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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 83 (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...
The Conservation Theorem revisited
, 1993
"... This paper describes a method of proving strong normalization based on an extension of the conservation theorem. We introduce a structural notion of reduction that we call fi S , and we prove that any term that has a fi I fi Snormal form is strongly finormalizable. We show how to use this result ..."
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Cited by 32 (0 self)
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This paper describes a method of proving strong normalization based on an extension of the conservation theorem. We introduce a structural notion of reduction that we call fi S , and we prove that any term that has a fi I fi Snormal form is strongly finormalizable. We show how to use this result to prove the strong normalization of different typed calculi.
Prelogical Relations
, 1999
"... this paper but which have some intriguing connections to some of our results and techniques, are [32] and [20]. We believe that the concept of prelogical relation would have a beneficial impact on the presentation and understanding of their results ..."
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Cited by 26 (5 self)
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this paper but which have some intriguing connections to some of our results and techniques, are [32] and [20]. We believe that the concept of prelogical relation would have a beneficial impact on the presentation and understanding of their results
A Relevant Analysis of Natural Deduction
 Journal of Logic and Computation
, 1999
"... Linear and other relevant logics have been studied widely in mathematical, philosophical and computational logic. We describe a logical framework, RLF, for defining natural deduction presentations of such logics. RLF consists in a language together, in a manner similar to that of Harper, Honsell and ..."
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Cited by 23 (7 self)
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Linear and other relevant logics have been studied widely in mathematical, philosophical and computational logic. We describe a logical framework, RLF, for defining natural deduction presentations of such logics. RLF consists in a language together, in a manner similar to that of Harper, Honsell and Plotkin's LF, with a representation mechanism: the language of RLF is the lLcalculus; the representation mechanism is judgementsastypes, developed for relevant logics. The lLcalculus type theory is a firstorder dependent type theory with two kinds of dependent function spaces: a linear one and an intuitionistic one. We study a natural deduction presentation of the type theory and establish the required prooftheoretic metatheory. The RLF framework is a conservative extension of LF. We show that RLF uniformly encodes (fragments of) intuitionistic linear logic, Curry's l I calculus and ML with references. We describe the CurryHowardde Bruijn correspondence of the lLcalculus with a s...
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 20 (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...
HigherOrder Rewriting
 12th Int. Conf. on Rewriting Techniques and Applications, LNCS 2051
, 1999
"... This paper will appear in the proceedings of the 10th international conference on rewriting techniques and applications (RTA'99). c flSpringer Verlag. ..."
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Cited by 20 (1 self)
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This paper will appear in the proceedings of the 10th international conference on rewriting techniques and applications (RTA'99). c flSpringer Verlag.
The Longest Perpetual Reductions in Orthogonal Expression Reduction Systems
 In: Proc. of the 3 rd International Conference on Logical Foundations of Computer Science, LFCS'94, A. Nerode and Yu.V. Matiyasevich, eds., Springer LNCS
, 1994
"... We consider reductions in Orthogonal Expression Reduction Systems (OERS), that is, Orthogonal Term Rewriting Systems with bound variables and substitutions, as in the calculus. We design a strategy that for any given term t constructs a longest reduction starting from t if t is strongly normaliza ..."
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Cited by 18 (8 self)
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We consider reductions in Orthogonal Expression Reduction Systems (OERS), that is, Orthogonal Term Rewriting Systems with bound variables and substitutions, as in the calculus. We design a strategy that for any given term t constructs a longest reduction starting from t if t is strongly normalizable, and constructs an infinite reduction otherwise. The Conservation Theorem for OERSs follows easily from the properties of the strategy. We develop a method for computing the length of a longest reduction starting from a strongly normalizable term. We study properties of pure substitutions and several kinds of similarity of redexes. We apply these results to construct an algorithm for computing lengths of longest reductions in strongly persistent OERSs that does not require actual transformation of the input term. As a corollary, we have an algorithm for computing lengths of longest developments in OERSs. 1 Introduction A strategy is perpetual if, given a term t, it constructs an infinit...
Addendum to `New notions of reduction and nonsemantic proofs of βstrong normalization in typed λcalculi
, 1995
"... ..."
Ensuring termination by typability
 In Proceedings of IFIP TCS 2004
, 2004
"... Abstract. A term terminates if all its reduction sequences are of finite length. We show four type systems that ensure termination of welltyped sscalculus processes. The systems are obtained by successive refinements of the types of the simply typed sscalculus. For all (but one of) the type syste ..."
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Cited by 14 (3 self)
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Abstract. A term terminates if all its reduction sequences are of finite length. We show four type systems that ensure termination of welltyped sscalculus processes. The systems are obtained by successive refinements of the types of the simply typed sscalculus. For all (but one of) the type systems we also present upper bounds to the number of steps welltyped processes take to terminate. The termination proofs use techniques from term rewriting systems. We show the usefulness of the type systems on some nontrivial examples: the encodings of primitive recursive functions, the protocol for encoding separate choice in terms of parallel composition, a symbol table implemented as a dynamic chain of cells. 1 Introduction A term terminates if all its reduction sequences are of finite length. As far as programminglanguages are concerned, termination means that computation in programs will eventually stop. In computer science termination has been extensively investigated in term rewritingsystems [7, 5] and *calculi [9, 4] (where strong normalization is a synonym more commonlyused). Termination has also been discussed in process calculi, notably the
Termination Proofs for Higherorder Rewrite Systems
 IN 1ST INTERNATIONAL WORKSHOP ON HIGHERORDER ALGEBRA, LOGIC AND TERM REWRITING
, 1994
"... This paper deals with termination proofs for HigherOrder Rewrite Systems (HRSs), introduced in [12]. This formalism combines the computational aspects of term rewriting and simply typed lambda calculus. The result is a proof technique for the termination of a HRS, similar to the proof technique &qu ..."
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Cited by 13 (0 self)
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This paper deals with termination proofs for HigherOrder Rewrite Systems (HRSs), introduced in [12]. This formalism combines the computational aspects of term rewriting and simply typed lambda calculus. The result is a proof technique for the termination of a HRS, similar to the proof technique "Termination by interpretation in a wellfounded monotone algebra", described in [8, 19]. The resulting technique is as follows: Choose a higherorder algebra with operations for each function symbol in the HRS, equipped with some wellfounded partial ordering. The operations must be strictly monotonic in this ordering. This choice generates a model for the HRS. If the choice can be made in such a way that for each rule the interpretation of the left hand side is greater than the interpretation of the right hand side, then the HRS is terminating. At the end of the paper some applications of this technique are given, which show that this technique is natural and can easily be applied.