We introduce an efficient combination of polyhedral analysis and predicate partitioning. Template polyhedral analysis abstracts numerical variables inside a program by one polyhedron per control location, with a priori fixed directions for the faces. The strongest inductive invariant in such an abstract domain may be computed by upward strategy iteration. If the transition relation includes disjunctions and existential quantifiers (a succinct representation for an exponential set of paths), this invariant can be computed by a combination of strategy iteration and satisfiability modulo theory (SMT) solving. Unfortunately, the above approaches lead to unacceptable space and time costs if applied to a program whose control states have been partitioned according to predicates. We therefore propose a modification of the strategy iteration algorithm where the strategies are stored succinctly, and the linear programs to be solved at each iteration step are simplified according to an equivalence relation. We have implemented the technique in a prototype tool and we demonstrate on a series of examples that the approach performs significantly better than previous strategy iteration techniques.
Early work in compilation and verification of reactive systems @cite_3 advocated quotienting the Boolean state space according to some form of bisimulation. In contrast, we compute coarser equivalences according to per-constraint semantics. In the industrial-strength analyzer analyzer , static heuristics determine reasonably small packs of related'' Booleans and numerical variables, such that the values of the numerical variables are analyzed separately for each Boolean valuation [ .2.4] BlanchetCousotEtAl_PLDI03 . In contrast, our equivalence classes are computed dynamically and per-constraint.
@cite_3
39834257 2282479846 2952801764 81626549
Geometric heuristics for the quantifier elimination approach presented by Kapur (2004) are investigated to automatically derive loop invariants expressing weakly relational numerical properties (such as l≤x≤h or l≤±x ±y≤h) for imperative programs. Such properties have been successfully used to analyze commercial software consisting of hundreds of thousands of lines of code (using for example, the Astree tool based on abstract interpretation framework proposed by Cousot and his group). The main attraction of the proposed approach is its much lower complexity in contrast to the abstract interpretation approach (O(n2) in contrast to O(n4), where n is the number of variables) with the ability to still generate invariants of comparable strength. This approach has been generalized to consider disjunctive invariants of the similar form, expressed using maximum function (such as max (x+a,y+b,z+c,d)≤max (x+e,y+f,z+g,h)), thus enabling automatic generation of a subclass of disjunctive invariants for imperative programs as well. This paper proposes algorithms for learning two-level Boolean rules in Conjunctive Normal Form (CNF, i.e. AND-of-ORs) or Disjunctive Normal Form (DNF, i.e. OR-of-ANDs) as a type of human-interpretable classification model, aiming for a favorable trade-off between the classification accuracy and the simplicity of the rule. Two formulations are proposed. The first is an integer program whose objective function is a combination of the total number of errors and the total number of features used in the rule. We generalize a previously proposed linear programming (LP) relaxation from one-level to two-level rules. The second formulation replaces the 0-1 classification error with the Hamming distance from the current two-level rule to the closest rule that correctly classifies a sample. Based on this second formulation, block coordinate descent and alternating minimization algorithms are developed. Experiments show that the two-level rules can yield noticeably better performance than one-level rules due to their dramatically larger modeling capacity, and the two algorithms based on the Hamming distance formulation are generally superior to the other two-level rule learning methods in our comparison. A proposed approach to binarize any fractional values in the optimal solutions of LP relaxations is also shown to be effective. In this paper, we introduce a new adaptive data analysis method to study trend and instantaneous frequency of nonlinear and non-stationary data. This method is inspired by the Empirical Mode Decomposition method (EMD) and the recently developed compressed (compressive) sensing theory. The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions of the form @math , where @math , @math consists of the functions smoother than @math and @math . This problem can be formulated as a nonlinear @math optimization problem. In order to solve this optimization problem, we propose a nonlinear matching pursuit method by generalizing the classical matching pursuit for the @math optimization problem. One important advantage of this nonlinear matching pursuit method is it can be implemented very efficiently and is very stable to noise. Further, we provide a convergence analysis of our nonlinear matching pursuit method under certain scale separation assumptions. Extensive numerical examples will be given to demonstrate the robustness of our method and comparison will be made with the EMD EEMD method. We also apply our method to study data without scale separation, data with intra-wave frequency modulation, and data with incomplete or under-sampled data. The IC3 algorithm was recently introduced for proving properties of finite state reactive systems. It has been applied very successfully to hardware model checking. We provide a specification of the algorithm using an abstract transition system and highlight its dual operation: model search and conflict resolution. We then generalize it along two dimensions. Along one dimension we address nonlinear fixed-point operators (push-down systems) and evaluate the algorithm on Boolean programs. In the second dimension we leverage proofs and models and generalize the method to Boolean constraints involving theories.