The results from our generalising n-bit parity checking experiments are given in Table 7.5. We list the MEs required to solve the problem and the IQR in evaluations. We also give the generalisation success rate (SR) of each algorithm, which is the proportion of runs in which the found solution perfectly generalised to 14-bit sequences. We test for statistical significance with the two-tailed Mann–Whitney U test producing the p values shown. Where p < 0.054 , the effect size from the Vargha–Delaney A test is shown.
We find that on 3 of the problems (n-EP, n-EPhand n-OPh) R-EGGP found solutions more
quickly than RCGP with respect to evaluations used (measured by median) with statistical significance (p < 0.054 ). On 1 problem, n-OP, we see lower MEs used but without statistical significance. In no problems do we see a large effect size. From these results we can infer that R-EGGP is in general more effective than RCGP at synthesising generalising n-bit parity checks, although on the problems studied the difference in performance is not large.
It is interesting to examine the rate of successful generalisation of solutions found by both algorithms. On all problems, the success rate (SR) was close to 100%. There are no significant differences between SRs, but it is interesting to see that both algorithms are capable of finding
n-bit parity checks which generalise to bit sequences longer than those they were trained on.
7.11 Conclusions and Future Work
In this chapter we have presented R-EGGP. We have presented an initialisation procedure which generates RFGs and parameterised by probability prec which controls the rate of recur-
rent edges in the initial solution. We have also given two edge mutations, one which mutates an edge to be non-recurrent while maintaining the acyclicity of the subgraph induced by non-recurrent edges, and the other which mutates an edge to be recurrent.
We have extensively compared R-EGGP to RCGP on various benchmark problems. On digital counter synthesis problems, we found that R-EGGP significantly outperforms RCGP on many problems, particularly the most difficult problems. On mathematical sequence synthesis problems, we found few statistical differences, but did observe that R-EGGP signif- icantly outperforms RCGP on one problem. On three of the n-bit parity problems, we found that R-EGGP significantly outperforms RCGP.
Overall, we have described and rigorously evaluated an approach for evolving RFGs which has genetic operators described as P-GP 2 programs. We have found that this technique can effectively learn solutions to a variety of recurrent program synthesis tasks and often outperforms RCGP. In particular we have seen that R-EGGP can synthesise recurrent digital circuits which generalise to solve problems they were not trained on.
There are a number of areas for future work on R-EGGP. Firstly, in our experiments we have fixed the rate of recurrent edges prec = 0.1. It would be interesting to carry out
experiments varying this parameter. We expect that increasing prec would lead to a larger
solution size, as recurrent edges may add entire new subgraphs to the active component. Whether this helps or hinders the evolutionary process is a matter for empirical analysis.
Interesting behaviour occurs at the two extremes of parameterisation of prec. When prec=
0, R-EGGP is equivalent to EGGP. However, when prec= 1, then all constraints of acyclicity
of the individual are removed. This leads to some interesting insights, for example that the initialisation procedure might be viewed as a variant of the directed random graph model from Section 3.3.4 with a fixed degree sequence (see [159]). Observations such as this may then yield new understanding of the biases of initialisation. R-EGGP with prec = 1 may be
used as a model of of search in many interesting domains, for example, in the search for a topology of an echo state network [108] or a random Boolean network [211].
Drift
Abstract
We introduce the concept of Semantic Neutral Drift (SND) for Evolving Graphs by Graph Programming (EGGP), where we exploit equivalence laws to design semantics-preserving mu- tations guaranteed to preserve individuals’ fitness scores. A number of digital circuit bench- mark problems are implemented with rule-based graph programs and empirically evaluated, demonstrating quantitative improvements in evolutionary performance. Analysis reveals that the benefits of the designed SND reside in more complex processes than simple growth of individuals, and that there are circumstances where it is beneficial to choose otherwise detri- mental parameters for a Genetic Programming (GP) system if that facilitates the inclusion of SND.
Relevant Publications
Content from the following publications is used in this chapter:
[11] T. Atkinson, D. Plump, and S. Stepney, “Evolving graphs with semantic neutral drift,” Natural Computing, 2019.
8.1 Introduction
In Genetic Programming (GP) the ability to escape local optima is key to finding globally optimal solutions. Neutral drift, a mechanism whereby individuals with fitness-equivalent phenotypes to the existing population may be generated by mutation [72] offers the search of new neighbourhoods for sampling thus increasing the chance of leaving local optima. A number of studies on neutrality in Cartesian Genetic Programming (CGP) [156,244,251] find it to be an almost always beneficial property for studied problems. In general, comparative studies [155] find that CGP using only mutation and neutral drift is able to compete with traditional Tree-Based GP (TGP) which uses more familiar crossover operators (see [129]) to introduce genetic variation.
A distinction has been made [244] between implicit neutral drift, where a genetic operator yields a semantically equivalent child, and explicit neutral drift, where a genetic operator only modifies intronic code. We note that many comparative studies largely focus on the role of both types of neutral drift as byproducts of existing genetic operators and neutrality within the representation [19,156,244,251] rather than as deliberately designed features of an evolutionary system. We propose the opposite; to employ domain knowledge of equivalence laws to specify mutation operators on the active components of individuals which always induce neutral drift. Hence our work can be viewed as an attempt to explicitly induce additional implicit neutral drift in the sense of [244].
We build on our approach, Evolving Graphs by Graph Programming (EGGP), by im- plementing semantics-preserving mutations to directly achieve neutral drift on the active components of individual solutions. Here, we implement logical equivalence laws as muta- tions on the active components of candidate solutions to digital circuit problems to produce semantically equivalent, equally fit, children. While our semantics-preserving mutations pro- duce semantically equivalent children they do not guarantee preservation of size; our fitness measures evaluate semantics only, not, for example, size or complexity.
We describe and implement Semantic Neutral Drift (SND) straightforwardly by using rule- based graph programs in P-GP 2. This continues from Chapter 5 where we use P-GP 2 to design acyclicity-preserving edge mutations for digital circuits that correctly identify the set of all possible valid mutations. The use of P-GP 2 here enables concise description of complex transformations such as De Morgan’s laws by identifying and rewriting potential matches for these laws in the existing formalism of graph transformation. This reinforces the notion that the direct encoding of solutions as graphs is useful as it allows immediate access to the
phenotype of individual solutions and makes it possible to design complex mutations by using powerful algorithmic concepts from graph programming.
We investigate four sets of semantics-preserving mutations for digital circuit design, three built upon logical equivalence laws and a fourth taken from term-graph rewriting. We run EGGP with each rule-set on a set of benchmark problems and establish statistically significant improvements in performance for most of our visited problems. An analysis of our results re- veals evidence that it is the semantic transformations, beyond simple ‘neutral growth’, which are aiding performance. We then combine our two best performing sets of mutation operators and evaluate this new set under the same conditions, achieving further improvements. We also provide evidence that, although operators implementing semantics-preserving mutations may be more difficult to use, the inclusion of those semantics-preserving mutations may allow evolution to out-perform equivalent processes that use ‘easier’ operators.
The rest of this chapter is organised as follows. In Section 8.2 we review existing literature on neutral drift in GP. In Section 8.3 we describe our extension to EGGP where we incor- porate deliberate neutral drifts into the evolutionary process. In Section 8.4 we describe our experimental setup and in Section 8.5 we give the results from these experiments. In Section 8.6 we provide in-depth analysis of these results to establish precisely what components of our approach are aiding performance. In Section 8.7 we conclude our work and propose potential future work on this topic.