A polytypic function definition is a function definition that is parametrised with a datatype. It embraces a class of algorithms. As an example we define a simple polytypic "crush" combinator that can be used to calculate polytypically. The ability to define functions polytypically adds another level of flexibility in the reusability of programming idioms and in the design of libraries of interoperable components.

Dynamic programming is an important algorithm design technique. It is used for solving problems whose solutions involve recursively solving subproblems that share subsubproblems. While a straightforward recursive program solves common subsubproblems repeatedly and often takes exponential time, a dynamic programming algorithm solves every subsubproblem just once, saves the result, reuses it when the subsubproblem is encountered again, and takes polynomial time. This paper describes a systematic method for transforming programs written as straightforward recursions into programs that use dynamic programming. The method extends the original program to cache all possibly computed values, incrementalizes the extended program with respect to an input increment to use and maintain all cached results, prunes out cached results that are not used in the incremental computation, and uses the resulting incremental program to form an optimized new program. Incrementalization statically exploits semantics of both control structures and data structures and maintains as invariants equalities characterizing cached results. The principle underlying incrementalization is general for achieving drastic program speedups. Compared with previous methods that perform memoization or tabulation, the method based on incrementalization is more powerful and systematic. It has been implemented and applied to numerous problems and succeeded on all of them. 1

Parallel primitives (skeletons) intend to encourage programmers to build a parallel program from ready-made components for which efficient implementations are known to exist, making the parallelization process easier. However, programmers often suffer from the difficulty to choose a combination of proper parallel primitives so as to construct efficient parallel programs. To overcome this difficulty, we shall propose a new transformation, called diffusion, which can efficiently decompose a recursive definition into several functions such that each function can be described by some parallel primitive. This allows programmers to describe algorithms in a more natural recursive form. We demonstrate our idea with several interesting examples. Our diffusion transformation should be significant not only in development of new parallel algorithms, but also in construction of parallelizing compilers.

This paper describes a systematic method for optimizing recursive functions using both indexed and recursive data structures. The method is based on two critical ideas: first, determining a minimal input increment operation so as to compute a function on repeatedly incremented input; second, determining appropriate additional values to maintain in appropriate data structures, based on what values are needed in computation on an incremented input and how these values can be established and accessed. Once these two are determined, the method extends the original program to return the additional values, derives an incremental version of the extended program, and forms an optimized program that repeatedly calls the incremental program. The method can derive all dynamic programming algorithms found in standard algorithm textbooks. There are many previous methods for deriving efficient algorithms, but none is as simple, general, and systematic as ours.

Abstract Many functions have to be written over and over again for different datatypes, either because datatypes change during the development of programs, or because functions with similar functionality are needed on different datatypes. Examples of such functions are pretty printers, pattern matchers, equality functions, unifiers, rewriting functions, etc. Such functions are called polytypic functions. A polytypic function is a function that is defined by induction on the structure of user-defined datatypes. This thesis introduces polytypic functions, shows how to construct and reason about polytypic functions and describes the implementation of the polytypic programming system PolyP. PolyP extends a functional language (a subset of Haskell) with a construct for writing polytypic functions. The extended language type checks definitions of polytypic functions, and infers the types of all other expressions. Programs in the extended language are translated to Haskell.

We describe a unified, lazy, declarative framework for solving constraint satisfaction problems, an important subclass of combinatorial search problems. These problems are both practically significant and hard. Finding good solutions involves combining good general-purpose search algorithms with problemspecific heuristics. Conventional imperative algorithms are usually implemented and presented monolithically, which makes them hard to understand and reuse, even though new algorithms often are combinations of simpler ones. Lazy functional languages, such as Haskell, encourage modular structuring of search algorithms by separating the generation and testing of potential solutions into distinct functions communicating through an explicit, lazy intermediate data structure. But only relatively simple search algorithms have been treated in this way in the past. Our framework uses a generic generation and pruning algorithm parameterized by a labeling function that annotates search t...

In the constructive programming community it is commonplace to see formal developments of textbook algorithms. In the algorithm design community, on the other hand, it may be well known that the textbook solution to a problem is not the most efficient possible. However, in presenting the more efficient solution, the algorithm designer will usually omit some of the implementation details, thus creating an algorithm gap between the abstract algorithm and its concrete implementation. This is in contrast to the formal development, which usually presents the complete concrete implementation of the less efficient solution. We claim that the algorithm designer is forced to omit some of the details by the relative expressive poverty of the Pascal-like languages typically used to present the solution; the greater expressiveness provided by a functional language allows the whole story to be told in a reasonable amount of space. We therefore hope to bridge the algorithm gap between ab...

Data parallelism is currently one of the most successful models for programming massively parallel computers. The central idea is to evaluate a uniform collection of data in parallel by simultaneously manipulating each data element in the collection. Despite many of its promising features, the current approach suffers from two problems. First, the main parallel data structures that most data parallel languages currently support are restricted to simple collection data types like lists, arrays or similar structures. But other useful data structures like trees have not been well addressed. Second, parallel programming relies on a set of parallel primitives that capture parallel skeletons of interest. However, these primitives are not well structured, and efficient parallel programming with these primitives is difficult. In this paper, we propose a polytypic framework for developing efficient parallel programs on most data structures. We showhow a set of polytypic parallel primitives can be formally defined for manipulating most data structures, how these primitives can be successfully structured into a uniform recursive definition, and how an efficient combination of primitives can be derived from a naive specification program. Our framework should be significant not only in development of new parallel algorithms, but also in construction of parallelizing compilers.

Program generation has seen an important role in a wide range of software development processes, where effective calculation rules are critical. In this paper, we propose a more general calculation rule for generation of efficient programs for solving maximum marking problems. Easy to use and implement, our new rule gives a significant extension of the rule proposed by Sasano et al., allowing multiple kinds of marks as well as more general description of the property of acceptable markings. We illustrate its effectiveness using several interesting problems.

In existing work on graph algorithms, it is known that a linear time algorithm can be derived mechanically from a logical formula for a class of optimization problems. But this has a serious problem that the derived algorithm has huge constant factor. In this work, we redene this problem on recursive data structures as a maximum marking problem and propose method for deriving a linear time algorithm for that. In this method, speci cation is given using recursive functions instead of logical formula, which results in a practical linear time algorithm. This method is mechanical and in fact, based on this deriving method, we make a system which automatically generates a practical linear time algorithm from specication for a maximum marking problem.