WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X) u21(ackout(X),Y) -> u22(ackin(Y,X)) - Signature: {ackin/2,u21/2} / {ackout/1,s/1,u22/1} - Obligation: innermost runtime complexity wrt. defined symbols {ackin,u21} and constructors {ackout,s,u22} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X) u21(ackout(X),Y) -> u22(ackin(Y,X)) - Signature: {ackin/2,u21/2} / {ackout/1,s/1,u22/1} - Obligation: innermost runtime complexity wrt. defined symbols {ackin,u21} and constructors {ackout,s,u22} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () *** Step 1.a:1.a:1: Ara. MAYBE + Considered Problem: - Strict TRS: ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X) u21(ackout(X),Y) -> u22(ackin(Y,X)) - Signature: {ackin/2,u21/2} / {ackout/1,s/1,u22/1} - Obligation: innermost runtime complexity wrt. defined symbols {ackin,u21} and constructors {ackout,s,u22} + Applied Processor: Ara {minDegree = 1, maxDegree = 3, araTimeout = 15, araRuleShifting = Just 1, isBestCase = True, mkCompletelyDefined = False, verboseOutput = False} + Details: Signatures used: ---------------- F (TrsFun "ackin") :: ["A"(1, 1, 1) x "A"(0, 0, 0)] -(0)-> "A"(0, 0, 0) F (TrsFun "ackout") :: ["A"(0, 0, 0)] -(0)-> "A"(0, 0, 0) F (TrsFun "main") :: ["A"(0, 0, 0) x "A"(0, 0, 1)] -(1)-> "A"(0, 0, 0) F (TrsFun "s") :: ["A"(2, 2, 1)] -(1)-> "A"(1, 1, 1) F (TrsFun "s") :: ["A"(0, 0, 0)] -(0)-> "A"(0, 0, 0) F (TrsFun "s") :: ["A"(1, 1, 1)] -(1)-> "A"(1, 0, 1) F (TrsFun "u21") :: ["A"(0, 0, 0) x "A"(1, 1, 1)] -(1)-> "A"(0, 0, 0) F (TrsFun "u22") :: ["A"(0, 0, 0)] -(0)-> "A"(0, 0, 0) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X) u21(ackout(X),Y) -> u22(ackin(Y,X)) main(x1,x2) -> u21(x1,x2) 2. Weak: *** Step 1.a:1.b:1: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X) u21(ackout(X),Y) -> u22(ackin(Y,X)) - Signature: {ackin/2,u21/2} / {ackout/1,s/1,u22/1} - Obligation: innermost runtime complexity wrt. defined symbols {ackin,u21} and constructors {ackout,s,u22} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: ackin(s(x),y){y -> s(y)} = ackin(s(x),s(y)) ->^+ u21(ackin(s(x),y),x) = C[ackin(s(x),y) = ackin(s(x),y){}] ** Step 1.b:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X) u21(ackout(X),Y) -> u22(ackin(Y,X)) - Signature: {ackin/2,u21/2} / {ackout/1,s/1,u22/1} - Obligation: innermost runtime complexity wrt. defined symbols {ackin,u21} and constructors {ackout,s,u22} + 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(u21) = {1}, uargs(u22) = {1} Following symbols are considered usable: {ackin,u21} TcT has computed the following interpretation: p(ackin) = [1] p(ackout) = [15] p(s) = [1] p(u21) = [1] x1 + [0] p(u22) = [1] x1 + [6] Following rules are strictly oriented: u21(ackout(X),Y) = [15] > [7] = u22(ackin(Y,X)) Following rules are (at-least) weakly oriented: ackin(s(X),s(Y)) = [1] >= [1] = u21(ackin(s(X),Y),X) ** Step 1.b:2: NaturalPI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X) - Weak TRS: u21(ackout(X),Y) -> u22(ackin(Y,X)) - Signature: {ackin/2,u21/2} / {ackout/1,s/1,u22/1} - Obligation: innermost runtime complexity wrt. defined symbols {ackin,u21} and constructors {ackout,s,u22} + 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(u21) = {1}, uargs(u22) = {1} Following symbols are considered usable: {ackin,u21} TcT has computed the following interpretation: p(ackin) = x2 p(ackout) = 10 + x1 p(s) = 2 + x1 p(u21) = x1 p(u22) = 10 + x1 Following rules are strictly oriented: ackin(s(X),s(Y)) = 2 + Y > Y = u21(ackin(s(X),Y),X) Following rules are (at-least) weakly oriented: u21(ackout(X),Y) = 10 + X >= 10 + X = u22(ackin(Y,X)) ** Step 1.b:3: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X) u21(ackout(X),Y) -> u22(ackin(Y,X)) - Signature: {ackin/2,u21/2} / {ackout/1,s/1,u22/1} - Obligation: innermost runtime complexity wrt. defined symbols {ackin,u21} and constructors {ackout,s,u22} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))