/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: active(f(f(a()))) -> mark(c(f(g(f(a()))))) c(active(X)) -> c(X) c(mark(X)) -> c(X) f(active(X)) -> f(X) f(mark(X)) -> f(X) g(active(X)) -> g(X) g(mark(X)) -> g(X) mark(a()) -> active(a()) mark(c(X)) -> active(c(X)) mark(f(X)) -> active(f(mark(X))) mark(g(X)) -> active(g(mark(X))) - Signature: {active/1,c/1,f/1,g/1,mark/1} / {a/0} - Obligation: innermost derivational complexity wrt. signature {a,active,c,f,g,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) = [1] p(active) = [1] x1 + [1] p(c) = [1] x1 + [0] p(f) = [1] x1 + [8] p(g) = [1] x1 + [0] p(mark) = [1] x1 + [0] Following rules are strictly oriented: active(f(f(a()))) = [18] > [17] = mark(c(f(g(f(a()))))) c(active(X)) = [1] X + [1] > [1] X + [0] = c(X) f(active(X)) = [1] X + [9] > [1] X + [8] = f(X) g(active(X)) = [1] X + [1] > [1] X + [0] = g(X) Following rules are (at-least) weakly oriented: c(mark(X)) = [1] X + [0] >= [1] X + [0] = c(X) f(mark(X)) = [1] X + [8] >= [1] X + [8] = f(X) g(mark(X)) = [1] X + [0] >= [1] X + [0] = g(X) mark(a()) = [1] >= [2] = active(a()) mark(c(X)) = [1] X + [0] >= [1] X + [1] = active(c(X)) mark(f(X)) = [1] X + [8] >= [1] X + [9] = active(f(mark(X))) mark(g(X)) = [1] X + [0] >= [1] X + [1] = active(g(mark(X))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: c(mark(X)) -> c(X) f(mark(X)) -> f(X) g(mark(X)) -> g(X) mark(a()) -> active(a()) mark(c(X)) -> active(c(X)) mark(f(X)) -> active(f(mark(X))) mark(g(X)) -> active(g(mark(X))) - Weak TRS: active(f(f(a()))) -> mark(c(f(g(f(a()))))) c(active(X)) -> c(X) f(active(X)) -> f(X) g(active(X)) -> g(X) - Signature: {active/1,c/1,f/1,g/1,mark/1} / {a/0} - Obligation: innermost derivational complexity wrt. signature {a,active,c,f,g,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) = [1] p(active) = [1] x1 + [6] p(c) = [1] x1 + [2] p(f) = [1] x1 + [0] p(g) = [1] x1 + [0] p(mark) = [1] x1 + [3] Following rules are strictly oriented: c(mark(X)) = [1] X + [5] > [1] X + [2] = c(X) f(mark(X)) = [1] X + [3] > [1] X + [0] = f(X) g(mark(X)) = [1] X + [3] > [1] X + [0] = g(X) Following rules are (at-least) weakly oriented: active(f(f(a()))) = [7] >= [6] = mark(c(f(g(f(a()))))) c(active(X)) = [1] X + [8] >= [1] X + [2] = c(X) f(active(X)) = [1] X + [6] >= [1] X + [0] = f(X) g(active(X)) = [1] X + [6] >= [1] X + [0] = g(X) mark(a()) = [4] >= [7] = active(a()) mark(c(X)) = [1] X + [5] >= [1] X + [8] = active(c(X)) mark(f(X)) = [1] X + [3] >= [1] X + [9] = active(f(mark(X))) mark(g(X)) = [1] X + [3] >= [1] X + [9] = active(g(mark(X))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(a()) -> active(a()) mark(c(X)) -> active(c(X)) mark(f(X)) -> active(f(mark(X))) mark(g(X)) -> active(g(mark(X))) - Weak TRS: active(f(f(a()))) -> mark(c(f(g(f(a()))))) c(active(X)) -> c(X) c(mark(X)) -> c(X) f(active(X)) -> f(X) f(mark(X)) -> f(X) g(active(X)) -> g(X) g(mark(X)) -> g(X) - Signature: {active/1,c/1,f/1,g/1,mark/1} / {a/0} - Obligation: innermost derivational complexity wrt. signature {a,active,c,f,g,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) = [0] [0] p(active) = [1 4] x1 + [0] [0 0] [0] p(c) = [1 0] x1 + [0] [0 0] [0] p(f) = [1 4] x1 + [0] [0 0] [1] p(g) = [1 4] x1 + [0] [0 0] [0] p(mark) = [1 4] x1 + [4] [0 0] [0] Following rules are strictly oriented: mark(a()) = [4] [0] > [0] [0] = active(a()) mark(c(X)) = [1 0] X + [4] [0 0] [0] > [1 0] X + [0] [0 0] [0] = active(c(X)) Following rules are (at-least) weakly oriented: active(f(f(a()))) = [8] [0] >= [8] [0] = mark(c(f(g(f(a()))))) c(active(X)) = [1 4] X + [0] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = c(X) c(mark(X)) = [1 4] X + [4] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = c(X) f(active(X)) = [1 4] X + [0] [0 0] [1] >= [1 4] X + [0] [0 0] [1] = f(X) f(mark(X)) = [1 4] X + [4] [0 0] [1] >= [1 4] X + [0] [0 0] [1] = f(X) g(active(X)) = [1 4] X + [0] [0 0] [0] >= [1 4] X + [0] [0 0] [0] = g(X) g(mark(X)) = [1 4] X + [4] [0 0] [0] >= [1 4] X + [0] [0 0] [0] = g(X) mark(f(X)) = [1 4] X + [8] [0 0] [0] >= [1 4] X + [8] [0 0] [0] = active(f(mark(X))) mark(g(X)) = [1 4] X + [4] [0 0] [0] >= [1 4] X + [4] [0 0] [0] = active(g(mark(X))) * Step 4: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(f(X)) -> active(f(mark(X))) mark(g(X)) -> active(g(mark(X))) - Weak TRS: active(f(f(a()))) -> mark(c(f(g(f(a()))))) c(active(X)) -> c(X) c(mark(X)) -> c(X) f(active(X)) -> f(X) f(mark(X)) -> f(X) g(active(X)) -> g(X) g(mark(X)) -> g(X) mark(a()) -> active(a()) mark(c(X)) -> active(c(X)) - Signature: {active/1,c/1,f/1,g/1,mark/1} / {a/0} - Obligation: innermost derivational complexity wrt. signature {a,active,c,f,g,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) = [3] [4] p(active) = [1 0] x1 + [0] [0 1] [0] p(c) = [1 0] x1 + [0] [0 0] [0] p(f) = [1 0] x1 + [5] [0 1] [1] p(g) = [1 0] x1 + [0] [0 1] [0] p(mark) = [1 2] x1 + [0] [0 1] [3] Following rules are strictly oriented: mark(f(X)) = [1 2] X + [7] [0 1] [4] > [1 2] X + [5] [0 1] [4] = active(f(mark(X))) Following rules are (at-least) weakly oriented: active(f(f(a()))) = [13] [6] >= [13] [3] = mark(c(f(g(f(a()))))) c(active(X)) = [1 0] X + [0] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = c(X) c(mark(X)) = [1 2] X + [0] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = c(X) f(active(X)) = [1 0] X + [5] [0 1] [1] >= [1 0] X + [5] [0 1] [1] = f(X) f(mark(X)) = [1 2] X + [5] [0 1] [4] >= [1 0] X + [5] [0 1] [1] = f(X) g(active(X)) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = g(X) g(mark(X)) = [1 2] X + [0] [0 1] [3] >= [1 0] X + [0] [0 1] [0] = g(X) mark(a()) = [11] [7] >= [3] [4] = active(a()) mark(c(X)) = [1 0] X + [0] [0 0] [3] >= [1 0] X + [0] [0 0] [0] = active(c(X)) mark(g(X)) = [1 2] X + [0] [0 1] [3] >= [1 2] X + [0] [0 1] [3] = active(g(mark(X))) * Step 5: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(g(X)) -> active(g(mark(X))) - Weak TRS: active(f(f(a()))) -> mark(c(f(g(f(a()))))) c(active(X)) -> c(X) c(mark(X)) -> c(X) f(active(X)) -> f(X) f(mark(X)) -> f(X) g(active(X)) -> g(X) g(mark(X)) -> g(X) mark(a()) -> active(a()) mark(c(X)) -> active(c(X)) mark(f(X)) -> active(f(mark(X))) - Signature: {active/1,c/1,f/1,g/1,mark/1} / {a/0} - Obligation: innermost derivational complexity wrt. signature {a,active,c,f,g,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) = [2] [1] p(active) = [1 0] x1 + [0] [0 1] [0] p(c) = [1 0] x1 + [0] [0 0] [0] p(f) = [1 0] x1 + [4] [0 1] [0] p(g) = [1 0] x1 + [0] [0 1] [3] p(mark) = [1 1] x1 + [0] [0 1] [1] Following rules are strictly oriented: mark(g(X)) = [1 1] X + [3] [0 1] [4] > [1 1] X + [0] [0 1] [4] = active(g(mark(X))) Following rules are (at-least) weakly oriented: active(f(f(a()))) = [10] [1] >= [10] [1] = mark(c(f(g(f(a()))))) c(active(X)) = [1 0] X + [0] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = c(X) c(mark(X)) = [1 1] X + [0] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = c(X) f(active(X)) = [1 0] X + [4] [0 1] [0] >= [1 0] X + [4] [0 1] [0] = f(X) f(mark(X)) = [1 1] X + [4] [0 1] [1] >= [1 0] X + [4] [0 1] [0] = f(X) g(active(X)) = [1 0] X + [0] [0 1] [3] >= [1 0] X + [0] [0 1] [3] = g(X) g(mark(X)) = [1 1] X + [0] [0 1] [4] >= [1 0] X + [0] [0 1] [3] = g(X) mark(a()) = [3] [2] >= [2] [1] = active(a()) mark(c(X)) = [1 0] X + [0] [0 0] [1] >= [1 0] X + [0] [0 0] [0] = active(c(X)) mark(f(X)) = [1 1] X + [4] [0 1] [1] >= [1 1] X + [4] [0 1] [1] = active(f(mark(X))) * Step 6: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: active(f(f(a()))) -> mark(c(f(g(f(a()))))) c(active(X)) -> c(X) c(mark(X)) -> c(X) f(active(X)) -> f(X) f(mark(X)) -> f(X) g(active(X)) -> g(X) g(mark(X)) -> g(X) mark(a()) -> active(a()) mark(c(X)) -> active(c(X)) mark(f(X)) -> active(f(mark(X))) mark(g(X)) -> active(g(mark(X))) - Signature: {active/1,c/1,f/1,g/1,mark/1} / {a/0} - Obligation: innermost derivational complexity wrt. signature {a,active,c,f,g,mark} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))