/export/starexec/sandbox/solver/bin/starexec_run_tct_dci /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?,O(n^2)) * Step 1: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) +(s(x),y) -> s(+(x,y)) not(false()) -> true() not(true()) -> false() odd(0()) -> false() odd(s(x)) -> not(odd(x)) - Signature: {+/2,not/1,odd/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost derivational complexity wrt. signature {+,0,false,not,odd,s,true} + 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(+) = [1] x1 + [1] x2 + [1] p(0) = [0] p(false) = [0] p(not) = [1] x1 + [0] p(odd) = [1] x1 + [0] p(s) = [1] x1 + [0] p(true) = [0] Following rules are strictly oriented: +(x,0()) = [1] x + [1] > [1] x + [0] = x Following rules are (at-least) weakly oriented: +(x,s(y)) = [1] x + [1] y + [1] >= [1] x + [1] y + [1] = s(+(x,y)) +(s(x),y) = [1] x + [1] y + [1] >= [1] x + [1] y + [1] = s(+(x,y)) not(false()) = [0] >= [0] = true() not(true()) = [0] >= [0] = false() odd(0()) = [0] >= [0] = false() odd(s(x)) = [1] x + [0] >= [1] x + [0] = not(odd(x)) * Step 2: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: +(x,s(y)) -> s(+(x,y)) +(s(x),y) -> s(+(x,y)) not(false()) -> true() not(true()) -> false() odd(0()) -> false() odd(s(x)) -> not(odd(x)) - Weak TRS: +(x,0()) -> x - Signature: {+/2,not/1,odd/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost derivational complexity wrt. signature {+,0,false,not,odd,s,true} + 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(+) = [1] x1 + [1] x2 + [2] p(0) = [8] p(false) = [9] p(not) = [1] x1 + [5] p(odd) = [1] x1 + [6] p(s) = [1] x1 + [8] p(true) = [12] Following rules are strictly oriented: not(false()) = [14] > [12] = true() not(true()) = [17] > [9] = false() odd(0()) = [14] > [9] = false() odd(s(x)) = [1] x + [14] > [1] x + [11] = not(odd(x)) Following rules are (at-least) weakly oriented: +(x,0()) = [1] x + [10] >= [1] x + [0] = x +(x,s(y)) = [1] x + [1] y + [10] >= [1] x + [1] y + [10] = s(+(x,y)) +(s(x),y) = [1] x + [1] y + [10] >= [1] x + [1] y + [10] = s(+(x,y)) * Step 3: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: +(x,s(y)) -> s(+(x,y)) +(s(x),y) -> s(+(x,y)) - Weak TRS: +(x,0()) -> x not(false()) -> true() not(true()) -> false() odd(0()) -> false() odd(s(x)) -> not(odd(x)) - Signature: {+/2,not/1,odd/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost derivational complexity wrt. signature {+,0,false,not,odd,s,true} + 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(+) = [1 0] x1 + [1 1] x2 + [0] [0 1] [0 1] [0] p(0) = [1] [0] p(false) = [1] [0] p(not) = [1 0] x1 + [0] [0 1] [1] p(odd) = [1 0] x1 + [0] [0 1] [0] p(s) = [1 0] x1 + [0] [0 1] [2] p(true) = [1] [0] Following rules are strictly oriented: +(x,s(y)) = [1 0] x + [1 1] y + [2] [0 1] [0 1] [2] > [1 0] x + [1 1] y + [0] [0 1] [0 1] [2] = s(+(x,y)) Following rules are (at-least) weakly oriented: +(x,0()) = [1 0] x + [1] [0 1] [0] >= [1 0] x + [0] [0 1] [0] = x +(s(x),y) = [1 0] x + [1 1] y + [0] [0 1] [0 1] [2] >= [1 0] x + [1 1] y + [0] [0 1] [0 1] [2] = s(+(x,y)) not(false()) = [1] [1] >= [1] [0] = true() not(true()) = [1] [1] >= [1] [0] = false() odd(0()) = [1] [0] >= [1] [0] = false() odd(s(x)) = [1 0] x + [0] [0 1] [2] >= [1 0] x + [0] [0 1] [1] = not(odd(x)) * Step 4: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: +(s(x),y) -> s(+(x,y)) - Weak TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) not(false()) -> true() not(true()) -> false() odd(0()) -> false() odd(s(x)) -> not(odd(x)) - Signature: {+/2,not/1,odd/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost derivational complexity wrt. signature {+,0,false,not,odd,s,true} + 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(+) = [1 1] x1 + [1 0] x2 + [0] [0 1] [0 1] [6] p(0) = [4] [0] p(false) = [2] [0] p(not) = [1 0] x1 + [7] [0 0] [4] p(odd) = [1 2] x1 + [0] [0 1] [4] p(s) = [1 0] x1 + [4] [0 1] [2] p(true) = [1] [0] Following rules are strictly oriented: +(s(x),y) = [1 1] x + [1 0] y + [6] [0 1] [0 1] [8] > [1 1] x + [1 0] y + [4] [0 1] [0 1] [8] = s(+(x,y)) Following rules are (at-least) weakly oriented: +(x,0()) = [1 1] x + [4] [0 1] [6] >= [1 0] x + [0] [0 1] [0] = x +(x,s(y)) = [1 1] x + [1 0] y + [4] [0 1] [0 1] [8] >= [1 1] x + [1 0] y + [4] [0 1] [0 1] [8] = s(+(x,y)) not(false()) = [9] [4] >= [1] [0] = true() not(true()) = [8] [4] >= [2] [0] = false() odd(0()) = [4] [4] >= [2] [0] = false() odd(s(x)) = [1 2] x + [8] [0 1] [6] >= [1 2] x + [7] [0 0] [4] = not(odd(x)) * Step 5: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) +(s(x),y) -> s(+(x,y)) not(false()) -> true() not(true()) -> false() odd(0()) -> false() odd(s(x)) -> not(odd(x)) - Signature: {+/2,not/1,odd/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost derivational complexity wrt. signature {+,0,false,not,odd,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))