Multi Expression Programming

Multi Expression Programming #

Name #

Multi Expression Programming (MEP)

Taxonomy #

Multi Expression Programming is a Genetic Programming variant that evolves linear chromosomes encoding complex expressions. It is closely related to Gene Expression Programming (GEP) and Linear Genetic Programming (LGP).

  • Artificial Intelligence
    • Computational Intelligence
      • Evolutionary Computation
        • Evolutionary Algorithms
          • Genetic Programming
            • Linear Genetic Programming
              • Multi Expression Programming

Strategy #

MEP individuals are represented as linear chromosomes encoding multiple expressions, allowing the evolution of more complex solutions compared to traditional tree-based GP. The expressions are encoded in a linear fashion, with each gene representing a terminal or a function. The expressions are linked together by a special linking function, which enables the formation of multiple subexpressions within a single chromosome.

The evolutionary process in MEP follows the standard steps of selection, reproduction, and replacement. During selection, individuals are chosen based on their fitness to undergo genetic operations. The reproduction phase applies genetic operators such as crossover and mutation to create new offspring. The offspring then replace less fit individuals in the population, driving the evolution towards better solutions.

MEP employs a unique encoding scheme that allows for the effective evolution of complex expressions. Each gene in the chromosome can encode either a terminal (e.g., a variable or constant) or a function (e.g., arithmetic or logical operators). The linking function connects the expressions encoded by different genes, enabling the formation of multiple subexpressions within a single individual.

Procedure #

  1. Initialize the population
    1. Set the population size and the maximum number of generations
    2. Create random individuals by generating linear chromosomes of a fixed length
      1. For each gene in the chromosome, randomly assign either a terminal or a function
      2. Ensure the chromosome is syntactically valid and can be parsed into expressions
    3. Evaluate the fitness of each individual in the population
  2. Perform the evolutionary process until the termination criterion is met
    1. Select parents for reproduction using a selection method (e.g., tournament selection)
    2. Apply genetic operators to the selected parents
      1. Crossover: Exchange genetic material between two parents to create offspring
      2. Mutation: Randomly modify genes in the offspring with a certain probability
    3. Evaluate the fitness of the offspring
    4. Replace less fit individuals in the population with the offspring
  3. Return the best individual found during the evolution

Data Structures #

  • Individual: Represents a candidate solution in MEP, encoded as a linear chromosome of fixed length. Each gene in the chromosome can be either a terminal or a function.
  • Population: A collection of individuals that evolve over generations. The population size remains constant throughout the evolution.
  • Fitness Function: Evaluates the quality of an individual by measuring how well it solves the given problem. The fitness function is problem-specific and guides the evolution towards better solutions.

Parameters #

  • Population Size: The number of individuals in the population. A larger population size increases diversity but also requires more computational resources.
  • Max Generations: The maximum number of generations allowed for the evolution. It determines the stopping criterion for the algorithm.
  • Crossover Probability: The probability of applying the crossover operator to selected parents. Higher values promote exploration of the search space.
  • Mutation Probability: The probability of applying the mutation operator to offspring. It introduces random variations and maintains diversity in the population.

Considerations #

Advantages #

  • Expressiveness: MEP can evolve complex expressions by combining multiple subexpressions, allowing for the discovery of intricate solutions.
  • Flexibility: The linear chromosome representation in MEP allows for easy implementation of various genetic operators and modifications to the algorithm.
  • Efficiency: MEP’s encoding scheme enables the evolution of compact and efficient solutions compared to traditional tree-based GP.

Disadvantages #

  • Chromosome Length: The fixed length of MEP chromosomes may limit the complexity of the evolved solutions. Determining the optimal chromosome length can be challenging.
  • Function Set: The choice of functions included in the function set significantly impacts the search space and the algorithm’s performance. Selecting an appropriate function set requires domain knowledge.
  • Bloat: MEP may suffer from bloat, where the evolved expressions become unnecessarily large without improving fitness. Bloat control techniques may be needed to mitigate this issue.

Heuristics #

Function Set Selection #

  • Include functions that are relevant to the problem domain and can effectively express the desired solutions.
  • Start with a minimal function set and gradually expand it if necessary, as a larger function set increases the search space.
  • Consider the tradeoff between expressiveness and complexity when choosing the function set.

Chromosome Length #

  • Set the chromosome length based on the expected complexity of the solutions and the available computational resources.
  • Experiment with different chromosome lengths to find a balance between solution quality and computational efficiency.
  • Longer chromosomes allow for more complex solutions but may require more computational effort.

Population Size and Generations #

  • Determine the population size based on the available computational resources and the problem complexity.
  • A larger population size promotes diversity but requires more evaluations per generation.
  • Set the maximum number of generations based on the problem complexity and the available computational budget.
  • Monitor the convergence of the population and consider early stopping if the fitness improvement stagnates.

Genetic Operator Probabilities #

  • Set the crossover probability relatively high (e.g., 0.8-0.9) to encourage exploration of the search space.
  • Set the mutation probability relatively low (e.g., 0.01-0.1) to introduce small variations and maintain diversity.
  • Adjust the probabilities based on the problem characteristics and the desired balance between exploration and exploitation.

Fitness Function Design #

  • Design a fitness function that accurately measures the quality of the evolved solutions in the context of the problem.
  • Consider incorporating multiple objectives or constraints into the fitness function if necessary.
  • Normalize the fitness values to ensure fair comparison between individuals.

Elitism and Diversity Maintenance #

  • Implement elitism to preserve the best individuals across generations, preventing the loss of high-quality solutions.
  • Apply diversity maintenance techniques, such as fitness sharing or crowding, to prevent premature convergence and maintain a diverse population.
  • Monitor the diversity of the population and consider introducing new randomly generated individuals if diversity drops significantly.