/export/starexec/sandbox2/solver/bin/starexec_run_tct_dci /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?,O(n^2)) * Step 1: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: B(B(x1)) -> b(x1) B(b(x1)) -> x1 C(x1) -> c(x1) C(c(x1)) -> x1 b(B(x1)) -> x1 b(b(x1)) -> B(x1) c(B(c(b(c(x1))))) -> B(c(b(c(B(c(b(x1))))))) c(C(x1)) -> x1 c(c(x1)) -> x1 - Signature: {B/1,C/1,b/1,c/1} / {} - Obligation: innermost derivational complexity wrt. signature {B,C,b,c} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind triangular matrix interpretation: Following symbols are considered usable: all TcT has computed the following interpretation: p(B) = [1] x1 + [0] p(C) = [1] x1 + [2] p(b) = [1] x1 + [0] p(c) = [1] x1 + [0] Following rules are strictly oriented: C(x1) = [1] x1 + [2] > [1] x1 + [0] = c(x1) C(c(x1)) = [1] x1 + [2] > [1] x1 + [0] = x1 c(C(x1)) = [1] x1 + [2] > [1] x1 + [0] = x1 Following rules are (at-least) weakly oriented: B(B(x1)) = [1] x1 + [0] >= [1] x1 + [0] = b(x1) B(b(x1)) = [1] x1 + [0] >= [1] x1 + [0] = x1 b(B(x1)) = [1] x1 + [0] >= [1] x1 + [0] = x1 b(b(x1)) = [1] x1 + [0] >= [1] x1 + [0] = B(x1) c(B(c(b(c(x1))))) = [1] x1 + [0] >= [1] x1 + [0] = B(c(b(c(B(c(b(x1))))))) c(c(x1)) = [1] x1 + [0] >= [1] x1 + [0] = x1 * Step 2: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: B(B(x1)) -> b(x1) B(b(x1)) -> x1 b(B(x1)) -> x1 b(b(x1)) -> B(x1) c(B(c(b(c(x1))))) -> B(c(b(c(B(c(b(x1))))))) c(c(x1)) -> x1 - Weak TRS: C(x1) -> c(x1) C(c(x1)) -> x1 c(C(x1)) -> x1 - Signature: {B/1,C/1,b/1,c/1} / {} - Obligation: innermost derivational complexity wrt. signature {B,C,b,c} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind triangular matrix interpretation: Following symbols are considered usable: all TcT has computed the following interpretation: p(B) = [1] x1 + [0] p(C) = [1] x1 + [13] p(b) = [1] x1 + [0] p(c) = [1] x1 + [2] Following rules are strictly oriented: c(c(x1)) = [1] x1 + [4] > [1] x1 + [0] = x1 Following rules are (at-least) weakly oriented: B(B(x1)) = [1] x1 + [0] >= [1] x1 + [0] = b(x1) B(b(x1)) = [1] x1 + [0] >= [1] x1 + [0] = x1 C(x1) = [1] x1 + [13] >= [1] x1 + [2] = c(x1) C(c(x1)) = [1] x1 + [15] >= [1] x1 + [0] = x1 b(B(x1)) = [1] x1 + [0] >= [1] x1 + [0] = x1 b(b(x1)) = [1] x1 + [0] >= [1] x1 + [0] = B(x1) c(B(c(b(c(x1))))) = [1] x1 + [6] >= [1] x1 + [6] = B(c(b(c(B(c(b(x1))))))) c(C(x1)) = [1] x1 + [15] >= [1] x1 + [0] = x1 * Step 3: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: B(B(x1)) -> b(x1) B(b(x1)) -> x1 b(B(x1)) -> x1 b(b(x1)) -> B(x1) c(B(c(b(c(x1))))) -> B(c(b(c(B(c(b(x1))))))) - Weak TRS: C(x1) -> c(x1) C(c(x1)) -> x1 c(C(x1)) -> x1 c(c(x1)) -> x1 - Signature: {B/1,C/1,b/1,c/1} / {} - Obligation: innermost derivational complexity wrt. signature {B,C,b,c} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind triangular matrix interpretation: Following symbols are considered usable: all TcT has computed the following interpretation: p(B) = [1] x1 + [3] p(C) = [1] x1 + [9] p(b) = [1] x1 + [3] p(c) = [1] x1 + [4] Following rules are strictly oriented: B(B(x1)) = [1] x1 + [6] > [1] x1 + [3] = b(x1) B(b(x1)) = [1] x1 + [6] > [1] x1 + [0] = x1 b(B(x1)) = [1] x1 + [6] > [1] x1 + [0] = x1 b(b(x1)) = [1] x1 + [6] > [1] x1 + [3] = B(x1) Following rules are (at-least) weakly oriented: C(x1) = [1] x1 + [9] >= [1] x1 + [4] = c(x1) C(c(x1)) = [1] x1 + [13] >= [1] x1 + [0] = x1 c(B(c(b(c(x1))))) = [1] x1 + [18] >= [1] x1 + [24] = B(c(b(c(B(c(b(x1))))))) c(C(x1)) = [1] x1 + [13] >= [1] x1 + [0] = x1 c(c(x1)) = [1] x1 + [8] >= [1] x1 + [0] = x1 Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: MI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: c(B(c(b(c(x1))))) -> B(c(b(c(B(c(b(x1))))))) - Weak TRS: B(B(x1)) -> b(x1) B(b(x1)) -> x1 C(x1) -> c(x1) C(c(x1)) -> x1 b(B(x1)) -> x1 b(b(x1)) -> B(x1) c(C(x1)) -> x1 c(c(x1)) -> x1 - Signature: {B/1,C/1,b/1,c/1} / {} - Obligation: innermost derivational complexity wrt. signature {B,C,b,c} + Applied Processor: MI {miKind = Automaton Nothing, miDimension = 4, miUArgs = NoUArgs, miURules = NoURules, miSelector = Just any strict-rules} + Details: We apply a matrix interpretation of kind Automaton Nothing: Following symbols are considered usable: all TcT has computed the following interpretation: p(B) = [1 0 0 0] [0] [0 0 1 0] x_1 + [0] [0 0 0 1] [0] [0 1 0 0] [0] p(C) = [1 0 0 1] [1] [0 0 1 0] x_1 + [1] [0 1 0 0] [0] [0 0 0 1] [0] p(b) = [1 0 0 0] [0] [0 0 0 1] x_1 + [0] [0 1 0 0] [0] [0 0 1 0] [0] p(c) = [1 0 0 1] [0] [0 0 1 0] x_1 + [1] [0 1 0 0] [0] [0 0 0 1] [0] Following rules are strictly oriented: c(B(c(b(c(x1))))) = [1 1 1 1] [2] [0 1 0 0] x1 + [1] [0 0 0 1] [0] [0 0 1 0] [2] > [1 1 1 1] [1] [0 1 0 0] x1 + [1] [0 0 0 1] [0] [0 0 1 0] [2] = B(c(b(c(B(c(b(x1))))))) Following rules are (at-least) weakly oriented: B(B(x1)) = [1 0 0 0] [0] [0 0 0 1] x1 + [0] [0 1 0 0] [0] [0 0 1 0] [0] >= [1 0 0 0] [0] [0 0 0 1] x1 + [0] [0 1 0 0] [0] [0 0 1 0] [0] = b(x1) B(b(x1)) = [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 1 0] [0] [0 0 0 1] [0] >= [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 1 0] [0] [0 0 0 1] [0] = x1 C(x1) = [1 0 0 1] [1] [0 0 1 0] x1 + [1] [0 1 0 0] [0] [0 0 0 1] [0] >= [1 0 0 1] [0] [0 0 1 0] x1 + [1] [0 1 0 0] [0] [0 0 0 1] [0] = c(x1) C(c(x1)) = [1 0 0 2] [1] [0 1 0 0] x1 + [1] [0 0 1 0] [1] [0 0 0 1] [0] >= [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 1 0] [0] [0 0 0 1] [0] = x1 b(B(x1)) = [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 1 0] [0] [0 0 0 1] [0] >= [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 1 0] [0] [0 0 0 1] [0] = x1 b(b(x1)) = [1 0 0 0] [0] [0 0 1 0] x1 + [0] [0 0 0 1] [0] [0 1 0 0] [0] >= [1 0 0 0] [0] [0 0 1 0] x1 + [0] [0 0 0 1] [0] [0 1 0 0] [0] = B(x1) c(C(x1)) = [1 0 0 2] [1] [0 1 0 0] x1 + [1] [0 0 1 0] [1] [0 0 0 1] [0] >= [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 1 0] [0] [0 0 0 1] [0] = x1 c(c(x1)) = [1 0 0 2] [0] [0 1 0 0] x1 + [1] [0 0 1 0] [1] [0 0 0 1] [0] >= [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 1 0] [0] [0 0 0 1] [0] = x1 * Step 5: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: B(B(x1)) -> b(x1) B(b(x1)) -> x1 C(x1) -> c(x1) C(c(x1)) -> x1 b(B(x1)) -> x1 b(b(x1)) -> B(x1) c(B(c(b(c(x1))))) -> B(c(b(c(B(c(b(x1))))))) c(C(x1)) -> x1 c(c(x1)) -> x1 - Signature: {B/1,C/1,b/1,c/1} / {} - Obligation: innermost derivational complexity wrt. signature {B,C,b,c} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))