1. Application of machine learning to RCF decision procedures Zongyan Huang

2. What is MetiTarski  Automatic theorem prover  Prove universally quantified inequalities involving special functions (ln, exp, sin, etc.,.) e.g. e.g. Prove within a second!

3. Why reasoning about special functions  Wide ranges of engineering applications Mechanical systemsMechanical systems Electrical circuitsElectrical circuits Chemical process controlChemical process control Embedded computation systemsEmbedded computation systems  Hybrid systems are dynamic system which exhibits both continuous and discrete dynamic behavior  Properties are expressed by formula involving special functions

4. How MetiTarski works  Combines a resolution theorem prover (Metis) with RCF decision procedures  The theory of RCF concerns boolean combinations of polynomial equations and inequalities over the real numbers  Eliminate special functions (upper and lower bounds)  Transform parts of the problem into polynomial inequalities  Apply a RCF decision procedure

5. RCF decision procedures  Proof search generates a series of RCF subproblems  Simplify clauses by deleting literals that are inconsistent with other algebraic facts  RCF Strategies used QEPCADQEPCAD MathematicaMathematica Z3Z3  No single RCF decision procedure always gives the fastest runtime  Use machine learning to find the “ best ” RCF strategy

6. Machine Learning  Statistical methods to infer information from training examples  Information applied to new problems  The Support Vector Machine (Joachims ’ SVMLight) SVM learn: generate modelSVM learn: generate model SVM classify: predict the class label and output the margin valuesSVM classify: predict the class label and output the margin values

7. Methodology  Identify features of the problems  Select the best kernel function and parameter values for SVM-Light base on F 1 maximization  Combine the models for decision procedures  Compare the margin values. The classifier with most positive (or least negative) margin was selected.

8. Results  The experiment was done on 825 MetiTarski problems  The total number of problems proved out of 194 testing problems was used to measure the efficacy  Machine learned selection yields better results than any individual fixed decision procedure Machine learned Fix Z3 Fix Mathematica Fix QEPCAD 163160153158

9. Future work  Extend to the heuristic selection within decision procedures  Extend the range of features used and apply feature selection  Provide feedback for development of RCF decision procedures

10. Thank you