Large induced subgraphs via triangulations and CMSO

We obtain an algorithmic meta-theorem for the following optimization problem. Let \phi\ be a Counting Monadic Second Order Logic (CMSO) formula and t be an integer. For a given graph G, the task is to maximize |X| subject to the following: there is a set of vertices F of G, containing X, such that the subgraph G[F] induced by F is of treewidth at most t, and structure (G[F],X) models \phi. Some special cases of this optimization problem are the following generic examples. Each of these cases contains various problems as a special subcase: 1) "Maximum induced subgraph with at most l copies of cycles of length 0 modulo m", where for fixed nonnegative integers m and l, the task is to find a maximum induced subgraph of a given graph with at most l vertex-disjoint cycles of length 0 modulo m. 2) "Minimum \Gamma-deletion", where for a fixed finite set of graphs \Gamma\ containing a planar graph, the task is to find a maximum induced subgraph of a given graph containing no graph from \Gamma\ as a minor. 3) "Independent \Pi-packing", where for a fixed finite set of connected graphs \Pi, the task is to find an induced subgraph G[F] of a given graph G with the maximum number of connected components, such that each connected component of G[F] is isomorphic to some graph from \Pi. We give an algorithm solving the optimization problem on an n-vertex graph G in time O(#pmc n^{t+4} f(t,\phi)), where #pmc is the number of all potential maximal cliques in G and f is a function depending of t and \phi\ only. We also show how a similar running time can be obtained for the weighted version of the problem. Pipelined with known bounds on the number of potential maximal cliques, we deduce that our optimization problem can be solved in time O(1.7347^n) for arbitrary graphs, and in polynomial time for graph classes with polynomial number of minimal separators

Delta-Decision Procedures for Exists-Forall Problems over the Reals

Solving nonlinear SMT problems over real numbers has wide applications in robotics and AI. While significant progress is made in solving quantifier-free SMT formulas in the domain, quantified formulas have been much less investigated. We propose the first delta-complete algorithm for solving satisfiability of nonlinear SMT over real numbers with universal quantification and a wide range of nonlinear functions. Our methods combine ideas from counterexample-guided synthesis, interval constraint propagation, and local optimization. In particular, we show how special care is required in handling the interleaving of numerical and symbolic reasoning to ensure delta-completeness. In experiments, we show that the proposed algorithms can handle many new problems beyond the reach of existing SMT solvers

Verification of advanced controllers for safety-critical systems

In order to design and deploy a feedback controller in a real application, one must determine suitable specifications that the design must meet ("validate"), and then ensure that the chosen specifications have been met ("verify"). In this thesis, we investigate a verification paradigm based on formal methods, such as the Satisfiability Modulo Theories (SMT) and quantifier elimination (Weispfenning’s virtual term substitution and quantifier elimination by cylindrical algebraic decomposition) algorithms. Any control design requirement (such as satisfactory performance, robustness to uncertainties, stability, etc.) that can be expressed in a first order logic formula can be (in principle) verified by using one of these methods. Consequently, in principle, this allows us to consider problems like general non-convex optimisation, exact computation of structured singular value, and synthesis of non-convex feasible parameter sets. In practice, the generality of algorithms like quantifier elimination by cylindrical algebraic decomposition come with a downside of high running time when applied to more complex systems with more parameters. This, in some cases, limits the complexity of the system that we could consider. Therefore, we focused our attention on control problems such as obtaining an explicit MPC law for a linear time invariant system with a quadratic objective and polytopic constraints, or computation of the structured singular value for a system under parametric (and not norm-bounded) uncertainty. Such problems can be expressed as quantifier elimination problems with a particular quantification structure that allows us to take advantage of a specialised quantifier elimination algorithm - the quantifier elimination by Weispfenning’s virtual term substitution procedure that has much lower worst-case running time on these types of problems than quantifier elimination by cylindrical algebraic decomposition algorithm. Despite these constraints, we were able to apply a quantifier-elimination-based verification framework to clearance of a flight control law developed for a real world industrial system from the aerospace field not only at particular combination of parameters but throughout the whole flight envelope. In conclusion, while in principle formal methods are applicable to a large body of problems arising in control theory, more widespread practical application depends on further research in efficiency and running time improvement in the implementation of these algorithms.Full EC Project Title: Reconfiguration of control in flight for integral global upset recovery (RECONFIGURE) EC Project #: 314544 RG # & UFS Project Code: RG66745, NMZN/04

Backward Reachability Analysis of Perturbed Continuous-Time Linear Systems Using Set Propagation

Backward reachability analysis computes the set of states that reach a target set under the competing influence of control input and disturbances. Depending on their interplay, the backward reachable set either represents all states that can be steered into the target set or all states that cannot avoid entering it -- the corresponding solutions can be used for controller synthesis and safety verification, respectively. A popular technique for backward reachable set computation solves Hamilton-Jacobi-Isaacs equations, which scales exponentially with the state dimension due to gridding the state space. In this work, we instead use set propagation techniques to design backward reachability algorithms for linear time-invariant systems. Crucially, the proposed algorithms scale only polynomially with the state dimension. Our numerical examples demonstrate the tightness of the obtained backward reachable sets and show an overwhelming improvement of our proposed algorithms over state-of-the-art methods regarding scalability, as systems with well over a hundred states can now be analyzed.Comment: 16 page

Computer Aided Verification

This open access two-volume set LNCS 10980 and 10981 constitutes the refereed proceedings of the 30th International Conference on Computer Aided Verification, CAV 2018, held in Oxford, UK, in July 2018. The 52 full and 13 tool papers presented together with 3 invited papers and 2 tutorials were carefully reviewed and selected from 215 submissions. The papers cover a wide range of topics and techniques, from algorithmic and logical foundations of verification to practical applications in distributed, networked, cyber-physical, and autonomous systems. They are organized in topical sections on model checking, program analysis using polyhedra, synthesis, learning, runtime verification, hybrid and timed systems, tools, probabilistic systems, static analysis, theory and security, SAT, SMT and decisions procedures, concurrency, and CPS, hardware, industrial applications