/export/starexec/sandbox/solver/bin/starexec_run_tct_dci /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?,O(n^2)) * Step 1: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: +(x,+(y,z)) -> +(+(x,y),z) +(x,0()) -> x +(x,i(x)) -> 0() +(+(x,y),i(y)) -> x +(+(x,i(y)),y) -> x +(0(),y) -> y +(i(x),x) -> 0() i(+(x,y)) -> +(i(x),i(y)) i(0()) -> 0() i(i(x)) -> x - Signature: {+/2,i/1} / {0/0} - Obligation: innermost derivational complexity wrt. signature {+,0,i} + 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(+) = [1] x1 + [1] x2 + [0] p(0) = [0] p(i) = [1] x1 + [8] Following rules are strictly oriented: +(x,i(x)) = [2] x + [8] > [0] = 0() +(+(x,y),i(y)) = [1] x + [2] y + [8] > [1] x + [0] = x +(+(x,i(y)),y) = [1] x + [2] y + [8] > [1] x + [0] = x +(i(x),x) = [2] x + [8] > [0] = 0() i(0()) = [8] > [0] = 0() i(i(x)) = [1] x + [16] > [1] x + [0] = x Following rules are (at-least) weakly oriented: +(x,+(y,z)) = [1] x + [1] y + [1] z + [0] >= [1] x + [1] y + [1] z + [0] = +(+(x,y),z) +(x,0()) = [1] x + [0] >= [1] x + [0] = x +(0(),y) = [1] y + [0] >= [1] y + [0] = y i(+(x,y)) = [1] x + [1] y + [8] >= [1] x + [1] y + [16] = +(i(x),i(y)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: +(x,+(y,z)) -> +(+(x,y),z) +(x,0()) -> x +(0(),y) -> y i(+(x,y)) -> +(i(x),i(y)) - Weak TRS: +(x,i(x)) -> 0() +(+(x,y),i(y)) -> x +(+(x,i(y)),y) -> x +(i(x),x) -> 0() i(0()) -> 0() i(i(x)) -> x - Signature: {+/2,i/1} / {0/0} - Obligation: innermost derivational complexity wrt. signature {+,0,i} + 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(+) = [1] x1 + [1] x2 + [1] p(0) = [0] p(i) = [1] x1 + [0] Following rules are strictly oriented: +(x,0()) = [1] x + [1] > [1] x + [0] = x +(0(),y) = [1] y + [1] > [1] y + [0] = y Following rules are (at-least) weakly oriented: +(x,+(y,z)) = [1] x + [1] y + [1] z + [2] >= [1] x + [1] y + [1] z + [2] = +(+(x,y),z) +(x,i(x)) = [2] x + [1] >= [0] = 0() +(+(x,y),i(y)) = [1] x + [2] y + [2] >= [1] x + [0] = x +(+(x,i(y)),y) = [1] x + [2] y + [2] >= [1] x + [0] = x +(i(x),x) = [2] x + [1] >= [0] = 0() i(+(x,y)) = [1] x + [1] y + [1] >= [1] x + [1] y + [1] = +(i(x),i(y)) i(0()) = [0] >= [0] = 0() i(i(x)) = [1] x + [0] >= [1] x + [0] = x * Step 3: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: +(x,+(y,z)) -> +(+(x,y),z) i(+(x,y)) -> +(i(x),i(y)) - Weak TRS: +(x,0()) -> x +(x,i(x)) -> 0() +(+(x,y),i(y)) -> x +(+(x,i(y)),y) -> x +(0(),y) -> y +(i(x),x) -> 0() i(0()) -> 0() i(i(x)) -> x - Signature: {+/2,i/1} / {0/0} - Obligation: innermost derivational complexity wrt. signature {+,0,i} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, 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(+) = [1 0] x1 + [1 1] x2 + [5] [0 1] [0 1] [2] p(0) = [6] [2] p(i) = [1 4] x1 + [1] [0 1] [0] Following rules are strictly oriented: +(x,+(y,z)) = [1 0] x + [1 1] y + [1 2] z + [12] [0 1] [0 1] [0 1] [4] > [1 0] x + [1 1] y + [1 1] z + [10] [0 1] [0 1] [0 1] [4] = +(+(x,y),z) i(+(x,y)) = [1 4] x + [1 5] y + [14] [0 1] [0 1] [2] > [1 4] x + [1 5] y + [7] [0 1] [0 1] [2] = +(i(x),i(y)) Following rules are (at-least) weakly oriented: +(x,0()) = [1 0] x + [13] [0 1] [4] >= [1 0] x + [0] [0 1] [0] = x +(x,i(x)) = [2 5] x + [6] [0 2] [2] >= [6] [2] = 0() +(+(x,y),i(y)) = [1 0] x + [2 6] y + [11] [0 1] [0 2] [4] >= [1 0] x + [0] [0 1] [0] = x +(+(x,i(y)),y) = [1 0] x + [2 6] y + [11] [0 1] [0 2] [4] >= [1 0] x + [0] [0 1] [0] = x +(0(),y) = [1 1] y + [11] [0 1] [4] >= [1 0] y + [0] [0 1] [0] = y +(i(x),x) = [2 5] x + [6] [0 2] [2] >= [6] [2] = 0() i(0()) = [15] [2] >= [6] [2] = 0() i(i(x)) = [1 8] x + [2] [0 1] [0] >= [1 0] x + [0] [0 1] [0] = x * Step 4: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: +(x,+(y,z)) -> +(+(x,y),z) +(x,0()) -> x +(x,i(x)) -> 0() +(+(x,y),i(y)) -> x +(+(x,i(y)),y) -> x +(0(),y) -> y +(i(x),x) -> 0() i(+(x,y)) -> +(i(x),i(y)) i(0()) -> 0() i(i(x)) -> x - Signature: {+/2,i/1} / {0/0} - Obligation: innermost derivational complexity wrt. signature {+,0,i} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))