H. Barendregt. The type free lambda calculus. In Barwise [6], chapter D.7, pages 1091-1132.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is a variable) As a consequence, the result of the diverging term (#x.xx)#x.xx is an infinite sequence of repetition rules. arises by a coinductive interpretation of the grammar of the wellfounded # calculus [Joa01] and provides a sound framework to model non wellfounded # terms and Bohm trees [Bar77]. To dispense with the intricate mechanism of bound variable renaming in semiformal term systems, it is based on a deBruijn style management of bound variables [Bru72] Apart from all terms of the usual # calculus, # harbors interesting objects like the directly defined (and even well typed) ....

Henk Barendregt. The type free lambda calculus. In Jon Barwise, editor, Handbook of Mathematical Logik, chapter D.7, pages 1091--1132. NorthHolland Publishing Company, 1977.

.... B ohm s Theorem [4] states that two di erent normal terms r; s can be separated by a context C, i.e. Cfrg x 6= y Cfsg, so that the addition of the equality r = s to the axioms of becomes inconsistent (all terms become provably equal) The standard references for B ohm s theorem [1, 2] derive it as an instance of observations on B ohm trees, while B ohm s original proof as well as [3] do not appeal to such in nitary objects. In this short note we give an exposition of a short and nitary proof of the theorem (without claiming originality) Notation. We use r; s; t for terms of ....

Henk Barendregt. The type free lambda calculus. In Jon Barwise, editor, Handbook of Mathematical Logik, chapter D.7, pages 1091-1132. NorthHolland Publishing Company, 1977.

....following reference books as background reading. Note of course, that only a few of the basic concepts described in these books will be needed. Lecture notes on functional programming by Paulson [Pau00] Several reference articles by Barendregt on lambda calculi and functional programming [Bar77,Bar90,Bar92] We use both dependent type theory and Standard ML (SML) as implementation languages, and assume that the reader is familiar with these languages. Knowl6 edge of other typed functional languages such as OCAML or Haskell is of course also useful. Tutorials on Standard ML can be ....

Henk Barendregt. The type free lambda calculus. In Handbook of Mathematical Logic, pages 1092-1132. North-Holland, 1977.

....is a variable) As a consequence, the result of the diverging term ( x:xx) x:xx is an in nite sequence of repetition rules. arises by a coinductive interpretation of the grammar of the wellfounded calculus [Joa01] and provides a sound framework to model non wellfounded terms and B ohm trees [Bar77]. To dispense with the intricate mechanism of bound variable renaming in semiformal term systems, it is based on a deBruijn style management of bound variables [Bru72] Apart from all terms of the usual calculus, harbors interesting objects like the directly de ned (and even well typed) xed ....

Henk Barendregt. The type free lambda calculus. In Jon Barwise, editor, Handbook of Mathematical Logik, chapter D.7, pages 1091-1132. NorthHolland Publishing Company, 1977.

....first order theory of equality constraints over trees [18, 19, 12] since its axioms also hold over non empty sets of trees (but don t over possibly empty sets) There exists a natural interpretation of INES constraint over tree like structures that we call tree prefixes. In a different context [6] tree prefixes are called Bohm trees (without binders) Tree prefixes come with a natural ordering relation where the empty tree prefix is the greatest element. We prove that an INES constraint is satisfiable over non empty sets of trees if and only if it is satisfiable over tree prefixes (where ....

....structures P (Tree) and Prefix. A formula that holds over Prefix but not over (Tree) is given by 8x(a x b x 8y(y x) where a 6= b. Another formula distinguishing both structures comes with a constraint based reformulation of the coherence property (defined for complete partial orders in [6]) We say that an ordering relation satisfies the coherence property if it satisfies the following formulae for all finite sets I (where inclusion symbol is interpreted as the given ordering) i;j2I 9z(z x i z x j ) 9z( i2I z x i ) This formula states that for all variable assignment ....

H. P. Barendregt. The Type Free Lambda Calculus. In Barwise [7], 1977.

....al. 1989] are, as models of computation with real numbers, naturally closer to numerical computations. However, in the theory of computation there exist models of computation which are given as algebraic structures, e.g. combinatory algebras. A popular model of combinatory algebras is calculus [Barendregt, 1977]. In this work we consider other models of combinatory algebras, namely graph models [Engeler, 1981A, Engeler, 1981B] It was shown that any algebraic structure can be embedded in a graph model [Engeler, 1988] Hence graph models give rise to an algebraic model of computation in algebraic ....

Barendregt, H.(1977). The type free lambda calculus. Handbook of Mathematical Logic, ed. Jon Barwise, North Holland.

....( 2 ) cannot have been applied and ( 1 ) is only used in the form of ( Corollary. r s 2 NF = #[ r] s. 8 3.5. B ohm trees. A closer inspection reveals that lazy evaluation is also sucient to successively develop non normalizing terms. The coinductive notion of B ohm trees [Bar77] is usually invoked to model iterative headnormalization: the B ohm tree of r is either unde ned (if r has no head normal form) or x(x ) if are the B ohm trees of r where x(x r ) is a head normal form of r. We will show below that (the de ned parts of) B ohm trees can be computed ....

....Abs (Abs (Var 1) Main nbe i Abs (Var 0) Main nbe (k App i App oomega) Abs (Var 0) 6.5. B ohm Trees. Theorem 3.5 states that our approach can be used to calculate B ohm trees as well. The xpoint combinator Y : f ( x (f(xx) x (f(xx) 17 has the in nite B ohm tree f(f(f( Bar77] The de nition of Y reads y = Abs ( Abs (Var 1 App (Var 0 App Var 0) App (Abs (Var 1 App (Var 0 App Var 0) Since Haskell prints the calculated part of a result as it proceeds, we can have a look at the beginning of the B ohm tree of Y . Main nbe y Abs (App (Var 0) App (Var 0) ....

Henk Barendregt. The type free lambda calculus. In Jon Barwise, editor, Handbook of Mathematical Logic, chapter D.7, pages 1091{ 1132. North-Holland, 1977.

.... process can modify its environment via substitutions of a placeholder (represented by a variable) with something else (a process) We show here that designing systems in a multiset style and equipping them with a simple interaction mechanism is enough to express the computations of the calculus [2]. In fact, the request for a computation is represented in the calculus by the application of a function to its argument and the actual computation is basically carried out by the substitution mechanism. The cited works show that substitution plays a central role in computation based on linear ....

Henk P. Barendregt. The type free lambda calculus. In J. Barwise, editor, Handbook of Mathematical Logic, volume 90 of Studies in Logic and the Foundations of Mathematics, chapter D.7, pages 1091-1132. North-Holland, Amsterdam, 1977.

.... process can modify its environment via substitutions of a placeholder (represented by a variable) with something else (a process) We show here that designing systems in a multiset style and equipping them with a simple interaction mechanism is enough to express the computations of the calculus [2], of the fusion calculus [11] and hence [9] fl [12] and ae [10] calculi) and of the proof nets [7] 2 The basic calculus In this section we introduce our calculus. We first define its syntax and then its operational semantics. Finally, we discuss confluence properties of the calculus. 2.1 ....

Henk P. Barendregt. The type free lambda calculus. In J. Barwise, editor, Handbook of Mathematical Logic, volume 90 of Studies in Logic and the Foundations of Mathematics, chapter D.7, pages 1091--1132. NorthHolland, Amsterdam, 1977.

....untyped calculus) is the equality of the iin nite j contractionj of BTs. BTs naturally give rise to a tree topology that has been used for the proof of some seminal results of the calculus like Berry s sequentiality theorem (brieAEy discussed in Section 2. 2) BTs were introduced by Barendregt [Bar77] and called so after B#hm s proof and theorem about separability of terms. The proof technique for this theorem, called the B#hm out technique, roughly consists in nding a context capable of isolating a given subtree of a BT; in this way, certain terms that have dioeerent BTs may be ....

H. Barendregt. The type free lambda calculus. In J. Barwise, editor, Handbook of Mathematical Logic, Studies in Logic 90, pages 1092 1132. North Holland, 1977.

....were, and then shown that it is sufficient to describe existing mathematics. For both the goal is to describe mathematics whatever the 30 semantics ascribed to this description might be. 2.3. 3 The calculus The calculus [12] and combinatory logic [16] were developed as devices to study rules [2], 3]a, where the rules are represented as functions. Looking at the functional definition as embodied in the terms led to many results from the perspective of what is computable via these functional definitions what terminates and what can be represented. Again, as in the logics above, the ....

....at the functional definition as embodied in the terms led to many results from the perspective of what is computable via these functional definitions what terminates and what can be represented. Again, as in the logics above, the system of terms was intended as a foundation for mathematics [2]. This goal was not entirely realized, though many more limited goals were. In the context of the calculus, many properties of functions have been examined, such as the correspondence of terms with functions (in particular the partial recursive functions) the normal form properties of the ....

Barendregt, H. P. The type free lambda calculus. In Mathematical Logic, J. Barwise, Ed. North Holland, 1977, pp. 1091--1132.

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H. Barendregt. The type free lambda calculus. In Barwise [6], chapter D.7, pages 1091-1132.

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