Continuous and Interval Constraints

Lyapunov Function Synthesis using Handelman Representations.

“…However, since interval evaluation is often conservative, it is frequently applied with a branch-and-bound approach that repeatedly subdivides the domain of interest and evaluates the function on each subdomain. Techniques such as interval constraint propagation allow us to subdivide efficiently without necessarily examining a large number of subdivisions (see Benhamou and Granvilliers [2006] for a survey). Finally, tools such as iSAT implement interval constraint propagation techniques to solve non linear constraints over intervals ( Fränzle et al [2007]).…”

SAT Modulo ODE: A Direct SAT Approach to Hybrid Systems

“…Additionally flow invariants of the form l ≤ x i ≤ u can be given that constrain the range of the variables in V during a continuous flow. 2 An input model comprises predicative encodings of the initial state set init, the transition relation trans over current-step (x) and next-step variables (x ′ ), and the (unsafe) target state. ODE constraints can only occur in the transition relation where they define the relationship between two successive valuations of the variables in V by constraining the possible trajectories in between the steps.…”

“…By the front-end of our solver, constraint formulae are rewritten into equi-satisfiable quantifier-free formulae in conjunctive normal form, with atomic propositions ranging over propositional variables and (in-)equational constraints confined to a form resembling three-address code. This rewriting is based on the standard mechanism of introducing auxiliary variables for the values of arithmetic sub-expressions and of logical sub-formulae, thereby eliminating common sub-expressions and sub-formulae through re-use of the auxiliary variables, thus reducing the search space of the SAT solver and enhancing the reasoning power of the interval contractors used in arithmetic reasoning [2]. Thus, the internal syntax of constraint formulae is as follows: where ∼∈ {<, ≤, >, ≥}, the non-terminals bop, uop denote the binary and unary operator symbols (including arithmetic operators such as + or sin), and term the terms over real-valued variables built using these.…”

“…The essential characteristics of these narrowing operators, which are borrowed from the context of hull consistency [2], is that they narrow the valuation ρ by pruning away only parts of the search space that cannot contain any satisfying valuations. In order to extend the concept of interval-based narrowing operators to ODEs, we first introduce the following definitions.…”

“…On the theory solver side, iSAT is based on interval constraint propagation (ICP) for arithmetic (in-)equations [2], which we extend to ODEs as follows. Interval-based safe numeric approximation of ODEs is used as an interval contractor being able to narrow candidate sets in phase space in both temporal directions: post-images of ODEs (i.e., sets of states reachable from a set of initial values) are narrowed based on partial information about the initial values and, vice versa, pre-images are narrowed based on partial knowledge about post-sets.…”