/export/starexec/sandbox2/solver/bin/starexec_run_tct_rci /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1),O(n^3)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^3)) + Considered Problem: - Strict TRS: a__f(X1,X2) -> f(X1,X2) a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y)) mark(f(X1,X2)) -> a__f(mark(X1),X2) mark(g(X)) -> g(mark(X)) - Signature: {a__f/2,mark/1} / {f/2,g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: a__f(X1,X2) -> f(X1,X2) a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y)) mark(f(X1,X2)) -> a__f(mark(X1),X2) mark(g(X)) -> g(mark(X)) - Signature: {a__f/2,mark/1} / {f/2,g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: a__f(X1,X2) -> f(X1,X2) a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y)) mark(f(X1,X2)) -> a__f(mark(X1),X2) mark(g(X)) -> g(mark(X)) - Signature: {a__f/2,mark/1} / {f/2,g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: mark(x){x -> f(x,y)} = mark(f(x,y)) ->^+ a__f(mark(x),y) = C[mark(x) = mark(x){}] ** Step 1.b:1: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__f(X1,X2) -> f(X1,X2) a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y)) mark(f(X1,X2)) -> a__f(mark(X1),X2) mark(g(X)) -> g(mark(X)) - Signature: {a__f/2,mark/1} / {f/2,g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__f) = {1}, uargs(g) = {1} Following symbols are considered usable: {a__f,mark} TcT has computed the following interpretation: p(a__f) = [1] x1 + [0] p(f) = [1] x1 + [0] p(g) = [1] x1 + [2] p(mark) = [1] x1 + [0] Following rules are strictly oriented: a__f(g(X),Y) = [1] X + [2] > [1] X + [0] = a__f(mark(X),f(g(X),Y)) Following rules are (at-least) weakly oriented: a__f(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = f(X1,X2) mark(f(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = a__f(mark(X1),X2) mark(g(X)) = [1] X + [2] >= [1] X + [2] = g(mark(X)) ** Step 1.b:2: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__f(X1,X2) -> f(X1,X2) mark(f(X1,X2)) -> a__f(mark(X1),X2) mark(g(X)) -> g(mark(X)) - Weak TRS: a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y)) - Signature: {a__f/2,mark/1} / {f/2,g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__f) = {1}, uargs(g) = {1} Following symbols are considered usable: {a__f,mark} TcT has computed the following interpretation: p(a__f) = [1 2 6] [0] [0 1 4] x1 + [0] [0 0 1] [2] p(f) = [1 2 0] [0] [0 1 4] x1 + [0] [0 0 1] [2] p(g) = [1 2 1] [0] [0 1 4] x1 + [2] [0 0 1] [0] p(mark) = [1 2 0] [2] [0 1 1] x1 + [0] [0 0 1] [0] Following rules are strictly oriented: mark(g(X)) = [1 4 9] [6] [0 1 5] X + [2] [0 0 1] [0] > [1 4 3] [2] [0 1 5] X + [2] [0 0 1] [0] = g(mark(X)) Following rules are (at-least) weakly oriented: a__f(X1,X2) = [1 2 6] [0] [0 1 4] X1 + [0] [0 0 1] [2] >= [1 2 0] [0] [0 1 4] X1 + [0] [0 0 1] [2] = f(X1,X2) a__f(g(X),Y) = [1 4 15] [4] [0 1 8] X + [2] [0 0 1] [2] >= [1 4 8] [2] [0 1 5] X + [0] [0 0 1] [2] = a__f(mark(X),f(g(X),Y)) mark(f(X1,X2)) = [1 4 8] [2] [0 1 5] X1 + [2] [0 0 1] [2] >= [1 4 8] [2] [0 1 5] X1 + [0] [0 0 1] [2] = a__f(mark(X1),X2) ** Step 1.b:3: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__f(X1,X2) -> f(X1,X2) mark(f(X1,X2)) -> a__f(mark(X1),X2) - Weak TRS: a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y)) mark(g(X)) -> g(mark(X)) - Signature: {a__f/2,mark/1} / {f/2,g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__f) = {1}, uargs(g) = {1} Following symbols are considered usable: {a__f,mark} TcT has computed the following interpretation: p(a__f) = [1 0 2] [0 0 1] [3] [0 1 0] x1 + [0 0 0] x2 + [2] [0 0 1] [0 0 0] [2] p(f) = [1 0 2] [0 0 1] [0] [0 1 0] x1 + [0 0 0] x2 + [2] [0 0 1] [0 0 0] [2] p(g) = [1 4 1] [2] [0 0 1] x1 + [1] [0 0 1] [5] p(mark) = [1 4 0] [0] [0 0 1] x1 + [0] [0 0 1] [2] Following rules are strictly oriented: a__f(X1,X2) = [1 0 2] [0 0 1] [3] [0 1 0] X1 + [0 0 0] X2 + [2] [0 0 1] [0 0 0] [2] > [1 0 2] [0 0 1] [0] [0 1 0] X1 + [0 0 0] X2 + [2] [0 0 1] [0 0 0] [2] = f(X1,X2) mark(f(X1,X2)) = [1 4 2] [0 0 1] [8] [0 0 1] X1 + [0 0 0] X2 + [2] [0 0 1] [0 0 0] [4] > [1 4 2] [0 0 1] [7] [0 0 1] X1 + [0 0 0] X2 + [2] [0 0 1] [0 0 0] [4] = a__f(mark(X1),X2) Following rules are (at-least) weakly oriented: a__f(g(X),Y) = [1 4 3] [0 0 1] [15] [0 0 1] X + [0 0 0] Y + [3] [0 0 1] [0 0 0] [7] >= [1 4 3] [14] [0 0 1] X + [2] [0 0 1] [4] = a__f(mark(X),f(g(X),Y)) mark(g(X)) = [1 4 5] [6] [0 0 1] X + [5] [0 0 1] [7] >= [1 4 5] [4] [0 0 1] X + [3] [0 0 1] [7] = g(mark(X)) ** Step 1.b:4: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a__f(X1,X2) -> f(X1,X2) a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y)) mark(f(X1,X2)) -> a__f(mark(X1),X2) mark(g(X)) -> g(mark(X)) - Signature: {a__f/2,mark/1} / {f/2,g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^3))