1. Deductive and inductive methods for program synthesis Jelena Sanko, Jaan Penjam Institute of Cybernetics October 29, 2005

2. Agenda Program construction Inductive program construction Experiments Conclusions

3. Program construction Set of behaviours Requirements specification program Deductive programming Inductive programming algorithm

4. Problem Statement x R1 R2 R3 R4 R5 R6 z v y u t composition

5. Scoring of programs s - measure of fitness p i =  R 1 ; R 2 ; R 3 ; R 4 ; R 5 ;... ;R 6  Square root fitness measure of the program p t2t2 x2x2 tntn xnxn …… t1t1 x1x1 OUTIN

6. Program synthesis as optimization s z 1 z0z0 P=  R 2 ; R 3 ; R 5 ; R 6 ; R 2 ; …; R 6  h

7. Coding of programs z = 0,23562…7 h p=  R 2 ; R 3 ; R 5 ; R 6 ; R 2 ; …; R 6 

8. PR6PR6 Coding of programs (2) x R1 R2 R3 R4 R5 R6 z v y u t M(P) - State Transition Machine computational model

9. Coding of programs (3) 0 1 0 1 z = 0.912 p=  R 3 ;...  p=  R 3 ; R 4 ;...  P=  R 3 ;R 4 ;R 1 ;R 5 ;R 6  x 1 6 4;5 5 5 4 4 3 3 1, 2 1, 2 1, 2, 2 xyz vxyz uvxz uvxyz tuvxyz vxz xz xy 0.330.66

10. Function to be optimized (f(z)) 1 z s

11. DE is a method for finding extreme points of real-valued multi-modal functions Optimization technique -Differential Evolution, Rainer Storn and Kenneth Price DE is a heuristic method that can be used for optimization of non-differentiable functions in continuous spaces The convergence rate of the floating-point encoded DE is more than 10 times higher than the convergence rate of the traditional binary encoded GA

12. Experimental Data P={(0.75,1.87*10^-07), (0.70, 1.04*10^-07) (0.65,5.44*10^-08), (0.60,2.61*10^-08) (0.55,1.17*10^-08),(0.50, 3.40*10^-09)} P.M. Murphy and D.W. Aha, Uci repository of machine learning databases x R1 R2 R3 R4 R5 R6 z v y u t I(R1)=“y:=ln(x)” I(R2)=“y:=exp(-x)” I(R3)=“z:=x^3+6x” I(R4)=“v:=2(x^2+z)” I(R5)=“(u,z):=(v+6,v/2)” I(R6)=“v:=0.16yu/7.85^4”

13. The optimal solutions like 0.37362, 0.40419, 0.795112 correspond to the following sequence of relations “R2;R3;R4;R5;R4;R5;R4;R5;R6” Experimental Data (2) I(R1)=“y:=ln(x)” I(R2)=“y:=exp(-x)” I(R3)=“z:=x^3+6x” I(R4)=“v:=2(x^2+z)” I(R5)=“(u,z):=(v+6,v/2)” I(R6)=“v:=0.16yu/7.85^4” where The optimal solution of the problem obtained by the DE method corresponds to the proposed by Knut Angström functional relation:

14. Conclusions: The inductive approach to program construction can promote control over construction process and take an appropriate decision when several solutions are available IPS uses interpretation of relations and a fitting function IPS considers all programs and selects the best The research is in progress... IPS strategy need to be checked in practice and extension to high order constraints is required

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