Numerical Simplification and its Effect on Fragment Distributions in Genetic Programming

<p>In tree-based genetic programming (GP) there is a tendency for the program trees to increase in size from one generation to the next. If this increase in program size is not accompanied by an improvement in fitness then this unproductive increase is known as bloat. It is standard practice to place some form of control on program size. This can be done by limiting the number of nodes or the depth of the program trees, or by adding a component to the fitness function that rewards smaller programs (parsimony pressure) or by simplifying individual programs using algebraic methods. This thesis proposes a novel program simplification method called numerical simplification that uses only the range of values the nodes take during fitness evaluation. The effect of online program simplification, both algebraic and numerical, on program size and resource usage is examined. This thesis also examines the distribution of program fragments within a genetic programming population and how this is changed by using simplification. It is shown that both simplification approaches result in reductions in average program size, memory used and computation time and that numerical simplification performs at least as well as algebraic simplification, and in some cases will outperform algebraic simplification. This reduction in program size and the resources required to process the GP run come without any significant reduction in accuracy. It is also shown that although the two online simplification methods destroy some existing program fragments, they generate new fragments during evolution, which compensates for any negative effects from the disruption of existing fragments. It is also shown that, after the first few generations, the rate new fragments are created, the rate fragments are lost from the population, and the number of distinct (different) fragments in the population remain within a very narrow range of values for the remainder of the run.</p>

Numerical Simplification and its Effect on Fragment Distributions in Genetic Programming

<p>In tree-based genetic programming (GP) there is a tendency for the program trees to increase in size from one generation to the next. If this increase in program size is not accompanied by an improvement in fitness then this unproductive increase is known as bloat. It is standard practice to place some form of control on program size. This can be done by limiting the number of nodes or the depth of the program trees, or by adding a component to the fitness function that rewards smaller programs (parsimony pressure) or by simplifying individual programs using algebraic methods. This thesis proposes a novel program simplification method called numerical simplification that uses only the range of values the nodes take during fitness evaluation. The effect of online program simplification, both algebraic and numerical, on program size and resource usage is examined. This thesis also examines the distribution of program fragments within a genetic programming population and how this is changed by using simplification. It is shown that both simplification approaches result in reductions in average program size, memory used and computation time and that numerical simplification performs at least as well as algebraic simplification, and in some cases will outperform algebraic simplification. This reduction in program size and the resources required to process the GP run come without any significant reduction in accuracy. It is also shown that although the two online simplification methods destroy some existing program fragments, they generate new fragments during evolution, which compensates for any negative effects from the disruption of existing fragments. It is also shown that, after the first few generations, the rate new fragments are created, the rate fragments are lost from the population, and the number of distinct (different) fragments in the population remain within a very narrow range of values for the remainder of the run.</p>

Lambda calculus with algebraic simplification for reduction parallelisation: Extended study

Parallel reduction is a major component of parallel programming and widely used for summarisation and aggregation. It is not well understood, however, what sorts of non-trivial summarisations can be implemented as parallel reductions. This paper develops a calculus named λAS, a simply typed lambda calculus with algebraic simplification. This calculus provides a foundation for studying a parallelisation of complex reductions by equational reasoning. Its key feature is δ abstraction. A δ abstraction is observationally equivalent to the standard λ abstraction, but its body is simplified before the arrival of its arguments using algebraic properties such as associativity and commutativity. In addition, the type system of λAS guarantees that simplifications due to δ abstractions do not lead to serious overheads. The usefulness of λAS is demonstrated on examples of developing complex parallel reductions, including those containing more than one reduction operator, loops with conditional jumps, prefix sum patterns and even tree manipulations.

Pruning of genetic programming trees using permutation tests

We present a novel approach based on statistical permutation tests for pruning redundant subtrees from genetic programming (GP) trees that allows us to explore the extent of effective redundancy . We observe that over a range of regression problems, median tree sizes are reduced by around 20% largely independent of test function, and that while some large subtrees are removed, the median pruned subtree comprises just three nodes; most take the form of an exact algebraic simplification. Our statistically-based pruning technique has allowed us to explore the hypothesis that a given subtree can be replaced with a constant if this substitution results in no statistical change to the behavior of the parent tree—what we term approximate simplification. In the eventuality, we infer that more than 95% of the accepted pruning proposals are the result of algebraic simplifications, which provides some practical insight into the scope of removing redundancies in GP trees.

KESULITAN SISWA DALAM MENYELESAIKAN OPERASI HITUNG ALJABAR BENTUK PECAHAN DITINJAU DARI KEMAMPUAN MATEMATIKA SISWA SMP KELAS VII SMP NEGERI 1 SALATIGA

Abstrak: Penelitian ini bertujuan untuk mendiskripsikan kesalahan dan kesulitan siswa dalam menyelesaikan soal operasi hitung aljabar bentuk pecahan serta mengetahui factor-faktor penyebabnya ditinjau dari kemampuan matematika siswa di SMP N 1 Salatiga. Metode pengambilan subjek yaitu berdasarkan purposive sampling yang terdiri dari tiga subjek. Berdasarkan hasil dan pembahasan penelitian dapat disimpulkan bahwa 1) subjek berkemampuan rendah dan sedang melakukan tiga kesalahan konsep, prinsip, dan operasi hitung, 2) subjek berkemampuan tinggi tidak melakukan kesalahan prinsip. Kesulitan yang paling banyak dialami siswa adalah kesulitan pada konsep dan penyelesaian operasi. Faktor penyebabnya adalah pemahaman siswa yang kurang terkait aturan penyederhanaan aljabar bentuk pecahan.Abstract: This research has a purpose to describe errors and difficulties of students in solving algebra arithmetic counting problems and to know the factors of student’s math ability in SMP N 1 Salatiga. The data of this study were collected through purposive sampling that consisted of three subjects. The findings and resukt showed that of the research can be concluded that 1) the low-ability subjects and are doing three errors of concepts, principles, and operations count, 2) high-ability subjects not to make a mistake principle. The most difficulties that student’s faced in terms of the operation. It could be happened because the factor of students lack of understanding of the rules of algebraic simplification of fractional forms.