Stefan Kupferschmid

Incremental preprocessing methods for use in BMC

Traditional incremental SAT solvers have achieved great success in the domain of Bounded Model Checking (BMC). Recently, modern solvers have introduced advanced preprocessing procedures that have allowed them to obtain high levels of performance. Unfortunately, many preprocessing techniques such as variable and (blocked) clause elimination cannot be directly used in an incremental manner. This work focuses on extending these techniques and Craig interpolation so that they can be used effectively together in incremental SAT solving (in the context of BMC). The techniques introduced here doubled the performance of our BMC solver on both SAT and UNSAT problems. For UNSAT problems, preprocessing had the added advantage that Craig interpolation was able to find the fixed point sooner, reducing the number of incremental SAT iterations. Furthermore, our ideas seem to perform better as the benchmarks become larger, and/or deeper, which is exactly when they are needed. Lastly, our methods can be integrated into other SAT based BMC tools to achieve similar speedups.

Functional test of small-delay faults using SAT and Craig interpolation

We present SATSEQ, a timing-aware ATPG system for small-delay faults in non-scan circuits. The tool identifies the longest paths suitable for functional fault propagation and generates the shortest possible sub-sequences per fault. Based on advanced model-checking techniques, SATSEQ provides detection of small-delay faults through the longest functional paths. All test sequences start at the circuits initial state; therefore, overtesting is avoided. Moreover, potential invalidation of the fault detection is taken into account. Experimental results show high detection and better performance than scan testing in terms of test application time and overtesting-avoidance.

Encoding techniques, craig interpolants and bounded model checking for incomplete designs

This paper focuses on bounded invariant checking for partially specified circuits – designs containing so-called blackboxes – using the well known 01X- and QBF-encoding techniques. For detecting counterexamples, modeling the behavior of a blackbox using 01X-encoding is fast, but rather coarse as it limits what problems can be verified. We introduce the idea of 01X-hardness, mainly the classification of problems for which this encoding technique does not provide any useful information about the existence of a counterexample. Furthermore, we provide a proof for 01X-hardness based on Craig interpolation, and show how the information contained within the Craig interpolant or unsat-core can be used to determine heuristically which blackbox outputs to model in a more precise way. We then compare 01X, QBF and multiple hybrid modeling methods. Finally, our total workflow along with multiple state-of-the-art QBF-solvers are shown to perform well on a range of industrial blackbox circuit problems.

Computing Optimized Representations for Non-convex Polyhedra by Detection and Removal of Redundant Linear Constraints

We present a method which computes optimized representations for non-convex polyhedra. Our method detects so-called redundant linear constraints in these representations by using an incremental SMT (Satisfiability Modulo Theories) solver and then removes the redundant constraints based on Craig interpolation. The approach is motivated by applications in the context of model checking for Linear Hybrid Automata. Basically, it can be seen as an optimization method for formulas including arbitrary boolean combinations of linear constraints and boolean variables. We show that our method provides an essential step making quantifier elimination for linear arithmetic much more efficient. Experimental results clearly show the advantages of our approach in comparison to state-of-the-art solvers.

Craig interpolation in the presence of non-linear constraints

An increasing number of applications in particular in the verification area leverages Craig interpolation. Craig interpolants (CIs) can be computed for many different theories such as: propositional logic, linear inequalities over the reals, and the combination of the preceding theories with uninterpreted function symbols. To the best of our knowledge all previous tools that provide CIs are addressing decidable theories. With this paper we make Craig interpolation available for an in general undecidable theory that contains Boolean combinations of linear and non-linear constraints including transcendental functions like sin(ċ) and cos(ċ). Such formulae arise e.g. during the verification of hybrid systems. We show how the construction rules for CIs can be extended to handle non-linear constraints. To do so, an existing SMT solver based on a close integration of SAT and Interval Constraint Propagation is enhanced to construct CIs on the basis of proof trees. We provide first experimental results demonstrating the usefulness of our approach: With the help of Craig interpolation we succeed in proving safety in cases where the basic solver could not provide a complete answer. Furthermore, we point out the (heuristic) decisions we made to obtain suitable CIs and discuss further possibilities to increase the flexibility of the CI construction.

Using an SMT solver and Craig interpolation to detect and remove redundant linear constraints in representations of non-convex polyhedra

We present a method which computes optimized representations for non-convex polyhedra. Our method detects so-called redundant linear constraints in these representations by using an incremental SMT solver and then removes the redundant constraints based on Craig interpolation. The approach is evaluated both for formulas from the model checking context including boolean combinations of linear constraints and boolean variables and for random trees composed of quantifiers, AND-, OR-, NOT-operators, and linear constraints produced by a generator. The results clearly show the advantages of our approach in comparison to state-of-the-art solvers.

Analysis of Reachable Sensitisable Paths in Sequential Circuits with SAT and Craig Interpolation

Test pattern generation for sequential circuits benefits from scanning strategies as these allow the justification of arbitrary circuit states. However, some of these states may be unreachable during normal operation. This results in non-functional operation which may lead to abnormal circuit behaviour and result in over-testing. In this work, we present a versatile approach that combines a highly adaptable SAT-based path-enumeration algorithm with a model-checking solver for invariant properties that relies on the theory of Craig interpolants to prove the unreachability of circuit states. The method enumerates a set of longest sensitisable paths and yields test sequences of minimal length able to sensitise the found paths starting from a given circuit state. We present detailed experimental results on the reach ability of sensitisable paths in ITC 99 circuits.

Proof certificates and non-linear arithmetic constraints

Symbolic methods in computer-aided verification rely heavily on constraint solvers. The correctness and reliability of these solvers are of vital importance in the analysis of safety-critical systems, e.g., in the automotive context. Satisfiability results of a solver can usually be checked by probing the computed solution. This is in general not the case for un-satisfiability results. In this paper, we propose a certification method for unsatisfiability results for mixed Boolean and non-linear arithmetic constraint formulae. Such formulae arise in the analysis of hybrid discrete/continuous systems. Furthermore, we test our approach by enhancing the iSAT constraint solver to generate unsatisfiability proofs, and implemented a tool that can efficiently validate such proofs. Finally, some experimental results showing the effectiveness of our techniques are given.

Challenges in Constraint-Based Analysis of Hybrid Systems

In the analysis of hybrid discrete-continuous systems, rich arithmetic constraint formulae with complex Boolean structure arise naturally. The iSAT algorithm, a solver for such formulae, is aimed at bounded model checking of hybrid systems. In this paper, we identify challenges emerging from planned and ongoing work to enhance the iSAT algorithm. First, we propose an extension of iSAT to directly handle ordinary differential equations as constraints. Second, we outline the recently introduced generalization of the iSAT algorithm to deal with probabilistic hybrid systems and some open research issues in that context. Third, we present ideas on how to move from bounded to unbounded model checking by using the concept of interpolation. Finally, we discuss the adaption of some parallelization techniques to the iSAT case, which will hopefully lead to performance gains in the future. By presenting these open research questions, this paper aims at fostering discussions on these extensions of constraint solving.

Integration of an LP solver into interval constraint propagation

This paper describes the integration of an LP solver into iSAT, a Satisfiability Modulo Theories solver that can solve Boolean combinations of linear and nonlinear constraints. iSAT is a tight integration of the well-known DPLL algorithm and interval constraint propagation allowing it to reason about linear and nonlinear constraints. As interval arithmetic is known to be less efficient on solving linear programs, we will demonstrate how the integration of an LP solver can improve the overall solving performance of iSAT.