/export/starexec/sandbox/solver/bin/starexec_run_tct_rc /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/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: a__b() -> a() a__b() -> b() a__f(X,X) -> a__f(a(),b()) a__f(X1,X2) -> f(X1,X2) mark(a()) -> a() mark(b()) -> a__b() mark(f(X1,X2)) -> a__f(mark(X1),X2) - Signature: {a__b/0,a__f/2,mark/1} / {a/0,b/0,f/2} - Obligation: runtime complexity wrt. defined symbols {a__b,a__f,mark} and constructors {a,b,f} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: a__b() -> a() a__b() -> b() a__f(X,X) -> a__f(a(),b()) a__f(X1,X2) -> f(X1,X2) mark(a()) -> a() mark(b()) -> a__b() mark(f(X1,X2)) -> a__f(mark(X1),X2) - Signature: {a__b/0,a__f/2,mark/1} / {a/0,b/0,f/2} - Obligation: runtime complexity wrt. defined symbols {a__b,a__f,mark} and constructors {a,b,f} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: a__b() -> a() a__b() -> b() a__f(X,X) -> a__f(a(),b()) a__f(X1,X2) -> f(X1,X2) mark(a()) -> a() mark(b()) -> a__b() mark(f(X1,X2)) -> a__f(mark(X1),X2) - Signature: {a__b/0,a__f/2,mark/1} / {a/0,b/0,f/2} - Obligation: runtime complexity wrt. defined symbols {a__b,a__f,mark} and constructors {a,b,f} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: mark(x){x -> f(x,y)} = mark(f(x,y)) ->^+ a__f(mark(x),y) = C[mark(x) = mark(x){}] ** Step 1.b:1: ToInnermost. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__b() -> a() a__b() -> b() a__f(X,X) -> a__f(a(),b()) a__f(X1,X2) -> f(X1,X2) mark(a()) -> a() mark(b()) -> a__b() mark(f(X1,X2)) -> a__f(mark(X1),X2) - Signature: {a__b/0,a__f/2,mark/1} / {a/0,b/0,f/2} - Obligation: runtime complexity wrt. defined symbols {a__b,a__f,mark} and constructors {a,b,f} + 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: a__b() -> a() a__b() -> b() a__f(X,X) -> a__f(a(),b()) a__f(X1,X2) -> f(X1,X2) mark(a()) -> a() mark(b()) -> a__b() mark(f(X1,X2)) -> a__f(mark(X1),X2) - Signature: {a__b/0,a__f/2,mark/1} / {a/0,b/0,f/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__b,a__f,mark} and constructors {a,b,f} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs a__b#() -> c_1() a__b#() -> c_2() a__f#(X,X) -> c_3(a__f#(a(),b())) a__f#(X1,X2) -> c_4() mark#(a()) -> c_5() mark#(b()) -> c_6(a__b#()) mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1)) Weak DPs and mark the set of starting terms. ** Step 1.b:3: PredecessorEstimation. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a__b#() -> c_1() a__b#() -> c_2() a__f#(X,X) -> c_3(a__f#(a(),b())) a__f#(X1,X2) -> c_4() mark#(a()) -> c_5() mark#(b()) -> c_6(a__b#()) mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1)) - Weak TRS: a__b() -> a() a__b() -> b() a__f(X,X) -> a__f(a(),b()) a__f(X1,X2) -> f(X1,X2) mark(a()) -> a() mark(b()) -> a__b() mark(f(X1,X2)) -> a__f(mark(X1),X2) - Signature: {a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,4,5} by application of Pre({1,2,4,5}) = {3,6,7}. Here rules are labelled as follows: 1: a__b#() -> c_1() 2: a__b#() -> c_2() 3: a__f#(X,X) -> c_3(a__f#(a(),b())) 4: a__f#(X1,X2) -> c_4() 5: mark#(a()) -> c_5() 6: mark#(b()) -> c_6(a__b#()) 7: mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1)) ** Step 1.b:4: PredecessorEstimation. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a__f#(X,X) -> c_3(a__f#(a(),b())) mark#(b()) -> c_6(a__b#()) mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1)) - Weak DPs: a__b#() -> c_1() a__b#() -> c_2() a__f#(X1,X2) -> c_4() mark#(a()) -> c_5() - Weak TRS: a__b() -> a() a__b() -> b() a__f(X,X) -> a__f(a(),b()) a__f(X1,X2) -> f(X1,X2) mark(a()) -> a() mark(b()) -> a__b() mark(f(X1,X2)) -> a__f(mark(X1),X2) - Signature: {a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2} by application of Pre({1,2}) = {3}. Here rules are labelled as follows: 1: a__f#(X,X) -> c_3(a__f#(a(),b())) 2: mark#(b()) -> c_6(a__b#()) 3: mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1)) 4: a__b#() -> c_1() 5: a__b#() -> c_2() 6: a__f#(X1,X2) -> c_4() 7: mark#(a()) -> c_5() ** Step 1.b:5: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1)) - Weak DPs: a__b#() -> c_1() a__b#() -> c_2() a__f#(X,X) -> c_3(a__f#(a(),b())) a__f#(X1,X2) -> c_4() mark#(a()) -> c_5() mark#(b()) -> c_6(a__b#()) - Weak TRS: a__b() -> a() a__b() -> b() a__f(X,X) -> a__f(a(),b()) a__f(X1,X2) -> f(X1,X2) mark(a()) -> a() mark(b()) -> a__b() mark(f(X1,X2)) -> a__f(mark(X1),X2) - Signature: {a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1)) -->_2 mark#(b()) -> c_6(a__b#()):7 -->_1 a__f#(X,X) -> c_3(a__f#(a(),b())):4 -->_2 mark#(a()) -> c_5():6 -->_1 a__f#(X1,X2) -> c_4():5 -->_2 mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1)):1 2:W:a__b#() -> c_1() 3:W:a__b#() -> c_2() 4:W:a__f#(X,X) -> c_3(a__f#(a(),b())) -->_1 a__f#(X1,X2) -> c_4():5 5:W:a__f#(X1,X2) -> c_4() 6:W:mark#(a()) -> c_5() 7:W:mark#(b()) -> c_6(a__b#()) -->_1 a__b#() -> c_2():3 -->_1 a__b#() -> c_1():2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: mark#(a()) -> c_5() 4: a__f#(X,X) -> c_3(a__f#(a(),b())) 5: a__f#(X1,X2) -> c_4() 7: mark#(b()) -> c_6(a__b#()) 2: a__b#() -> c_1() 3: a__b#() -> c_2() ** Step 1.b:6: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1)) - Weak TRS: a__b() -> a() a__b() -> b() a__f(X,X) -> a__f(a(),b()) a__f(X1,X2) -> f(X1,X2) mark(a()) -> a() mark(b()) -> a__b() mark(f(X1,X2)) -> a__f(mark(X1),X2) - Signature: {a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1)) -->_2 mark#(f(X1,X2)) -> c_7(a__f#(mark(X1),X2),mark#(X1)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: mark#(f(X1,X2)) -> c_7(mark#(X1)) ** Step 1.b:7: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mark#(f(X1,X2)) -> c_7(mark#(X1)) - Weak TRS: a__b() -> a() a__b() -> b() a__f(X,X) -> a__f(a(),b()) a__f(X1,X2) -> f(X1,X2) mark(a()) -> a() mark(b()) -> a__b() mark(f(X1,X2)) -> a__f(mark(X1),X2) - Signature: {a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: mark#(f(X1,X2)) -> c_7(mark#(X1)) ** Step 1.b:8: PredecessorEstimationCP. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mark#(f(X1,X2)) -> c_7(mark#(X1)) - Signature: {a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f} + 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: mark#(f(X1,X2)) -> c_7(mark#(X1)) The strictly oriented rules are moved into the weak component. *** Step 1.b:8.a:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mark#(f(X1,X2)) -> c_7(mark#(X1)) - Signature: {a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f} + 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_7) = {1} Following symbols are considered usable: {a__b#,a__f#,mark#} TcT has computed the following interpretation: p(a) = [2] p(a__b) = [1] p(a__f) = [1] p(b) = [0] p(f) = [1] x1 + [1] p(mark) = [2] p(a__b#) = [1] p(a__f#) = [8] x2 + [1] p(mark#) = [1] x1 + [10] p(c_1) = [1] p(c_2) = [0] p(c_3) = [1] x1 + [1] p(c_4) = [0] p(c_5) = [1] p(c_6) = [2] x1 + [4] p(c_7) = [1] x1 + [0] Following rules are strictly oriented: mark#(f(X1,X2)) = [1] X1 + [11] > [1] X1 + [10] = c_7(mark#(X1)) Following rules are (at-least) weakly oriented: *** Step 1.b:8.a:2: Assumption. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: mark#(f(X1,X2)) -> c_7(mark#(X1)) - Signature: {a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}} + Details: () *** Step 1.b:8.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: mark#(f(X1,X2)) -> c_7(mark#(X1)) - Signature: {a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:mark#(f(X1,X2)) -> c_7(mark#(X1)) -->_1 mark#(f(X1,X2)) -> c_7(mark#(X1)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: mark#(f(X1,X2)) -> c_7(mark#(X1)) *** Step 1.b:8.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Signature: {a__b/0,a__f/2,mark/1,a__b#/0,a__f#/2,mark#/1} / {a/0,b/0,f/2,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1,c_7/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__b#,a__f#,mark#} and constructors {a,b,f} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))