arXiv subjects

Stefan Ratschan

Publications and source records attributed to Stefan Ratschan.

At least 19 recordsLinked to original sources

arXiv2026

Deciding Predicate Logical Theories of Real-Valued Functions

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.

cs.LO↗

arXiv2026

Inductive Satisfiability Certification for Universal Quantifiers and Uninterpreted Function Symbols

The combination of uninterpreted function symbols and universal quantification occurs in many applications of automated reasoning, for example, due to their ability to reason about arrays. Yet the satisfiability of such formulas is, in general, undecidable. In practice, SMT solvers are often successful in the unsatisfiable case, using heuristics. However, in the satisfiable case, they rely on explicit model construction, which fails for formulas whose smallest model is not small enough. We introduce an alternative approach that certifies satisfiability using induction arguments, and apply it to the case of linear integer arithmetic. The resulting algorithm is able to prove satisfiability of formulas that are out of reach for current SMT solvers.

cs.LO↗

arXiv2025

Input-based Three-valued Abstraction Refinement

Unlike Counterexample-Guided Abstraction Refinement (CEGAR), Three-Valued Abstraction Refinement (TVAR) is able to verify all properties of the mu-calculus. We present a novel algorithmic framework for TVAR that employs a simulator-like approach to build and refine the abstract state space with input-based splitting. This leads to a state space formalism that is much simpler than in previous TVAR frameworks, which use modal transitions. We implemented the framework in our open-source tool machine-check and verified properties of machine-code systems for the AVR architecture, showing the ability to verify systems and mu-calculus properties not verifiable by naive model checking or CEGAR, respectively. This is the first practical use of TVAR for machine-code verification.

cs.LO↗

arXiv2025

Applications of Quantified Constraint Solving over the Reals -- Bibliography

Quantified constraints over the reals appear in numerous contexts. Usually existential quantification occurs when some parameter can be chosen by the user of a system, and univeral quantification when the exact value of a parameter is either unknown, or when it occurs in infinitely many, similar versions. The following is a list of application areas and publications that contain applications for solving quantified constraints over the reals. The list is certainly not complete, but grows as the author encounters new items. Contributions are very welcome!

cs.LO↗

arXiv2025

SMT and Functional Equation Solving over the Reals: Challenges from the IMO

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.

cs.LO↗

arXiv2025

Satisfiability of Non-Linear Transcendental Arithmetic as a Certificate Search Problem

For typical first-order logical theories, satisfying assignments have a straightforward finite representation that can directly serve as a certificate that a given assignment satisfies the given formula. For non-linear real arithmetic augmented with trigonometric and exponential functions (NTA), however, there is no known direct representation of satisfying assignments that allows for a simple independent check of whether the represented numbers exist and satisfy the given formula. Hence, in this paper, we introduce a different form of satisfiability certificate for NTA, and formulate the satisfiability problem as the problem of searching for such a certificate. This does not only ease the independent verification of satisfiability, but also allows the design of new algorithms that show satisfiability by systematically searching for such certificates. Computational experiments document that the resulting algorithms are able to prove satisfiability of a substantially higher number of benchmark problems than existing methods. We also characterize the formulas whose satisfiability can be demonstrated by such a certificate, by providing lower and upper bounds in terms of relevant well-known classes. Finally we show the existence of a procedure for checking the satisfiability of NTA-formulas that terminates for formulas that satisfy certain robustness assumptions.

cs.LO↗

arXiv2024

Computation of Feedback Control Laws Based on Switched Tracking of Demonstrations

A common approach in robotics is to learn tasks by generalizing from special cases given by a so-called demonstrator. In this paper, we apply this paradigm and present an algorithm that uses a demonstrator (typically given by a trajectory optimizer) to automatically synthesize feedback controllers for steering a system described by ordinary differential equations into a goal set. The resulting feedback control law switches between the demonstrations that it uses as reference trajectories. In comparison to the direct use of trajectory optimization as a control law, for example, in the form of model predictive control, this allows for a much simpler and more efficient implementation of the controller. The synthesis algorithm comes with rigorous convergence and optimality results, and computational experiments confirm its efficiency.

eess.SY↗

arXiv2023

Multi-Agent Path Finding with Continuous Time Using SAT Modulo Linear Real Arithmetic

This paper introduces a new approach to solving a continuous-time version of the multi-agent path finding problem. The algorithm translates the problem into an extension of the classical Boolean satisfiability problem, satisfiability modulo theories (SMT), that can be solved by off-the-shelf solvers. This enables the exploitation of conflict generalization techniques that such solvers can handle. Computational experiments show that the new approach scales better with respect to the available computation time than state-of-the art approaches and is usually able to avoid their exponential behavior on a class of benchmark problems modeling a typical bottleneck situation.

cs.MA↗

arXiv2023

LQR-trees with Sampling Based Exploration of the State Space

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 trajectories, 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.

eess.SY↗

arXiv2022

Railway Scheduling Using Boolean Satisfiability Modulo Simulations

Railway scheduling is a problem that exhibits both non-trivial discrete and continuous behavior. In this paper, we simulate train networks at a low level, where a number of timing and ordering constraints can appear. We model this problem using a combination of SAT and ordinary differential equations (SAT modulo ODE). In addition, we adapt our existing method for solving such problems in such a way that the resulting solver is competitive with methods based on dedicated railway simulators while being more general and extensible.

eess.SY↗

arXiv2022

Computing Funnels Using Numerical Optimization Based Falsifiers

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.

cs.RO↗

arXiv2021

Proceedings 6th International Workshop on Symbolic-Numeric methods for Reasoning about CPS and IoT

The proceedings of the 6th International Workshop on Symbolic-Numeric Methods for Reasoning about CPS and IoT (SNR 2020) contains papers underlying talks presented at the workshop. SNR focuses on the combination of symbolic and numeric methods for reasoning about Cyber-Physical Systems and the Internet of Things to facilitate model identification, specification, verification, and control synthesis for these systems.

cs.LO↗

arXiv2018

Simulation Based Computation of Certificates for Safety of Hybrid Dynamical Systems

In this paper, we present an algorithm for synthesizing certificates---so-called barrier certificates---for safety of hybrid dynamical systems. Unlike the usual approach of using constraint solvers to compute the certificate from the system dynamics, we synthesize the certificate from system simulations. This makes the algorithm applicable even in cases where the dynamics is either not explicitly available, or too complicated to be analyzed by constraint solvers, for example, due to the presence of transcendental function symbols. The algorithm itself allows the usage of heuristic techniques in which case it does not formally guarantee correctness of the result. However, in cases that do allow rigorous constraint solving, the computed barrier certificate can be rigorously verified, if desired. Hence, in such cases, our algorithm reduces the problem of finding a barrier certificate to the problem of formally verifying a given barrier certificate.

eess.SY↗

arXiv2018

Converse Theorems for Safety and Barrier Certificates

An important tool for proving safety of dynamical systems is the notion of a barrier certificate. In this paper we prove that every robustly safe ordinary differential equation has a barrier certificate. Moreover, we show a construction of such a barrier certificate based on a set of states that is reachable in finite time.

eess.SY↗

arXiv2017

Solving Reachability Problems by a Scalable Constrained Optimization Method

In this paper we consider the problem of finding an evolution of a dynamical system that originates and terminates in given sets of states. However, if such an evolution exists then it is usually not unique. We investigate this problem and find a scalable approach for solving it. To this end we formulate an equality constrained nonlinear program that addresses the non-uniqueness of the solution of the original problem. In addition, the resulting saddle-point matrix is sparse. We exploit the structure in order to reach an efficient implementation of our method. In computational experiments we compare line search and trust-region methods as well as various updates for the Hessian.

math.OC↗

arXiv2016

Solving Underdetermined Boundary Value Problems By Sequential Quadratic Programming

Ordinary differential equations that model technical systems often contain states, that are considered dangerous for the system. A trajectory that reaches such a state usually indicates a flaw in the design. In this paper, we present and study the properties of an algorithm for finding such trajectories. That is, for a given ordinary differential equation, the algorithm finds a trajectory that originates in one set of states and reaches another one. The algorithm is based on sequential quadratic programming applied to a regularized optimization problem obtained by multiple shooting.

math.OC↗

arXiv2016

Quasi-decidability of a Fragment of the First-order Theory of Real Numbers

In this paper we consider a fragment of the first-order theory of the real numbers that includes systems of equations of continuous functions in bounded domains, and for which all functions are computable in the sense that it is possible to compute arbitrarily close piece-wise interval approximations. Even though this fragment is undecidable, we prove that there is a (possibly non-terminating) algorithm for checking satisfiability such that (1) whenever it terminates, it computes a correct answer, and (2) it always terminates when the input is robust. A formula is robust, if its satisfiability does not change under small perturbations. As a basic tool for our algorithm we use the notion of degree from the field of (differential) topology.

cs.CC↗

arXiv2014

Efficient Solution of a Class of Quantified Constraints with Quantifier Prefix Exists-Forall

In various applications the search for certificates for certain properties (e.g., stability of dynamical systems, program termination) can be formulated as a quantified constraint solving problem with quantifier prefix exists-forall. In this paper, we present an algorithm for solving a certain class of such problems based on interval techniques in combination with conservative linear programming approximation. In comparison with previous work, the method is more general - allowing general Boolean structure in the input constraint, and more efficient - using splitting heuristics that learn from the success of previous linear programming approximations.

cs.LO↗