Advances in high energy physics have created the need to increase computational capacity. Project HEPGAME was composed to address this challenge. One of the issues is that numerical integration of expressions of current interest have millions of terms and takes weeks to compute. We have investigated ways to simplify these expressions, using Horner schemes and common subexpression elimination. Our approach applies MCTS, a search procedure that has been successful in AI. We use it to find near-optimal Horner schemes. Although MCTS finds better solutions, this approach gives rise to two further challenges. (1) MCTS (with UCT) introduces a constant, @math that governs the balance between exploration and exploitation. This constant has to be tuned manually. (2) There should be more guided exploration at the bottom of the tree, since the current approach reduces the quality of the solution towards the end of the expression. We investigate NMCS (Nested Monte Carlo Search) to address both issues, but find that NMCS is computationally unfeasible for our problem. Then, we modify the MCTS formula by introducing a dynamic exploration-exploitation parameter @math that decreases linearly with the iteration number. Consequently, we provide a performance analysis. We observe that a variable @math solves our domain: it yields more exploration at the bottom and as a result the tuning problem has been simplified. The region in @math for which good values are found is increased by more than a tenfold. This result encourages us to continue our research to solve other prominent problems in High Energy Physics.
Computer algebra systems, expression simplification, and boolean problems are closely related. General purpose packages such as Mathematica and Maple have evolved out of early systems created by physicists and artificial intelligence in the 1960s. A prime example for the first development was the work by the later Nobel Prize laureate Martinus Veltman, who designed a program for symbolic mathematics, especially High Energy Physics, called Schoonschip (Dutch for clean ship,'' or clean up'') in 1963. In the course of the 1960s Gerard 't Hoofd accompanied Veltman with whom he shared the Nobel Prize. The first popular computer algebra systems were muMATH, Reduce, Derive (based on muMATH), and Macsyma. It is interesting to note that FORM @cite_1 , the system that we use in our work, is a direct successor to Schoonschip.
@cite_1
1963991920
Abstract We present version 4.0 of the symbolic manipulation system Form . The most important new features are manipulation of rational polynomials and the factorization of expressions. Many other new functions and commands are also added; some of them are very general, while others are designed for building specific high level packages, such as one for Grobner bases. New is also the checkpoint facility, that allows for periodic backups during long calculations. Finally, Form 4.0 has become available as open source under the GNU General Public License version 3. Program summary Program title: FORM. Catalogue identifier: AEOT_v1_0 Program summary URL: http: cpc.cs.qub.ac.uk summaries AEOT_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: GNU General Public License, version 3 No. of lines in distributed program, including test data, etc.: 151599 No. of bytes in distributed program, including test data, etc.: 1 078 748 Distribution format: tar.gz Programming language: The FORM language. FORM itself is programmed in a mixture of C and C++. Computer: All. Operating system: UNIX, LINUX, Mac OS, Windows. Classification: 5. Nature of problem: FORM defines a symbolic manipulation language in which the emphasis lies on fast processing of very large formulas. It has been used successfully for many calculations in Quantum Field Theory and mathematics. In speed and size of formulas that can be handled it outperforms other systems typically by an order of magnitude. Special in this version: The version 4.0 contains many new features. Most important are factorization and rational arithmetic. The program has also become open source under the GPL. The code in CPC is for reference. You are encouraged to upload the most recent sources from www.nikhef.nl form formcvs.php because of frequent bug fixes. Solution method: See “Nature of Problem”, above. Additional comments: NOTE: The code in CPC is for reference. You are encouraged to upload the most recent sources from www.nikhef.nl form formcvs.php because of frequent bug fixes.