/export/starexec/sandbox/solver/bin/starexec_run_tct_rc /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1),O(n^2)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^2)) + Considered Problem: - Strict TRS: activate(X) -> X and(tt(),X) -> activate(X) plus(N,0()) -> N plus(N,s(M)) -> s(plus(N,M)) x(N,0()) -> 0() x(N,s(M)) -> plus(x(N,M),N) - Signature: {activate/1,and/2,plus/2,x/2} / {0/0,s/1,tt/0} - Obligation: runtime complexity wrt. defined symbols {activate,and,plus,x} and constructors {0,s,tt} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: activate(X) -> X and(tt(),X) -> activate(X) plus(N,0()) -> N plus(N,s(M)) -> s(plus(N,M)) x(N,0()) -> 0() x(N,s(M)) -> plus(x(N,M),N) - Signature: {activate/1,and/2,plus/2,x/2} / {0/0,s/1,tt/0} - Obligation: runtime complexity wrt. defined symbols {activate,and,plus,x} and constructors {0,s,tt} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: activate(X) -> X and(tt(),X) -> activate(X) plus(N,0()) -> N plus(N,s(M)) -> s(plus(N,M)) x(N,0()) -> 0() x(N,s(M)) -> plus(x(N,M),N) - Signature: {activate/1,and/2,plus/2,x/2} / {0/0,s/1,tt/0} - Obligation: runtime complexity wrt. defined symbols {activate,and,plus,x} and constructors {0,s,tt} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: plus(x,y){y -> s(y)} = plus(x,s(y)) ->^+ s(plus(x,y)) = C[plus(x,y) = plus(x,y){}] ** Step 1.b:1: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: activate(X) -> X and(tt(),X) -> activate(X) plus(N,0()) -> N plus(N,s(M)) -> s(plus(N,M)) x(N,0()) -> 0() x(N,s(M)) -> plus(x(N,M),N) - Signature: {activate/1,and/2,plus/2,x/2} / {0/0,s/1,tt/0} - Obligation: runtime complexity wrt. defined symbols {activate,and,plus,x} and constructors {0,s,tt} + 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(plus) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [9] x1 + [4] p(and) = [3] x1 + [9] x2 + [2] p(plus) = [4] x1 + [0] p(s) = [1] x1 + [0] p(tt) = [7] p(x) = [0] Following rules are strictly oriented: activate(X) = [9] X + [4] > [1] X + [0] = X and(tt(),X) = [9] X + [23] > [9] X + [4] = activate(X) Following rules are (at-least) weakly oriented: plus(N,0()) = [4] N + [0] >= [1] N + [0] = N plus(N,s(M)) = [4] N + [0] >= [4] N + [0] = s(plus(N,M)) x(N,0()) = [0] >= [0] = 0() x(N,s(M)) = [0] >= [0] = plus(x(N,M),N) ** Step 1.b:2: NaturalPI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: plus(N,0()) -> N plus(N,s(M)) -> s(plus(N,M)) x(N,0()) -> 0() x(N,s(M)) -> plus(x(N,M),N) - Weak TRS: activate(X) -> X and(tt(),X) -> activate(X) - Signature: {activate/1,and/2,plus/2,x/2} / {0/0,s/1,tt/0} - Obligation: runtime complexity wrt. defined symbols {activate,and,plus,x} and constructors {0,s,tt} + Applied Processor: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(plus) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = 0 p(activate) = 8 + 14*x1 p(and) = 8 + 14*x2 p(plus) = x1 p(s) = x1 p(tt) = 0 p(x) = 1 Following rules are strictly oriented: x(N,0()) = 1 > 0 = 0() Following rules are (at-least) weakly oriented: activate(X) = 8 + 14*X >= X = X and(tt(),X) = 8 + 14*X >= 8 + 14*X = activate(X) plus(N,0()) = N >= N = N plus(N,s(M)) = N >= N = s(plus(N,M)) x(N,s(M)) = 1 >= 1 = plus(x(N,M),N) ** Step 1.b:3: NaturalPI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: plus(N,0()) -> N plus(N,s(M)) -> s(plus(N,M)) x(N,s(M)) -> plus(x(N,M),N) - Weak TRS: activate(X) -> X and(tt(),X) -> activate(X) x(N,0()) -> 0() - Signature: {activate/1,and/2,plus/2,x/2} / {0/0,s/1,tt/0} - Obligation: runtime complexity wrt. defined symbols {activate,and,plus,x} and constructors {0,s,tt} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(plus) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = 1 p(activate) = 7 + 2*x1 p(and) = 1 + 5*x1 + x1^2 + 4*x2 + 3*x2^2 p(plus) = x1 + 4*x2 p(s) = 2 + x1 p(tt) = 2 p(x) = 1 + x1 + 2*x1*x2 + x1^2 Following rules are strictly oriented: plus(N,0()) = 4 + N > N = N plus(N,s(M)) = 8 + 4*M + N > 2 + 4*M + N = s(plus(N,M)) Following rules are (at-least) weakly oriented: activate(X) = 7 + 2*X >= X = X and(tt(),X) = 15 + 4*X + 3*X^2 >= 7 + 2*X = activate(X) x(N,0()) = 1 + 3*N + N^2 >= 1 = 0() x(N,s(M)) = 1 + 2*M*N + 5*N + N^2 >= 1 + 2*M*N + 5*N + N^2 = plus(x(N,M),N) ** Step 1.b:4: NaturalPI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: x(N,s(M)) -> plus(x(N,M),N) - Weak TRS: activate(X) -> X and(tt(),X) -> activate(X) plus(N,0()) -> N plus(N,s(M)) -> s(plus(N,M)) x(N,0()) -> 0() - Signature: {activate/1,and/2,plus/2,x/2} / {0/0,s/1,tt/0} - Obligation: runtime complexity wrt. defined symbols {activate,and,plus,x} and constructors {0,s,tt} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(plus) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = 0 p(activate) = 2 + 4*x1 + x1^2 p(and) = 2 + x1 + x1^2 + 4*x2 + 2*x2^2 p(plus) = 4 + x1 + 5*x2 p(s) = 2 + x1 p(tt) = 0 p(x) = 2*x1 + 3*x1*x2 + 2*x2^2 Following rules are strictly oriented: x(N,s(M)) = 8 + 8*M + 3*M*N + 2*M^2 + 8*N > 4 + 3*M*N + 2*M^2 + 7*N = plus(x(N,M),N) Following rules are (at-least) weakly oriented: activate(X) = 2 + 4*X + X^2 >= X = X and(tt(),X) = 2 + 4*X + 2*X^2 >= 2 + 4*X + X^2 = activate(X) plus(N,0()) = 4 + N >= N = N plus(N,s(M)) = 14 + 5*M + N >= 6 + 5*M + N = s(plus(N,M)) x(N,0()) = 2*N >= 0 = 0() ** Step 1.b:5: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X and(tt(),X) -> activate(X) plus(N,0()) -> N plus(N,s(M)) -> s(plus(N,M)) x(N,0()) -> 0() x(N,s(M)) -> plus(x(N,M),N) - Signature: {activate/1,and/2,plus/2,x/2} / {0/0,s/1,tt/0} - Obligation: runtime complexity wrt. defined symbols {activate,and,plus,x} and constructors {0,s,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^2))