Special Algorithm for Stability Analysis of Multistable Biological Regulatory Systems

We consider the problem of counting (stable) equilibriums of an important family of algebraic differential equations modeling multistable biological regulatory systems. The problem can be solved, in principle, using real quantifier elimination algorithms, in particular real root classification algorithms. However, it is well known that they can handle only very small cases due to the enormous computing time requirements. In this paper, we present a special algorithm which is much more efficient than the general methods. Its efficiency comes from the exploitation of certain interesting structures of the family of differential equations.Comment: 24 pages, 5 algorithms, 10 figure

Verification of model predictive control laws using weispfenning's quantifier elimination by virtual substitution algorithm

© 2016 IEEE. A method based on a quantifier elimination algorithm is suggested for obtaining explicit model predictive control (MPC) laws for linear time invariant systems with quadratic objective and polytopic constraints. The structure of the control problem considered allows Weispfenning's 'quantifier elimination by virtual substitution' algorithm to be used. This is applicable to first order formulas in which quantified variables appear at most quadratically. It has much better practical computational complexity than general quantifier elimination algorithms, such as cylindrical algebraic decomposition. We show how this explicit MPC solution, together with Weispfenning's algorithm, can be used to check recursive feasibility of the system, for both nominal and disturbed systems. Extension to cases beyond linear MPC using Weispfenning's algorithm is part of future work.Engineering and Physical Sciences Research Council, and European Union Seventh Framework Programme FP7/2007-2013 grant agreement number 314 544, project “RECONFIGURE”

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

On the complexity of nonlinear mixed-integer optimization

This is a survey on the computational complexity of nonlinear mixed-integer optimization. It highlights a selection of important topics, ranging from incomputability results that arise from number theory and logic, to recently obtained fully polynomial time approximation schemes in fixed dimension, and to strongly polynomial-time algorithms for special cases.Comment: 26 pages, 5 figures; to appear in: Mixed-Integer Nonlinear Optimization, IMA Volumes, Springer-Verla

Approaching Arithmetic Theories with Finite-State Automata

The automata-theoretic approach provides an elegant method for deciding linear arithmetic theories. This approach has recently been instrumental for settling long-standing open problems about the complexity of deciding the existential fragments of Büchi arithmetic and linear arithmetic over p-adic fields. In this article, which accompanies an invited talk, we give a high-level exposition of the NP upper bound for existential Büchi arithmetic, obtain some derived results, and further discuss some open problems