/export/starexec/sandbox2/solver/bin/starexec_run_tct_rc /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) - Signature: {U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0} - Obligation: runtime complexity wrt. defined symbols {U11,U12,activate,plus} and constructors {0,s,tt} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) - Signature: {U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0} - Obligation: runtime complexity wrt. defined symbols {U11,U12,activate,plus} and constructors {0,s,tt} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) - Signature: {U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0} - Obligation: runtime complexity wrt. defined symbols {U11,U12,activate,plus} and constructors {0,s,tt} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: U11(tt(),z,x_31){z -> s(z)} = U11(tt(),s(z),x_31) ->^+ s(U11(tt(),z,activate(activate(x_31)))) = C[U11(tt(),z,activate(activate(x_31))) = U11(tt(),z,x_31){x_31 -> activate(activate(x_31))}] ** Step 1.b:1: ToInnermost. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) - Signature: {U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0} - Obligation: runtime complexity wrt. defined symbols {U11,U12,activate,plus} and constructors {0,s,tt} + Applied Processor: ToInnermost + Details: switch to innermost, as the system is overlay and right linear and does not contain weak rules ** Step 1.b:2: DependencyPairs. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) - Signature: {U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {U11,U12,activate,plus} and constructors {0,s,tt} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) activate#(X) -> c_3() plus#(N,0()) -> c_4() plus#(N,s(M)) -> c_5(U11#(tt(),M,N)) Weak DPs and mark the set of starting terms. ** Step 1.b:3: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) activate#(X) -> c_3() plus#(N,0()) -> c_4() plus#(N,s(M)) -> c_5(U11#(tt(),M,N)) - Strict TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) - Signature: {U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0 ,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate(X) -> X U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) activate#(X) -> c_3() plus#(N,0()) -> c_4() plus#(N,s(M)) -> c_5(U11#(tt(),M,N)) ** Step 1.b:4: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) activate#(X) -> c_3() plus#(N,0()) -> c_4() plus#(N,s(M)) -> c_5(U11#(tt(),M,N)) - Strict TRS: activate(X) -> X - Signature: {U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0 ,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(U12#) = {2,3}, uargs(plus#) = {1,2}, uargs(c_1) = {1}, uargs(c_2) = {1}, uargs(c_5) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(U11) = [0] p(U12) = [0] p(activate) = [1] x1 + [1] p(plus) = [0] p(s) = [1] x1 + [0] p(tt) = [3] p(U11#) = [8] x1 + [1] x2 + [1] x3 + [0] p(U12#) = [1] x2 + [1] x3 + [0] p(activate#) = [1] x1 + [8] p(plus#) = [1] x1 + [1] x2 + [0] p(c_1) = [1] x1 + [0] p(c_2) = [1] x1 + [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [1] x1 + [0] Following rules are strictly oriented: U11#(tt(),M,N) = [1] M + [1] N + [24] > [1] M + [1] N + [2] = c_1(U12#(tt(),activate(M),activate(N))) activate#(X) = [1] X + [8] > [0] = c_3() activate(X) = [1] X + [1] > [1] X + [0] = X Following rules are (at-least) weakly oriented: U12#(tt(),M,N) = [1] M + [1] N + [0] >= [1] M + [1] N + [2] = c_2(plus#(activate(N),activate(M))) plus#(N,0()) = [1] N + [0] >= [0] = c_4() plus#(N,s(M)) = [1] M + [1] N + [0] >= [1] M + [1] N + [24] = c_5(U11#(tt(),M,N)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 1.b:5: PredecessorEstimation. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) plus#(N,0()) -> c_4() plus#(N,s(M)) -> c_5(U11#(tt(),M,N)) - Weak DPs: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) activate#(X) -> c_3() - Weak TRS: activate(X) -> X - Signature: {U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0 ,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,3} by application of Pre({2,3}) = {1}. Here rules are labelled as follows: 1: U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) 2: plus#(N,0()) -> c_4() 3: plus#(N,s(M)) -> c_5(U11#(tt(),M,N)) 4: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) 5: activate#(X) -> c_3() ** Step 1.b:6: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) - Weak DPs: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) activate#(X) -> c_3() plus#(N,0()) -> c_4() plus#(N,s(M)) -> c_5(U11#(tt(),M,N)) - Weak TRS: activate(X) -> X - Signature: {U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0 ,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) -->_1 plus#(N,s(M)) -> c_5(U11#(tt(),M,N)):5 -->_1 plus#(N,0()) -> c_4():4 2:W:U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) -->_1 U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))):1 3:W:activate#(X) -> c_3() 4:W:plus#(N,0()) -> c_4() 5:W:plus#(N,s(M)) -> c_5(U11#(tt(),M,N)) -->_1 U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: activate#(X) -> c_3() 4: plus#(N,0()) -> c_4() ** Step 1.b:7: PredecessorEstimationCP. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) - Weak DPs: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) plus#(N,s(M)) -> c_5(U11#(tt(),M,N)) - Weak TRS: activate(X) -> X - Signature: {U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0 ,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) Consider the set of all dependency pairs 1: U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) 2: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) 5: plus#(N,s(M)) -> c_5(U11#(tt(),M,N)) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) BEST_CASE TIME (?,?) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2,5} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. *** Step 1.b:7.a:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) - Weak DPs: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) plus#(N,s(M)) -> c_5(U11#(tt(),M,N)) - Weak TRS: activate(X) -> X - Signature: {U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0 ,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_1) = {1}, uargs(c_2) = {1}, uargs(c_5) = {1} Following symbols are considered usable: {activate,U11#,U12#,activate#,plus#} TcT has computed the following interpretation: p(0) = [4] p(U11) = [1] x1 + [1] p(U12) = [1] x1 + [1] x3 + [2] p(activate) = [1] x1 + [1] p(plus) = [1] x1 + [1] x2 + [2] p(s) = [1] x1 + [5] p(tt) = [3] p(U11#) = [4] x1 + [4] x2 + [1] x3 + [4] p(U12#) = [2] x1 + [4] x2 + [1] x3 + [5] p(activate#) = [0] p(plus#) = [1] x1 + [4] x2 + [4] p(c_1) = [1] x1 + [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] p(c_4) = [1] p(c_5) = [1] x1 + [3] Following rules are strictly oriented: U12#(tt(),M,N) = [4] M + [1] N + [11] > [4] M + [1] N + [9] = c_2(plus#(activate(N),activate(M))) Following rules are (at-least) weakly oriented: U11#(tt(),M,N) = [4] M + [1] N + [16] >= [4] M + [1] N + [16] = c_1(U12#(tt(),activate(M),activate(N))) plus#(N,s(M)) = [4] M + [1] N + [24] >= [4] M + [1] N + [19] = c_5(U11#(tt(),M,N)) activate(X) = [1] X + [1] >= [1] X + [0] = X *** Step 1.b:7.a:2: Assumption. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) plus#(N,s(M)) -> c_5(U11#(tt(),M,N)) - Weak TRS: activate(X) -> X - Signature: {U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0 ,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}} + Details: () *** Step 1.b:7.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) plus#(N,s(M)) -> c_5(U11#(tt(),M,N)) - Weak TRS: activate(X) -> X - Signature: {U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0 ,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) -->_1 U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))):2 2:W:U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) -->_1 plus#(N,s(M)) -> c_5(U11#(tt(),M,N)):3 3:W:plus#(N,s(M)) -> c_5(U11#(tt(),M,N)) -->_1 U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))) 3: plus#(N,s(M)) -> c_5(U11#(tt(),M,N)) 2: U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))) *** Step 1.b:7.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X - Signature: {U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0 ,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))