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
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The seminal works of Wilf and Nijenhuis in the late 70s have led to efficient algorithms for counting and generating uniformly at random a variety of combinatorial structures. In 1994, Flajolet, Zimmermann and Van Cutsem have widely generalised and systematised the approach. This paper presents several applications of these powerful results to software random testing, and random model exploration. During the 60s and 70s, AI researchers explored intuitions about intelligence by writing programs that displayed intelligent behavior. Many good ideas came out from this work but programs written by hand were not robust or general. After the 80s, research increasingly shifted to the development of learners capable of inferring behavior and functions from experience and data, and solvers capable of tackling well-defined but intractable models like SAT, classical planning, Bayesian networks, and POMDPs. The learning approach has achieved considerable success but results in black boxes that do not have the flexibility, transparency, and generality of their model-based counterparts. Model-based approaches, on the other hand, require models and scalable algorithms. Model-free learners and model-based solvers have close parallels with Systems 1 and 2 in current theories of the human mind: the first, a fast, opaque, and inflexible intuitive mind; the second, a slow, transparent, and flexible analytical mind. In this paper, I review developments in AI and draw on these theories to discuss the gap between model-free learners and model-based solvers, a gap that needs to be bridged in order to have intelligent systems that are robust and general. Abstract A computer program capable of acting intelligently in the world must have a general representation of the world in terms of which its inputs are interpreted. Designing such a program requires commitments about what knowledge is and how it is obtained. Thus, some of the major traditional problems of philosophy arise in artificial intelligence. More specifically, we want a computer program that decides what to do by inferring in a formal language that a certain strategy will achieve its assigned goal. This requires formalizing concepts of causality, ability, and knowledge. Such formalisms are also considered in philosophical logic. The first part of the paper begins with a philosophical point of view that seems to arise naturally once we take seriously the idea of actually making an intelligent machine. We go on to the notions of metaphysically and epistemo-logically adequate representations of the world and then to an explanation of can, causes , and knows in terms of a representation of the world by a system of interacting automata. A proposed resolution of the problem of freewill in a deterministic universe and of counterfactual conditional sentences is presented. The second part is mainly concerned with formalisms within which it can be proved that a strategy will achieve a goal. Concepts of situation, fluent, future operator, action, strategy, result of a strategy and knowledge are formalized. A method is given of constructing a sentence of first-order logic which will be true in all models of certain axioms if and only if a certain strategy will achieve a certain goal. The formalism of this paper represents an advance over McCarthy (1963) and Green (1969) in that it permits proof of the correctness of strategies that contain loops and strategies that involve the acquisition of knowledge; and it is also somewhat more concise. The third part discusses open problems in extending the formalism of part 2. The fourth part is a review of work in philosophical logic in relation to problems of artificial intelligence and a discussion of previous efforts to program ‘general intelligence’ from the point of view of this paper. The Tale of a Brainteaser Francis Guthrie certainly did it, when he coined his innocent little coloring puzzle in 1852. He managed to embarrass successively his mathematician brother, his brother’s professor, Augustus de Morgan, and all of de Morgan’s visitors, who couldn’t solve it; the Royal Society, who only realized ten years later that Alfred Kempe’s 1879 solution was wrong; and the three following generations of mathematicians who couldn’t fix it [19]. Even Appel and Haken’s 1976 triumph [2] had a hint of defeat: they’d had a computer do the proof for them! Perhaps the mathematical controversy around the proof died down with their book [3] and with the elegant 1995 revision [13] by Robertson, Saunders, Seymour, and Thomas. However something was still amiss: both proofs combined a textual argument, which could reasonably be checked by inspection, with computer code that could not. Worse, the empirical evidence provided by running code several times with the same input is weak, as it is blind to the most common cause of “computer” error: programmer error. For some thirty years, computer science has been working out a solution to this problem: formal program proofs. The idea is to write code that describes not only what the machine should do, but also why it should be doing it—a formal proof of correctness. The validity of the proof is an objective mathematical fact that can be checked by a different program, whose own validity can be ascertained empirically because it does run on many inputs. The main technical difficulty is that formal proofs are very difficult to produce,