This chapter asserts that, in current state-of-the-art symbolic regression engines, accuracy is poor. That is to say that state-of-the-art symbolic regression engines return a champion with good fitness; however, obtaining a champion with the correct formula is not forthcoming even in cases of only one basis function with minimally complex grammar depth.Ideally, users expect that for test problems created with no noise, using only functions in the specified grammar, with only one basis function and some minimal grammar depth, that state-of-the-art symbolic regression systems should return the exact formula (or at least an isomorph) used to create the test data. Unfortunately, this expectation cannot currently be achieved using published state-of-the-art symbolic regression techniques.Several classes of test formulas, which prove intractable, are examined and an understanding of why they are intractable is developed. Techniques in Abstract Expression Grammars are employed to render these problems tractable, including manipulation of the epigenome during the evolutionary process, together with breeding of multiple targeted epigenomes in separate population islands.A selected set of currently intractable problems are shown to be solvable, using these techniques, and a proposal is put forward for a discipline-wide program of improving accuracy in state-of-the-art symbolic regression systems.

This chapter examines the use of Abstract Expression Grammars to perform the entire Symbolic Regression process without the use of Genetic Programming per se. The techniques explored produce a symbolic regression engine which has absolutely no bloat, which allows total user control of the search space and output formulas, which is faster, and more accurate than the engines produced in our previous papers using Genetic Programming. The genome is an all vector structure with four chromosomes plus additional epigenetic and constraint vectors, allowing total user control of the search space and the final output formulas. A combination of specialized compiler techniques, genetic algorithms, particle swarm, aged layered populations, plus discrete and continuous differential evolution are used to produce an improved symbolic regression sytem. Nine base test cases, from the literature, are used to test the improvement in speed and accuracy. The improved results indicate that these techniques move us a big step closer toward future industrial strength symbolic regression systems. While the techniques, described in detail in (Korns, 2009), produce a symbolic regression system of breadth and strength, lack of user control of the search space, bloated unreadable output formulas, accuracy, and slow convergence speed are all issues keeping an industrial strength symbolic regression system tantalizingly out of reach. In this chapter abstract expression grammars become the main focus and are promoted as the sole means of performing symbolic regression. Using the nine base test cases from (Korns, 2007) as a training set, to test for improvements in accuracy, we constructed our symbolic regression system using these important techniques:Abstract expression grammars Universal abstract goal expression Standard single point vector-based mutation Standard two point vector-based cross over Continuous vector differential evolution Discrete vector differential evolution Continuous particle swarm evolution Pessimal vertical slicing and out-of-sample scoring during training Age-layered populations User defined epigenetic factors User defined constraints

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