/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^1)) * Step 1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: evenodd(x,0()) -> not(evenodd(x,s(0()))) evenodd(0(),s(0())) -> false() evenodd(s(x),s(0())) -> evenodd(x,0()) not(false()) -> true() not(true()) -> false() - Signature: {evenodd/2,not/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost derivational complexity wrt. signature {0,evenodd,false,not,s,true} + 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(0) = [0] p(evenodd) = [1] x1 + [1] x2 + [4] p(false) = [0] p(not) = [1] x1 + [0] p(s) = [1] x1 + [0] p(true) = [0] Following rules are strictly oriented: evenodd(0(),s(0())) = [4] > [0] = false() Following rules are (at-least) weakly oriented: evenodd(x,0()) = [1] x + [4] >= [1] x + [4] = not(evenodd(x,s(0()))) evenodd(s(x),s(0())) = [1] x + [4] >= [1] x + [4] = evenodd(x,0()) not(false()) = [0] >= [0] = true() not(true()) = [0] >= [0] = false() * Step 2: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: evenodd(x,0()) -> not(evenodd(x,s(0()))) evenodd(s(x),s(0())) -> evenodd(x,0()) not(false()) -> true() not(true()) -> false() - Weak TRS: evenodd(0(),s(0())) -> false() - Signature: {evenodd/2,not/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost derivational complexity wrt. signature {0,evenodd,false,not,s,true} + 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(0) = [0] p(evenodd) = [1] x1 + [1] x2 + [2] p(false) = [2] p(not) = [1] x1 + [8] p(s) = [1] x1 + [0] p(true) = [8] Following rules are strictly oriented: not(false()) = [10] > [8] = true() not(true()) = [16] > [2] = false() Following rules are (at-least) weakly oriented: evenodd(x,0()) = [1] x + [2] >= [1] x + [10] = not(evenodd(x,s(0()))) evenodd(0(),s(0())) = [2] >= [2] = false() evenodd(s(x),s(0())) = [1] x + [2] >= [1] x + [2] = evenodd(x,0()) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: evenodd(x,0()) -> not(evenodd(x,s(0()))) evenodd(s(x),s(0())) -> evenodd(x,0()) - Weak TRS: evenodd(0(),s(0())) -> false() not(false()) -> true() not(true()) -> false() - Signature: {evenodd/2,not/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost derivational complexity wrt. signature {0,evenodd,false,not,s,true} + 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(0) = [2] p(evenodd) = [1] x1 + [1] x2 + [12] p(false) = [0] p(not) = [1] x1 + [5] p(s) = [1] x1 + [4] p(true) = [1] Following rules are strictly oriented: evenodd(s(x),s(0())) = [1] x + [22] > [1] x + [14] = evenodd(x,0()) Following rules are (at-least) weakly oriented: evenodd(x,0()) = [1] x + [14] >= [1] x + [23] = not(evenodd(x,s(0()))) evenodd(0(),s(0())) = [20] >= [0] = false() not(false()) = [5] >= [1] = true() not(true()) = [6] >= [0] = false() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: evenodd(x,0()) -> not(evenodd(x,s(0()))) - Weak TRS: evenodd(0(),s(0())) -> false() evenodd(s(x),s(0())) -> evenodd(x,0()) not(false()) -> true() not(true()) -> false() - Signature: {evenodd/2,not/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost derivational complexity wrt. signature {0,evenodd,false,not,s,true} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind triangular matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima): Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] [6] p(evenodd) = [1 1] x1 + [1 4] x2 + [0] [0 0] [0 0] [1] p(false) = [2] [0] p(not) = [1 1] x1 + [2] [0 0] [0] p(s) = [1 1] x1 + [10] [0 0] [0] p(true) = [1] [0] Following rules are strictly oriented: evenodd(x,0()) = [1 1] x + [24] [0 0] [1] > [1 1] x + [19] [0 0] [0] = not(evenodd(x,s(0()))) Following rules are (at-least) weakly oriented: evenodd(0(),s(0())) = [22] [1] >= [2] [0] = false() evenodd(s(x),s(0())) = [1 1] x + [26] [0 0] [1] >= [1 1] x + [24] [0 0] [1] = evenodd(x,0()) not(false()) = [4] [0] >= [1] [0] = true() not(true()) = [3] [0] >= [2] [0] = false() * Step 5: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: evenodd(x,0()) -> not(evenodd(x,s(0()))) evenodd(0(),s(0())) -> false() evenodd(s(x),s(0())) -> evenodd(x,0()) not(false()) -> true() not(true()) -> false() - Signature: {evenodd/2,not/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost derivational complexity wrt. signature {0,evenodd,false,not,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))