Zena M. Ariola and Jan Willem Klop. Lambda calculus with explicit recursion. Inf. & Comput., 139:154--233, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....3.3, 6.1, and 3.4 hold. 2. WB Std( 3. NM( 4. If t 1 t 2 , then t [t 1 ] t [t 2 ] for any context t . Proof. Many standard proofs by induction which are left to the reader. The only di#cult bit is BE( part of WB Std( First, Trm( is proven by a known argument (e.g. see [5]) of rearranging the mark values so that rewriting decreases the multiset of all marks in the term in the multiset extension of . Because the rewriting system is finitely branching, this is equivalent to Bnd( which in turn implies BE( Theorem 7.2 (Meaning Preservation) MP. Proof. ....

....of the work of Ariola and Blom [2] The only previously known methods for reasoning about the correctness of Ariola Blom style letrec rules seem more di#cult to us. The development of # is nearing completion. Because # is nonconfluent (due to using rules for letrec that Ariola and Klop [5] proved nonconfluent) we were using the lift project method to prove meaning preservation. It does not have finite developments, but has a number of rule subsets whose associated rewrite step sets satisfy the BE property. The last barrier to completing the proof of meaning preservation was ....

Z. M. Ariola, J. W. Klop. Lambda calculus with explicit recursion. Inform. & Comput., 139, 1997.

....term graphs is to use the letrec syntax. By using this syntax, we can derive term graph rewrite systems from term rewrite systems. For example, from the # rule in the lambda calculus M [x : N ] we can derive letrec x = N in M ; letrec D in M)N lift letrec D in (MN) 1. 1) See [AK97] for details on exactly how this rewrite system may be derived. We will refer to these derived systems as cyclic extensions. The Bohm tree and the Levy Longo tree are nice notions of semantics for the lambda calculus. It is easy to extend the theory of these trees to the simple cyclic extension ....

Z. M. Ariola and J. W. Klop. Lambda calculus with explicit recursion. Information and computation, 139(2):154--233, December 1997.

....We believe that in this framework one can have a fresh look at typing issues and primitives for code analysis, but above all it should describe how to add staging to your favorite programming language. We are mostly interested in adding staging (and computational e#ects) to pure calculi (like [AK97, AZ02, WV00] for describing mutual dependencies and sharing, as provided by mixins. A very important principle of Haskell [PHA 97] is that pure functional evaluation (and all the optimization techniques that come with it) should not be corrupted by the addition of computational e#ects. In ....

Zena M. Ariola and Jan Willem Klop. Lambda calculus with explicit recursion. Inform. & Comput., 139:154--233, 1997.

....term graph rewriting and event structures [62, 17] and the term generating power of context free jungle grammars [35] The area of graph reduction for the lambda calculus is related to term graph rewriting, but is beyond the scope of this survey. For information about this topic, we refer to [8, 5] and the references given there. 54 ....

Zena M. Ariola and Jan Willem Klop. Lambda calculus with explicit recursion. Information and Computation, 139:154--233, 1997.

....optimal reductions e.g. 4 A. K. Moran and D. Sands [Field 1990; Maranget 1991; Yoshida 1993] One limitation of the original work by Ariola et al. is in the treatment of recursive cycles ; na ve extension of the calculi to deal with recursive lets leads to a loss of con uence [Je rey 1993; Ariola and Klop 1997]. The original call by need calculus considers recursive lets only brie y. To recover con uence, one can simply disallow reductions under cycles, as in e.g. Benaissa et al. 1996; Niehren 1996] Ariola and Blom give a full study of cyclic recursion in [Ariola and Blom 1997; Ariola and Blom 1998] ....

....con uence [Je rey 1993; Ariola and Klop 1997] The original call by need calculus considers recursive lets only brie y. To recover con uence, one can simply disallow reductions under cycles, as in e.g. Benaissa et al. 1996; Niehren 1996] Ariola and Blom give a full study of cyclic recursion in [Ariola and Blom 1997; Ariola and Blom 1998] and show that an approximation to con uence can be obtained by equating terms with the same in nite normal form. Their share calculus can be seen as the natural successor to the call by need calculi. In general, reduction calculi appear to be a good vehicle for exploring ....

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Ariola, Z. M. and Klop, J. W. 1997. Lambda calculus with explicit recursion. Information and Computation 139, 2, 154-233.

....term graph rewriting and event structures [63,17] and the term generating power of context free jungle grammars [35] The area of graph reduction for the lambda calculus is related to term graph rewriting, but is beyond the scope of this survey. For information about this topic, we refer to [8,5] and the references given there. ....

Zena M. Ariola and Jan Willem Klop. Lambda calculus with explicit recursion. Information and Computation, 139:154--233, 1997.

....1. The computational rewrite rules. 8 the desired equations between programs. To avoid these difficulties, unlike the calculus of Ancona and Zucca [4] the m calculus substitutes for one target at a time (via the (subst) and (subst letrec) rules) in a style pioneered by Ariola, Blom, and Klop [7, 5, 6]. The m calculus letrec contruct is, in a sense, a delayed substitution that allows avoiding duplication when a component is selected from a module. The (subst) rule in Figure 1 uses the notion of one component of a collection depending on another to exclude certain rewriting possibilities. ....

....Felleisen present a calculus with an operational semantics for mixins and classes in the context of Java [21] 5.3 Calculi for Cycles. Inspiring much of our formulation, Ariola and Klop did ground breaking work on reasoning about terms combined with a construct for mutually recursive definitions [7]. Ariola and Blom refined this work to prove consistency in the absence of confluence [5, 6] 5.4 ML style Modules vs. Types. Crary, Harper, and Puri describe how to extend the ML module system to deal with recursion [14] Earlier work to add first class modules (i.e. higher order functors) to ....

Z. M. Ariola and J. W. Klop. Lambda calculus with explicit recursion. Inf. & Comput., 139:154--233, 1997. 14

....The second case follows from the definition of marked contraction as unmarked contraction with superimposed marks. 2 20 4. 3 Strong Normalization of the Marked m Calculus The technique used here to prove the strong normalization of the marked m calculus is very similar to a technique used in [AK97]. Definition 10 (Weight) The weight of a marked raw term is the multiset of all of the redex marks occurring in it. Definition 11 Given two marked raw terms, M and N , M is more marked than N , written M N , iff b Mc = b Nc and for each mark occurring in N there is a (possibly ....

....with an operational semantics for mixins and classes in the context of Java [FKF98] 5. 3 Calculi for Cycles Inspiring a large part of our formulation, Ariola and Klop did ground breaking work on reasoning about terms combined with a cycle forming construct for mutually recursive definitions [AK97]. Ariola and Blom refined this work to deal with proving consistency in the absence of confluence [AB97a, AB97b] 5.4 ML style Modules vs. Types Much of the work on ML style modules has focused on typing issues. For ML style modules, relevant work includes that of Crary, Harper, and Puri in ....

Zena M. Ariola and Jan Willem Klop. Lambda calculus with explicit recursion. Inf. & Comput., 139:154--233, 1997.

....Strategies have been well studied in this framework, in particular for optimal reduction. Related to this work is also the encoding of interaction nets as combinatory reduction systems given in [8] The notations that we introduce in the present work are inspired by formalisms for cyclic rewriting [2], and proof expressions for linear logic [1] Overview. The rest of this paper is structured as follows: In the next section we recall some basic preliminaries on interaction nets and present several textual languages for interaction nets. Section 3 gives a thorough study of the calculus and ....

....which is well known in graph reduction for functional languages. Two examples of this are the following configurations: hy j x = ffi(x; y)i, where ffi is the duplicator agent, and h j x = xi, which both can be thought of as representing the cyclic term ff = ff in cyclic term rewriting [2]. Both of these configurations are irreducible. The latter is a net without an interface, and the first is the same thing with an interface. Hence our interaction net configurations allow us to distinguish these two cases. Proposition 4. The rules Indirection and Collect are terminating. Proof. ....

Z. M. Ariola and J. W. Klop. Lambda calculus with explicit recursion. Information and Computation, 139(2):154--233, 1997.

....constructor. CN k (e 1 , Delta Delta Delta, e k ) Gamma f t 1 = e 1 ; Delta Delta Delta ; t k = e k in CN k (t 1 , Delta Delta Delta, t k ) g The ffi and conditional rules from the calculus carry over unmodified to let . 4. 4 (Non )Confluence The following theorem by Ariola and Klop [6] states that the introduction of let blocks has destroyed confluence: Proposition 1 (Ariola and Klop) let is not confluent. To see this, consider the following program: Term0: f odd = n. cond(n = 0, False, even(n 1) even = n. cond(n = 0, True, odd(n 1) in . g Suppose we substitute odd ....

....and thus should be equal but are syntactically very different. It is common to introduce one or more garbage collection rules in the calculus to attempt to remove the irrelevant clutter (see, for example, 3] But even this is inadequate: as the (non )confluence results of Ariola and Klop ([6]) show, there are pairs of terms which ought to be considered equal which can never reduce to compatible syntactic forms. We therefore introduce the concept of a printable value . This is a rudimentary notion of value for example, in let al..l abstractions are represented by the single symbol ....

Zena M. Ariola and J. W. Klop. Lambda calculus with explicit recursion. Technical Report CIS-TR-96-04, Dept. of Computer and Information Sciences, Univ. of Oregon, Eugene OR, USA, 1996.

....and execution of programs. Presentation of such a theory is the goal of this paper. What makes a theory of cycles difficult to develop is that confluence is lost once lambda abstraction and cycles are admitted, unless the theory is powerful enough to represent irregular structures as shown in [6, 7]. To regain confluence, current formulations of cycles either impose restrictions, such as disallowing reduction under a lambda abstraction or on a cycle [6, 7, 15, 31, 35] or adopt a framework based on interaction nets [24] As discussed in [28] and [12] cycles do not destroy confluence in the ....

.... lost once lambda abstraction and cycles are admitted, unless the theory is powerful enough to represent irregular structures as shown in [6, 7] To regain confluence, current formulations of cycles either impose restrictions, such as disallowing reduction under a lambda abstraction or on a cycle [6, 7, 15, 31, 35], or adopt a framework based on interaction nets [24] As discussed in [28] and [12] cycles do not destroy confluence in the context of interaction nets, but only at the expense of greater complexity. In this paper, we limit our attention to cyclic lambda graphs that occur in current ....

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Z. M. Ariola and J. W. Klop. Lambda calculus with explicit recursion. Technical Report CIS-TR-96-04, Department of computer and information science, University of Oregon. To appear in Information and computation.

....necessarily a fi let redex. Suppose M j (x:x) x:x) N j let z be (let w be x:x in w) in zz; then M N , yet N does not contain a fi let redex. This is so because the language of dags captures only the sharing in a term but not its let structure. We thus enrich dags with boxes and labeled edges (Ariola Klop, 1996). Dag2 (M ) is the decorated dag associated with a term M . A box can be thought of as a refined version of a node; the label associated with an edge is just a sequence of let bound variable names. The label can be thought of as a direction to be followed in order to get to a particular node. Each ....

....language no longer reflects the sharing in the evaluator. For example, the term M j Y(y:cons(1; y) evaluates to a term containing two distinct cons cells even though a lazy evaluator will only allocate one cell and will represent M as a cyclic structure. To cope with recursion, we transform Ariola and Klop s (1994; 1996) call byname calculus with cycles to a call by need calculus with cycles. The first step is to add a letrec construct of the form letrec x 1 be N 1 ; Delta Delta Delta ; xn be Nn in M to the syntax. No ordering among the bindings is assumed. The set of evaluation contexts is the natural ....

Ariola, Z. M., & Klop, J. W. (1996). Lambda calculus with explicit recursion. Tech. rept.

....j y = 0; 2] 0i The flattening is necessary to allow reductions of the form: hy j y = hread 2 j di; vi hy j y = v; 2] d; vi At this point, the only thing that we require is that the calculus be consistent. Unfortunately, this calculus turns out to be non confluent [5]. The culprit is the substitution inside the bindings of a box, which is essential in a call byneed setting. One solution is to restrict this rule to evaluation contexts [3] That is, we could replace the third lambda graph rule with the rule: hE[x1 ] j x1 = E[x2 ] xn = E[x] x = v; di ....

Ariola, Z. M., and Klop, J. W. Lambda calculus with explicit recursion. Tech. Rep. CIS-TR-96-04, Dept. of Computer and Information Science, University of Oregon, 1996. To appear in Inf. Comp..

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Zena M. Ariola and Jan Willem Klop. Lambda calculus with explicit recursion. Inf. & Comput., 139:154--233, 1997.

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Z. M. Ariola and J. W. Klop. Lambda calculus with explicit recursion. Inf. & Comput., 139(2):154--233, 15 Dec. 1997.

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Z. M. Ariola, J. W. Klop. Lambda calculus with explicit recursion. Inform. & Comput., 139, 1997.

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Z. M. Ariola and J. W. Klop. Lambda calculus with explicit recursion. Inform. & Comput., 139(2):154--233, 15 Dec. 1997.

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Z. M. Ariola, J. W. Klop. Lambda calculus with explicit recursion. Inform. & Comput., 139, 1997.

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Ariola, Z. M. and Klop, J. W. (1995). Lambda calculus with explicit recursion. Personal Communication.

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Ariola, Z. M. and Klop, J. W. (1995). Lambda calculus with explicit recursion. Personal Communication.

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Ariola, Z. M. and Klop, J. W. (1995). Lambda calculus with explicit recursion. Personal Communication.

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