/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^3)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^3)) + Considered Problem: - Strict TRS: log(s(0())) -> 0() log(s(s(X))) -> s(log(s(quot(X,s(s(0())))))) min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Signature: {log/1,min/2,quot/2} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: log(s(0())) -> 0() log(s(s(X))) -> s(log(s(quot(X,s(s(0())))))) min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Signature: {log/1,min/2,quot/2} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: log(s(0())) -> 0() log(s(s(X))) -> s(log(s(quot(X,s(s(0())))))) min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Signature: {log/1,min/2,quot/2} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: min(x,y){x -> s(x),y -> s(y)} = min(s(x),s(y)) ->^+ min(x,y) = C[min(x,y) = min(x,y){}] ** Step 1.b:1: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: log(s(0())) -> 0() log(s(s(X))) -> s(log(s(quot(X,s(s(0())))))) min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Signature: {log/1,min/2,quot/2} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s} + 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(log) = {1}, uargs(min) = {1}, uargs(quot) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(log) = [1] x1 + [2] p(min) = [1] x1 + [0] p(quot) = [1] x1 + [0] p(s) = [1] x1 + [1] Following rules are strictly oriented: log(s(0())) = [3] > [0] = 0() min(s(X),s(Y)) = [1] X + [1] > [1] X + [0] = min(X,Y) Following rules are (at-least) weakly oriented: log(s(s(X))) = [1] X + [4] >= [1] X + [4] = s(log(s(quot(X,s(s(0())))))) min(X,0()) = [1] X + [0] >= [1] X + [0] = X quot(0(),s(Y)) = [0] >= [0] = 0() quot(s(X),s(Y)) = [1] X + [1] >= [1] X + [1] = s(quot(min(X,Y),s(Y))) ** Step 1.b:2: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: log(s(s(X))) -> s(log(s(quot(X,s(s(0())))))) min(X,0()) -> X quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Weak TRS: log(s(0())) -> 0() min(s(X),s(Y)) -> min(X,Y) - Signature: {log/1,min/2,quot/2} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s} + 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(log) = {1}, uargs(min) = {1}, uargs(quot) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(log) = [8] x1 + [13] p(min) = [1] x1 + [0] p(quot) = [1] x1 + [0] p(s) = [1] x1 + [1] Following rules are strictly oriented: log(s(s(X))) = [8] X + [29] > [8] X + [22] = s(log(s(quot(X,s(s(0())))))) Following rules are (at-least) weakly oriented: log(s(0())) = [21] >= [0] = 0() min(X,0()) = [1] X + [0] >= [1] X + [0] = X min(s(X),s(Y)) = [1] X + [1] >= [1] X + [0] = min(X,Y) quot(0(),s(Y)) = [0] >= [0] = 0() quot(s(X),s(Y)) = [1] X + [1] >= [1] X + [1] = s(quot(min(X,Y),s(Y))) ** Step 1.b:3: NaturalPI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: min(X,0()) -> X quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Weak TRS: log(s(0())) -> 0() log(s(s(X))) -> s(log(s(quot(X,s(s(0())))))) min(s(X),s(Y)) -> min(X,Y) - Signature: {log/1,min/2,quot/2} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s} + 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(log) = {1}, uargs(min) = {1}, uargs(quot) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = 12 p(log) = 1 + 2*x1 p(min) = x1 p(quot) = 1 + x1 p(s) = 2 + x1 Following rules are strictly oriented: quot(0(),s(Y)) = 13 > 12 = 0() Following rules are (at-least) weakly oriented: log(s(0())) = 29 >= 12 = 0() log(s(s(X))) = 9 + 2*X >= 9 + 2*X = s(log(s(quot(X,s(s(0())))))) min(X,0()) = X >= X = X min(s(X),s(Y)) = 2 + X >= X = min(X,Y) quot(s(X),s(Y)) = 3 + X >= 3 + X = s(quot(min(X,Y),s(Y))) ** Step 1.b:4: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: min(X,0()) -> X quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Weak TRS: log(s(0())) -> 0() log(s(s(X))) -> s(log(s(quot(X,s(s(0())))))) min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() - Signature: {log/1,min/2,quot/2} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s} + 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(log) = {1}, uargs(min) = {1}, uargs(quot) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] [0] [0] p(log) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(min) = [1 0 0] [1] [0 1 0] x1 + [0] [0 0 1] [0] p(quot) = [1 0 1] [0 0 0] [0] [0 1 0] x1 + [1 0 0] x2 + [0] [0 0 1] [0 0 0] [0] p(s) = [1 0 0] [0] [0 1 1] x1 + [0] [0 0 1] [1] Following rules are strictly oriented: min(X,0()) = [1 0 0] [1] [0 1 0] X + [0] [0 0 1] [0] > [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] = X Following rules are (at-least) weakly oriented: log(s(0())) = [0] [0] [1] >= [0] [0] [0] = 0() log(s(s(X))) = [1 1 2] [1] [0 1 2] X + [1] [0 0 1] [2] >= [1 1 2] [0] [0 1 2] X + [1] [0 0 1] [2] = s(log(s(quot(X,s(s(0())))))) min(s(X),s(Y)) = [1 0 0] [1] [0 1 1] X + [0] [0 0 1] [1] >= [1 0 0] [1] [0 1 0] X + [0] [0 0 1] [0] = min(X,Y) quot(0(),s(Y)) = [0 0 0] [0] [1 0 0] Y + [0] [0 0 0] [0] >= [0] [0] [0] = 0() quot(s(X),s(Y)) = [1 0 1] [0 0 0] [1] [0 1 1] X + [1 0 0] Y + [0] [0 0 1] [0 0 0] [1] >= [1 0 1] [0 0 0] [1] [0 1 1] X + [1 0 0] Y + [0] [0 0 1] [0 0 0] [1] = s(quot(min(X,Y),s(Y))) ** Step 1.b:5: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Weak TRS: log(s(0())) -> 0() log(s(s(X))) -> s(log(s(quot(X,s(s(0())))))) min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() - Signature: {log/1,min/2,quot/2} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s} + 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(log) = {1}, uargs(min) = {1}, uargs(quot) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] [0] [1] p(log) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(min) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(quot) = [1 0 1] [1 0 0] [0] [0 1 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 0] [0] p(s) = [1 0 0] [0] [0 1 1] x1 + [0] [0 0 1] [1] Following rules are strictly oriented: quot(s(X),s(Y)) = [1 0 1] [1 0 0] [1] [0 1 1] X + [0 0 0] Y + [0] [0 0 1] [0 0 0] [1] > [1 0 1] [1 0 0] [0] [0 1 1] X + [0 0 0] Y + [0] [0 0 1] [0 0 0] [1] = s(quot(min(X,Y),s(Y))) Following rules are (at-least) weakly oriented: log(s(0())) = [2] [1] [2] >= [1] [0] [1] = 0() log(s(s(X))) = [1 1 2] [1] [0 1 2] X + [1] [0 0 1] [2] >= [1 1 2] [1] [0 1 2] X + [1] [0 0 1] [2] = s(log(s(quot(X,s(s(0())))))) min(X,0()) = [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] = X min(s(X),s(Y)) = [1 0 0] [0] [0 1 1] X + [0] [0 0 1] [1] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] = min(X,Y) quot(0(),s(Y)) = [1 0 0] [2] [0 0 0] Y + [0] [0 0 0] [1] >= [1] [0] [1] = 0() ** Step 1.b:6: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: log(s(0())) -> 0() log(s(s(X))) -> s(log(s(quot(X,s(s(0())))))) min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Signature: {log/1,min/2,quot/2} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^3))