Classifying and Solving Horn Clauses for Verification - Lab for ...

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12 Rümmer, Hojjat, Kuncaktruen 8 ≤ 0n 9 ≤ −1 ∨ (rec 9 = 1 ∧ n 9 = 0)x 2 ≥ 0n f ≤ −1 ∨ (rec f = 1 ∧ n f = 0)res 3 = x 3 + 1falseFig. 5. Tree interpolant solving the interpolation problem in Fig. 4The UFO verification system [3] is able to compute DAG interpolants, based onthe interpolation functionality of MathSAT [9]. We can observe that DAG interpolants(despite their name) are incomparable in expressiveness to tree interpolation. This isbecause DAG interpolants correspond to linear Horn clauses, and might have sharedrelation symbol in bodies, while tree interpolants correspond to possibly nonlinear treelikeHorn clauses, but do not allow shared relation symbols in bodies.Encoding of restricted DAG interpolants as linear Horn clauses. For every v ∈ V, let{ ¯x v } = ( ⋃fv(L E (a, v)) ) ∩ ( ⋃fv(L E (v, a)) )(a,v)∈E(v,a)∈Ebe the variables allowed in the interpolant to be computed for v, and p v be a freshrelation symbol of arity | ¯x v |. The interpolation problem is then defined by the followingset of linear Horn clauses:For each (v, w) ∈ E: L V (v) ∧ L E (v, w) ∧ p v ( ¯x v ) → p w ( ¯x w ),L V (v) ∧ ¬L V (w) ∧ L E (v, w) ∧ p v ( ¯x v ) → false,For en, ex ∈ V: true → p en ( ¯x en ), p ex ( ¯x ex ) → falseEncoding of linear Horn clauses as DAG interpolants. Suppose HC is a finite, recursionfree,and linear set of Horn clauses. We can solve the system of Horn clauses by computinga DAG interpolant for every connected component of the → HC -graph. As inSect. 5.2, we normalise Horn clauses according to Def. 2. We also assume that multipleclauses C ∧ p( ¯x p ) → q( ¯x q ) and D ∧ p( ¯x p ) → q( ¯x q ) with the same relation symbols aremerged to (C ∨ D) ∧ p( ¯x p ) → q( ¯x q ).Let {p 1 , . . . , p n } be all relation symbols of one connected component. We then definethe DAG interpolation problem (V, E, en, ex), L E , L V by
Classifying and Solving Horn Clauses for Verification 13– the vertices V = {p 1 , . . . , p n } ∪ {en, ex}, including two fresh nodes en, ex,– the edge relationE ={(p, q) | there is a clause C ∧ p( ¯x p ) → q( ¯x q ) ∈ HC}∪ {(en, p) | there is a clause D → p( ¯x p ) ∈ HC}∪ {(p, ex) | there is a clause E ∧ p( ¯x p ) → false ∈ HC} ,– for each (v, w) ∈ E, the edge labelling⎧C ∧ ¯x v = ¯x v ∧ ¯x w = ¯x w⎪⎨L E (v, w) = D ∧ ¯x w = ¯x w⎪⎩ E ∧ ¯x v = ¯x vif C ∧ v( ¯x v ) → w( ¯x w ) ∈ HCif v = en and D → w( ¯x w ) ∈ HCif w = ex and E ∧ v( ¯x v ) → false ∈ HCNote that the labels include equations like ¯x v = ¯x v to ensure that the right variablesare allowed to occur in interpolants.– for each v ∈ V, the node labelling L V (v) = true.By checking the definition of DAG interpolants, it can be verified that every interpolantsolving the problem (V, E, en, ex), L E , L V is also a solution of the linear Horn clauses.5.5 Disjunctive Interpolants [30]Disjunctive interpolants were introduced in [30] as a generalisation of tree interpolants.Disjunctive interpolants resemble tree interpolants in the sense that the relationship ofthe components of an interpolant is defined by a tree; in contrast to tree interpolants,however, this tree is an and/or-tree: branching in the tree can represent either conjunctionsor disjunctions. Disjunctive interpolants correspond to sets of body-disjoint Hornclauses; in this representation, and-branching is encoded by clauses with multiple bodyliterals (like with tree interpolants), while or-branching is interpreted as multiple clausessharing the same head symbol. For a detailed account on disjunctive interpolants, werefer the reader to [30].The solution of body-disjoint Horn clauses can be computed by solving a sequenceof tree-like sets of Horn clauses:Lemma 2. Let HC be a finite set of recursion-free body-disjoint Horn clauses. HChas a syntactic/semantic solution if and only if every maximum tree-like subset of HChas a syntactic/semantic solution.Proof. We outline direction “⇐” for syntactic solutions. Solving the tree-like subsets ofHC yields, for each relation symbol p ∈ R, a set SC p of solution constraints. A globalsolution of HC can be constructed by forming a positive Boolean combination of theconstraints in SC p for each p ∈ R.⊓⊔Example 2. We consider a recursion-free unwinding of the Horn clauses in Fig. 2. Tomake the set of clauses body-disjoint, the clauses (6), (9), (11), (12) were duplicated,introducing primed copies of all relation symbols involved. The clauses are not headdisjoint,since (10) and (11) share the same head symbol:
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