A constraint-based approach to invariant generation in programs translates a program into constraints that are solved using off-theshelf constraint solvers to yield desired program invariants.In this paper we show how the constraint-based approach can be used to model a wide spectrum of program analyses in an expressive domain containing disjunctions and conjunctions of linear inequalities. In particular, we show how to model the problem of context-sensitive interprocedural program verification. We also present the first constraint-based approach to weakest precondition and strongest postcondition inference. The constraints we generate are boolean combinations of quadratic inequalities over integer variables. We reduce these constraints to SAT formulae using bitvector modeling and use off-the-shelf SAT solvers to solve them.Furthermore, we present interesting applications of the above analyses, namely bounds analysis and generation of most-general counter-examples for both safety and termination properties. We also present encouraging preliminary experimental results demonstrating the feasibility of our technique on a variety of challenging examples.

To solve a problem with a dynamic programming algorithm, one must reformulate the problem such that its solution can be formed from solutions to overlapping subproblems. Because overlapping subproblems may not be apparent in the specification, it is desirable to obtain the algorithm directly from the specification. We describe a semi-automatic synthesizer of linear-time dynamic programming algorithms. The programmer supplies a declarative specification of the problem and the operators that might appear in the solution. The synthesizer obtains the algorithm by searching a space of candidate algorithms; internally, the search is implemented with constraint solving. The space of candidate algorithms is defined with a program template reusable across all linear-time dynamic programming algorithms, which we characterize as first-order recurrences. This paper focuses on how to write the template so that the constraint solving process scales to real-world linear-time dynamic programming algorithms. We show how to reduce the space with (i)~symmetry reduction and (ii)~domain knowledge of dynamic programming algorithms. We have synthesized algorithms for variants of maximal substring matching, an assembly-line optimization, and the extended Euclid algorithm. We have also synthesized a problem outside the class of first-order recurrences, by composing three instances of the algorithm template.

We address the problem of automatically generating invariants with quantified and boolean structure for proving the validity of given assertions or generating pre-conditions under which the assertions are valid. We present three novel algorithms, having different strengths, that combine template and predicate abstraction based formalisms to discover required sophisticated program invariants using SMT solvers.Two of these algorithms use an iterative approach to compute fixed-points (one computes a least fixed-point and the other computes a greatest fixed-point), while the third algorithm uses a constraint based approach to encode the fixed-point. The key idea in all these algorithms is to reduce the problem of invariant discovery to that of finding optimal solutions for unknowns (over conjunctions of some predicates from a given set) in a template formula such that the formula is valid.Preliminary experiments using our implementation of these algorithms show encouraging results over a benchmark of small but complicated programs. Our algorithms can verify program properties that, to our knowledge, have not been automatically verified before. In particular, our algorithms can generate full correctness proofs for sorting algorithms (which requires nested universallyexistentially quantified invariants) and can also generate preconditions required to establish worst-case upper bounds of sorting algorithms. Furthermore, for the case of previously considered properties, in particular sortedness in sorting algorithms, our algorithms take less time than reported by previous techniques.