Improving Interpolants for Linear Arithmetic

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.

Appendices

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 and $B^i = \left\{ x \in \mathbb {R}^m |\, \mathcal {B}^i x \le \beta ^i \right\}$, with . 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.