/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: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__b() -> a() a__b() -> b() a__f(X1,X2,X3) -> f(X1,X2,X3) a__f(a(),X,X) -> a__f(X,a__b(),b()) mark(a()) -> a() mark(b()) -> a__b() mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3) - Signature: {a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3} - Obligation: innermost derivational complexity wrt. signature {a,a__b,a__f,b,f,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) = [0] p(a__b) = [0] p(a__f) = [1] x1 + [1] x2 + [1] x3 + [0] p(b) = [0] p(f) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [1] x1 + [8] Following rules are strictly oriented: mark(a()) = [8] > [0] = a() mark(b()) = [8] > [0] = a__b() Following rules are (at-least) weakly oriented: a__b() = [0] >= [0] = a() a__b() = [0] >= [0] = b() a__f(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = f(X1,X2,X3) a__f(a(),X,X) = [2] X + [0] >= [1] X + [0] = a__f(X,a__b(),b()) mark(f(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [8] >= [1] X1 + [1] X2 + [1] X3 + [8] = a__f(X1,mark(X2),X3) * Step 2: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__b() -> a() a__b() -> b() a__f(X1,X2,X3) -> f(X1,X2,X3) a__f(a(),X,X) -> a__f(X,a__b(),b()) mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3) - Weak TRS: mark(a()) -> a() mark(b()) -> a__b() - Signature: {a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3} - Obligation: innermost derivational complexity wrt. signature {a,a__b,a__f,b,f,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) = [1] p(a__b) = [1] p(a__f) = [1] x1 + [1] x2 + [1] x3 + [0] p(b) = [0] p(f) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [1] x1 + [1] Following rules are strictly oriented: a__b() = [1] > [0] = b() Following rules are (at-least) weakly oriented: a__b() = [1] >= [1] = a() a__f(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = f(X1,X2,X3) a__f(a(),X,X) = [2] X + [1] >= [1] X + [1] = a__f(X,a__b(),b()) mark(a()) = [2] >= [1] = a() mark(b()) = [1] >= [1] = a__b() mark(f(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [1] >= [1] X1 + [1] X2 + [1] X3 + [1] = a__f(X1,mark(X2),X3) * Step 3: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__b() -> a() a__f(X1,X2,X3) -> f(X1,X2,X3) a__f(a(),X,X) -> a__f(X,a__b(),b()) mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3) - Weak TRS: a__b() -> b() mark(a()) -> a() mark(b()) -> a__b() - Signature: {a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3} - Obligation: innermost derivational complexity wrt. signature {a,a__b,a__f,b,f,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) = [11] p(a__b) = [0] p(a__f) = [1] x1 + [1] x2 + [1] x3 + [0] p(b) = [0] p(f) = [1] x1 + [1] x2 + [1] x3 + [1] p(mark) = [1] x1 + [0] Following rules are strictly oriented: a__f(a(),X,X) = [2] X + [11] > [1] X + [0] = a__f(X,a__b(),b()) mark(f(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [1] > [1] X1 + [1] X2 + [1] X3 + [0] = a__f(X1,mark(X2),X3) Following rules are (at-least) weakly oriented: a__b() = [0] >= [11] = a() a__b() = [0] >= [0] = b() a__f(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [1] = f(X1,X2,X3) mark(a()) = [11] >= [11] = a() mark(b()) = [0] >= [0] = a__b() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__b() -> a() a__f(X1,X2,X3) -> f(X1,X2,X3) - Weak TRS: a__b() -> b() a__f(a(),X,X) -> a__f(X,a__b(),b()) mark(a()) -> a() mark(b()) -> a__b() mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3) - Signature: {a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3} - Obligation: innermost derivational complexity wrt. signature {a,a__b,a__f,b,f,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] [1] p(a__b) = [1] [1] p(a__f) = [1 1] x1 + [1 0] x2 + [1 1] x3 + [0] [0 0] [0 0] [0 0] [1] p(b) = [0] [0] p(f) = [1 1] x1 + [1 0] x2 + [1 1] x3 + [0] [0 0] [0 0] [0 0] [0] p(mark) = [1 0] x1 + [1] [0 0] [1] Following rules are strictly oriented: a__b() = [1] [1] > [0] [1] = a() Following rules are (at-least) weakly oriented: a__b() = [1] [1] >= [0] [0] = b() a__f(X1,X2,X3) = [1 1] X1 + [1 0] X2 + [1 1] X3 + [0] [0 0] [0 0] [0 0] [1] >= [1 1] X1 + [1 0] X2 + [1 1] X3 + [0] [0 0] [0 0] [0 0] [0] = f(X1,X2,X3) a__f(a(),X,X) = [2 1] X + [1] [0 0] [1] >= [1 1] X + [1] [0 0] [1] = a__f(X,a__b(),b()) mark(a()) = [1] [1] >= [0] [1] = a() mark(b()) = [1] [1] >= [1] [1] = a__b() mark(f(X1,X2,X3)) = [1 1] X1 + [1 0] X2 + [1 1] X3 + [1] [0 0] [0 0] [0 0] [1] >= [1 1] X1 + [1 0] X2 + [1 1] X3 + [1] [0 0] [0 0] [0 0] [1] = a__f(X1,mark(X2),X3) * Step 5: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__f(X1,X2,X3) -> f(X1,X2,X3) - Weak TRS: a__b() -> a() a__b() -> b() a__f(a(),X,X) -> a__f(X,a__b(),b()) mark(a()) -> a() mark(b()) -> a__b() mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3) - Signature: {a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3} - Obligation: innermost derivational complexity wrt. signature {a,a__b,a__f,b,f,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) = [0] [0] p(a__b) = [0] [0] p(a__f) = [1 3] x1 + [1 0] x2 + [1 3] x3 + [2] [0 1] [0 1] [0 0] [2] p(b) = [0] [0] p(f) = [1 0] x1 + [1 0] x2 + [1 3] x3 + [0] [0 1] [0 1] [0 0] [2] p(mark) = [1 4] x1 + [0] [0 1] [4] Following rules are strictly oriented: a__f(X1,X2,X3) = [1 3] X1 + [1 0] X2 + [1 3] X3 + [2] [0 1] [0 1] [0 0] [2] > [1 0] X1 + [1 0] X2 + [1 3] X3 + [0] [0 1] [0 1] [0 0] [2] = f(X1,X2,X3) Following rules are (at-least) weakly oriented: a__b() = [0] [0] >= [0] [0] = a() a__b() = [0] [0] >= [0] [0] = b() a__f(a(),X,X) = [2 3] X + [2] [0 1] [2] >= [1 3] X + [2] [0 1] [2] = a__f(X,a__b(),b()) mark(a()) = [0] [4] >= [0] [0] = a() mark(b()) = [0] [4] >= [0] [0] = a__b() mark(f(X1,X2,X3)) = [1 4] X1 + [1 4] X2 + [1 3] X3 + [8] [0 1] [0 1] [0 0] [6] >= [1 3] X1 + [1 4] X2 + [1 3] X3 + [2] [0 1] [0 1] [0 0] [6] = a__f(X1,mark(X2),X3) * Step 6: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a__b() -> a() a__b() -> b() a__f(X1,X2,X3) -> f(X1,X2,X3) a__f(a(),X,X) -> a__f(X,a__b(),b()) mark(a()) -> a() mark(b()) -> a__b() mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3) - Signature: {a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3} - Obligation: innermost derivational complexity wrt. signature {a,a__b,a__f,b,f,mark} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))