Farkas-Based Tree Interpolation

Abstract

Linear arithmetic over reals (LRA) underlies a wide range of SMT-based modeling approaches, and, strengthened with Craig interpolation using Farkas’ lemma, is a central tool for efficient over-approximation. Recent advances in LRA interpolation have resulted in a range of promising interpolation algorithms with so far poorly understood properties. In this work we study the Farkas-based algorithms with respect to tree interpolation, a practically important approach where a set of interpolants is constructed following a given tree structure. We classify the algorithms based on whether they guarantee the tree interpolation property, and present how to lift a recently introduced approach producing conjunctive LRA interpolants to tree interpolation in the quantifier-free LRA fragment of first-order logic. Our experiments show that the standard interpolation and the approach using conjunctive interpolants are complementary in tree interpolation, and suggest that their combination would be very powerful in practice.

This work was supported by Swiss National Science Foundation grant 200021_185031 and by Czech Science Foundation grant 20-07487S.

A Appendix A

In this appendix we give auxiliary material for more formal treatment of the connection between propositional and theory interpolation. The propositional resolution rule state that an assignment satisfying the clauses $cl^+ \vee p$ and $cl^- \vee \overline{p}$ also satisfies $cl^+ \vee cl^-$.

Our propositional interpolation works on a refutation of a formula $A \wedge B$. We denote atoms of A and B as $Atoms (A, B)$. Note that each $At \in Atoms (A, B)$ may appear only in A, only in B, or in both conjuncts; Similarly to the notation in [11], we assign a color among $\{a,b,ab\}$ independently to each $At$, depending on whether $At$ occurs only in A, only in B, or in both, respectively.

Table 2 describes the Pudlák interpolation algorithm, where the notation $p\text {:}\varepsilon$ indicates that a literal p has color $\varepsilon$.

Table 2. Pudlák’s interpolation algorithm

Lemma 2 (source clause, base case)

The strong tree-interpolation property holds for Pudlák’s interpolation algorithm $Itp ^{P}{}$ for source clauses.

Proof

Let cl be a source clause. There are three cases: $cl \in X$, $cl \in Y$, or $cl \in Z$. We consider the three interpolation instances $(X \,| \, Y\wedge Z)$, $(Y \,| \, X\wedge Z)$, and $(X\wedge Y \,| \, Z)$, and check whether TI holds, i.e., whether

$$\begin{aligned} Itp ^{P}{(X \,| \, Y \wedge Z)} \wedge Itp ^{P}{(Y \,| \, X\wedge Z)} \implies Itp ^{P}{(X\wedge Y \,| \, Z)}. \end{aligned}$$

(3)

The relevant part in the algorithm is shown in Table 2 (left).

• $cl \in X$: When $cl \in X$, using Pudlák’s interpolation algorithm and substituting the interpolants in Eq. (3), we have $( {\bot } \wedge {\top } ) \implies {\bot }$, which is valid.

• $cl \in Y$: The case $cl \in Y$ is symmetric to the case when $cl \in X$, and thus valid.

• $cl \in Z$: When $cl \in Z$, we have again by substiting in Eq. (3) $( {\top } \wedge {\top } ) \implies {\top }$, which is valid.

   $\Box$

Lemma 3 (inner node)

Let p be a variable. In refutation $\mathbb {P}$, if partial interpolants for nodes $cl^+ \vee p$ and $cl^- \vee \hat{p}$ satisfy the strong tree-interpolation property, then the partial interpolant for $cl^+ \vee cl^-$ satisfy the strong tree-interpolation property.

Proof

We show that for all resolvents in refutation $\mathbb {P}$, the implication $(I_X \wedge I_Y) \implies I_{ XY }$ holds, where $I_X = (X \,| \, Y\wedge Z), I_Y = (Y \,| \, X\wedge Z)$, and $I_{ XY } = ( XY \,| \, Z).$

we consider a node $cl^+ \vee cl^-$ representing resolution over a variable p with parent nodes $p \vee cl^+$ and $\bar{p} \vee cl^-$. From the inductive hypotheses, we have partial interpolants $I^+_X$, $I^+_Y$, and $I^+_{XY}$ for the node $p \vee cl^+$ so that $(I^+_X \wedge I^+_Y) \implies I^+_{XY}$ and partial interpolants $I^-_X$, $I^-_Y$, and $I^-_{XY}$ for the node $\bar{p} \vee cl^-$ so that $(I^-_X \wedge I^-_Y) \implies I^-_{XY}$.

We consider different cases of coloring of p. Depending on presence of p in the three partitions, i.e., X, Y, and Z, and also depending on interpolation instances $(X \,| \, Y \wedge Z)$, $(Y \,| \, X \wedge Z)$, and $(X\wedge Y \,| \, Z)$, p is colored a, b, or ab (Table 3).

Table 3. Coloring of variable p for each partial interpolant.

In case of $p \in X$, based on Pudlák’s algorithm 2, $I_X \equiv I^+_X \vee I^-_X , \quad I_Y \equiv I^+_Y \wedge I^-_Y , \quad I_{XY} \equiv I^+_{XY} \vee I^-_{XY}$.

Using the inductive hypothesis, we have $((I^+_X \vee I^-_X) \wedge I^+_Y \wedge I^-_Y) \implies (I^+_{XY} \vee I^-_{XY})$, which is the required claim $(I_X \wedge I_Y) \implies I_{XY}$. The case $p \in Y$ is symmetric.

In case of $p \in Z$, we have $I_X \equiv I^+_X \wedge I^-_X , \quad I_Y \equiv I^+_Y \wedge I^-_Y , \quad I_{XY} \equiv I^+_{XY} \wedge I^-_{XY}$. Using the inductive hypothesis, we have $(I^+_X \wedge I^-_X \wedge I^+_Y \wedge I^-_Y) \implies (I^+_{XY} \wedge I^-_{XY})$, which is the required claim $(I_X \wedge I_Y) \implies I_{XY}$.

In case of $p \in X \cap Y \cap Z$, using sel(p, P, Q) as a shortcut for $(p \vee P) \wedge (\bar{p} \vee Q)$, we get: $I_X = sel(p, I^+_X, I^-_X) , \quad I_Y = sel(p, I^+_Y, I^-_Y) , \quad I_{XY} = sel(p, I^+_{XY}, I^-_{XY}).$ Using the inductive hypothesis and considering both possible values of p, we have $(sel(p, I^+_X, I^-_X) \wedge sel(p, I^+_Y, I^-_Y)) \implies sel(p, I^+_{XY}, I^-_{XY})$, which is the desired claim $(I_X \wedge I_Y) \implies I_{XY}$. The other cases where $p \in X \cap Y$ or $p \in X \cap Z$ or $p \in Y \cap Z$ are subsumed by this case as $(P \wedge Q) \implies sel(p, P, Q) \implies (P \vee Q)$.   $\Box$