/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: a__c(X) -> c(X) a__c(X) -> d(X) a__f(X) -> f(X) a__f(f(X)) -> a__c(f(g(f(X)))) a__h(X) -> a__c(d(X)) a__h(X) -> h(X) mark(c(X)) -> a__c(X) mark(d(X)) -> d(X) mark(f(X)) -> a__f(mark(X)) mark(g(X)) -> g(X) mark(h(X)) -> a__h(mark(X)) - Signature: {a__c/1,a__f/1,a__h/1,mark/1} / {c/1,d/1,f/1,g/1,h/1} - Obligation: innermost derivational complexity wrt. signature {a__c,a__f,a__h,c,d,f,g,h,mark} + 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(a__c) = [1] x1 + [0] p(a__f) = [1] x1 + [0] p(a__h) = [1] x1 + [2] p(c) = [1] x1 + [0] p(d) = [1] x1 + [0] p(f) = [1] x1 + [0] p(g) = [1] x1 + [0] p(h) = [1] x1 + [2] p(mark) = [1] x1 + [0] Following rules are strictly oriented: a__h(X) = [1] X + [2] > [1] X + [0] = a__c(d(X)) Following rules are (at-least) weakly oriented: a__c(X) = [1] X + [0] >= [1] X + [0] = c(X) a__c(X) = [1] X + [0] >= [1] X + [0] = d(X) a__f(X) = [1] X + [0] >= [1] X + [0] = f(X) a__f(f(X)) = [1] X + [0] >= [1] X + [0] = a__c(f(g(f(X)))) a__h(X) = [1] X + [2] >= [1] X + [2] = h(X) mark(c(X)) = [1] X + [0] >= [1] X + [0] = a__c(X) mark(d(X)) = [1] X + [0] >= [1] X + [0] = d(X) mark(f(X)) = [1] X + [0] >= [1] X + [0] = a__f(mark(X)) mark(g(X)) = [1] X + [0] >= [1] X + [0] = g(X) mark(h(X)) = [1] X + [2] >= [1] X + [2] = a__h(mark(X)) * Step 2: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__c(X) -> c(X) a__c(X) -> d(X) a__f(X) -> f(X) a__f(f(X)) -> a__c(f(g(f(X)))) a__h(X) -> h(X) mark(c(X)) -> a__c(X) mark(d(X)) -> d(X) mark(f(X)) -> a__f(mark(X)) mark(g(X)) -> g(X) mark(h(X)) -> a__h(mark(X)) - Weak TRS: a__h(X) -> a__c(d(X)) - Signature: {a__c/1,a__f/1,a__h/1,mark/1} / {c/1,d/1,f/1,g/1,h/1} - Obligation: innermost derivational complexity wrt. signature {a__c,a__f,a__h,c,d,f,g,h,mark} + 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(a__c) = [1] x1 + [0] p(a__f) = [1] x1 + [4] p(a__h) = [1] x1 + [0] p(c) = [1] x1 + [0] p(d) = [1] x1 + [0] p(f) = [1] x1 + [7] p(g) = [1] x1 + [9] p(h) = [1] x1 + [0] p(mark) = [1] x1 + [1] Following rules are strictly oriented: mark(c(X)) = [1] X + [1] > [1] X + [0] = a__c(X) mark(d(X)) = [1] X + [1] > [1] X + [0] = d(X) mark(f(X)) = [1] X + [8] > [1] X + [5] = a__f(mark(X)) mark(g(X)) = [1] X + [10] > [1] X + [9] = g(X) Following rules are (at-least) weakly oriented: a__c(X) = [1] X + [0] >= [1] X + [0] = c(X) a__c(X) = [1] X + [0] >= [1] X + [0] = d(X) a__f(X) = [1] X + [4] >= [1] X + [7] = f(X) a__f(f(X)) = [1] X + [11] >= [1] X + [23] = a__c(f(g(f(X)))) a__h(X) = [1] X + [0] >= [1] X + [0] = a__c(d(X)) a__h(X) = [1] X + [0] >= [1] X + [0] = h(X) mark(h(X)) = [1] X + [1] >= [1] X + [1] = a__h(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__c(X) -> c(X) a__c(X) -> d(X) a__f(X) -> f(X) a__f(f(X)) -> a__c(f(g(f(X)))) a__h(X) -> h(X) mark(h(X)) -> a__h(mark(X)) - Weak TRS: a__h(X) -> a__c(d(X)) mark(c(X)) -> a__c(X) mark(d(X)) -> d(X) mark(f(X)) -> a__f(mark(X)) mark(g(X)) -> g(X) - Signature: {a__c/1,a__f/1,a__h/1,mark/1} / {c/1,d/1,f/1,g/1,h/1} - Obligation: innermost derivational complexity wrt. signature {a__c,a__f,a__h,c,d,f,g,h,mark} + 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(a__c) = [1] x1 + [0] p(a__f) = [1] x1 + [5] p(a__h) = [1] x1 + [8] p(c) = [1] x1 + [0] p(d) = [1] x1 + [8] p(f) = [1] x1 + [5] p(g) = [1] x1 + [0] p(h) = [1] x1 + [13] p(mark) = [1] x1 + [0] Following rules are strictly oriented: mark(h(X)) = [1] X + [13] > [1] X + [8] = a__h(mark(X)) Following rules are (at-least) weakly oriented: a__c(X) = [1] X + [0] >= [1] X + [0] = c(X) a__c(X) = [1] X + [0] >= [1] X + [8] = d(X) a__f(X) = [1] X + [5] >= [1] X + [5] = f(X) a__f(f(X)) = [1] X + [10] >= [1] X + [10] = a__c(f(g(f(X)))) a__h(X) = [1] X + [8] >= [1] X + [8] = a__c(d(X)) a__h(X) = [1] X + [8] >= [1] X + [13] = h(X) mark(c(X)) = [1] X + [0] >= [1] X + [0] = a__c(X) mark(d(X)) = [1] X + [8] >= [1] X + [8] = d(X) mark(f(X)) = [1] X + [5] >= [1] X + [5] = a__f(mark(X)) mark(g(X)) = [1] X + [0] >= [1] X + [0] = g(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__c(X) -> c(X) a__c(X) -> d(X) a__f(X) -> f(X) a__f(f(X)) -> a__c(f(g(f(X)))) a__h(X) -> h(X) - Weak TRS: a__h(X) -> a__c(d(X)) mark(c(X)) -> a__c(X) mark(d(X)) -> d(X) mark(f(X)) -> a__f(mark(X)) mark(g(X)) -> g(X) mark(h(X)) -> a__h(mark(X)) - Signature: {a__c/1,a__f/1,a__h/1,mark/1} / {c/1,d/1,f/1,g/1,h/1} - Obligation: innermost derivational complexity wrt. signature {a__c,a__f,a__h,c,d,f,g,h,mark} + 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(a__c) = [1] x1 + [1] p(a__f) = [1] x1 + [0] p(a__h) = [1] x1 + [2] p(c) = [1] x1 + [3] p(d) = [1] x1 + [0] p(f) = [1] x1 + [0] p(g) = [1] x1 + [0] p(h) = [1] x1 + [4] p(mark) = [1] x1 + [0] Following rules are strictly oriented: a__c(X) = [1] X + [1] > [1] X + [0] = d(X) Following rules are (at-least) weakly oriented: a__c(X) = [1] X + [1] >= [1] X + [3] = c(X) a__f(X) = [1] X + [0] >= [1] X + [0] = f(X) a__f(f(X)) = [1] X + [0] >= [1] X + [1] = a__c(f(g(f(X)))) a__h(X) = [1] X + [2] >= [1] X + [1] = a__c(d(X)) a__h(X) = [1] X + [2] >= [1] X + [4] = h(X) mark(c(X)) = [1] X + [3] >= [1] X + [1] = a__c(X) mark(d(X)) = [1] X + [0] >= [1] X + [0] = d(X) mark(f(X)) = [1] X + [0] >= [1] X + [0] = a__f(mark(X)) mark(g(X)) = [1] X + [0] >= [1] X + [0] = g(X) mark(h(X)) = [1] X + [4] >= [1] X + [2] = a__h(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__c(X) -> c(X) a__f(X) -> f(X) a__f(f(X)) -> a__c(f(g(f(X)))) a__h(X) -> h(X) - Weak TRS: a__c(X) -> d(X) a__h(X) -> a__c(d(X)) mark(c(X)) -> a__c(X) mark(d(X)) -> d(X) mark(f(X)) -> a__f(mark(X)) mark(g(X)) -> g(X) mark(h(X)) -> a__h(mark(X)) - Signature: {a__c/1,a__f/1,a__h/1,mark/1} / {c/1,d/1,f/1,g/1,h/1} - Obligation: innermost derivational complexity wrt. signature {a__c,a__f,a__h,c,d,f,g,h,mark} + 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(a__c) = [1] x1 + [1] p(a__f) = [1] x1 + [0] p(a__h) = [1] x1 + [1] p(c) = [1] x1 + [0] p(d) = [1] x1 + [0] p(f) = [1] x1 + [1] p(g) = [1] x1 + [13] p(h) = [1] x1 + [1] p(mark) = [1] x1 + [8] Following rules are strictly oriented: a__c(X) = [1] X + [1] > [1] X + [0] = c(X) Following rules are (at-least) weakly oriented: a__c(X) = [1] X + [1] >= [1] X + [0] = d(X) a__f(X) = [1] X + [0] >= [1] X + [1] = f(X) a__f(f(X)) = [1] X + [1] >= [1] X + [16] = a__c(f(g(f(X)))) a__h(X) = [1] X + [1] >= [1] X + [1] = a__c(d(X)) a__h(X) = [1] X + [1] >= [1] X + [1] = h(X) mark(c(X)) = [1] X + [8] >= [1] X + [1] = a__c(X) mark(d(X)) = [1] X + [8] >= [1] X + [0] = d(X) mark(f(X)) = [1] X + [9] >= [1] X + [8] = a__f(mark(X)) mark(g(X)) = [1] X + [21] >= [1] X + [13] = g(X) mark(h(X)) = [1] X + [9] >= [1] X + [9] = a__h(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__f(X) -> f(X) a__f(f(X)) -> a__c(f(g(f(X)))) a__h(X) -> h(X) - Weak TRS: a__c(X) -> c(X) a__c(X) -> d(X) a__h(X) -> a__c(d(X)) mark(c(X)) -> a__c(X) mark(d(X)) -> d(X) mark(f(X)) -> a__f(mark(X)) mark(g(X)) -> g(X) mark(h(X)) -> a__h(mark(X)) - Signature: {a__c/1,a__f/1,a__h/1,mark/1} / {c/1,d/1,f/1,g/1,h/1} - Obligation: innermost derivational complexity wrt. signature {a__c,a__f,a__h,c,d,f,g,h,mark} + 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(a__c) = [1 0] x1 + [0] [0 0] [0] p(a__f) = [1 1] x1 + [0] [0 0] [1] p(a__h) = [1 1] x1 + [0] [0 0] [1] p(c) = [1 0] x1 + [0] [0 0] [0] p(d) = [1 0] x1 + [0] [0 0] [0] p(f) = [1 1] x1 + [0] [0 0] [1] p(g) = [1 0] x1 + [0] [0 0] [0] p(h) = [1 1] x1 + [0] [0 0] [1] p(mark) = [1 1] x1 + [0] [0 0] [1] Following rules are strictly oriented: a__f(f(X)) = [1 1] X + [1] [0 0] [1] > [1 1] X + [0] [0 0] [0] = a__c(f(g(f(X)))) Following rules are (at-least) weakly oriented: a__c(X) = [1 0] X + [0] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = c(X) a__c(X) = [1 0] X + [0] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = d(X) a__f(X) = [1 1] X + [0] [0 0] [1] >= [1 1] X + [0] [0 0] [1] = f(X) a__h(X) = [1 1] X + [0] [0 0] [1] >= [1 0] X + [0] [0 0] [0] = a__c(d(X)) a__h(X) = [1 1] X + [0] [0 0] [1] >= [1 1] X + [0] [0 0] [1] = h(X) mark(c(X)) = [1 0] X + [0] [0 0] [1] >= [1 0] X + [0] [0 0] [0] = a__c(X) mark(d(X)) = [1 0] X + [0] [0 0] [1] >= [1 0] X + [0] [0 0] [0] = d(X) mark(f(X)) = [1 1] X + [1] [0 0] [1] >= [1 1] X + [1] [0 0] [1] = a__f(mark(X)) mark(g(X)) = [1 0] X + [0] [0 0] [1] >= [1 0] X + [0] [0 0] [0] = g(X) mark(h(X)) = [1 1] X + [1] [0 0] [1] >= [1 1] X + [1] [0 0] [1] = a__h(mark(X)) * Step 7: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__f(X) -> f(X) a__h(X) -> h(X) - Weak TRS: a__c(X) -> c(X) a__c(X) -> d(X) a__f(f(X)) -> a__c(f(g(f(X)))) a__h(X) -> a__c(d(X)) mark(c(X)) -> a__c(X) mark(d(X)) -> d(X) mark(f(X)) -> a__f(mark(X)) mark(g(X)) -> g(X) mark(h(X)) -> a__h(mark(X)) - Signature: {a__c/1,a__f/1,a__h/1,mark/1} / {c/1,d/1,f/1,g/1,h/1} - Obligation: innermost derivational complexity wrt. signature {a__c,a__f,a__h,c,d,f,g,h,mark} + 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(a__c) = [1 1] x1 + [0] [0 0] [0] p(a__f) = [1 0] x1 + [1] [0 1] [1] p(a__h) = [1 4] x1 + [0] [0 1] [0] p(c) = [1 1] x1 + [0] [0 0] [0] p(d) = [1 0] x1 + [0] [0 0] [0] p(f) = [1 0] x1 + [0] [0 1] [1] p(g) = [1 0] x1 + [0] [0 0] [0] p(h) = [1 4] x1 + [0] [0 1] [0] p(mark) = [1 1] x1 + [0] [0 1] [0] Following rules are strictly oriented: a__f(X) = [1 0] X + [1] [0 1] [1] > [1 0] X + [0] [0 1] [1] = f(X) Following rules are (at-least) weakly oriented: a__c(X) = [1 1] X + [0] [0 0] [0] >= [1 1] X + [0] [0 0] [0] = c(X) a__c(X) = [1 1] X + [0] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = d(X) a__f(f(X)) = [1 0] X + [1] [0 1] [2] >= [1 0] X + [1] [0 0] [0] = a__c(f(g(f(X)))) a__h(X) = [1 4] X + [0] [0 1] [0] >= [1 0] X + [0] [0 0] [0] = a__c(d(X)) a__h(X) = [1 4] X + [0] [0 1] [0] >= [1 4] X + [0] [0 1] [0] = h(X) mark(c(X)) = [1 1] X + [0] [0 0] [0] >= [1 1] X + [0] [0 0] [0] = a__c(X) mark(d(X)) = [1 0] X + [0] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = d(X) mark(f(X)) = [1 1] X + [1] [0 1] [1] >= [1 1] X + [1] [0 1] [1] = a__f(mark(X)) mark(g(X)) = [1 0] X + [0] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = g(X) mark(h(X)) = [1 5] X + [0] [0 1] [0] >= [1 5] X + [0] [0 1] [0] = a__h(mark(X)) * Step 8: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__h(X) -> h(X) - Weak TRS: a__c(X) -> c(X) a__c(X) -> d(X) a__f(X) -> f(X) a__f(f(X)) -> a__c(f(g(f(X)))) a__h(X) -> a__c(d(X)) mark(c(X)) -> a__c(X) mark(d(X)) -> d(X) mark(f(X)) -> a__f(mark(X)) mark(g(X)) -> g(X) mark(h(X)) -> a__h(mark(X)) - Signature: {a__c/1,a__f/1,a__h/1,mark/1} / {c/1,d/1,f/1,g/1,h/1} - Obligation: innermost derivational complexity wrt. signature {a__c,a__f,a__h,c,d,f,g,h,mark} + 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(a__c) = [1 0] x1 + [0] [0 0] [0] p(a__f) = [1 0] x1 + [0] [0 1] [0] p(a__h) = [1 3] x1 + [5] [0 1] [4] p(c) = [1 0] x1 + [0] [0 0] [0] p(d) = [1 0] x1 + [0] [0 0] [0] p(f) = [1 0] x1 + [0] [0 1] [0] p(g) = [1 0] x1 + [0] [0 1] [0] p(h) = [1 3] x1 + [1] [0 1] [4] p(mark) = [1 2] x1 + [0] [0 1] [1] Following rules are strictly oriented: a__h(X) = [1 3] X + [5] [0 1] [4] > [1 3] X + [1] [0 1] [4] = h(X) Following rules are (at-least) weakly oriented: a__c(X) = [1 0] X + [0] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = c(X) a__c(X) = [1 0] X + [0] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = d(X) a__f(X) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = f(X) a__f(f(X)) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 0] [0] = a__c(f(g(f(X)))) a__h(X) = [1 3] X + [5] [0 1] [4] >= [1 0] X + [0] [0 0] [0] = a__c(d(X)) mark(c(X)) = [1 0] X + [0] [0 0] [1] >= [1 0] X + [0] [0 0] [0] = a__c(X) mark(d(X)) = [1 0] X + [0] [0 0] [1] >= [1 0] X + [0] [0 0] [0] = d(X) mark(f(X)) = [1 2] X + [0] [0 1] [1] >= [1 2] X + [0] [0 1] [1] = a__f(mark(X)) mark(g(X)) = [1 2] X + [0] [0 1] [1] >= [1 0] X + [0] [0 1] [0] = g(X) mark(h(X)) = [1 5] X + [9] [0 1] [5] >= [1 5] X + [8] [0 1] [5] = a__h(mark(X)) * Step 9: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a__c(X) -> c(X) a__c(X) -> d(X) a__f(X) -> f(X) a__f(f(X)) -> a__c(f(g(f(X)))) a__h(X) -> a__c(d(X)) a__h(X) -> h(X) mark(c(X)) -> a__c(X) mark(d(X)) -> d(X) mark(f(X)) -> a__f(mark(X)) mark(g(X)) -> g(X) mark(h(X)) -> a__h(mark(X)) - Signature: {a__c/1,a__f/1,a__h/1,mark/1} / {c/1,d/1,f/1,g/1,h/1} - Obligation: innermost derivational complexity wrt. signature {a__c,a__f,a__h,c,d,f,g,h,mark} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))