Abstract We use SMT technology to address a class of problems involving uninterpreted functions and nonlinear real arithmetic. In particular, we focus on problems commonly found in mathematical competitions, such as the International Mathematical Olympiad (IMO), where the task is to determine all solutions to constraints on an uninterpreted function. Although these problems require only high-school-level mathematics, state-of-the-art SMT solvers often struggle with them. We propose several techniques to improve SMT performance in this setting.

This paper introduces an extension of the LQR-tree algorithm, which is a feedback-motion-planning algorithm for stabilizing a system of ordinary differential equations from a bounded set of initial conditions to a goal. The constructed policies are represented by a tree of exemplary system trajec-tories, so called demonstrations, and linear-quadratic regulator (LQR) feedback controllers. Consequently, the crucial component of any LQR-tree algorithm is a demonstrator that provides suitable demonstrations. In previous work, such a demonstrator was given by a local trajectory optimizer. However, these require appropriate initial guesses of solutions to provide valid results, which was pointed out, but largely unresolved in previous implementations. In this paper, we augment the LQR-tree algorithm with a randomized motion-planning procedure to discover new valid demonstration candidates to initialize the demonstrator in parts of state space not yet covered by the LQR-tree. In comparison to the previous versions of the LQR-tree algorithm, the resulting exploring LQR-tree algorithm reliably synthesizes feedback control laws for a far more general set of problems.

The notion of a real-valued function is central to mathematics, computer science, and many other scientific fields. Despite this importance, there are hardly any positive results on decision procedures for predicate logical theories that reason about real-valued functions. This paper defines a first-order predicate language for reasoning about multi-dimensional smooth real-valued functions and their derivatives, and demonstrates that - despite the obvious undecidability barriers - certain positive decidability results for such a language are indeed possible.

In this paper, we present an algorithm that computes funnels along trajectories of systems of ordinary differential equations. A funnel is a time-varying set of states containing the given trajectory, for which the evolution from within the set at any given time stays in the funnel. Hence it generalizes the behavior of single trajectories to sets around them, which is an important task, for example, in robot motion planning. In contrast to approaches based on sum-of-squares programming, which poorly scale to high dimensions, our approach is based on falsification and tackles the funnel computation task directly, through numerical optimization. This approach computes accurate funnel estimates far more efficiently and leaves formal verification to the end, outside all funnel size optimization loops.

Let a quantified inequality constraint over the reals be a formula in the first-order predicate language over the structure of the real numbers, where the allowed predicate symbols are ≤ and <. Solving such constraints is an undecidable problem when allowing function symbols such sin or cos. In this article, we give an algorithm that terminates with a solution for all, except for very special, pathological inputs. We ensure the practical efficiency of this algorithm by employing constraint programming techniques.

Exactly solving first-order constraints (i.e., first-order formulas over a certain predefined structure) can be a very hard, or even undecidable problem. In continuous structures like the real numbers it is promising to compute approximate solutions instead of exact ones. However, the quantifiers of the first-order predicate language are an obstacle to allowing approximations to arbitrary small error bounds. In this article, we remove this obstacle by modifying the first-order language and replacing the classical quantifiers with approximate quantifiers. These also have two additional advantages: First, they are tunable, in the sense that they allow the user to decide on the trade-off between precision and efficiency. Second, they introduce additional expressivity into the first-order language by allowing reasoning over the size of solution sets.