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

We illustrate a variety of programming problems that seemingly require two separate list traversals, but that can be efficiently solved in one recursive descent, without any other auxiliary storage but what can be expected from a control stack. The idea is to perform the second traversal when returning from the first.

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.

This paper describes a formal derivation of an optimized Ackermann's function following a general and systematic method based on incrementalization. The method identifies an appropriate input increment operation and computes the function by repeatedly performing an incremental computation at the step of the increment. This eliminates repeated subcomputations in executions that follow the straightforward recursive definition of Ackermann's function, yielding an optimized program that is drastically faster and takes extremely little space. This case study uniquely shows the power and limitation of the incrementalization method, as well as both the iterative and recursive nature of computation underlying the optimized Ackermann's function.

This paper presents program analyses and transformations for strengthening invariants for the purpose of efficient computation. Finding the stronger invariants corresponds to discovering a general class of auxiliary information for any incremental computation problem. Combining the techniques with previous techniques for caching intermediate results, we obtain a systematic approach that transforms non-incremental programs into ecient incremental programs that use and maintain useful auxiliary information as well as useful intermediate results. The use of auxiliary information allows us to achieve a greater degree of incrementality than otherwise possible. Applications of the approach include strength reduction in optimizing compilers and finite differencing in transformational programming.

ABSTRACT This paper presents the first automatic scheme to allocate local (stack) data in recursive functions to scratch-pad memory (SPM) in embedded systems. A scratch-pad is a fast directly addressed compiler-managed SRAM memory that replaces the hardware-managed cache. It is motivated by its significantly lower access time, energy consumption, real-time bounds, area and overall runtime. Existing compiler methods for allocating data to scratch-pad are able to place only code, global, heap and non-recursive stack data in scratch-pad memory; stack data for recursive functions is allocated entirely in DRAM, resulting in poor performance. In this paper we present a dynamic yet compiler-directed allocation method for recursive function stack data that for the first time, is able to place a portion of recursive stack data in scratch-pad. It has almost no software-caching overhead, and is able to move recursive function data back and forth between scratchpad and DRAM to better track the program’s locality characteristics. With our method, all code, global, stack and heap variables can share the same scratch-pad. When compared to placing all recursive function data in DRAM and all other variables in scratch-pad, our results show that our method reduces the average runtime of our benchmarks by 29.3%, and the average power consumption by 31.1%, for the same size of scratch-pad fixed at 5 % of total data size. Furthermore, significant savings were observed when comparing our method against cache-based alternatives for SPM allocation. Finally, we show results that analyze the effects of profile variation on our allocation approach and present a modified version of our method which minimizes variation for profile-based allocations. 1

For agents to fulfill their potential of being intelligent and adaptive, it is useful to model their interaction protocols as executable entities that can be referenced, inspected, composed, shared and invoked between agents, all at runtime. We use the term first-class protocol to refer to such protocols. Rather than having hard-coded decision making mechanisms for choosing their next move, agents can inspect the protocol specification at runtime to do so, increasing their flexibility. In this paper, we show that propositional dynamic logic (PDL) can be used to represent and reason about the outcomes of first-class protocols. We define a proof system for PDL that permits reasoning about recursively-defined protocols. The proof system is divided into two parts: one for reasoning about terminating protocols, and one for reasoning about non-terminating protocols. We prove that proofs about terminating protocols can be automated, while proofs about non-terminating protocols are unable to be automated in some cases. We prove that, for a restricted class of non-terminating protocols, proofs about them can be transformed to proofs about terminating protocols, making them automatable. Key words: multi-agent systems, agent interaction protocols, propositional dynamic logic, first-class protocols.

Building program generators that do not duplicate generated code can be challenging. At the same time, code duplication can easily increase both generation time and runtime of generated programs by an exponential factor. We identify an instance of this problem that can arise when memoized functions are staged. Without addressing this problem, it would be impossible to effectively stage dynamic programming algorithms. Intuitively, direct staging undoes the effect of memoization. To solve this problem once and for all, and for any function that uses memoization, we propose a staged monadic combinator library. Experimental results confirm that the library works as expected. Preliminary results also indicate that the library is useful even when memoization is not used.

Abstract. High-level languages offer abstraction mechanisms that can reduce development time and improve software quality. But abstraction mechanisms often have an accumulative runtime overhead that can discourage their use. Multi-stage programming (MSP) languages offer constructs that make it possible to use abstraction mechanisms without paying a runtime overhead. This paper studies applying MSP to implementing dynamic programming (DP) problems. The study reveals that staging high-level implementations of DP algorithms naturally leads to a code explosion problem. In addition, it is common that high-level languages are not designed to deliver the kind of performance that is desirable in implementations of such algorithms. The paper proposes a solution to each of these two problems. Staged memoization is used for code explosion, and a kind of “offshoring ” translation is used to address the second. For basic DP problems, the performance of the resulting specialized C implementations is almost always better than the hand-written generic C implementations. 1

Recursive programs may require large numbers of procedure calls and stack operations, and many such recursive programs exhibit exponential time complexity, due to the time spent re-calculating already computed sub-problems. As a result, methods which transform a given recursive program to an iterative one have been intensively studied. We propose here a new framework for transforming programs by removing recursion. The framework includes a unified method of deriving low time-complexity programs by solving recurrences extracted from the program sources. Our prototype system, ������, is an initial implementation of the framework, automatically finding simpler “closed form ” versions of a class of recursive programs. Though in general the solution of recurrences is easier if the functions have only a single recursion parameter, we show a practical technique for solving those with multiple recursion parameters.