Using Program Synthesis for Program Analysis

In this paper, we identify a fragment of second-order logic with restricted quantification that is expressive enough to capture numerous static analysis problems (e.g. safety proving, bug finding, termination and non-termination proving, superoptimisation). We call this fragment the {\it synthesis fragment}. Satisfiability of a formula in the synthesis fragment is decidable over finite domains; specifically the decision problem is NEXPTIME-complete. If a formula in this fragment is satisfiable, a solution consists of a satisfying assignment from the second order variables to \emph{functions over finite domains}. To concretely find these solutions, we synthesise \emph{programs} that compute the functions. Our program synthesis algorithm is complete for finite state programs, i.e. every \emph{function} over finite domains is computed by some \emph{program} that we can synthesise. We can therefore use our synthesiser as a decision procedure for the synthesis fragment of second-order logic, which in turn allows us to use it as a powerful backend for many program analysis tasks. To show the tractability of our approach, we evaluate the program synthesiser on several static analysis problems.Comment: 19 pages, to appear in LPAR 2015. arXiv admin note: text overlap with arXiv:1409.492

On complexity of optimized crossover for binary representations

We consider the computational complexity of producing the best possible offspring in a crossover, given two solutions of the parents. The crossover operators are studied on the class of Boolean linear programming problems, where the Boolean vector of variables is used as the solution representation. By means of efficient reductions of the optimized gene transmitting crossover problems (OGTC) we show the polynomial solvability of the OGTC for the maximum weight set packing problem, the minimum weight set partition problem and for one of the versions of the simple plant location problem. We study a connection between the OGTC for linear Boolean programming problem and the maximum weight independent set problem on 2-colorable hypergraph and prove the NP-hardness of several special cases of the OGTC problem in Boolean linear programming.Comment: Dagstuhl Seminar 06061 "Theory of Evolutionary Algorithms", 200

Combinatorial optimization and metaheuristics

Today, combinatorial optimization is one of the youngest and most active areas of discrete mathematics. It is a branch of optimization in applied mathematics and computer science, related to operational research, algorithm theory and computational complexity theory. It sits at the intersection of several fields, including artificial intelligence, mathematics and software engineering. Its increasing interest arises for the fact that a large number of scientific and industrial problems can be formulated as abstract combinatorial optimization problems, through graphs and/or (integer) linear programs. Some of these problems have polynomial-time (“efficient”) algorithms, while most of them are NP-hard, i.e. it is not proved that they can be solved in polynomial-time. Mainly, it means that it is not possible to guarantee that an exact solution to the problem can be found and one has to settle for an approximate solution with known performance guarantees. Indeed, the goal of approximate methods is to find “quickly” (reasonable run-times), with “high” probability, provable “good” solutions (low error from the real optimal solution). In the last 20 years, a new kind of algorithm commonly called metaheuristics have emerged in this class, which basically try to combine heuristics in high level frameworks aimed at efficiently and effectively exploring the search space. This report briefly outlines the components, concepts, advantages and disadvantages of different metaheuristic approaches from a conceptual point of view, in order to analyze their similarities and differences. The two very significant forces of intensification and diversification, that mainly determine the behavior of a metaheuristic, will be pointed out. The report concludes by exploring the importance of hybridization and integration methods

Protein Design is NP-hard

Biologists working in the area of computational protein design have never doubted the seriousness of the algorithmic challenges that face them in attempting in silico sequence selection. It turns out that in the language of the computer science community, this discrete optimization problem is NP-hard. The purpose of this paper is to explain the context of this observation, to provide a simple illustrative proof and to discuss the implications for future progress on algorithms for computational protein design

Heuristics for The Whitehead Minimization Problem

In this paper we discuss several heuristic strategies which allow one to solve the Whitehead's minimization problem much faster (on most inputs) than the classical Whitehead algorithm. The mere fact that these strategies work in practice leads to several interesting mathematical conjectures. In particular, we conjecture that the length of most non-minimal elements in a free group can be reduced by a Nielsen automorphism which can be identified by inspecting the structure of the corresponding Whitehead Graph

Improved RNA pseudoknots prediction and classification using a new topological invariant

We propose a new topological characterization of RNA secondary structures with pseudoknots based on two topological invariants. Starting from the classic arc-representation of RNA secondary structures, we consider a model that couples both I) the topological genus of the graph and II) the number of crossing arcs of the corresponding primitive graph. We add a term proportional to these topological invariants to the standard free energy of the RNA molecule, thus obtaining a novel free energy parametrization which takes into account the abundance of topologies of RNA pseudoknots observed in RNA databases.Comment: 9 pages, 6 figure