/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: average(x,s(s(s(y)))) -> s(average(s(x),y)) average(0(),0()) -> 0() average(0(),s(0())) -> 0() average(0(),s(s(0()))) -> s(0()) average(s(x),y) -> average(x,s(y)) - Signature: {average/2} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {average} and constructors {0,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: average(x,s(s(s(y)))) -> s(average(s(x),y)) average(0(),0()) -> 0() average(0(),s(0())) -> 0() average(0(),s(s(0()))) -> s(0()) average(s(x),y) -> average(x,s(y)) - Signature: {average/2} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {average} and constructors {0,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: average(x,s(s(s(y)))) -> s(average(s(x),y)) average(0(),0()) -> 0() average(0(),s(0())) -> 0() average(0(),s(s(0()))) -> s(0()) average(s(x),y) -> average(x,s(y)) - Signature: {average/2} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {average} and constructors {0,s} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: average(x,y){y -> s(s(s(y)))} = average(x,s(s(s(y)))) ->^+ s(average(s(x),y)) = C[average(s(x),y) = average(x,y){x -> s(x)}] ** Step 1.b:1: ToInnermost. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: average(x,s(s(s(y)))) -> s(average(s(x),y)) average(0(),0()) -> 0() average(0(),s(0())) -> 0() average(0(),s(s(0()))) -> s(0()) average(s(x),y) -> average(x,s(y)) - Signature: {average/2} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {average} and constructors {0,s} + 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: average(x,s(s(s(y)))) -> s(average(s(x),y)) average(0(),0()) -> 0() average(0(),s(0())) -> 0() average(0(),s(s(0()))) -> s(0()) average(s(x),y) -> average(x,s(y)) - Signature: {average/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {average} and constructors {0,s} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs average#(x,s(s(s(y)))) -> c_1(average#(s(x),y)) average#(0(),0()) -> c_2() average#(0(),s(0())) -> c_3() average#(0(),s(s(0()))) -> c_4() average#(s(x),y) -> c_5(average#(x,s(y))) Weak DPs and mark the set of starting terms. ** Step 1.b:3: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: average#(x,s(s(s(y)))) -> c_1(average#(s(x),y)) average#(0(),0()) -> c_2() average#(0(),s(0())) -> c_3() average#(0(),s(s(0()))) -> c_4() average#(s(x),y) -> c_5(average#(x,s(y))) - Strict TRS: average(x,s(s(s(y)))) -> s(average(s(x),y)) average(0(),0()) -> 0() average(0(),s(0())) -> 0() average(0(),s(s(0()))) -> s(0()) average(s(x),y) -> average(x,s(y)) - Signature: {average/2,average#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {average#} and constructors {0,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: average#(x,s(s(s(y)))) -> c_1(average#(s(x),y)) average#(0(),0()) -> c_2() average#(0(),s(0())) -> c_3() average#(0(),s(s(0()))) -> c_4() average#(s(x),y) -> c_5(average#(x,s(y))) ** Step 1.b:4: PredecessorEstimation. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: average#(x,s(s(s(y)))) -> c_1(average#(s(x),y)) average#(0(),0()) -> c_2() average#(0(),s(0())) -> c_3() average#(0(),s(s(0()))) -> c_4() average#(s(x),y) -> c_5(average#(x,s(y))) - Signature: {average/2,average#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {average#} and constructors {0,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,3,4} by application of Pre({2,3,4}) = {5}. Here rules are labelled as follows: 1: average#(x,s(s(s(y)))) -> c_1(average#(s(x),y)) 2: average#(0(),0()) -> c_2() 3: average#(0(),s(0())) -> c_3() 4: average#(0(),s(s(0()))) -> c_4() 5: average#(s(x),y) -> c_5(average#(x,s(y))) ** Step 1.b:5: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: average#(x,s(s(s(y)))) -> c_1(average#(s(x),y)) average#(s(x),y) -> c_5(average#(x,s(y))) - Weak DPs: average#(0(),0()) -> c_2() average#(0(),s(0())) -> c_3() average#(0(),s(s(0()))) -> c_4() - Signature: {average/2,average#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {average#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:average#(x,s(s(s(y)))) -> c_1(average#(s(x),y)) -->_1 average#(s(x),y) -> c_5(average#(x,s(y))):2 -->_1 average#(x,s(s(s(y)))) -> c_1(average#(s(x),y)):1 2:S:average#(s(x),y) -> c_5(average#(x,s(y))) -->_1 average#(0(),s(s(0()))) -> c_4():5 -->_1 average#(0(),s(0())) -> c_3():4 -->_1 average#(s(x),y) -> c_5(average#(x,s(y))):2 -->_1 average#(x,s(s(s(y)))) -> c_1(average#(s(x),y)):1 3:W:average#(0(),0()) -> c_2() 4:W:average#(0(),s(0())) -> c_3() 5:W:average#(0(),s(s(0()))) -> c_4() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: average#(0(),0()) -> c_2() 4: average#(0(),s(0())) -> c_3() 5: average#(0(),s(s(0()))) -> c_4() ** Step 1.b:6: PredecessorEstimationCP. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: average#(x,s(s(s(y)))) -> c_1(average#(s(x),y)) average#(s(x),y) -> c_5(average#(x,s(y))) - Signature: {average/2,average#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {average#} and constructors {0,s} + 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: average#(x,s(s(s(y)))) -> c_1(average#(s(x),y)) 2: average#(s(x),y) -> c_5(average#(x,s(y))) The strictly oriented rules are moved into the weak component. *** Step 1.b:6.a:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: average#(x,s(s(s(y)))) -> c_1(average#(s(x),y)) average#(s(x),y) -> c_5(average#(x,s(y))) - Signature: {average/2,average#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {average#} and constructors {0,s} + 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_5) = {1} Following symbols are considered usable: {average#} TcT has computed the following interpretation: p(0) = [2] p(average) = [1] x2 + [1] p(s) = [1] x1 + [1] p(average#) = [7] x1 + [4] x2 + [6] p(c_1) = [1] x1 + [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [1] x1 + [0] Following rules are strictly oriented: average#(x,s(s(s(y)))) = [7] x + [4] y + [18] > [7] x + [4] y + [13] = c_1(average#(s(x),y)) average#(s(x),y) = [7] x + [4] y + [13] > [7] x + [4] y + [10] = c_5(average#(x,s(y))) Following rules are (at-least) weakly oriented: *** Step 1.b:6.a:2: Assumption. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: average#(x,s(s(s(y)))) -> c_1(average#(s(x),y)) average#(s(x),y) -> c_5(average#(x,s(y))) - Signature: {average/2,average#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {average#} and constructors {0,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}} + Details: () *** Step 1.b:6.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: average#(x,s(s(s(y)))) -> c_1(average#(s(x),y)) average#(s(x),y) -> c_5(average#(x,s(y))) - Signature: {average/2,average#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {average#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:average#(x,s(s(s(y)))) -> c_1(average#(s(x),y)) -->_1 average#(s(x),y) -> c_5(average#(x,s(y))):2 -->_1 average#(x,s(s(s(y)))) -> c_1(average#(s(x),y)):1 2:W:average#(s(x),y) -> c_5(average#(x,s(y))) -->_1 average#(s(x),y) -> c_5(average#(x,s(y))):2 -->_1 average#(x,s(s(s(y)))) -> c_1(average#(s(x),y)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: average#(x,s(s(s(y)))) -> c_1(average#(s(x),y)) 2: average#(s(x),y) -> c_5(average#(x,s(y))) *** Step 1.b:6.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Signature: {average/2,average#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {average#} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))