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

14 Rümmer, Hojjat, Kuncak(1) r1(X, Res) ← true(2) r2(X’, Res) ← r1(X, Res) ∧ X’ ≥ 0(3) r3(X, Res’) ← r2(X, Res) ∧ rf(X, Res’)(5) false ← r3(X, Res) ∧ Res X + 1(6) r5(N, Rec, Tmp) ← true(7) r6(N, Rec, Tmp) ← r5’(N, Rec, Tmp) ∧ N > 0(8) r7(N, Rec, Tmp’) ← r6(N, Rec, Tmp) ∧ rf’(N − 1, Tmp’)(9) r8(N, Rec, Tmp) ← r5(N, Rec, Tmp) ∧ N ≤ 0(10) r9(N, Rec’, Tmp) ← r7(N, Rec, Tmp) ∧ Rec’ = Tmp + 1(11) r9(N, Rec’, Tmp) ← r8(N, Rec, Tmp) ∧ Rec’ = 1(12) rf (N, Rec) ← r9(N, Rec, Tmp)(6’) r5 ’( N, Rec, Tmp) ← true(6’’) r5’’(N, Rec, Tmp) ← true(9’) r8 ’( N, Rec, Tmp) ← r5’’(N, Rec, Tmp) ∧ N ≤ 0(11’) r9 ’( N, Rec’, Tmp) ← r8’(N, Rec, Tmp) ∧ Rec’ = 1(12’) rf ’( N, Rec) ← r9’(N, Rec, Tmp)There are two maximum tree-like subsets: T 1 = {(1), (2), (3), (5), (6), (9), (11), (12)},and T 2 = {(1), (2), (3), (5), (7), (8), (10), (12), (6 ′ ), (6 ′′ ), (9 ′ ), (11 ′ ), (12 ′ )}. The subset T 1has been discussed in Example 1. In the same way, it is possible to construct a solutionfor T 2 by solving a tree interpolation problem. The two solutions can be combined toconstruct a solution of T 1 ∪ T 2 :T 1 T 2 T 1 ∪ T 2r 1 (x, r) true true truer 2 (x, r) x ≥ 0 true x ≥ 0r 3 (x, r) r = x + 1 r = x + 1 r = x + 1r 5 (n, c, t) true true truer 6 (n, c, t) − n ≥ 1 n ≥ 1r 7 (n, c, t) − t = n t = nr 8 (n, c, t) n ≤ 0 − n ≤ 0r 9 (n, c, t) n ≤ −1 ∨ (c = 1 ∧ n = 0) c = n + 1 n ≤ −1 ∨ c = n + 1r f (n, c) n ≤ −1 ∨ (c = 1 ∧ n = 0) c = n + 1 n ≤ −1 ∨ c = n + 1r ′ 5r ′′5r ′ 8r ′ 9r ′ f(n, c, t) − true true(n, c, t) − true true(n, c, t) − n ≤ 0 n ≤ 0(n, c, t) − n ≤ −1 ∨ (c = 1 ∧ n = 0) n ≤ −1 ∨ (c = 1 ∧ n = 0)(n, c, t) − n ≤ −1 ∨ (c = 1 ∧ n = 0) n ≤ −1 ∨ (c = 1 ∧ n = 0)In particular, the disjunction of the two interpretations of r 9 (n, c, t) has to be used,in order to satisfy both (10) and (11) (similarly for r f (n, c)). In contrast, the conjunctionof the interpretations of r 2 (x, r) is needed to satisfy (3).6 The Complexity of Recursion-free Horn Clausesover Quantifier-free Presburger ArithmeticWe give an overview of the considered fragments of recursion-free Horn clauses, andthe corresponding interpolation problem, in Fig. 6. The diagram also shows the com-