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Memoizing polymorphic functions via unmemoization
Last year I wrote two posts on memoizing polymorphic functions. The first post gave a solution of sorts that uses a combination of patricia trees (for integer-keyed maps), stable names, and type...
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Adding numbers
IntroductionI’m starting to think about exact numeric computation. As a first step in getting into issues, I’ve been playing with addition on number representations, particularly carry look-ahead...
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Type-bounded numbers
I’ve been thinking a lot lately about how to derive low-level massively parallel programs from high-level specifications. One of the recurrent tools is folds (reductions) with an associative operator....
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Fixing lists
In the post Memoizing polymorphic functions via unmemoization, I toyed with the idea of lists as tries. I don’t think [a] is a trie, simply because [a] is a sum type (being either nil or a cons),...
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Doing more with length-typed vectors
The post Fixing lists defined a (commonly used) type of vectors, whose lengths are determined statically, by type. In Vec n a, the length is n, and the elements have type a, where n is a type-encoded...
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Reverse-engineering length-typed vectors
The last few posts posts followed a winding path toward a formulation of a type for length-typed vectors. In Fixing lists, I mused how something like lists could be a trie type. The Stream functor...
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A trie for length-typed vectors
As you might have noticed, I’ve been thinking and writing about memo tries lately. I don’t mean to; they just keep coming up. Memoization is the conversion of functions to data structures. A simple,...
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From tries to trees
This post is the last of a series of six relating numbers, vectors, and trees, revolving around the themes of static size-typing and memo tries. We’ve seen that length-typed vectors form a trie for...
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Deriving list scans
I’ve been playing with deriving efficient parallel, imperative implementations of "prefix sum" or more generally "left scan". Following posts will explore the parallel & imperative derivations,...
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Deriving parallel tree scans
The post Deriving list scans explored folds and scans on lists and showed how the usual, efficient scan implementations can be derived from simpler specifications. Let’s see now how to apply the same...
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Composable parallel scanning
The post Deriving list scans gave a simple specification of the list-scanning functions scanl and scanr, and then transformed those specifications into the standard optimized implementations. Next,...
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Parallel tree scanning by composition
My last few blog posts have been on the theme of scans, and particularly on parallel scans. In Composable parallel scanning, I tackled parallel scanning in a very general setting. There are five...
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A third view on trees
A few recent posts have played with trees from two perspectives. The more commonly used I call "top-down", because the top-level structure is most immediately apparent. A top-down binary tree is...
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Parallel speculative addition via memoization
I’ve been thinking much more about parallel computation for the last couple of years, especially since starting to work at Tabula a year ago. Until getting into parallelism explicitly, I’d naïvely...
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Reimagining matrices
The function of the imagination is notto make strange things settled, so much asto make settled things strange.- G.K. Chesterton Why is matrix multiplication defined so very differently from matrix...
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From Haskell to hardware via cartesian closed categories
Since fall of last year, I’ve been working at Tabula, a Silicon Valley start-up developing an innovative programmable hardware architecture called “Spacetime”, somewhat similar to an FPGA, but much...
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Overloading lambda
Haskell’s type class facility is a powerful abstraction mechanism. Using it, we can overload multiple interpretations onto a single vocabulary, with each interpretation corresponding to a different...
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Optimizing CCCs
In the post Overloading lambda, I gave a translation from a typed lambda calculus into the vocabulary of cartesian closed categories (CCCs). This simple translation leads to unnecessarily complex...
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Circuits as a bicartesian closed category
My previous few posts have been about cartesian closed categories (CCCs). In From Haskell to hardware via cartesian closed categories, I gave a brief motivation: typed lambda expressions and the CCC...
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