduced in ....

....of witnessing is due to Sieg in his important work on Herbrand analyses , cf. Sieg [60, 61] and also Buchholz and Sieg [7] In our definition of realizability below we will make use of the open term model . This model is based on the usual ## reduction of the untyped lambda calculus (cf. [1, 38]) and exploits the well known equivalence between combinatory logic with extensionality and ##. In order to deal with the constants di#erent from k and s, one extends ## reduction by the obvious reduction clauses for these new constants and checks that the so obtained new reduction relation enjoys ....

Hindley, J. R., and Seldin, J. P. Introduction to Combinators and #-Calculus. Cambridge University Press, 1986.

....a = # a a = # b b = # a a = # bP(b) P (a) The type information can be extracted. If a = b then both terms have the relevant type. a : # a = # b 11.2 Types Functions These rules for abstraction and application are typical of type inference systems: see Chapter 15 of Hindley and Seldin [17]. Applications are written with an explicit operator: f a. #x : #.b(x) # ##a: f a : # We have # and # conversion: #x : #.b(x) a = # b(a) #x : #.f x = ### f In # conversion, variable x may not be free in f . All rules that discharge the assumption x : # are subject to ....

J. Roger Hindley and Jonathon P. Seldin. Introduction to Combinators and #-Calculus. Cambridge University Press, 1986.

....in the typed # calculus are identical to those in the # calculus with one important difference: types must agree. For example, the above example could only take an argument for the variable x which has # as its type. For a useful introduction to the # calculus, including the typed # calculus, see [13]. 3. The event subscription language Here we describe a language of event expressions which is based on the typed # calculus. Types in this language are built using the following grammar T : t TSeq T 1 T 2 TDisj T 1 T 2 TConj T 1 T 2 The type language consists of the types of ....

J. Hindley and J. Seldin. Introduction to combinators and #-calculus. Cambridge, 1986.

....B[N x] is provable. Lemma 2.11 (Type Preservation) If A M: A is provable and M N, then A [ N: A is provable. Type preservation is sometimes called subject reduction, but we prefer to reserve that name for its original meaning as a similar property of untyped A terms [Curry Feys 1958, Hindley Seldin 1986] Lemma 2.12 (Unique Typing) If A [ M: A and A M: B are provable, then A . B. Static typing. Let any type convertible to the form II: be called a type. A term with a type A is either a type, if A = or a function returning a type, i.e. a type generator. If A M: A and A is ....

J. Roger Hindley and Jonathan P. Seldin. Introduction to Combinators and )-Calculus. Volume 1 of London Mathematical Society Student Texts, Cambridge University Press, 1986.

....the # calculus, where one substitutes channels by channels. To make sure we got the concept right, we start with a general notion of substitution: a total function on names (sites, channels, located channels) defined along the lines of the substitution for the # calculus, by Hindley and Seldin [9]. This function is then used to define name replacement (used in alphacongruence) name instantiation (the substitution that arises from communication, as in a (x)P ) and name translation (the substitution that happens during migration, as in a s #v# or a s (x)P ) For the type system, we ....

....n) P . Let the operator # extend as far to the right as possible. In (P Q) we omit the parentheses when the meaning is clear. 2. 2 Free and bound names We envisage a natural definition for the free names of a process or of a network, according to the classical definitions for the # calculus [3, 9], and meeting the intuitions of any # calculist. Notation 2.2 (Useful sets) Let A, B . 5 N fn(N) bn(N) M) fn(N) fn(M) bn(N) bn(M) # s) N fn(N) fn(N) s) bn(N) bn(N) s (# a s) N fn(N) bn(N) s[P ] locate(fn(P ) s) locate(bn(P ) s) Figure 2: Free and bound names in ....

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J. Roger Hindley and Jonathan P. Seldin. Introduction to Combinators and #-Calculus. Cambridge University Press, 1986.

....object) In order to give a model for the lambda calculus we must produce a structure whose objects can be interpreted both as function and argument. A model for the untyped lambda calculus is a reflexive object in a cartesian closed category (ccc) with enough points. We refer the reader to [5, 18, 2] for a complete exposition of these notions. Let us recall that a reflexive object D in a ccc C is a solution of D ) D . D (i.e there exist a retraction pair (F; G) between D and D ) D with F ffi G = id D)D ) where D ) D is the Work partially supported by the program Bourse Lavoisier du Ministere ....

....constraints given in the preceding section. Section five summarizes the obtained incompleteness and incomparability results. 6 2 Preliminaries We assume that the reader is familiar with the lambda calculus, in particular for what concern the syntax and basic notions in semantics. We refer to [2, 5, 9, 18, 26, 27]. Anyway here we summarize without proofs the relevant material used in this paper in order to render it as self contained as possible. Terminology and notations We designate by the quotient of the set of terms by ff equivalence, and by 0 the subset of closed terms. We use the applicative ....

J.R. Hindley and J.P. Seldin, Introduction to Combinators and -Calculus, Cambridge University Press, 1986.

....reducible term also rewrites to another when we instantiate its variable occurrences. This property plays an important role in studying the computational properties of rewriting such as confluence and termination. We also show that several common reduction strategies used in calculus [4, 5], such as weak reduction and lazy reduction, can be viewed as an instance of the context sensitive rewriting. The paper is organized as follows. In Section 2, we briefly recall the technical concepts and results used in the remainder of the paper. In Section 3, we shows the basic properties of the ....

.... application as a binary symbol and each variable as a symbol of arity zero [5] This is expressed by means of the signature = where = and is a set of arbitrary constants (one for each variable symbol) The immediate reduction of calculus , can be obtained from the following inference system [4, 5], which we refer to as (in spite of the introduction of , we express in the standard notation) We can see that any instance of the axiom scheme ( can be taken as a rewrite rule ( In the sequel, is the set of rules which are instances of the ( axiom scheme. 1) ....

J.R. Hindley and J.P. Seldin. Introduction to Combinators and-Calculus. Cambridge University Press, 1986.

....because the higher order logics offer increased expressive power, and partly due to a wish to extend the applicative programming paradigm by adding a rewriting relation to fi reduction. The focus of this research is on the combination of terms and its formal theory, the typed lambda calculus [1, 2, 8], with algebraic terms and term rewriting [6] There exist several different formalisms on the integration of typed lambda calculus and rewrite systems, and on the study of the interaction between algebraic rewriting and fi reduction. The earliest one seems to be the work on combinatory ....

R. Hindley and J. Seldin. Introduction to Combinators and -calculus. Cambridge University Press, 1986.

.... (TLC for short) is a very basic functional framework built around the concepts of abstraction and function application, and it forms the backbone of most modern functional programming languages (e.g. ML [G78, Mil85] Miranda [Tur85] and Haskell [HW88] Let us briefly review its definition (cf. [Bar84, HS86]) The syntax of TLC types is given by the grammar T j t j (T T ) where t ranges over a set of type variables. Thus, ff is a type, as are (ff fi) and (ff (ff ff) As usual, the type ff fi fl stands for ff (fi fl) The order of a type is defined as order (t) 0 for a type ....

....of terms P ; Q of type , we have Eq o i o j P Q ae P if i = j, Q if i 6= j. 5 Thus, the term (Eq o i o j P Q) is the equivalent of if o i = o j then P else Q . Just as TLC, TLC = is Church Rosser and strongly normalizing, and principal types can be inferred from unadorned terms. See [Bar84, HS86, KMM91] for a discussion of these concepts. 3.2 Representing Databases and Queries Relations are represented in TLC = as follows. Let O = fo 1 ; o 2 ; g be the set of constants of the TLC = calculus. For convenience, we assume that this set of constants also serves as the universe over which ....

J. R. Hindley and J. P. Seldin. Introduction to Combinators and -Calculus. Cambridge University Press, 1986.

....is a general concept informing different areas of computer science, suchas symbolic computation and term rewriting systems, abstract data type specifi functional programming languages and concurrency. The abstract concept of reduction affects different objects: terms [8] combinators and terms [6], proofs [11] graphs [5] polynomials [1] In [8] the concept of Abstract Reduction System is proposed, that, abstracting from the particular objects to which reduction apply, presents a unifying treatment of reduction, where some important concepts, such as confluence, noetherianity, normal form ....

J. Hindley and J. Seldin. Introduction to combinators and -calculus. Cambridge University Press, 1986.

....1.3.3 Continuous and other models We finish this section by mentioning some continuous models of our theories, without giving their definition. Among them are Scott s D1 , Plotkin s P , Engeler s DA , Barendregt s Bohm tree model, and Scott s information systems; for details and references, cf. [2, 3, 41, 71]. All these models satisfy the axiom of totality (Tot) Another interesting class of partial models are the so called generated models of Feferman [20] we will see a particular example of such a model construction in Chapter 3, where we will discuss polynomial time operations in applicative ....

....an f 2 U so that fm 1 : m n = F (m 1 ; m n ) holds in M for all m 1 ; m n 2 U . The following theorem is a generalization of the Scott Curry undecidability theorem to arbitrary, not necessarily term models. Its proof very closely follows the one given in Hindley and Seldin [41]. Theorem 12 Let M = hU; App; Eq; Nat; i be a number theoretic model of TON so that App is numeralwise definable in M. Assume further that A and B are sets of natural numbers so that (i) A; B ae U ; ii) A 6= B 6= iii) A; B are closed under equality Eq in M. Then A and B are ....

Hindley, J. R., and Seldin, J. P. Introduction to Combinators and - Calculus. Cambridge University Press, 1986.

....cannot be terms. Trivial, because the absence of functionals such as map from TRSs seems to make partial applications pointless. Both thoughts miss an important point: the type problem has to do with the fact that the type system one usually considers for TRSs comes from the simply typed calculus (Hindley and Seldin, 1986) and one could well consider more powerful type systems; the absence of functionals is disturbed by collapsing rules (rules with a variable as right hand side) because they can act as functionals in a curried untyped TRS. We can even observe both aspects in ML: fun head(CONS(x,xs) x The type ....

....of currying. The disjoint union of applicative TRSs is known not to preserve confluence. Applicative TRSs have a binary symbol Apply (shared among all ATRSs) and all other symbols have arity 0. The symbol Apply is notationally suppressed using the conventions of calculus and Combinatory Logic (Hindley and Seldin, 1986). Consider the following ATRS M : M x x 0 M (Succ x) x 1 M is a confluent ATRS, because (as a TRS) it has no critical pairs and it is terminating. If we combine it with the rewrite system of Combinatory Logic (which is confluent in itself) we lose confluence. The ATRS of combinatory logic ....

Hindley, J.R., Seldin, J.P. (1986). Introduction to Combinators and -Calculus. Cambridge University Press.

....as in Hindley [Hin69] rather than polymorphic types as in Damas [DM82] The extension of T to polymorphic types is possible in the same way that Damas extended Milner s work, though we do not pursue this extension. For an enlightening discussion of Curry s philosophy see Hindley and Seldin [HS86] For a discussion of an alternative Church philosophy, where terms are always introduced with their types, see the discussion in Harper and Mitchell on the type structure of standard ML [HM93] For a comparison of the Church and Curry philosophies see Pierce, et al. [PDM89] Our study actually ....

....distinction between this Curry philosophy and a Church philosophy where terms are always introduced with their types. We imagine that we are looking for well typed terms in a larger universe of untyped terms. See the presentations in Hindley and Seldin, Harper and Mitchell, and Pierce et al. [HS86, HM93, PDM89] for discussions and comparisons of the Curry versus Church philosophies and explicit versus implicit typing. In this first section we give rules for types and rules for deriving well typed terms, that is, what we know as the T type inference system. In the next section we present ....

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J. Roger Hindley and Jonathan P. Seldin. Introduction to Combinators and -Calculus. Cambridge University Press, Cambridge, 1986.

....We proof soundness and completeness results in Section 6 and an optimality result in Section 7. Finally, Section 8 discusses criteria to avoid the sometimes operationally complex higher order unification features of LNT. 2 Preliminaries We briefly introduce the simply typed calculus (see e.g. [10]) We assume the following variable conventions: ffl F; G; H;P;X;Y denote free variables, ffl a; b; c; f; g (function) constants, and ffl x; y; z bound variables. 4 Type judgments are written as t : Further, we often use s and t for terms and u; v; w for constants or bound variables. The ....

....are defined as: ffl ff conversion: x:t = ff y: fx 7 ygt) ffl fi conversion: x:s)t = fi fx 7 tgs, and ffl j conversion: if x = 2 FV(t) then x: tx) j t. The long fij normal form [26] of a term t, denoted by tl j fi , is the j expanded form of the fi normal form of t. It is well known [10] that s = fffij t iff sl j fi = ff tl j fi . As long fij normal forms exist for typed terms, we will in general assume that terms are in long fij normal form. For brevity, we may write variables in j normal form, e.g. X instead of x n :X(x n ) We assume that the transformation into long ....

J.R. Hindley and J. P. Seldin. Introduction to Combinators and -Calculus. Cambridge University Press, 1986.

....meaning of our calculus by showing the simulation of fi reduction, disjunction and catch throw. 1 Introduction The Curry Howard isomorphism[6] is one of the foundation in logic and computer science. It says that normalization of proof in intuitionistic logic corresponds to fi reduction of calculus[2, 3]. After the work of Griffin[1] there have been many works to extend the isomorphism for classical logic and to understand computational meaning of classical logic[8 11, 14] Most of these research uses sequent calculus with multiple conclusion. However, in sequent calculus, we need a new concept ....

J. R. Hindley and J. P. Seldin: "Introduction to Combinators and -Calculus", Cambridge University Press, 1986.

....this is very powerful, we show that the incurring higher order unification can sometimes be avoided by techniques similar to [4] Due to lack of space, some details and the proofs are omitted. They can be found in [9] 2 Preliminaries We briefly introduce the simply typed calculus (see e.g. [10]) We assume the following variable conventions: F; G; H;P;X;Y denote free variables, a; b; c; f; g (function) constants, and x; y; z bound variables. Type judgments are written as t : Further, we often use s and t for terms and u; v; w for constants or bound variables. The set of ....

....are defined as: ff conversion: x:t = ff y: fx 7 ygt) fi conversion: x:s)t = fi fx 7 tgs, and j conversion: if x = 2 FV(t) then x: tx) j t. The long fij normal form [14] of a term t, denoted by tl j fi , is the j expanded form of the fi normal form of t. It is well known [10] that s = fffij t iff sl j fi = ff tl j fi . As long fij normal forms exist for typed terms, we will in general assume that terms are in long fij normal form. For brevity, we may write variables in j normal form, e.g. X instead of xn:X(xn ) We assume that the transformation into long ....

J.R. Hindley and J. P. Seldin. Introduction to Combinators and -Calculus. Cambridge University Press, 1986.

....The terms M is constructed as M = x j x:M j MM j ff:M j [ff]M: The calculus has the following basic reduction rules. fi reduction: x:M )N M [x : N ] Structural reduction: ff:M)N ff:M [ ff]w : ff] wN ) Renaming: fi] ff:M) M [ff : fi] We assume some familiarity to calculus [2, 7, 8]. In the structural reduction, the substitution is defined as follows: 2 1. x[ ff]w : ff] wN) x 2. x:M ) ff]w : ff] wN ) x:M [ ff]w : ff] wN) 3. MM ) ff]w : ff] wN) M [ ff]w : ff] wN ) M [ ff]w : ff] wN ) 4. fi:M ) ff]w : ff] wN ) fi:M [ ff]w : ff] wN ) ....

J. R. Hindley, J. P. Seldin: "Introduction to Combinators and -Calculus", Cambridge University Press, 1986.

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J. Roger Hindley and Jonathan P. Seldin. Introduction to Combinators and #-calculus. Cambridge University Press, 1986.

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J. Roger Hindley and Jonathan P. Seldin. Introduction to Combinators and #-Calculus. Cambridge University Press, 1986. 17 Ravara, Matos et al.

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J. Roger Hindley and Jonathan P. Seldin. Introduction to Combinators and #- Calculus. Cambridge University Press, 1986. ISBN 0521268966.

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Hindley, J. R. and Seldin J. P., Introduction to Combinators and -Calculus. Cambridge University Press, 1986.

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J. R. Hindley and J. P. Seldin. Introduction to Combinators and #-Calculus. Cambridge University Press, 1986.

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J. Roger Hindley and Jonathan P. Seldin. Introduction to Combinators and -calculus. Volume 1 of London Mathematical Society Student texts. Cambridge University Press, 1986.

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Hindley, J. R. & J. P. Seldin, (1986). Introduction to Combinators and #-calculus. London Mathematical Society Student Texts, 1. Cambride University Press: Cambridge.

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J. Roger Hindley and Jonathan P. Seldin. Introduction to Combinators and #-Calculus. Cambridge University Press, 1986. 17 Ravara, Matos et al.

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J.R. Hindley, J.P. Seldin, Introduction to Combinators and -Calculus, Cambridge University Press, 1986.

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J.R. Hindley and J. P. Seldin. Introduction to Combinators and -Calculus. Cambridge University Press, 1986.

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J. Roger Hindley and Jonathan P. Seldin. Introduction to Combinators and #- Calculus. Cambridge University Press, 1986. ISBN 0521268966.

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J. Roger Hindley and Jonathan P. Seldin. Introduction to Combinators and #- Calculus. Cambridge University Press, 1986. ISBN 0521268966.

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J.R. Hindley and Jonathan P. Seldin. Introduction to Combinators and -Calculus. Cambridge University Press, 1986.

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J.R. Hindley and Jonathan P. Seldin. Introduction to Combinators and -Calculus.Cambridge University Press, 1986.

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J. Hindley and R. Seldin. Introduction to Combinators and -Calculus. Cambridge Univ. Press, 1986.

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J. Roger Hindley and Jonathan P. Seldin. Introduction to Combinators and -Calculus. Cambridge University Press, 1986.

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J. Roger Hindley and Jonathan P. Seldin. Introduction to Combinators and #-Calculus, page 63. Cambridge University Press, 1986.

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J.R. Hindley and J. P. Seldin. Introduction to Combinators and -Calculus. Cambridge University Press, 1986.

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J. R. Hindley and J. P Seldin. Introduction to Combinators and the -calculus. Cambridge University Press, 1986.

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R. Hindley and J. Seldin. Introduction to Combinators and -calculus. Cambridge University Press, 1986.

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Hindley, J.R., Seldin, J.P., \Introduction to Combinators and -Calculus" Cambridge University Press, 1986.

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J.R. Hindley and J.P. Seldin. Introduction to Combinators and -calculus, volume 1 of London Mathematical Society Student Texts. Cambridge University Press, 1986.

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J.R. Hindley and J.P. Seldin. Introduction to Combinators and -calculus, volume 1 of London Mathematical Society Student Texts. Cambridge University Press, 1986.

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James Hindley and Jonathan Seldin. Introduction to Combinators and #-calculus. Cambridge University Press, 1986.

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J.R.Hindley, J.P.Seldin. Introduction to Combinators and -Calculus. Cambridge University Press, 1986

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Hindley, R. J., and Seldin, J. P. Introduction to Combinators and - Calculus. Cambridge University Press, 1986.

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J.R. Hindley and J.P. Seldin. Introduction to Combinators and -calculus. Cambridge University Press, 1986.

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Hindley, R. J., and Seldin, J. P. Introduction to Combinators and - Calculus. Cambridge University Press, 1986.

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Hindley J.R., Seldin J.P., Introduction to Combinators and -calculus, Cambridge Univ. Press., Cambridge, 1986.

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