/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^3)) * Step 1: WeightGap. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: P(x1) -> Q(Q(p(x1))) P(p(x1)) -> x1 Q(p(q(x1))) -> q(p(Q(x1))) Q(q(x1)) -> x1 p(P(x1)) -> x1 p(Q(Q(x1))) -> Q(Q(p(x1))) p(p(x1)) -> q(q(x1)) q(Q(x1)) -> x1 q(q(p(x1))) -> p(q(q(x1))) - Signature: {P/1,Q/1,p/1,q/1} / {} - Obligation: innermost derivational complexity wrt. signature {P,Q,p,q} + 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(P) = [1] x1 + [3] p(Q) = [1] x1 + [0] p(p) = [1] x1 + [0] p(q) = [1] x1 + [5] Following rules are strictly oriented: P(x1) = [1] x1 + [3] > [1] x1 + [0] = Q(Q(p(x1))) P(p(x1)) = [1] x1 + [3] > [1] x1 + [0] = x1 Q(q(x1)) = [1] x1 + [5] > [1] x1 + [0] = x1 p(P(x1)) = [1] x1 + [3] > [1] x1 + [0] = x1 q(Q(x1)) = [1] x1 + [5] > [1] x1 + [0] = x1 Following rules are (at-least) weakly oriented: Q(p(q(x1))) = [1] x1 + [5] >= [1] x1 + [5] = q(p(Q(x1))) p(Q(Q(x1))) = [1] x1 + [0] >= [1] x1 + [0] = Q(Q(p(x1))) p(p(x1)) = [1] x1 + [0] >= [1] x1 + [10] = q(q(x1)) q(q(p(x1))) = [1] x1 + [10] >= [1] x1 + [10] = p(q(q(x1))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: Q(p(q(x1))) -> q(p(Q(x1))) p(Q(Q(x1))) -> Q(Q(p(x1))) p(p(x1)) -> q(q(x1)) q(q(p(x1))) -> p(q(q(x1))) - Weak TRS: P(x1) -> Q(Q(p(x1))) P(p(x1)) -> x1 Q(q(x1)) -> x1 p(P(x1)) -> x1 q(Q(x1)) -> x1 - Signature: {P/1,Q/1,p/1,q/1} / {} - Obligation: innermost derivational complexity wrt. signature {P,Q,p,q} + 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(P) = [1] x1 + [10] p(Q) = [1] x1 + [0] p(p) = [1] x1 + [8] p(q) = [1] x1 + [0] Following rules are strictly oriented: p(p(x1)) = [1] x1 + [16] > [1] x1 + [0] = q(q(x1)) Following rules are (at-least) weakly oriented: P(x1) = [1] x1 + [10] >= [1] x1 + [8] = Q(Q(p(x1))) P(p(x1)) = [1] x1 + [18] >= [1] x1 + [0] = x1 Q(p(q(x1))) = [1] x1 + [8] >= [1] x1 + [8] = q(p(Q(x1))) Q(q(x1)) = [1] x1 + [0] >= [1] x1 + [0] = x1 p(P(x1)) = [1] x1 + [18] >= [1] x1 + [0] = x1 p(Q(Q(x1))) = [1] x1 + [8] >= [1] x1 + [8] = Q(Q(p(x1))) q(Q(x1)) = [1] x1 + [0] >= [1] x1 + [0] = x1 q(q(p(x1))) = [1] x1 + [8] >= [1] x1 + [8] = p(q(q(x1))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: Q(p(q(x1))) -> q(p(Q(x1))) p(Q(Q(x1))) -> Q(Q(p(x1))) q(q(p(x1))) -> p(q(q(x1))) - Weak TRS: P(x1) -> Q(Q(p(x1))) P(p(x1)) -> x1 Q(q(x1)) -> x1 p(P(x1)) -> x1 p(p(x1)) -> q(q(x1)) q(Q(x1)) -> x1 - Signature: {P/1,Q/1,p/1,q/1} / {} - Obligation: innermost derivational complexity wrt. signature {P,Q,p,q} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 3, 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(P) = [1 3 2] [2] [0 1 1] x1 + [1] [0 0 1] [2] p(Q) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 1] [1] p(p) = [1 1 2] [0] [0 1 0] x1 + [1] [0 0 1] [0] p(q) = [1 1 0] [0] [0 1 0] x1 + [1] [0 0 1] [0] Following rules are strictly oriented: p(Q(Q(x1))) = [1 3 2] [4] [0 1 0] x1 + [1] [0 0 1] [2] > [1 3 2] [2] [0 1 0] x1 + [1] [0 0 1] [2] = Q(Q(p(x1))) Following rules are (at-least) weakly oriented: P(x1) = [1 3 2] [2] [0 1 1] x1 + [1] [0 0 1] [2] >= [1 3 2] [2] [0 1 0] x1 + [1] [0 0 1] [2] = Q(Q(p(x1))) P(p(x1)) = [1 4 4] [5] [0 1 1] x1 + [2] [0 0 1] [2] >= [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] = x1 Q(p(q(x1))) = [1 3 2] [3] [0 1 0] x1 + [2] [0 0 1] [1] >= [1 3 2] [3] [0 1 0] x1 + [2] [0 0 1] [1] = q(p(Q(x1))) Q(q(x1)) = [1 2 0] [1] [0 1 0] x1 + [1] [0 0 1] [1] >= [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] = x1 p(P(x1)) = [1 4 5] [7] [0 1 1] x1 + [2] [0 0 1] [2] >= [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] = x1 p(p(x1)) = [1 2 4] [1] [0 1 0] x1 + [2] [0 0 1] [0] >= [1 2 0] [1] [0 1 0] x1 + [2] [0 0 1] [0] = q(q(x1)) q(Q(x1)) = [1 2 0] [0] [0 1 0] x1 + [1] [0 0 1] [1] >= [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] = x1 q(q(p(x1))) = [1 3 2] [3] [0 1 0] x1 + [3] [0 0 1] [0] >= [1 3 2] [3] [0 1 0] x1 + [3] [0 0 1] [0] = p(q(q(x1))) * Step 4: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: Q(p(q(x1))) -> q(p(Q(x1))) q(q(p(x1))) -> p(q(q(x1))) - Weak TRS: P(x1) -> Q(Q(p(x1))) P(p(x1)) -> x1 Q(q(x1)) -> x1 p(P(x1)) -> x1 p(Q(Q(x1))) -> Q(Q(p(x1))) p(p(x1)) -> q(q(x1)) q(Q(x1)) -> x1 - Signature: {P/1,Q/1,p/1,q/1} / {} - Obligation: innermost derivational complexity wrt. signature {P,Q,p,q} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 3, 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(P) = [1 3 1] [3] [0 1 0] x1 + [3] [0 0 1] [3] p(Q) = [1 1 0] [0] [0 1 0] x1 + [1] [0 0 1] [1] p(p) = [1 0 1] [0] [0 1 0] x1 + [1] [0 0 1] [0] p(q) = [1 0 0] [0] [0 1 0] x1 + [1] [0 0 1] [0] Following rules are strictly oriented: Q(p(q(x1))) = [1 1 1] [2] [0 1 0] x1 + [3] [0 0 1] [1] > [1 1 1] [1] [0 1 0] x1 + [3] [0 0 1] [1] = q(p(Q(x1))) Following rules are (at-least) weakly oriented: P(x1) = [1 3 1] [3] [0 1 0] x1 + [3] [0 0 1] [3] >= [1 2 1] [3] [0 1 0] x1 + [3] [0 0 1] [2] = Q(Q(p(x1))) P(p(x1)) = [1 3 2] [6] [0 1 0] x1 + [4] [0 0 1] [3] >= [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] = x1 Q(q(x1)) = [1 1 0] [1] [0 1 0] x1 + [2] [0 0 1] [1] >= [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] = x1 p(P(x1)) = [1 3 2] [6] [0 1 0] x1 + [4] [0 0 1] [3] >= [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] = x1 p(Q(Q(x1))) = [1 2 1] [3] [0 1 0] x1 + [3] [0 0 1] [2] >= [1 2 1] [3] [0 1 0] x1 + [3] [0 0 1] [2] = Q(Q(p(x1))) p(p(x1)) = [1 0 2] [0] [0 1 0] x1 + [2] [0 0 1] [0] >= [1 0 0] [0] [0 1 0] x1 + [2] [0 0 1] [0] = q(q(x1)) q(Q(x1)) = [1 1 0] [0] [0 1 0] x1 + [2] [0 0 1] [1] >= [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] = x1 q(q(p(x1))) = [1 0 1] [0] [0 1 0] x1 + [3] [0 0 1] [0] >= [1 0 1] [0] [0 1 0] x1 + [3] [0 0 1] [0] = p(q(q(x1))) * Step 5: MI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: q(q(p(x1))) -> p(q(q(x1))) - Weak TRS: P(x1) -> Q(Q(p(x1))) P(p(x1)) -> x1 Q(p(q(x1))) -> q(p(Q(x1))) Q(q(x1)) -> x1 p(P(x1)) -> x1 p(Q(Q(x1))) -> Q(Q(p(x1))) p(p(x1)) -> q(q(x1)) q(Q(x1)) -> x1 - Signature: {P/1,Q/1,p/1,q/1} / {} - Obligation: innermost derivational complexity wrt. signature {P,Q,p,q} + Applied Processor: MI {miKind = Automaton Nothing, miDimension = 3, miUArgs = NoUArgs, miURules = NoURules, miSelector = Just any strict-rules} + Details: We apply a matrix interpretation of kind Automaton Nothing: Following symbols are considered usable: all TcT has computed the following interpretation: p(P) = [1 0 6] [2] [0 0 1] x_1 + [3] [0 1 0] [0] p(Q) = [1 0 0] [0] [0 0 1] x_1 + [0] [0 1 0] [0] p(p) = [1 0 4] [1] [0 0 1] x_1 + [3] [0 1 0] [0] p(q) = [1 0 4] [0] [0 0 1] x_1 + [2] [0 1 0] [0] Following rules are strictly oriented: q(q(p(x1))) = [1 4 8] [13] [0 0 1] x1 + [5] [0 1 0] [2] > [1 4 8] [9] [0 0 1] x1 + [5] [0 1 0] [2] = p(q(q(x1))) Following rules are (at-least) weakly oriented: P(x1) = [1 0 6] [2] [0 0 1] x1 + [3] [0 1 0] [0] >= [1 0 4] [1] [0 0 1] x1 + [3] [0 1 0] [0] = Q(Q(p(x1))) P(p(x1)) = [1 6 4] [3] [0 1 0] x1 + [3] [0 0 1] [3] >= [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] = x1 Q(p(q(x1))) = [1 4 4] [1] [0 0 1] x1 + [2] [0 1 0] [3] >= [1 4 4] [1] [0 0 1] x1 + [2] [0 1 0] [3] = q(p(Q(x1))) Q(q(x1)) = [1 0 4] [0] [0 1 0] x1 + [0] [0 0 1] [2] >= [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] = x1 p(P(x1)) = [1 4 6] [3] [0 1 0] x1 + [3] [0 0 1] [3] >= [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] = x1 p(Q(Q(x1))) = [1 0 4] [1] [0 0 1] x1 + [3] [0 1 0] [0] >= [1 0 4] [1] [0 0 1] x1 + [3] [0 1 0] [0] = Q(Q(p(x1))) p(p(x1)) = [1 4 4] [2] [0 1 0] x1 + [3] [0 0 1] [3] >= [1 4 4] [0] [0 1 0] x1 + [2] [0 0 1] [2] = q(q(x1)) q(Q(x1)) = [1 4 0] [0] [0 1 0] x1 + [2] [0 0 1] [0] >= [1 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] = x1 * Step 6: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: P(x1) -> Q(Q(p(x1))) P(p(x1)) -> x1 Q(p(q(x1))) -> q(p(Q(x1))) Q(q(x1)) -> x1 p(P(x1)) -> x1 p(Q(Q(x1))) -> Q(Q(p(x1))) p(p(x1)) -> q(q(x1)) q(Q(x1)) -> x1 q(q(p(x1))) -> p(q(q(x1))) - Signature: {P/1,Q/1,p/1,q/1} / {} - Obligation: innermost derivational complexity wrt. signature {P,Q,p,q} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^3))