Cellular Automata (CA) are commonly investigated as a particular type of dynamical systems, defined by shift-invariant local rules. In this paper, we consider instead CA as algebraic systems, focusing on the combinatorial designs induced by their short-term behavior. Specifically, we review the main results published in the literature concerning the construction of mutually orthogonal Latin squares via bipermutive CA, considering both the linear and nonlinear cases. We then survey some significant applications of these results to cryptography, and conclude with a discussion of open problems to be addressed in future research on CA-based combinatorial designs.

Abstract Geometric Semantic Genetic Programming ( $$\mathsf {GSGP}$$ ) is a powerful variant of Genetic Programming (GP) that defines genetic operators inducing unimodal fitness landscapes. In recent years, a new mutation operator, Geometric Semantic Mutation with Local Search (GSM-LS), has been proposed to include a local search step in the mutation process. The core idea of GSM-LS is to incorporate a linear regression step during mutation, thereby accelerating convergence toward high-quality solutions. While GSM-LS helps the convergence of the evolutionary search, it is prone to overfitting. Thus, it was suggested to apply GSM-LS only for a limited number of generations and then revert to standard geometric semantic mutation. A more recently defined variant of $$\mathsf {GSGP}$$ (called $$\mathsf {GSGP}$$ -reg) also includes a local search step, but shares similar strengths and weaknesses with GSM-LS. Here, we investigate several strategies to mitigate overfitting in GSM-LS and $$\mathsf {GSGP}$$ -reg, ranging from simple regularized regression techniques to adaptive methods that estimate overfitting risk at each mutation. The latter approaches partition the training set into two subsets: one used to perform the mutation, and the other to evaluate the risk of overfitting based on the mutation’s impact on held-out data. Experimental evaluations across seven real-world regression benchmarks show that, while plain GSGP underperforms on all datasets, methods incorporating local search often achieve significantly better test performance. For example, on the Airfoil dataset, the GSM-LS variant achieves a median RMSE below 10 compared to 30 with standard GSGP. On the LD50 and Bioavailability datasets, the proposed gen and ridge-regularized variants effectively mitigate overfitting, reducing test RMSE by up to 40% relative to baseline GSGP. We conclude that local search, when used with regularization strategies, enhances GSGP’s performance and generalization capability across a diverse range of tasks.

Curriculum learning (CL) consists in using a diverse set of user-provided test cases, with varying levels of difficulty and organized in a suitable progression, for learning a policy. The quality of test cases is important to allow optimization techniques as genetic programming (GP) to solve policy search problems. In this work, we evaluate large language models (LLMs) as providers of test cases for GP-based policy search. We consider two policy search tasks, a single-player and a multi-player game, and four LLMs differing in complexity and specialization, which we prompt in order to generate suitable test cases for the two games. We experimentally assess the intrinsic quality of LLM-generated test cases and their utility when inserted in a curriculum consumed by a GP optimization. We evaluate the robustness of the approach with respect to the way cases are scheduled in curricula and with respect to the policy representation, for which we use both graphs and linear programs evolved by GP. We observe that the effectiveness of LLM-assisted CL depends on both the choice of LLM and the design of the prompting and scheduling strategies. These findings highlight important considerations for leveraging LLMs in automated curriculum design for GP-based optimization.

Despite their impressive performance in classification tasks, neural networks are known to be vulnerable to adversarial attacks, subtle perturbations of the input data designed to deceive the model. In this work, we investigate the correlation between these perturbations and the implicit bias of neural networks trained with gradient-based algorithms. To this end, we analyse a representation of the network’s implicit bias through the lens of the Fourier transform. Specifically, we identify unique fingerprints of implicit bias and adversarial attacks by calculating the minimal, essential frequencies needed for accurate classification of each image, as well as the frequencies that drive misclassification in its adversarially perturbed counterpart. This approach enables us to uncover and analyse the correlation between these essential frequencies, providing a precise map of how the network’s biases align or contrast with the frequency components exploited by adversarial attacks. To this end, among other methods, we use a newly introduced technique capable of detecting nonlinear correlations between high-dimensional datasets. Our results provide empirical evidence that the network bias in Fourier space and the target frequencies of adversarial attacks are highly correlated and suggest new potential strategies for adversarial defence. Code is available at https://github.com/lorenzobasile/ImplicitBiasAdversarial

We explore the computational complexity of deciding the existence of fixed points and cycles that can be reached from any other states (called global attractors ) in the dynamics of inhibitorless and reactantless reaction systems. The problems we consider are all known to be PSPACE -complete in the case of unconstrained reaction systems; in this paper, we show that some of them become polynomially solvable when limited to inhibitorless and reactantless reaction systems, while others remain PSPACE -complete. Specifically, we prove that the problems of deciding (i) if a given state belongs to a cycle, (ii) whether two reaction systems have at least one cycle in common, and (iii) whether they have the same set of cycles, remain PSPACE -complete even in the inhibitorless and reactantless classes, as well as the problem of deciding if a global cycle attractor exists in a reactantless reaction system. Interestingly, however, we demonstrate that no global cycle attractor of length at least 2 can exist in inhibitorless reaction systems; and no global cycle attractor of length greater than 2 can exist in reactantless reaction systems. Furthermore, we show that the problems of deciding whether a given state is a global attractor and whether a global fixed point attractor exists become polynomially solvable when restricted to inhibitorless and reactantless reaction systems.

Genetic programming (GP) is a popular problem-solving and optimization technique. However, generating effective test cases for training and evaluating GP programs requires strong domain knowledge. Furthermore, GP programs often prematurely converge on local optima when given excessively difficult problems early in their training. Curriculum learning (CL) has been effective in addressing similar issues across different reinforcement learning (RL) domains, but it requires the manual generation of progressively difficult test cases as well as their careful scheduling. In this work, we leverage the domain knowledge and the strong generative abilities of large language models (LLMs) to generate effective test cases of increasing difficulties and schedule them according to various curricula. We show that by integrating a curriculum scheduler with LLM-generated test cases we can effectively train a GP agent player with environments-based curricula for a single-player game and opponent-based curricula for a multi-player game. Finally, we discuss the benefits and challenges of implementing this method for other problem domains.

Abstract This work introduces the new class of pure reaction automata, as well as a new update manner, called maximal reactive manner, that can also be applied to standard reaction automata. Pure reaction automata differ from the standard model in that they don’t have permanence: the entities that are not consumed by the reactions happening at a certain state are not conserved in the result states. We prove that the set of languages accepted by the new class under the maximal reactive manner contains the set of languages accepted by standard reaction automata under the same manner or under the maximal parallel manner. We also prove that a strict subclass of pure reaction automata can compute any partial recursive function.

Abstract Reaction systems are discrete dynamical systems that simulate biological processes within living cells through finite sets of reactants, inhibitors, and products. In this paper, we study the computational complexity of deciding on the existence of fixed points and attractors in the restricted class of additive reaction systems, in which each reaction involves at most one reactant and no inhibitors. We prove that all the considered problems, that are known to be hard for other classes of reaction systems, are polynomially solvable in additive systems. To arrive at these results, we provide several non-trivial reductions to problems on a polynomially computable graph representation of reaction systems that might prove useful for addressing other related problems in the future.

Abstract Most of the approaches published in the literature to construct S-boxes via Cellular Automata (CA) work by either iterating a finite CA for several time steps, or by a one-shot application of the global rule. The main characteristic that brings together these works is that they employ a single CA rule to define the vectorial Boolean function of the S-box. In this work, we explore a different direction for the design of S-boxes that leverages on Orthogonal CA (OCA), i.e. pairs of CA rules giving rise to orthogonal Latin squares. The motivation stands on the facts that an OCA pair already defines a bijective transformation, and moreover the orthogonality property of the resulting Latin squares ensures a minimum amount of diffusion. We exhaustively enumerate all S-boxes generated by OCA pairs of diameter $$4 \le d \le 6$$ 4 ≤ d ≤ 6 , and measure their nonlinearity. Interestingly, we observe that for $$d=4$$ d = 4 and $$d=5$$ d = 5 all S-boxes are linear, despite the underlying CA local rules being nonlinear. The smallest nonlinear S-boxes emerges for $$d=6$$ d = 6 , but their nonlinearity is still too low to be used in practice. Nonetheless, we unearth an interesting structure of linear OCA S-boxes, proving that their Linear Components Space is itself the image of a linear CA, or equivalently a polynomial code. We finally classify all linear OCA S-boxes in terms of their generator polynomials.

Reaction systems are discrete dynamical systems that model biochemical processes in living cells using finite sets of reactants, inhibitors, and products. We investigate the computational complexity of a comprehensive set of problems related to the existence of fixed points and attractors in two constrained classes of reaction systems, in which either reactants or inhibitors are disallowed. These problems have biological relevance and have been extensively studied in the unconstrained case; however, they remain unexplored in the context of reactantless or inhibitorless systems. Interestingly, we demonstrate that although the absence of reactants or inhibitors simplifies the system's dynamics, it does not always lead to a reduction in the complexity of the considered problems.

Research has shown how data sets convey social bias in AI systems, especially those based on machine learning. A biased data set is not representative of reality and might contribute to perpetuate societal biases within the model. To tackle this problem, it is important to understand how to avoid biases, errors, and unethical practices while creating the data sets. In this work we offer a preliminary framework for the semi-automated evaluation of fairness in data sets, by combining statistical information about data with qualitative consideration. We address the issue of how much (un)fairness can be included in a data set used for machine learning research, focusing on classification issues. In order to provide guidance for the use of data sets in contexts of critical decision-making, such as health decisions, we identify six fundamental features (balance, numerosity, unevenness, compliance, quality, incompleteness) that could affect model fairness. We developed a rule-based approach based on fuzzy logic that combines these characteristics into a single score and enables a semi-automatic evaluation of a data set in algorithmic fairness research.

Graphs are a way to describe complex entities and their relations that apply to many practically relevant domains. However, domains often differ not only in the properties of nodes and edges, but also in the constraints imposed to the overall structure. This makes hard to define a general representation and genetic operators for graphs that permit the evolutionary optimization over many domains. In this paper, we tackle this challenge. We first propose a representation template that can be customized by users for specific domains: the constraints and the genetic operators are given in Prolog, a declarative programming language for operating with logic. Then, we define an adaptive evolutionary algorithm that can work with a large number of genetic operators by modifying their usage probability during the evolution: in this way, we relieve the user from the burden of selecting in advance only operators that are "good enough". We experimentally evaluate our proposal on two radically different domains to demonstrate its applicability and effectiveness: symbolic regression with trees and text extraction with finite-state automata. The results are promising: our approach does not trade effectiveness for versatility and is not worse than other domain-tailored approaches.

Abstract An interesting thread in the research of Boolean functions for cryptography and coding theory is the study of secondary constructions: given a known function with a good cryptographic profile, the aim is to extend it to a (usually larger) function possessing analogous properties. In this work, we continue the investigation of a secondary construction based on cellular automata (CA), focusing on the classes of bent and semi-bent functions. We prove that our construction preserves the algebraic degree of the local rule, and we narrow our attention to the subclass of quadratic functions, performing several experiments based on exhaustive combinatorial search and heuristic optimization through Evolutionary Strategies (ES). Finally, we classify the obtained results up to permutation equivalence, remarking that the number of equivalence classes that our CA-XOR construction can successfully extend grows very quickly with respect to the CA diameter.

We investigate the use of Genetic Programming (GP) as a convolutional predictor for missing pixels in images. The training phase is performed by sweeping a sliding window over an image, where the pixels on the border represent the inputs of a GP tree. The output of the tree is taken as the predicted value for the central pixel. We consider two topologies for the sliding window, namely the Moore and the Von Neumann neighborhood. The best GP tree scoring the lowest prediction error over the training set is then used to predict the pixels in the test set. We experimentally assess our approach through two experiments. In the first one, we train a GP tree over a subset of 1000 complete images from the MNIST dataset. The results show that GP can learn the distribution of the pixels with respect to a simple baseline predictor, with no significant differences observed between the two neighborhoods. In the second experiment, we train a GP convolutional predictor on two degraded images, removing around 20% of their pixels. In this case, we observe that the Moore neighborhood works better, although the Von Neumann neighborhood allows for a larger training set.

This supplement contains seven revised and extended papers selected from CIBB 2018 and CIBB 2019, the 15th and 16th editions of the international conference on Computational Intelligence Methods for Bioinformatics and Biostatistics. CIBB is a venue that embraces researchers with different backgrounds, ranging from mathematics to computer science, from materials science to medicine, and from engineering to biology, all interested in the investigation and application of computational intelligence methods to open problems in bioinformatics, biostatistics, systems biology, synthetic biology, and medical informatics.

Global optimization problems can be effectively solved by means of Computational Intelligence methods. However, there are several areas in which the effectiveness of these algorithms can be hampered by the computational costs of the fitness evaluations, or by specific features of the fitness landscape that can be characterized by noise and by the presence of several (even infinite) local optima. These issues bring about the necessity of defining specific techniques to replace the original problem with a surrogate representation. Fourier surrogate modeling represents a novel and effective approach to generate smoother, and possibly easier to explore, fitness landscapes, and to reduce the computational effort. Fourier surrogates require an initial sampling of the search space that must be performed to calculate the Fourier transforms. In this paper we investigate the impact on the quality of the surrogate models of the hyper-parameters of the methodology, and of several methods that can be employed for the initial sampling of the fitness landscape (i.e., pseudorandom numbers, low discrepancy sequences, a logistic map in chaotic regime, true random positions generated by a quantum computer, and point packing). Our results show that semistructured approaches like quasi-random sequences and point packing can outperform the other sampling methods.

One of the most complex problems in Systems Biology is Parameter estimation (PE), which consists in inferring the kinetic parameters of biochemical systems. The identification of an accurate parameterization, able to reproduce any observed experimental behavior, is fundamental for the definition of predictive models. PE is a non-convex, multi-modal, and non-separable problem that is usually tackled by using Computational Intelligence methods. When the biochemical species appear in the system in a very low amount, the intrinsic noise due to the randomness of molecular collisions cannot be neglected. In this case, stochastic simulation algorithms should be employed to properly reproduce the system dynamics. Stochastic fluctuations make the PE problem even more complicated, as they can lead to radically different values of the fitness function for the same candidate parameterization. In addition, the kinetic parameters generally follow a log-uniform distribution, so that global optima tend to be localized in the lowest orders of magnitude of the search space. To simultaneously tackle all the aforementioned issues, in this work we investigate a novel approach based on the combination of dilation functions with Fourier surrogate modeling and filtering on the fitness landscape. The results show that our approach is able to strongly simplify the PE problem for low-dimensional optimization instances.

Tasks related to Natural Language Processing (NLP) have recently been the focus of a large research endeavor by the machine learning community. The increased interest in this area is mainly due to the success of deep learning methods. Genetic Programming (GP), however, was not under the spotlight with respect to NLP tasks. Here, we propose a first proof-of-concept that combines GP with the well established NLP tool word2vec for the next word prediction task. The main idea is that, once words have been moved into a vector space, traditional GP operators can successfully work on vectors, thus producing meaningful words as the output. To assess the suitability of this approach, we perform an experimental evaluation on a set of existing newspaper headlines. Individuals resulting from this (pre-)training phase can be employed as the initial population in other NLP tasks, like sentence generation, which will be the focus of future investigations, possibly employing adversarial co-evolutionary approaches.

Computational Intelligence (CI) is a computer science discipline encompassing the theory, design, development and application of biologically and linguistically derived computational paradigms. Traditionally, the main elements of CI are Evolutionary Computation, Swarm Intelligence, Fuzzy Logic, and Neural Networks. CI aims at proposing new algorithms able to solve complex computational problems by taking inspiration from natural phenomena. In an intriguing turn of events, these nature-inspired methods have been widely adopted to investigate a plethora of problems related to nature itself. In this paper we present a variety of CI methods applied to three problems in life sciences, highlighting their effectiveness: we describe how protein folding can be faced by exploiting Genetic Programming, the inference of haplotypes can be tackled using Genetic Algorithms, and the estimation of biochemical kinetic parameters can be performed by means of Swarm Intelligence. We show that CI methods can generate very high quality solutions, providing a sound methodology to solve complex optimization problems in life sciences.

Reaction systems are a computational model inspired by the bio-chemical reactions that happen inside biological cells. They have been and currently are studied for their many nice theoretical properties. They are also a useful modeling tool for biochemical systems, but in order to be able to employ them effectively in the field the presence of efficient and widely available simulators is essential. Here we explore three different algorithms and implementations of the simulation, comparing them to the current state of the art. We also show that we can obtain performances comparable to GPU-based simulations on real-world systems by using a carefully tuned CPU-based simulator.

Abstract We continue the study of asynchrony immunity in cellular automata (CA), which can be considered as a generalization of correlation immunity in the case of vectorial Boolean functions. The property could have applications as a countermeasure for side-channel attacks in CA-based cryptographic primitives, such as S-boxes and pseudorandom number generators. We first give some theoretical results on the properties that a CA rule must satisfy in order to meet asynchrony immunity, like central permutivity. Next, we perform an exhaustive search of all asynchrony immune CA rules of neighborhood size up to 5, leveraging on the discovered theoretical properties to greatly reduce the size of the search space.

Context-free grammars are useful tools for modeling the solution space of problems that can be solved by optimization algorithms. For a given solution space, there exists an infinite number of grammars defining that space, and there are clues that changing the grammar may impact the effectiveness of the optimization. In this article, we investigate theoretically and experimentally the possibility of specializing a grammar in a problem, that is, of systematically improving the quality of the grammar for the given problem. To this end, we define the quality of a grammar for a problem in terms of the average fitness of the candidate solutions generated using that grammar. Theoretically, we demonstrate the following findings: 1) that a simple mutation operator employed in a (1 + 1)-EA setting can be used to specialize a grammar in a problem without changing the solution space defined by the grammar and 2) that three grammars of equal quality for a grammar-based version of the ONEMAX problem greatly vary in how they can be specialized with that (1 + 1)-EA, as the expected time required to obtain the same improvement in quality can vary exponentially among grammars. Then, experimentally, we validate the theoretical findings and extend them to other problems, grammars, and a more general version of the mutation operator.

Geometric semantic genetic programming is a hot topic in evolutionary computation and recently it has been used with success on several problems from Biology and Medicine. Given the young age of geometric semantic genetic programming, in the last few years theoretical research, aimed at improving the method, and applicative research proceeded rapidly and in parallel. As a result, the current state of the art is confused and presents some “holes”. For instance, some recent improvements of geometric semantic genetic programming have never been applied to some popular biomedical applications. The objective of this paper is to fill this gap. We consider the biomedical applications that have more frequently been used by genetic programming researchers in the last few years and we systematically test, in a consistent way, using the same parameter settings and configurations, all the most popular existing variants of geometric semantic genetic programming on all those applications. Analysing all these results, we obtain a much more homogeneous and clearer picture of the state of the art, that allows us to draw stronger conclusions.

Traditionally, P systems allow their membranes or cells to grow exponentially (or even more) in volume with respect to the size of the multiset of objects they contain in the initial configuration. This behaviour is, in general, biologically unrealistic, since large cells tend to divide in order to maintain a suitably large surface-area-to-volume ratio. On the other hand, it is usually the number of cells that needs to grow exponentially with time by binary division in order to solve NP-complete problems in polynomial time. In this paper we investigate families of tissue P systems with cell division where each cell has a small volume (i.e., sub-polynomial with respect to the input size), assuming that each bit of information contained in the cell, including both those needed to represent the multiset of objects and the cell label, occupies a unit of volume. We show that even a constant volume bound allows us to reach computational universality for families of tissue P systems with cell division, if we employ an exponential-time uniformity condition on the families. Furthermore, we also show that a sub-polynomial volume does not suffice to solve NP-complete problems in polynomial time, unless the satisfiability problem for Boolean formulae can be solved in sub-exponential time, and that solving an NP-complete problem in polynomial time with logarithmic cell volume implies P = NP.

Reaction systems represent a theoretical framework based on the regulation mechanisms of facilitation and inhibition of biochemical reactions. The dynamic process defined by a reaction system is typically derived by hand, starting from the set of reactions and a given context sequence. However, thi s procedure may be error-prone and time-consuming, especially when the size of the reaction system increases. Here we present HERESY, a simulator of reaction systems accelerated on Graphics Processing Units (GPUs). HERESY is based on a fine-grained parallelization strategy, whereby all reactions are simultaneously executed on the GPU, therefore reducing the overall running time of the simulation. HERESY is particularly advantageous for the simulation of large-scale reaction systems, consisting of hundreds or thousands of reactions. By considering as test case some reaction systems with an increasing number of reactions and entities, as well as an increasing number of entities per reaction, we show that HERESY allows up to 29× speed-up with respect to a CPU-based simulator of reaction systems. Finally, we provide some directions for the optimization of HERESY, considering minimal reaction systems in normal form.

Polynomial-time P systems with active membranes characterise PSPACE by exploiting membranes nested to a polynomial depth, which may be subject to membrane division rules. When only elementary (leaf) membrane division rules are allowed, the computing power decreases to P PP = P #P , the class of problems solvable in polynomial time by deterministic Turing machines equipped with oracles for counting (or majority) problems. In this paper we investigate a variant of intermediate power, limiting membrane nesting (hence membrane division) to constant depth, and we prove that the resulting P systems can solve all problems in the counting hierarchy CH, which is located between P PP and PSPACE. In particular, for each integer k ≥ 0 we provide a lower bound to the computing power of P systems of depth k.

We show that a constant amount of space is sufficient to simulate a polynomial-space bounded Turing machine by P systems with active membranes. We thus obtain a new characterisation of PSPACE, which raises interesting questions about the definition o

Geometric Semantic Genetic Programming (GSGP) is a recently introduced framework to design domain-specific search operators for Genetic Programming (GP) to search directly the semantic space of functions. The fitness landscape seen by GSGP is always - for any domain and for any problem - unimodal with a constant slope by construction. This makes the search for the optimum much easier than for traditional GP, and it opens the way to analyse theoretically in a easy manner the optimisation time of GSGP in a general setting. We design and analyse a mutation-based GSGP for the class of all classification tree learning problems, which is a classic GP application domain.

Geometric Semantic Genetic Programming (GSGP) is a recently introduced form of Genetic Programming (GP), rooted in a geometric theory of representations, that searches directly the semantic space of functions/programs, rather than the space of their syntactic representations (e.g., trees) as in traditional GP. Remarkably, the fitness landscape seen by GSGP is always -- for any domain and for any problem -- unimodal with a linear slope by construction. This has two important consequences: (i) it makes the search for the optimum much easier than for traditional GP; (ii) it opens the way to analyse theoretically in a easy manner the optimisation time of GSGP in a general setting. The runtime analysis of GP has been very hard to tackle, and only simplified forms of GP on specific, unrealistic problems have been studied so far. We present a runtime analysis of GSGP with various types of mutations on the class of all Boolean functions.

Reaction systems is a formalism inspired by chemical reactions introduced by Rozenberg and Ehrenfeucht. Recently, an evolutionary algorithm based on this formalism, called Evolutionary Reaction Systems, has been presented. This new algorithm proved to have comparable performances to other well-established machine learning methods, like genetic programming, neural networks and support vector machines on both artificial and real-life problems. Even if the results are encouraging, to make Evolutionary Reaction Systems an established evolutionary algorithm, an in depth analysis of the effect of its parameters on the search process is needed, with particular focus on those parameters that are typical of Evolutionary Reaction Systems and do not have a counterpart in traditional evolutionary algorithms. Here we address this problem for the first time. The results we present show that one particular parameter, between the ones tested, has a great influence on the performances of Evolutionary Reaction Systems, and thus its setting deserves practitioners' particular attention: the number of symbols used to represent the reactions that compose the system. Furthermore, this work represents a first step towards the definition of a set of default parameter values for Evolutionary Reaction Systems, that should facilitate their use for beginners or inexpert practitioners.

Genetic programming (GP) is not a field noted for the rigor of its benchmarking. Some of its benchmark problems are popular purely through historical contingency, and they can be criticized as too easy or as providing misleading information concerning real-world performance, but they persist largely because of inertia and the lack of good alternatives. Even where the problems themselves are impeccable, comparisons between studies are made more difficult by the lack of standardization. We argue that the definition of standard benchmarks is an essential step in the maturation of the field. We make several contributions towards this goal. We motivate the development of a benchmark suite and define its goals; we survey existing practice; we enumerate many candidate benchmarks; we report progress on reference implementations; and we set out a concrete plan for gathering feedback from the GP community that would, if adopted, lead to a standard set of benchmarks.

This work studies some aspects of the computational power of fully asynchronous cellular automata (ACA). We deal with some notions of simulation between ACA and Turing Machines. In particular, we characterize the updating sequences specifying which are “universal”, i.e., allowing a (specific family of) ACA to simulate any Turing machine on any input. We also consider the computational cost of such simulations. Finally, we deal with ACA equipped with peculiar updating sequences, namely those generated by random walks.

The NK landscapes are a well known benchmark for genetic algorithms (GAs) in which it is possible to tune the ruggedness of the fitness landscape by simply modifying the value of a parameter K. They have successfully been used in many theoretical studies, allowing researchers to discover interesting properties of the GAs dynamics in presence of rugged landscapes. A similar benchmark does not exist for genetic programming (GP) yet. Nevertheless, during the EuroGP conference debates of the last few years, the necessity of defining new benchmark problems for GP has repeatedly been expressed by a large part of the attendees. This paper is intended to fill this gap, by introducing an extension of the NK landscapes to tree based GP, that we call K landscapes. In this benchmark, epistasis are expressed as growing mutual interactions between the substructures of a tree as the parameter K increases. The fact that the problem becomes more and more difficult as the value of K increases is experimentally demonstrated. Interestingly, we also show that GP "bloats" more and more as K increases.

Generalization is an important issue in machine learning. In fact, in several applications good results over training data are not as important as good results over unseen data. While this problem was deeply studied in other machine learning techniques, it has become an important issue for genetic programming only in the last few years. In this paper we compare the generalization ability of several different genetic programming frameworks, including some variants of multi-objective genetic programming and operator equalization, a recently defined bloat free genetic programming system. The test problem used is a hard regression real-life application in the field of drug discovery and development, characterized by a high number of features and where the generalization ability of the proposed solutions is a crucial issue. The results we obtained show that, at least for the considered problem, multi-optimization is effective in improving genetic programming generalization ability, outperforming all the other methods on test data.

Distances that are bound to (or consistent with) genetic operators are measures that quantify the difficulty of reaching and individual (or a population) starting from another individual (or population) and applying the genetic operator iteratively. Defining distance measures bound to genetic operators is a very important task in evolutionary computation. In fact these distances usually make the analysis of some indicators of the the search process, like for instance population diversity or well-known measures of problem hardness such as fitness distance correlation, more accurate. In this paper, we introduce a distance measure bound to one point standard crossover for genetic algorithms. This measure quantifies the minimum number of crossover operations that have to be applied to a population to tranform it into another population. It is based on the definition of a lattice over some particular schemata that represent the individuals in the population and on the construction of a discrete dynamic system that models the dynamics of the genetic algorithm under the sole effect of crossover. Using this distance measure, it is also possible to build a family of distances between individuals.