Automated Invariant Generation by Algebraic Techniques for Imperative Program Verification in Theorema Automated Invariant Generation by Algebraic Techniques for Imperative Program Verification in Theorema

This thesis presents algebraic and combinatorial approaches for reasoning about imperative loops with assignments, sequencing and conditionals. A certain family of loops, called P-solvable, is defined for which the value of each program variable can be expressed as a polynomial of the initial values of variables, the loop counter, and some new variables where there are algebraic dependencies among the new variables. For such loops, a systematic method is developed for generating polynomial invariants. Further, if the bodies of these loops consist only of assignments and conditional branches, and test conditions in the loop and conditionals are ignored, the method is shown to be complete for some special cases. By completeness we mean that it generates a set of polynomials from which, under additional assumptions for loops with conditional branches, any polynomial invariant can be derived. Many non-trivial algorithms working on numbers can be naturally implemented using P-solvable loops. By combining advanced techniques from algorithmic combinatorics, symbolic summation, computer algebra and computational logic, a framework is developed for generating polynomial invariants for imperative programs operating on numbers. Exploiting the symbolic manipulation capabilities of the computer algebra system Mathematica, these techniques are implemented in a new software package called Aligator. By using several combinatorial packages developed at RISC, Aligator includes algorithms for solving special classes of recurrence relations (those that are either Gosper-summable or C-finite) and generating polynomial dependencies among algebraic exponential sequences. Using Aligator, a complete set of polynomial invariants is successfully generated for numerous imperative programs working on numbers. The automatically obtained invariant assertions are subsequently used for proving the partial correctness of programs by generating appropriate verification conditions as first-order logical formulas. Based on Hoare logic and the weakest precondition strategy, this verification process is supported in an imperative verification environment implemented in the Theorema system. Theorema is convenient for such an integration given that it is built on top of the computer algebra system Mathematica and includes automated methods for theorem proving in predicate logic, domain specific reasoning and proving by induction.