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Functors for Proofs and Programs
, 2003
"... This paper presents the formal verication with the Coq proof assistant of several applicative data structures implementing nite sets. These implementations are parameterized by an ordered type for the elements, using functors from the ML module system. The verication follows closely this scheme, ..."
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Cited by 20 (3 self)
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This paper presents the formal verication with the Coq proof assistant of several applicative data structures implementing nite sets. These implementations are parameterized by an ordered type for the elements, using functors from the ML module system. The verication follows closely this scheme, using the newly Coq module system.
Dependently Typed Data Structures
, 1999
"... The mechanism for declaring datatypes in functional programming languages such as ML and Haskell is of great use in practice. This mechanism, however, often suffers from its imprecision in capturing the invariants inherent in data structures. We remedy the situation with the introduction of dependen ..."
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Cited by 14 (3 self)
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The mechanism for declaring datatypes in functional programming languages such as ML and Haskell is of great use in practice. This mechanism, however, often suffers from its imprecision in capturing the invariants inherent in data structures. We remedy the situation with the introduction of dependent datatypes so that we can model data structures with significantly more accuracy. We present a few interesting examples such as implementations of redblack trees and binomial heaps to illustrate the use of dependent datatypes in capturing some sophisticated invariants in data structures. We claim that dependent datatypes can enable the programmer to implement algorithms in a way that is more robust and easier to understand.
An Algebra of Scans
 In Mathematics of Program Construction
, 2004
"... A parallel prefix circuit takes n inputs x1 , x2 , . . . , xn and produces the n outputs x1 , x1 x2 , . . . , x1 x2 xn , where `#' is an arbitrary associative binary operation. Parallel prefix circuits and their counterparts in software, parallel prefix computations or scans, have numerou ..."
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Cited by 13 (0 self)
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A parallel prefix circuit takes n inputs x1 , x2 , . . . , xn and produces the n outputs x1 , x1 x2 , . . . , x1 x2 xn , where `#' is an arbitrary associative binary operation. Parallel prefix circuits and their counterparts in software, parallel prefix computations or scans, have numerous applications ranging from fast integer addition over parallel sorting to convex hull problems. A parallel prefix circuit can be implemented in a variety of ways taking into account constraints on size, depth, or fanout. Traditionally, implementations are either defined graphically or by enumerating the underlying graph. Both approaches have their pros and cons. A figure if well drawn conveys the possibly recursive structure of the scan but it is not amenable to formal manipulation. A description in form of a graph while rigorous obscures the structure of a scan and is equally hard to manipulate. In this paper we show that parallel prefix circuits enjoy a very pleasant algebra. Using only two basic building blocks and four combinators all standard designs can be described succinctly and rigorously. The rules of the algebra allow us to prove the circuits correct and to derive circuit designs in a systematic manner. lord darlington. . . . [Sees a fan lying on the table.] And what a wonderful fan! May I look at it? lady windermere. Do. Pretty, isn't it! It's got my name on it, and everything. I have only just seen it myself. It's my husband's birthday present to me. You know today is my birthday?  Oscar Wilde, Lady Windermere's Fan 1
A Simple Implementation Technique for Priority Search Queues
 Proceedings of the 2001 International Conference on Functional Programming
, 2001
"... This paper presents a new implementation technique for priority search queues. This abstract data type is an amazing blend of finite maps and priority queues. Our implementation supports logarithmic access to a binding with a given key and constant access to a binding with the minimum value. Priorit ..."
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Cited by 10 (2 self)
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This paper presents a new implementation technique for priority search queues. This abstract data type is an amazing blend of finite maps and priority queues. Our implementation supports logarithmic access to a binding with a given key and constant access to a binding with the minimum value. Priority search queues can be used, for instance, to give a simple, purely functional implementation of Dijkstra's singlesource shortestpaths algorithm.
Fast treetrellis list Viterbi decoding
 IEEE Trans. Communications
, 2006
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Programming with Static Invariants in \Omega mega
, 2004
"... Abstract We present an implementation of binomial heaps that enforces the balancing invariants at compile time. We present an implementation of a typed CPS transform that enforces the typing invariants of both the source and target terms. 1 Introduction There has been a lot of interest recently in g ..."
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Abstract We present an implementation of binomial heaps that enforces the balancing invariants at compile time. We present an implementation of a typed CPS transform that enforces the typing invariants of both the source and target terms. 1 Introduction There has been a lot of interest recently in generalizing algebraic datatypes asfound in most functional programming languages. These generalizations have
FUNCTIONAL PEARL Deletion: The curse of the redblack tree
"... Okasaki introduced the canonical formulation of functional redblack trees when he gave a concise, elegant method of persistent element insertion. Persistent element deletion, on the other hand, has not enjoyed the same treatment. For this reason, many functional implementations simply omit persiste ..."
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Okasaki introduced the canonical formulation of functional redblack trees when he gave a concise, elegant method of persistent element insertion. Persistent element deletion, on the other hand, has not enjoyed the same treatment. For this reason, many functional implementations simply omit persistent deletion. Those that include deletion typically take one of two approaches. The morecommon approach is a superficial translation of the standard imperative algorithm. The resulting algorithm has functional airs but remains clumsy and verbose, characteristic of its imperative heritage. (Indeed, even the term insertion is a holdover from imperative origins, but is now established in functional contexts. Accordingly, we use the term deletion which has the same connotation.) The lesscommon approach leverages the features of advanced type systems, which obscures the essence of the algorithm. Nevertheless, foreignlanguage implementors reference such implementations and, apparently unable to tease apart the algorithm and its type specification, transliterate the entirety unnecessarily. Our goal is to provide for persistent deletion what Okasaki did for insertion: a succinct, comprehensible method that will liberate implementors. We conceptually simplify deletion by temporarily introducing a “doubleblack ” color into Okasaki’s tree type. This third color, with its natural interpretation, significantly simplifies the preservation of invariants during deletion. 1
ABSTRACT A Simple Implementation Technique for Priority Search Queues
"... This paper presents a new implementation technique for priority search queues. This abstract data type is an amazing blend of finite maps and priority queues. Our implementation supports logarithmic access to a binding with a given key and constant access to a binding with the minimum value. Priorit ..."
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This paper presents a new implementation technique for priority search queues. This abstract data type is an amazing blend of finite maps and priority queues. Our implementation supports logarithmic access to a binding with a given key and constant access to a binding with the minimum value. Priority search queues can be used, for instance, to give a simple, purely functional implementation of Dijkstra’s singlesource shortestpaths algorithm. data types and views. Priority search queues have been largely ignored by the functional programming community and we believe that they deserve to be known better. Views prove their worth both in defining a convenient interface to the abstract data type and in providing a readable implementation. Categories and Subject Descriptors
Programming with Dependently Typed Data Structures ∗
"... The mechanism for declaring datatypes in functional programming languages such as ML and Haskell is of great use in practice. This mechanism, however, often suffers from its imprecision in capturing various invariants inherent in data structures. The introduction of dependent datatypes in Dependent ..."
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The mechanism for declaring datatypes in functional programming languages such as ML and Haskell is of great use in practice. This mechanism, however, often suffers from its imprecision in capturing various invariants inherent in data structures. The introduction of dependent datatypes in Dependent ML (DML) can partly remedy the situation, allowing the programmer to model data structures with significantly more accuracy. In this paper, we present several programming examples (e.g., implementations of randomaccess lists and redblack trees) to illustrate some practical use of dependent datatypes in capturing relatively sophisticated invariants in data structures. We claim that dependent datatypes can enable the programmer to implement algorithms in a manner that is more efficient, more robust and easier to understand. 1