Inverse Laplace Transformer 6.871 Final Project Ashwin Deshpande May 12th, 2005 Professor Davis- 2 - 1 Abstract.......................................................................................................... - 4 - 2 Introduction ................................................................................................... - 4 - 2.1 Design Parameters ................................................................................... - 4 - 2.2 Problem and Approach ............................................................................ - 5 - 3 Program Use................................................................................................... - 6 - 3.1 Program Formatting ................................................................................ - 6 - 3.2 Program Demonstration .......................................................................... - 7 - 4 Rules .............................................................................................................. - 9 - 4.1 Rules as the Knowledge Representation ................................................ - 10 - 4.2 Rule Types ............................................................................................. - 10 - 4.2.1 Always Rules .................................................................................. - 10 - 4.2.2 Lookup Rule ................................................................................... - 12 - 4.2.3 Guess Rules .................................................................................... - 13 - 5 Architecture ................................................................................................. - 14 - 5.1 Costs...................................................................................................... - 14 - 5.1.1 Complexity Cost ............................................................................. - 15 - 5.1.2 Rule Cost ........................................................................................ - 15 - 5.2 Simplification Propagation .................................................................... - 17 - 5.3 Inverse Laplace Transform Propagation ................................................ - 18 - 6 Program Review........................................................................................... - 20 - 6.1 Positive Aspects..................................................................................... - 20 - 6.1.1 Successful Knowledge Representation ............................................ - 20 - 6.1.2 System Generality ........................................................................... - 20 - 6.1.3 System Scalability ........................................................................... - 21 - 6.2 Negative Aspects ................................................................................... - 22 - 6.2.1 Halting Problems ............................................................................ - 22 - 6.2.2 Resource Problems .......................................................................... - 22 - 6.3 Future Improvements ............................................................................ - 23 - 6.3.1 Caching/Case-Based Reasoning ..................................................... - 23 - 6.3.2 Cost Variations ............................................................................... - 23 - 7 Lessons Learned ........................................................................................... - 24 - 7.1 Future Improvements ............................................................................ - 24 - 7.2 Control Structure Implementation......................................................... - 25 - 8 Conclusion ................................................................................................... - 25 - 9 Acknowledgements ...................................................................................... - 26 - Appendix A: Complete Rule Listings.............................................................. - 26 - 7.1 Always Rules ......................................................................................... - 26 - 7.2 Lookup Rules......................................................................................... - 29 - 7.3 Guess Rules ........................................................................................... - 30 - Appendix B: Complete Program Output ......................................................... - 31 - References....................................................................................................... - 43 - - 2 -- 3 - TABLE OF FIGURES Figure 1: Forward Laplace Transform and Inverse Laplace Transform Integrals…………….....5 Figure 2: Symbols and their Meanings………………………………………………………………..6 Figure 3: Format of Solution …………………………………………………………………………. 7 Figure 4: Always Rule Examples………………………………………………………………………12 Figure 5: The Lookup Rule……………………………………………………………………………. 13 Figure 6: Guess Rule Examples………………………………………………………………………. 14 Figure 7: A Sample Expression for Complexity Cost Analysis……………………………………. 15 Figure 8: Simplification Architecture………………………………………………………………… 17 Figure 9: Inverse Laplace Transform Propagation Architecture……………………………………18 Figure 10: Unsolvable Rational Functions……………………………………………………………21 - 3 -- 4 - 1 Abstract The goal of the project was to design and implement a knowledge-based application system capable of being proficient in its domain. The Inverse Laplace Transformer (ILT) meets these specifications as an expert system in the field of symbolic inverse Laplace Transforms. The ILT performs this task by employing a customized rule-based system involving weight-based forward-propagation. While the ILT has many limitations, it contains the knowledge to perform elementary inverse Laplace transforms and the framework to easily expand into a more powerful system. 2 Introduction The ILT was designed to act as an aid in performing inverse Laplace transforms. In addition to finding the inverse Laplace