/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: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: f(x,a()) -> x f(a(),y) -> y f(g(x,y),g(u,v)) -> g(f(x,u),f(y,v)) g(a(),a()) -> a() s(a()) -> a() s(f(x,y)) -> f(s(y),s(x)) s(g(x,y)) -> g(s(x),s(y)) s(s(x)) -> x - Signature: {f/2,g/2,s/1} / {a/0} - Obligation: innermost derivational complexity wrt. signature {a,f,g,s} + 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) = [0] p(f) = [1] x1 + [1] x2 + [11] p(g) = [1] x1 + [1] x2 + [1] p(s) = [1] x1 + [8] Following rules are strictly oriented: f(x,a()) = [1] x + [11] > [1] x + [0] = x f(a(),y) = [1] y + [11] > [1] y + [0] = y g(a(),a()) = [1] > [0] = a() s(a()) = [8] > [0] = a() s(s(x)) = [1] x + [16] > [1] x + [0] = x Following rules are (at-least) weakly oriented: f(g(x,y),g(u,v)) = [1] u + [1] v + [1] x + [1] y + [13] >= [1] u + [1] v + [1] x + [1] y + [23] = g(f(x,u),f(y,v)) s(f(x,y)) = [1] x + [1] y + [19] >= [1] x + [1] y + [27] = f(s(y),s(x)) s(g(x,y)) = [1] x + [1] y + [9] >= [1] x + [1] y + [17] = g(s(x),s(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: f(g(x,y),g(u,v)) -> g(f(x,u),f(y,v)) s(f(x,y)) -> f(s(y),s(x)) s(g(x,y)) -> g(s(x),s(y)) - Weak TRS: f(x,a()) -> x f(a(),y) -> y g(a(),a()) -> a() s(a()) -> a() s(s(x)) -> x - Signature: {f/2,g/2,s/1} / {a/0} - Obligation: innermost derivational complexity wrt. signature {a,f,g,s} + 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(f) = [1] x1 + [1] x2 + [0] p(g) = [1] x1 + [1] x2 + [2] p(s) = [1] x1 + [0] Following rules are strictly oriented: f(g(x,y),g(u,v)) = [1] u + [1] v + [1] x + [1] y + [4] > [1] u + [1] v + [1] x + [1] y + [2] = g(f(x,u),f(y,v)) Following rules are (at-least) weakly oriented: f(x,a()) = [1] x + [1] >= [1] x + [0] = x f(a(),y) = [1] y + [1] >= [1] y + [0] = y g(a(),a()) = [4] >= [1] = a() s(a()) = [1] >= [1] = a() s(f(x,y)) = [1] x + [1] y + [0] >= [1] x + [1] y + [0] = f(s(y),s(x)) s(g(x,y)) = [1] x + [1] y + [2] >= [1] x + [1] y + [2] = g(s(x),s(y)) s(s(x)) = [1] x + [0] >= [1] x + [0] = x * Step 3: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: s(f(x,y)) -> f(s(y),s(x)) s(g(x,y)) -> g(s(x),s(y)) - Weak TRS: f(x,a()) -> x f(a(),y) -> y f(g(x,y),g(u,v)) -> g(f(x,u),f(y,v)) g(a(),a()) -> a() s(a()) -> a() s(s(x)) -> x - Signature: {f/2,g/2,s/1} / {a/0} - Obligation: innermost derivational complexity wrt. signature {a,f,g,s} + 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(f) = [1 0] x1 + [1 0] x2 + [1] [0 1] [0 1] [0] p(g) = [1 0] x1 + [1 0] x2 + [3] [0 1] [0 1] [4] p(s) = [1 1] x1 + [0] [0 1] [0] Following rules are strictly oriented: s(g(x,y)) = [1 1] x + [1 1] y + [7] [0 1] [0 1] [4] > [1 1] x + [1 1] y + [3] [0 1] [0 1] [4] = g(s(x),s(y)) Following rules are (at-least) weakly oriented: f(x,a()) = [1 0] x + [3] [0 1] [1] >= [1 0] x + [0] [0 1] [0] = x f(a(),y) = [1 0] y + [3] [0 1] [1] >= [1 0] y + [0] [0 1] [0] = y f(g(x,y),g(u,v)) = [1 0] u + [1 0] v + [1 0] x + [1 0] y + [7] [0 1] [0 1] [0 1] [0 1] [8] >= [1 0] u + [1 0] v + [1 0] x + [1 0] y + [5] [0 1] [0 1] [0 1] [0 1] [4] = g(f(x,u),f(y,v)) g(a(),a()) = [7] [6] >= [2] [1] = a() s(a()) = [3] [1] >= [2] [1] = a() s(f(x,y)) = [1 1] x + [1 1] y + [1] [0 1] [0 1] [0] >= [1 1] x + [1 1] y + [1] [0 1] [0 1] [0] = f(s(y),s(x)) s(s(x)) = [1 2] x + [0] [0 1] [0] >= [1 0] x + [0] [0 1] [0] = x * Step 4: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: s(f(x,y)) -> f(s(y),s(x)) - Weak TRS: f(x,a()) -> x f(a(),y) -> y f(g(x,y),g(u,v)) -> g(f(x,u),f(y,v)) g(a(),a()) -> a() s(a()) -> a() s(g(x,y)) -> g(s(x),s(y)) s(s(x)) -> x - Signature: {f/2,g/2,s/1} / {a/0} - Obligation: innermost derivational complexity wrt. signature {a,f,g,s} + 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] [3] p(f) = [1 4] x1 + [1 4] x2 + [0] [0 1] [0 1] [1] p(g) = [1 4] x1 + [1 0] x2 + [0] [0 1] [0 1] [1] p(s) = [1 5] x1 + [0] [0 1] [0] Following rules are strictly oriented: s(f(x,y)) = [1 9] x + [1 9] y + [5] [0 1] [0 1] [1] > [1 9] x + [1 9] y + [0] [0 1] [0 1] [1] = f(s(y),s(x)) Following rules are (at-least) weakly oriented: f(x,a()) = [1 4] x + [12] [0 1] [4] >= [1 0] x + [0] [0 1] [0] = x f(a(),y) = [1 4] y + [12] [0 1] [4] >= [1 0] y + [0] [0 1] [0] = y f(g(x,y),g(u,v)) = [1 8] u + [1 4] v + [1 8] x + [1 4] y + [8] [0 1] [0 1] [0 1] [0 1] [3] >= [1 8] u + [1 4] v + [1 8] x + [1 4] y + [4] [0 1] [0 1] [0 1] [0 1] [3] = g(f(x,u),f(y,v)) g(a(),a()) = [12] [7] >= [0] [3] = a() s(a()) = [15] [3] >= [0] [3] = a() s(g(x,y)) = [1 9] x + [1 5] y + [5] [0 1] [0 1] [1] >= [1 9] x + [1 5] y + [0] [0 1] [0 1] [1] = g(s(x),s(y)) s(s(x)) = [1 10] x + [0] [0 1] [0] >= [1 0] x + [0] [0 1] [0] = x * Step 5: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: f(x,a()) -> x f(a(),y) -> y f(g(x,y),g(u,v)) -> g(f(x,u),f(y,v)) g(a(),a()) -> a() s(a()) -> a() s(f(x,y)) -> f(s(y),s(x)) s(g(x,y)) -> g(s(x),s(y)) s(s(x)) -> x - Signature: {f/2,g/2,s/1} / {a/0} - Obligation: innermost derivational complexity wrt. signature {a,f,g,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))