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...sted. 1 Introduction The evolution of digital multipliers has proved to be very difficult for evolutionary techniques (particularly when the number of bits in the multiplicands is greater than three) =-=[4]-=-[7][11][12][13]. Cartesian Genetic Programming (CGP) [5][6] is one technique that has been used to attack such problems. Even though CGP does not have the equivalent of Automatically Defined Functions...

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...uch problems. Even though CGP does not have the equivalent of Automatically Defined Functions (ADFs) it was empirically demonstrated to be more computationally efficient than Genetic Programming (GP) =-=[3]-=- with Automatically Defined Functions (ADF’s) on the even parity and 2-bit multiplier problems [5]. Embedded Cartesian Genetic Programming (ECGP) is a development of CGP that allows the construction a...

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...s has proved to be very difficult for evolutionary techniques (particularly when the number of bits in the multiplicands is greater than three) [4][7][11][12][13]. Cartesian Genetic Programming (CGP) =-=[5]-=-[6] is one technique that has been used to attack such problems. Even though CGP does not have the equivalent of Automatically Defined Functions (ADFs) it was empirically demonstrated to be more compu...

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... comparisons for all three experiments in section 6. Section 7 gives conclusions and some suggestions for future work.s2 Module Acquisition and Automatically Defined Functions Module acquisition (MA) =-=[1]-=- adds two operators to the evolutionary process, compress that selects a section of the genotype to make it immune to manipulation from operators (the module) and expand which decompresses a module in...

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...oblems. Once the contents of modules are themselves allowed to evolve (as in ECGP) they become a form of ADF, however in contrast to Koza's form of ADFs [3] and Spector's Automatically Defined Macros =-=[9]-=-, there is no explicit specification of the number or internal structure of such modules. This freedom does exist in Spector's PushGP [10]. 3 Embedded Cartesian Genetic Programming (ECGP) 3.1 Represen...

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...uction The evolution of digital multipliers has proved to be very difficult for evolutionary techniques (particularly when the number of bits in the multiplicands is greater than three) [4][7][11][12]=-=[13]-=-. Cartesian Genetic Programming (CGP) [5][6] is one technique that has been used to attack such problems. Even though CGP does not have the equivalent of Automatically Defined Functions (ADFs) it was ...

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...'s form of ADFs [3] and Spector's Automatically Defined Macros [9], there is no explicit specification of the number or internal structure of such modules. This freedom does exist in Spector's PushGP =-=[10]-=-. 3 Embedded Cartesian Genetic Programming (ECGP) 3.1 Representation ECGP and CGP share the same structure and represent a program as a directed graph (that for feed-forward functions is acyclic). The...

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...lows the possibility of having modules within modules. These techniques have been shown to decrease the time taken to find a solution. Rosca's method of Adaptive Representation through Learning (ARL) =-=[8]-=- also extracted program segments that were encapsulated and used to augment the GP function set. However, recently Dessi et al [2] showed that random selection of program sub-code for re-use is more e...

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...evelopment of CGP that allows the construction and evolution of modules that can be called from the main CGP code and has been shown to perform better than standard CGP on a series of parity problems =-=[14]-=-. In this paper we apply ECGP to the multiplier problem. We also introduce a new approach called Product Reduction (PR), which is designed to make evolving digital multipliers easier. The plan for the...

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...1 Introduction The evolution of digital multipliers has proved to be very difficult for evolutionary techniques (particularly when the number of bits in the multiplicands is greater than three) [4][7]=-=[11]-=-[12][13]. Cartesian Genetic Programming (CGP) [5][6] is one technique that has been used to attack such problems. Even though CGP does not have the equivalent of Automatically Defined Functions (ADFs)...

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...as proved to be very difficult for evolutionary techniques (particularly when the number of bits in the multiplicands is greater than three) [4][7][11][12][13]. Cartesian Genetic Programming (CGP) [5]=-=[6]-=- is one technique that has been used to attack such problems. Even though CGP does not have the equivalent of Automatically Defined Functions (ADFs) it was empirically demonstrated to be more computat...

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...ion. Rosca's method of Adaptive Representation through Learning (ARL) [8] also extracted program segments that were encapsulated and used to augment the GP function set. However, recently Dessi et al =-=[2]-=- showed that random selection of program sub-code for re-use is more effective than other Rosca’s method across a range of problems. Once the contents of modules are themselves allowed to evolve (as i...

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...troduction The evolution of digital multipliers has proved to be very difficult for evolutionary techniques (particularly when the number of bits in the multiplicands is greater than three) [4][7][11]=-=[12]-=-[13]. Cartesian Genetic Programming (CGP) [5][6] is one technique that has been used to attack such problems. Even though CGP does not have the equivalent of Automatically Defined Functions (ADFs) it ...