WORST_CASE(?,O(n^2)) * Step 1: NaturalMI WORST_CASE(?,O(n^2)) + 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 + [8] p(f) = [1] x1 + [8] p(g) = [1] x1 + [4] p(mark) = [1] x1 + [0] Following rules are strictly oriented: a__f(g(X),Y) = [1] X + [12] > [1] X + [8] = a__f(mark(X),f(g(X),Y)) Following rules are (at-least) weakly oriented: a__f(X1,X2) = [1] X1 + [8] >= [1] X1 + [8] = f(X1,X2) mark(f(X1,X2)) = [1] X1 + [8] >= [1] X1 + [8] = a__f(mark(X1),X2) mark(g(X)) = [1] X + [4] >= [1] X + [4] = g(mark(X)) * Step 2: WeightGap WORST_CASE(?,O(n^2)) + 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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: 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: all TcT has computed the following interpretation: p(a__f) = [1] x1 + [1] p(f) = [1] x1 + [2] p(g) = [1] x1 + [2] p(mark) = [1] x1 + [1] Following rules are strictly oriented: mark(f(X1,X2)) = [1] X1 + [3] > [1] X1 + [2] = a__f(mark(X1),X2) Following rules are (at-least) weakly oriented: a__f(X1,X2) = [1] X1 + [1] >= [1] X1 + [2] = f(X1,X2) a__f(g(X),Y) = [1] X + [3] >= [1] X + [2] = a__f(mark(X),f(g(X),Y)) mark(g(X)) = [1] X + [3] >= [1] X + [3] = g(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: Ara WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__f(X1,X2) -> f(X1,X2) mark(g(X)) -> g(mark(X)) - Weak TRS: a__f(g(X),Y) -> a__f(mark(X),f(g(X),Y)) mark(f(X1,X2)) -> a__f(mark(X1),X2) - 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: Ara {heuristics_ = NoHeuristics, minDegree = 2, maxDegree = 2, araTimeout = 15, araFindStrictRules = Just 1} + Details: Signatures used: ---------------- a__f :: [A(8, 7) x A(0, 0)] -(2)-> A(1, 7) f :: [A(15, 7) x A(0, 0)] -(8)-> A(8, 7) f :: [A(8, 7) x A(0, 0)] -(1)-> A(1, 7) f :: [A(0, 0) x A(0, 0)] -(0)-> A(0, 0) g :: [A(15, 7)] -(8)-> A(8, 7) g :: [A(11, 7)] -(4)-> A(4, 7) g :: [A(0, 0)] -(0)-> A(0, 0) mark :: [A(8, 7)] -(7)-> A(4, 7) Cost-free Signatures used: -------------------------- a__f :: [A_cf(7, 0) x A_cf(0, 0)] -(7)-> A_cf(7, 0) a__f :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) a__f :: [A_cf(4, 0) x A_cf(0, 0)] -(4)-> A_cf(4, 0) a__f :: [A_cf(3, 0) x A_cf(0, 0)] -(3)-> A_cf(3, 0) f :: [A_cf(7, 0) x A_cf(0, 0)] -(7)-> A_cf(7, 0) f :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) f :: [A_cf(4, 0) x A_cf(0, 0)] -(4)-> A_cf(4, 0) f :: [A_cf(3, 0) x A_cf(0, 0)] -(3)-> A_cf(3, 0) g :: [A_cf(7, 0)] -(7)-> A_cf(7, 0) g :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) g :: [A_cf(4, 0)] -(4)-> A_cf(4, 0) g :: [A_cf(3, 0)] -(3)-> A_cf(3, 0) mark :: [A_cf(7, 0)] -(0)-> A_cf(7, 0) mark :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) mark :: [A_cf(4, 0)] -(0)-> A_cf(4, 0) mark :: [A_cf(3, 0)] -(0)-> A_cf(3, 0) mark :: [A_cf(7, 0)] -(1)-> A_cf(7, 0) Base Constructor Signatures used: --------------------------------- f_A :: [A(1, 0) x A(0, 0)] -(1)-> A(1, 0) f_A :: [A(1, 1) x A(0, 0)] -(0)-> A(0, 1) g_A :: [A(1, 0)] -(1)-> A(1, 0) g_A :: [A(1, 1)] -(0)-> A(0, 1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: a__f(X1,X2) -> f(X1,X2) 2. Weak: mark(g(X)) -> g(mark(X)) * Step 4: Ara WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(g(X)) -> g(mark(X)) - 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) - 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: Ara {heuristics_ = NoHeuristics, minDegree = 2, maxDegree = 2, araTimeout = 15, araFindStrictRules = Just 1} + Details: Signatures used: ---------------- a__f :: [A(12, 2) x A(0, 0)] -(2)-> A(12, 2) f :: [A(13, 2) x A(0, 0)] -(2)-> A(13, 2) f :: [A(12, 2) x A(0, 0)] -(2)-> A(12, 2) f :: [A(0, 0) x A(0, 0)] -(0)-> A(0, 0) g :: [A(15, 2)] -(13)-> A(13, 2) g :: [A(14, 2)] -(12)-> A(12, 2) g :: [A(0, 0)] -(0)-> A(0, 0) mark :: [A(13, 2)] -(7)-> A(12, 2) Cost-free Signatures used: -------------------------- a__f :: [A_cf(2, 0) x A_cf(0, 0)] -(0)-> A_cf(2, 0) a__f :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) f :: [A_cf(2, 0) x A_cf(0, 0)] -(0)-> A_cf(2, 0) f :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) g :: [A_cf(2, 0)] -(2)-> A_cf(2, 0) g :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) mark :: [A_cf(2, 0)] -(0)-> A_cf(2, 0) mark :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) Base Constructor Signatures used: --------------------------------- f_A :: [A(1, 0) x A(0, 0)] -(0)-> A(1, 0) f_A :: [A(0, 1) x A(0, 0)] -(1)-> A(0, 1) g_A :: [A(1, 0)] -(1)-> A(1, 0) g_A :: [A(1, 1)] -(0)-> A(0, 1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: mark(g(X)) -> g(mark(X)) 2. Weak: * Step 5: 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(?,O(n^2))