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Combinatory Reduction Systems: introduction and survey
 THEORETICAL COMPUTER SCIENCE
, 1993
"... Combinatory Reduction Systems, or CRSs for short, were designed to combine the usual firstorder format of term rewriting with the presence of bound variables as in pure λcalculus and various typed calculi. Bound variables are also present in many other rewrite systems, such as systems with simpl ..."
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Cited by 96 (9 self)
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Combinatory Reduction Systems, or CRSs for short, were designed to combine the usual firstorder format of term rewriting with the presence of bound variables as in pure λcalculus and various typed calculi. Bound variables are also present in many other rewrite systems, such as systems with simplification rules for proof normalization. The original idea of CRSs is due to Aczel, who introduced a restricted class of CRSs and, under the assumption of orthogonality, proved confluence. Orthogonality means that the rules are nonambiguous (no overlap leading to a critical pair) and leftlinear (no global comparison of terms necessary). We introduce the class of orthogonal CRSs, illustrated with many examples, discuss its expressive power, and give an outline of a short proof of confluence. This proof is a direct generalization of Aczel's original proof, which is close to the wellknown confluence proof for λcalculus by Tait and MartinLof. There is a wellknown connection between the para...
Some Lambda Calculi With Categorical Sums and Products
, 1993
"... . We consider the simply typed calculus with primitive recursion operators and types corresponding to categorical products and coproducts.. The standard equations corresponding to extensionality and to surjectivity of pairing and its dual are oriented as expansion rules. Strong normalization an ..."
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Cited by 22 (1 self)
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. We consider the simply typed calculus with primitive recursion operators and types corresponding to categorical products and coproducts.. The standard equations corresponding to extensionality and to surjectivity of pairing and its dual are oriented as expansion rules. Strong normalization and ground (basetype) confluence is proved for the full calculus; full confluence is proved for the calculus omitting the rule for strong sums. In the latter case, fixedpoint constructors may be added while retaining confluence. 1 Introduction The systems investigated here are simply typed caluli whose types include pairs, unit, sums, an empty type, and a type of natural numbers supporting constructions by primitive recursion. In the core system the types behave as categorical product and coproducts, so the subject at hand is equivalently ([LS86]) the equational theory of the free bicartesian closed category (generated by objects for the base types) with weak natural numbers object. Su...
MLISP: A RepresentationIndependent Dialect of LISP with Reduction Semantics
 ACM Transactions on Programming Languages and Systems
, 1992
"... In this paper we introduce MLISP, a simple new dialect of LISP which is designed with an eye toward reconciling LISP's metalinguistic power with the structural style of operational semantics advocated by Plotkin [Plo75]. We begin by reviewing the original denition of LISP [McC61] in an atte ..."
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Cited by 21 (2 self)
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In this paper we introduce MLISP, a simple new dialect of LISP which is designed with an eye toward reconciling LISP's metalinguistic power with the structural style of operational semantics advocated by Plotkin [Plo75]. We begin by reviewing the original denition of LISP [McC61] in an attempt to clarify the source of its metalinguistic power. We nd that it arises from a problematic clause in this denition. We then dene the abstract syntax and operational semantics of MLISP, essentially a hybrid of Mexpression LISP and Scheme. Next, we tie the operational semantics to the corresponding equational logic. As usual, provable equality in the logic implies operational equality. Having established this framework we then extend MLISP with the metalinguistic eval and reify operators (the latter is a nonstrict operator which converts its argument to its metalanguage representation.) These operators encapsulate the metalinguistic representation conversions that occur globall...
Operational and Axiomatic Semantics of PCF
 In Proceedings of the LISP and Functional Programming Conference
, 1990
"... PCF, as considered in this paper, is a lazy typed lambda calculus with functions, pairing, fixedpoint operators and arbitrary algebraic data types. The natural equational axioms for PCF include jequivalence and the socalled "surjective pairing" axiom for pairs. However, the reduction sy ..."
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Cited by 4 (2 self)
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PCF, as considered in this paper, is a lazy typed lambda calculus with functions, pairing, fixedpoint operators and arbitrary algebraic data types. The natural equational axioms for PCF include jequivalence and the socalled "surjective pairing" axiom for pairs. However, the reduction system pcf j;sp defined by directing each equational axiom is not confluent, for virtually any choice of algebraic data types. Moreover, neither jreduction nor surjective pairing seems to have a counterpart in ordinary execution. Therefore, we consider a smaller reduction system pcf without j reduction or surjective pairing. The system pcf is confluent when combined with any linear, confluent algebraic rewrite rules. The system is also computationally adequate, in the sense that whenever a closed term of "observable" type has a pcf j;sp normal form, this is also the unique pcf normal form. Moreover, the equational axioms for PCF, including (j) and surjective pairing, are sound for pcf observational e...
Unique normal forms and confluence of rewrite systems: Persistence
 In Proc. 14th IJCAI
, 1995
"... Programming language interpreters, proving theorems of the form A = 2?, abstract data types, and program optimization can all be represented by a finite set of rules called a rewrite system. In this paper, we study two fundamental concepts, uniqueness of normal forms and confluence, for nonlinear sy ..."
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Cited by 3 (1 self)
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Programming language interpreters, proving theorems of the form A = 2?, abstract data types, and program optimization can all be represented by a finite set of rules called a rewrite system. In this paper, we study two fundamental concepts, uniqueness of normal forms and confluence, for nonlinear systems in the absence of termination. This is a difficult topic with only a few results so far. Through a novel approach, we show that every persistent system has unique normal forms. This result is tight and a substantial generalization of previous work. In the process we derive a necessary and sufficient condition for persistence for the first time and give new classes of persistent systems. We also prove the confluence of the union (function symbols can be shared) of a nonlinear system with a leftlinear system under fairly general conditions. Again persistence plays a key role in this proof. We are not aware of any confluence result that allows the same level of function symbol sharing. 1
Conditional Linearization
"... A nonleftlinear term rewriting system lacking the ChurchRosser property can sometimes be shown to satisfy the unique normal form property by shifting attention to an associated conditional term rewriting system that is leftlinear. We call this the method of conditional linearization. In the presen ..."
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Cited by 2 (0 self)
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A nonleftlinear term rewriting system lacking the ChurchRosser property can sometimes be shown to satisfy the unique normal form property by shifting attention to an associated conditional term rewriting system that is leftlinear. We call this the method of conditional linearization. In the present paper the method is described in a general setting and some applications are discussed. In particular we present a simple proof of the unique normal form property for Combinatory Logic extended with 'Parallel Conditional', that is, with constants C, T and F (conditional, true, false) and extra reduction rules CTxy x, CFxy y and Czxx x. A special feature of this application is that it involves the use of negative conditions. Contents Introduction 1. Four nonleftlinar, nonconfluent TRSs 2. Conditional Term Rewriting Systems 3. Application of CTRSs to prove uniqueness of normal forms 4. The case of Combinatory Logic plus Parallel Conditional 5. Chew's theorem 6. Remarks and further ques...
MLISP: Its Natural Semantics and Equational Logic (Extended Abtract)
, 1991
"... The LISP evaluator is a virtual machine analog of the storedprogram computer on which it executes  it has universal power and dynamically constructed representations of programs can be converted by the eval operator into executable programs. In this paper we study the natural operational semantic ..."
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The LISP evaluator is a virtual machine analog of the storedprogram computer on which it executes  it has universal power and dynamically constructed representations of programs can be converted by the eval operator into executable programs. In this paper we study the natural operational semantics and equational logic of a dialect of pure LISP and an extension which includes the eval operator and fexprs (i.e., nonstrict functions whose arguments are passed byrepresentation). We begin by defining a natural operational semantics for the pure subset of MLISP, a simple, representationindependent hybrid of McCarthy's original Mexpression LISP and Scheme. We then establish the connection between the semantics and its equational logic in the usual way and prove that the logic is sound and consistent. With this as our setting we define the axioms and inference ...
Abstract Operational and Axiomatic Semantics of PCF
"... PCF, as considered in this paper, is a lazy typed lambda calculus with functions, pairing, fixedpoint operators and arbitrary algebraic data types. The natural equational axioms for PCF include ηequivalence and the socalled “surjective pairing ” axiom for pairs. However, the reduction system pcf ..."
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PCF, as considered in this paper, is a lazy typed lambda calculus with functions, pairing, fixedpoint operators and arbitrary algebraic data types. The natural equational axioms for PCF include ηequivalence and the socalled “surjective pairing ” axiom for pairs. However, the reduction system pcf η,sp defined by directing each equational axiom is not confluent, for virtually any choice of algebraic data types. Moreover, neither ηreduction nor surjective pairing seems to have a counterpart in ordinary execution. Therefore, we consider a smaller reduction system pcf without ηreduction or surjective pairing. The system pcf is confluent when combined with any linear, confluent algebraic rewrite rules. The system is also computationally adequate, in the sense that whenever a closed term of “observable ” type has a pcf η,sp normal form, this is also the unique pcf normal form. Moreover, the equational axioms for PCF, including (η) and surjective pairing, are sound for pcf observational equivalence. These results suggest that if we take the equational axioms as defining the language, the smaller reduction system gives an appropriate operational semantics. 1