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24
Primitive Recursion for HigherOrder Abstract Syntax
 Theoretical Computer Science
, 1997
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Automating the Meta Theory of Deductive Systems
, 2000
"... not be interpreted as representing the o cial policies, either expressed or implied, of NSF or the U.S. Government. This thesis describes the design of a metalogical framework that supports the representation and veri cation of deductive systems, its implementation as an automated theorem prover, a ..."
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Cited by 80 (17 self)
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not be interpreted as representing the o cial policies, either expressed or implied, of NSF or the U.S. Government. This thesis describes the design of a metalogical framework that supports the representation and veri cation of deductive systems, its implementation as an automated theorem prover, and experimental results related to the areas of programming languages, type theory, and logics. Design: The metalogical framework extends the logical framework LF [HHP93] by a metalogic M + 2. This design is novel and unique since it allows higherorder encodings of deductive systems and induction principles to coexist. On the one hand, higherorder representation techniques lead to concise and direct encodings of programming languages and logic calculi. Inductive de nitions on the other hand allow the formalization of properties about deductive systems, such as the proof that an operational semantics preserves types or the proof that a logic is is a proof calculus whose proof terms are recursive functions that may be consistent.M +
The ∇calculus. Functional programming with higherorder encodings
 In Proceedings of the 7th International Conference on Typed Lambda Calculi and Applications
, 2005
"... Abstract. Higherorder encodings use functions provided by one language to represent variable binders of another. They lead to concise and elegant representations, which historically have been difficult to analyze and manipulate. In this paper we present the ∇calculus, a calculus for defining gener ..."
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Cited by 23 (3 self)
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Abstract. Higherorder encodings use functions provided by one language to represent variable binders of another. They lead to concise and elegant representations, which historically have been difficult to analyze and manipulate. In this paper we present the ∇calculus, a calculus for defining general recursive functions over higherorder encodings. To avoid problems commonly associated with using the same function space for representations and computations, we separate one from the other. The simplytyped λcalculus plays the role of the representationlevel. The computationlevel contains not only the usual computational primitives but also an embedding of the representationlevel. It distinguishes itself from similar systems by allowing recursion under representationlevel λbinders while permitting a natural style of programming which we believe scales to other logical frameworks. Sample programs include bracket abstraction, parallel reduction, and an evaluator for a simple language with firstclass continuations. 1
Normalization and the Yoneda Embedding
"... this paper we describe a new, categorical approach to normalization in typed  ..."
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Cited by 22 (3 self)
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this paper we describe a new, categorical approach to normalization in typed 
A Logic Programming Approach to Implementing HigherOrder Term Rewriting
 Second International Workshop on Extensions to Logic Programming, volume 596 of Lecture Notes in Arti Intelligence
, 1992
"... Term rewriting has proven to be an important technique in theorem proving. In this paper, we illustrate that rewrite systems and strategies for higherorder term rewriting, which includes the usual notion of firstorder rewriting, can be naturally specified and implemented in a higherorder logic pr ..."
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Cited by 16 (2 self)
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Term rewriting has proven to be an important technique in theorem proving. In this paper, we illustrate that rewrite systems and strategies for higherorder term rewriting, which includes the usual notion of firstorder rewriting, can be naturally specified and implemented in a higherorder logic programming language. We adopt a notion of higherorder rewrite system which uses the simply typed calculus as the language for expressing rules, with a restriction on the occurrences of free variables on the left hand sides of rules so that matching of terms with rewrite templates is decidable. The logic programming language contains an implementation of the simplytyped lambda calculus including fij conversion and higherorder unification. In addition, universal quantification in queries and the bodies of clauses is permitted. For higherorder rewriting, we show how these operations implemented at the metalevel provide elegant mechanisms for the objectlevel operations of descending thro...
Combining First Order Algebraic Rewriting Systems, Recursion and Extensional Lambda Calculi
 Intern. Conf. on Automata, Languages and Programming (ICALP), volume 820 of Lecture Notes in Computer Science
, 1994
"... It is well known that confluence and strong normalization are preserved when combining leftlinear algebraic rewriting systems with the simply typed lambda calculus. It is equally well known that confluence fails when adding either the usual extensional rule for j, or recursion together with the usu ..."
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Cited by 16 (8 self)
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It is well known that confluence and strong normalization are preserved when combining leftlinear algebraic rewriting systems with the simply typed lambda calculus. It is equally well known that confluence fails when adding either the usual extensional rule for j, or recursion together with the usual contraction rule for surjective pairing. We show that confluence and normalization are modular properties for the combination of leftlinear algebraic rewriting systems with typed lambda calculi enriched with expansive extensional rules for j and surjective pairing. For that, we use a translation technique allowing to simulate expansions without expansion rules. We also show that confluence is maintained in a modular way when adding fixpoints. This result is also obtained by a simple translation technique allowing to simulate bounded recursion with fi reduction. 1 Introduction Confluence and strong normalization for the combination of lambda calculus and algebraic rewriting systems have...
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 "Te ..."
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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.
Finite Family Developments
"... Associate to a rewrite system R having rules l → r, its labelled version R ω having rules l ◦ m+1 → r • , for any natural number m m ∈ ω. These rules roughly express that a lefthand side l carrying labels all larger than m can be replaced by its righthand side r carrying labels all smaller than o ..."
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Cited by 13 (6 self)
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Associate to a rewrite system R having rules l → r, its labelled version R ω having rules l ◦ m+1 → r • , for any natural number m m ∈ ω. These rules roughly express that a lefthand side l carrying labels all larger than m can be replaced by its righthand side r carrying labels all smaller than or equal to m. A rewrite system R enjoys finite family developments (FFD) if R ω is terminating. We show that the class of higher order pattern rewrite systems enjoys FFD, extending earlier results for the lambda calculus and first order term rewrite systems.
Modularity of Confluence: A Simplified Proof
, 1994
"... In this note we present a simple proof of a result of Toyama which states that the disjoint union of confluent term rewriting systems is confluent. 1985 Mathematics Subject Classification: 68Q50 1987 CR Categories: F.4.2 Key Words and Phrases: theory of computation, term rewriting systems, modular ..."
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Cited by 12 (5 self)
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In this note we present a simple proof of a result of Toyama which states that the disjoint union of confluent term rewriting systems is confluent. 1985 Mathematics Subject Classification: 68Q50 1987 CR Categories: F.4.2 Key Words and Phrases: theory of computation, term rewriting systems, modularity, confluence Introduction The topic of modularity of properties of term rewriting systems has caught much attention recently. An introduction to this area can be found in Klop [6]. For an early survey one may consult Middeldorp [7]. Moreover, the topic has received a fruitful offspring in the study of the conservation of properties when adding algebraic rewrite rules to various (typed) lambda calculi, see e.g. BreazuTannen and Gallier [1, 2] and Jouannaud and Okada [5]. 5 Partially supported by ESPRIT Basic Research Action 3020, INTEGRATION. 6 Partially supported by ESPRIT Basic Research Action 3074, SEMAGRAPH. 7 Partially supported by grants from NWO, Vrije Universiteit Amsterdam...
A Combinatory Logic Approach to Higherorder Eunification
 in Proceedings of the Eleventh International Conference on Automated Deduction, SpringerVerlag LNAI 607
, 1992
"... Let E be a firstorder equational theory. A translation of typed higherorder Eunification problems into a typed combinatory logic framework is presented and justified. The case in which E admits presentation as a convergent term rewriting system is treated in detail: in this situation, a modifi ..."
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Cited by 9 (3 self)
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Let E be a firstorder equational theory. A translation of typed higherorder Eunification problems into a typed combinatory logic framework is presented and justified. The case in which E admits presentation as a convergent term rewriting system is treated in detail: in this situation, a modification of ordinary narrowing is shown to be a complete method for enumerating higherorder Eunifiers. In fact, we treat a more general problem, in which the types of terms contain type variables. 1 Introduction Investigation of the interaction between firstorder and higherorder equational reasoning has emerged as an active line of research. The collective import of a recent series of papers, originating with [Bre88] and including (among others) [Bar90], [BG91a], [BG91b], [Dou92], [JO91] and [Oka89], is that when various typed calculi are enriched by firstorder equational theories, the validity problem is wellbehaved, and furthermore that the respective computational approaches to ...