WORST_CASE(?,O(n^2)) * Step 1: Ara WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__f(X) -> f(X) a__f(f(a())) -> a__f(g(f(a()))) mark(a()) -> a() mark(f(X)) -> a__f(X) mark(g(X)) -> g(mark(X)) - Signature: {a__f/1,mark/1} / {a/0,f/1,g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {a,f,g} + Applied Processor: Ara {heuristics_ = NoHeuristics, minDegree = 1, maxDegree = 3, araTimeout = 60, araFindStrictRules = Nothing, araSmtSolver = MiniSMT} + Details: Signatures used: ---------------- a :: [] -(0)-> A(0, 0) a :: [] -(0)-> A(4, 4) a :: [] -(0)-> A(4, 2) a__f :: [A(1, 0)] -(4)-> A(0, 0) f :: [A(0, 0)] -(1)-> A(1, 0) f :: [A(4, 0)] -(4)-> A(4, 4) f :: [A(0, 0)] -(0)-> A(0, 0) g :: [A(4, 4)] -(4)-> A(4, 4) g :: [A(0, 0)] -(0)-> A(1, 0) mark :: [A(4, 4)] -(1)-> A(0, 0) Cost-free Signatures used: -------------------------- a :: [] -(0)-> A_cf(0, 0) a__f :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) f :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) g :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) mark :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) Base Constructor Signatures used: --------------------------------- a_A :: [] -(0)-> A(1, 0) a_A :: [] -(0)-> A(0, 1) f_A :: [A(0, 0)] -(1)-> A(1, 0) f_A :: [A(1, 0)] -(0)-> A(0, 1) g_A :: [A(0, 0)] -(0)-> A(1, 0) g_A :: [A(1, 1)] -(1)-> A(0, 1) * Step 2: Open MAYBE - Strict TRS: a__f(X) -> f(X) a__f(f(a())) -> a__f(g(f(a()))) mark(a()) -> a() mark(f(X)) -> a__f(X) mark(g(X)) -> g(mark(X)) - Signature: {a__f/1,mark/1} / {a/0,f/1,g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {a,f,g} Following problems could not be solved: - Strict TRS: a__f(X) -> f(X) a__f(f(a())) -> a__f(g(f(a()))) mark(a()) -> a() mark(f(X)) -> a__f(X) mark(g(X)) -> g(mark(X)) - Signature: {a__f/1,mark/1} / {a/0,f/1,g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {a,f,g} WORST_CASE(?,O(n^2))