Improving Interpolants for Linear Arithmetic

Ernst Althaus; Björn Beber; Joschka Kupilas; Christoph Scholl

doi:10.1007/978-3-319-24953-7_5

ATVA 2015: Automated Technology for Verification and Analysis pp 48-63 | Cite as

Improving Interpolants for Linear Arithmetic

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Ernst Althaus
Björn BeberEmail author
Joschka Kupilas
Christoph Scholl

Conference paper

First Online: 22 November 2015

Part of the Lecture Notes in Computer Science book series (LNCS, volume 9364)

Abstract

Craig interpolation for satisfiability modulo theory formulas have come more into focus for applications of formal verification. In this paper we, introduce a method to reduce the size of linear constraints used in the description of already computed interpolant in the theory of linear arithmetic with respect to the number of linear constraints. We successfully improve interpolants by combining satisfiability modulo theory and linear programming in a local search heuristic. Our experimental results suggest a lower running time and a larger reduction compared to other methods from the literature.

Keywords

Craig-interpolation Linear arithmetic Satisfiability modulo theory Linear programming

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Notes

Acknowledgment

The results presented in this paper were developed in the context of the Transregional Collaborative Research Center ‘Automatic Verification and Analysis of Complex Systems’ (SFB/TR 14 AVACS) supported by the German Research Council (DFG). We worked in close coorperation with our colleagues from the ’First Order Model Checking Team’ within the subproject H3 and we would like to thank W. Damm, B. Wirtz, W. Hagemann, and A. Rakow from the University of Oldenburg, U. Waldmann from the Max Planck Institute for Informatics at Saarbrücken and S. Disch from the University of Freiburg for numerous ideas and discussions

A Detailed Description of the Linear Program

Recall the variables given in Sect. 3.4. Let $A^i,B^i$ be the convex sets of the i-th iteration, constructed by $s_{\!A^i}$, respectively $s_{\!B^i}$, conjunctions of linear constraints. Then $A^i$ is formally defined by $A^i = \left\{ x \in \mathbb {R}^m |\, \mathcal {A}^i x \le \alpha ^i \right\} $, with Open image in new window and $B^i = \left\{ x \in \mathbb {R}^m |\, \mathcal {B}^i x \le \beta ^i \right\} $, with Open image in new window . We additionally introduce $s_{\!A^i}$ variables $\lambda ^i$ and $s_{\!B^i}$ variables $\mu ^i$ for every iteration $i\in \{1,\dots ,k\}$.

We look for an inequality that maximizes a simple measure of the distance of the constructed inequality to the convex regions. We do this by subtracting the $\varepsilon $ to the positive convex combination of the inequalities from $A^i$ for l, i.e. the convex combination leads to $d^Tx \le d_0 - \varepsilon $. As we can scale any LP-solution by an arbitrary positive scalar so far, we have to normalize the solution. Therefore, we restrict the linear combination of one region to be a convex combination.

Hence, we obtain the following LP, where all linear constraints except (6) and (11) are introduced for all $i \in \left\{ 1 ,\dots , k \right\} $:

$$\begin{aligned} {\max }\quad {\varepsilon } \end{aligned}$$

(3)

$$\begin{aligned} \mathrm{s.t.}\quad {(\mathcal {A}^i)^T\lambda ^i}{=}{d} \end{aligned}$$

(4)

$$\begin{aligned} {(\mathcal {B}^i)^T\mu ^i}{=}{d} \end{aligned}$$

(5)

$$\begin{aligned} {\sum \lambda ^1}{=}{1} \end{aligned}$$

(6)

$$\begin{aligned} {(\alpha ^i)^T\lambda ^i}{\le }{d_0-\varepsilon } \end{aligned}$$

(7)

$$\begin{aligned} {(\beta ^i)^T\mu ^i}{\ge }{d_0+\varepsilon } \end{aligned}$$

(8)

$$\begin{aligned} {\lambda ^i}{\ge }{0} \end{aligned}$$

(9)

$$\begin{aligned} {\mu ^i}{\le }{0} \end{aligned}$$

(10)

$$\begin{aligned} {\varepsilon }{\ge }{0} \end{aligned}$$

(11)

Constraints (4) and (5) force that the direction of the new constraint, described by d, is representable by the linear constraint of the convex regions. Conditions (7–11) verify that convex regions are on the right side of $l^*$. Condition (6) normalizes the solutions.

B Detailed Distinction for Non-Closed Polyhedra

The following proposition states when we have found a separating constraint in case of $\varepsilon = 0$.

Proposition 2

Assume the LP (4–11) has optimal value 0 and let $(\bar{d},\bar{d_0})$ be the solution of the LP for the variables d and $d_0$.

1.

If for all $i \in \left\{ 1, \dots , k \right\} $ either $(\beta ^i)^T\mu ^i - d_0 > 0$ or there exists a strict inequality $s\ne \mathbf {0}$ in $\mathcal {B}^i$ with variable $(\mu ^i)_s$ such that $(\mu ^i)_s<0$, then $\bar{a}^T x \le \bar{d_0}$ separates the regions.
2.

If for all $i \in \left\{ 1, \dots , k \right\} $ either $(\alpha ^i)^T\lambda ^i - d_0 < 0$ or there exists a strict inequality $s\ne \mathbf {0}$ in $\mathcal {A}^i$ with variable $(\lambda ^i)_s$ such that $(\lambda ^i)_s>0$, then $\bar{a}^Tx < \bar{d_0}$ separates the regions.

The proof for this proposition is straight forward and will not be given in the paper.

References

1.
Pudlák, P.: Lower bounds for resolution and cutting plane proofs and monotone computations. J. Symbolic Logic 62(3), 981–998 (1997)MathSciNetCrossRefMATHGoogle Scholar
2.
McMillan, K.L.: Interpolation and sat-based model checking. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 1–13. Springer, Heidelberg (2003) CrossRefGoogle Scholar
3.
Albarghouthi, A., McMillan, K.L.: Beautiful interpolants. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 313–329. Springer, Heidelberg (2013) CrossRefGoogle Scholar
4.
Damm, W., Dierks, H., Disch, S., Hagemann, W., Pigorsch, F., Scholl, C., Waldmann, U., Wirtz, B.: Exact and fully symbolic verification of linear hybrid automata with large discrete state spaces. Sci. Comput. Program. 77(10–11), 1122–1150 (2012)CrossRefMATHGoogle Scholar
5.
Megiddo, N.: On the complexity of polyhedral separability. Discrete Comput. Geom. 3(1), 325–337 (1988)MathSciNetCrossRefMATHGoogle Scholar
6.
Scholl, C., Pigorsch, F., Disch, S., Althaus, E.: Simple interpolants for linear arithmetic. In: Design, Automation and Test in Europe Conference and Exhibition (DATE), 2014, pp. 1–6. IEEE (2014)Google Scholar
7.
William, C.: Three uses of the Herbrand-Gentzen theorem in relating model theory and proof theory. J. Symbolic Logic 22(03), 269–285 (1957)MathSciNetCrossRefMATHGoogle Scholar
8.
McMillan, K.L.: An interpolating theorem prover. Theoret. Comput. Sci. 345(1), 101–121 (2005)MathSciNetCrossRefMATHGoogle Scholar
9.
Rybalchenko, A., Sofronie-Stokkermans, V.: Constraint solving for interpolation. In: Cook, B., Podelski, A. (eds.) VMCAI 2007. LNCS, vol. 4349, pp. 346–362. Springer, Heidelberg (2007) CrossRefGoogle Scholar
10.
Scholl, C., Disch, S., Pigorsch, F., Kupferschmid, S.: Using an SMT solver and craig interpolation to detect and remove redundant linear constraints in representations of non-convex polyhedra. In: Proceedings of the Joint Workshops of the 6th International Workshop on Satisfiability Modulo Theories and 1st International Workshop on Bit-Precise Reasoning, pp. 18–26. ACM (2008)Google Scholar
11.
Damm, W., Disch, S., Hungar, H., Jacobs, S., Pang, J., Pigorsch, F., Scholl, C., Waldmann, U., Wirtz, B.: Exact state set representations in the verification of linear hybrid systems with large discrete state space. In: Namjoshi, K.S., Yoneda, T., Higashino, T., Okamura, Y. (eds.) ATVA 2007. LNCS, vol. 4762, pp. 425–440. Springer, Heidelberg (2007) CrossRefGoogle Scholar
12.
Dutertre, B., De Moura, L.: The yices SMT solver (2006). http://yices.csl.sri.com/tool-paper.pdf
13.
Applegate, D.L., Cook, W., Dash, S., Espinoza, D.G.: Exact solutions to linear programming problems. Oper. Res. Lett. 35(6), 693–699 (2007)MathSciNetCrossRefMATHGoogle Scholar
14.
Griggio, A.: A practical approach to satisfiability modulo linear integer arithmetic. JSAT 8, 1–27 (2012)MathSciNetMATHGoogle Scholar
15.
Rakow, A.: Flap/Slat System. http://www.avacs.org/fileadmin/Benchmarks/Open/FlapSlatSystem.pdf

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Authors and Affiliations

Ernst Althaus
- 1
- 2
Björn Beber
- 1
Email author
Joschka Kupilas
- 1
Christoph Scholl
- 3

1.Max Planck Institute for InformaticsSaarbrückenGermany
2.Johannes Gutenberg UniversityMainzGermany
3.Albert-Ludwigs-UniversitätFreiburg im BreisgauGermany

Improving Interpolants for Linear Arithmetic

Abstract

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Acknowledgment

References

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