/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^2)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). (0) CpxTRS (1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxWeightedTrs (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxTypedWeightedTrs (7) CompletionProof [UPPER BOUND(ID), 0 ms] (8) CpxTypedWeightedCompleteTrs (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (10) CpxRNTS (11) CompleteCoflocoProof [FINISHED, 237 ms] (12) BOUNDS(1, n^2) (13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (14) TRS for Loop Detection (15) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (16) BEST (17) proven lower bound (18) LowerBoundPropagationProof [FINISHED, 0 ms] (19) BOUNDS(n^1, INF) (20) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: lt(0, s(x)) -> true lt(x, 0) -> false lt(s(x), s(y)) -> lt(x, y) minus(x, y) -> help(lt(y, x), x, y) help(true, x, y) -> s(minus(x, s(y))) help(false, x, y) -> 0 S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) RcToIrcProof (BOTH BOUNDS(ID, ID)) Converted rc-obligation to irc-obligation. The duplicating contexts are: minus(x, []) minus([], y) The defined contexts are: help([], x1, x2) [] just represents basic- or constructor-terms in the following defined contexts: help([], x1, x2) As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc. ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2). The TRS R consists of the following rules: lt(0, s(x)) -> true lt(x, 0) -> false lt(s(x), s(y)) -> lt(x, y) minus(x, y) -> help(lt(y, x), x, y) help(true, x, y) -> s(minus(x, s(y))) help(false, x, y) -> 0 S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). The TRS R consists of the following rules: lt(0, s(x)) -> true [1] lt(x, 0) -> false [1] lt(s(x), s(y)) -> lt(x, y) [1] minus(x, y) -> help(lt(y, x), x, y) [1] help(true, x, y) -> s(minus(x, s(y))) [1] help(false, x, y) -> 0 [1] Rewrite Strategy: INNERMOST ---------------------------------------- (5) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (6) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: lt(0, s(x)) -> true [1] lt(x, 0) -> false [1] lt(s(x), s(y)) -> lt(x, y) [1] minus(x, y) -> help(lt(y, x), x, y) [1] help(true, x, y) -> s(minus(x, s(y))) [1] help(false, x, y) -> 0 [1] The TRS has the following type information: lt :: 0:s -> 0:s -> true:false 0 :: 0:s s :: 0:s -> 0:s true :: true:false false :: true:false minus :: 0:s -> 0:s -> 0:s help :: true:false -> 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (7) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: none And the following fresh constants: none ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: lt(0, s(x)) -> true [1] lt(x, 0) -> false [1] lt(s(x), s(y)) -> lt(x, y) [1] minus(x, y) -> help(lt(y, x), x, y) [1] help(true, x, y) -> s(minus(x, s(y))) [1] help(false, x, y) -> 0 [1] The TRS has the following type information: lt :: 0:s -> 0:s -> true:false 0 :: 0:s s :: 0:s -> 0:s true :: true:false false :: true:false minus :: 0:s -> 0:s -> 0:s help :: true:false -> 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 1 false => 0 ---------------------------------------- (10) Obligation: Complexity RNTS consisting of the following rules: help(z, z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0 help(z, z', z'') -{ 1 }-> 1 + minus(x, 1 + y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 lt(z, z') -{ 1 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x lt(z, z') -{ 1 }-> 1 :|: z' = 1 + x, x >= 0, z = 0 lt(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 minus(z, z') -{ 1 }-> help(lt(y, x), x, y) :|: x >= 0, y >= 0, z = x, z' = y Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (11) CompleteCoflocoProof (FINISHED) Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: eq(start(V1, V, V10),0,[lt(V1, V, Out)],[V1 >= 0,V >= 0]). eq(start(V1, V, V10),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]). eq(start(V1, V, V10),0,[help(V1, V, V10, Out)],[V1 >= 0,V >= 0,V10 >= 0]). eq(lt(V1, V, Out),1,[],[Out = 1,V = 1 + V2,V2 >= 0,V1 = 0]). eq(lt(V1, V, Out),1,[],[Out = 0,V3 >= 0,V1 = V3,V = 0]). eq(lt(V1, V, Out),1,[lt(V4, V5, Ret)],[Out = Ret,V = 1 + V5,V4 >= 0,V5 >= 0,V1 = 1 + V4]). eq(minus(V1, V, Out),1,[lt(V7, V6, Ret0),help(Ret0, V6, V7, Ret1)],[Out = Ret1,V6 >= 0,V7 >= 0,V1 = V6,V = V7]). eq(help(V1, V, V10, Out),1,[minus(V8, 1 + V9, Ret11)],[Out = 1 + Ret11,V = V8,V10 = V9,V1 = 1,V8 >= 0,V9 >= 0]). eq(help(V1, V, V10, Out),1,[],[Out = 0,V = V12,V10 = V11,V12 >= 0,V11 >= 0,V1 = 0]). input_output_vars(lt(V1,V,Out),[V1,V],[Out]). input_output_vars(minus(V1,V,Out),[V1,V],[Out]). input_output_vars(help(V1,V,V10,Out),[V1,V,V10],[Out]). CoFloCo proof output: Preprocessing Cost Relations ===================================== #### Computed strongly connected components 0. recursive : [lt/3] 1. recursive : [help/4,minus/3] 2. non_recursive : [start/3] #### Obtained direct recursion through partial evaluation 0. SCC is partially evaluated into lt/3 1. SCC is partially evaluated into minus/3 2. SCC is partially evaluated into start/3 Control-Flow Refinement of Cost Relations ===================================== ### Specialization of cost equations lt/3 * CE 9 is refined into CE [10] * CE 8 is refined into CE [11] * CE 7 is refined into CE [12] ### Cost equations --> "Loop" of lt/3 * CEs [11] --> Loop 8 * CEs [12] --> Loop 9 * CEs [10] --> Loop 10 ### Ranking functions of CR lt(V1,V,Out) * RF of phase [10]: [V,V1] #### Partial ranking functions of CR lt(V1,V,Out) * Partial RF of phase [10]: - RF of loop [10:1]: V V1 ### Specialization of cost equations minus/3 * CE 6 is refined into CE [13,14] * CE 5 is refined into CE [15,16] ### Cost equations --> "Loop" of minus/3 * CEs [16] --> Loop 11 * CEs [15] --> Loop 12 * CEs [14] --> Loop 13 * CEs [13] --> Loop 14 ### Ranking functions of CR minus(V1,V,Out) * RF of phase [13]: [V1-V] #### Partial ranking functions of CR minus(V1,V,Out) * Partial RF of phase [13]: - RF of loop [13:1]: V1-V ### Specialization of cost equations start/3 * CE 2 is refined into CE [17,18,19] * CE 1 is refined into CE [20] * CE 3 is refined into CE [21,22,23,24] * CE 4 is refined into CE [25,26,27,28,29] ### Cost equations --> "Loop" of start/3 * CEs [23,29] --> Loop 15 * CEs [19] --> Loop 16 * CEs [18,24,28] --> Loop 17 * CEs [17,22,26,27] --> Loop 18 * CEs [20,21,25] --> Loop 19 ### Ranking functions of CR start(V1,V,V10) #### Partial ranking functions of CR start(V1,V,V10) Computing Bounds ===================================== #### Cost of chains of lt(V1,V,Out): * Chain [[10],9]: 1*it(10)+1 Such that:it(10) =< V1 with precondition: [Out=1,V1>=1,V>=V1+1] * Chain [[10],8]: 1*it(10)+1 Such that:it(10) =< V with precondition: [Out=0,V>=1,V1>=V] * Chain [9]: 1 with precondition: [V1=0,Out=1,V>=1] * Chain [8]: 1 with precondition: [V=0,Out=0,V1>=0] #### Cost of chains of minus(V1,V,Out): * Chain [[13],11]: 3*it(13)+1*s(1)+1*s(4)+3 Such that:it(13) =< Out aux(2) =< V+Out s(1) =< aux(2) s(4) =< it(13)*aux(2) with precondition: [V1=Out+V,V>=1,V1>=V+1] * Chain [14,[13],11]: 4*it(13)+1*s(4)+6 Such that:aux(3) =< Out it(13) =< aux(3) s(4) =< it(13)*aux(3) with precondition: [V=0,V1=Out,V1>=2] * Chain [14,11]: 1*s(1)+6 Such that:s(1) =< 1 with precondition: [V1=1,V=0,Out=1] * Chain [12]: 3 with precondition: [V1=0,Out=0,V>=0] * Chain [11]: 1*s(1)+3 Such that:s(1) =< V1 with precondition: [Out=0,V1>=1,V>=V1] #### Cost of chains of start(V1,V,V10): * Chain [19]: 3 with precondition: [V1=0,V>=0] * Chain [18]: 1*s(5)+4*s(7)+1*s(8)+6 Such that:s(5) =< 1 s(6) =< V1 s(7) =< s(6) s(8) =< s(7)*s(6) with precondition: [V=0,V1>=0] * Chain [17]: 1*s(9)+2*s(10)+4 Such that:s(9) =< V aux(4) =< V1 s(10) =< aux(4) with precondition: [V1>=1,V>=V1] * Chain [16]: 3*s(12)+1*s(14)+1*s(15)+4 Such that:s(13) =< V s(12) =< V-V10 s(14) =< s(13) s(15) =< s(12)*s(13) with precondition: [V1=1,V10>=0,V>=V10+2] * Chain [15]: 1*s(16)+3*s(17)+1*s(19)+1*s(20)+3 Such that:s(18) =< V1 s(17) =< V1-V s(16) =< V s(19) =< s(18) s(20) =< s(17)*s(18) with precondition: [V>=1,V1>=V] Closed-form bounds of start(V1,V,V10): ------------------------------------- * Chain [19] with precondition: [V1=0,V>=0] - Upper bound: 3 - Complexity: constant * Chain [18] with precondition: [V=0,V1>=0] - Upper bound: 4*V1+7+V1*V1 - Complexity: n^2 * Chain [17] with precondition: [V1>=1,V>=V1] - Upper bound: 2*V1+V+4 - Complexity: n * Chain [16] with precondition: [V1=1,V10>=0,V>=V10+2] - Upper bound: 3*V-3*V10+(V+4+(V-V10)*V) - Complexity: n^2 * Chain [15] with precondition: [V>=1,V1>=V] - Upper bound: 3*V1-3*V+(V1+3+(V1-V)*V1+V) - Complexity: n^2 ### Maximum cost of start(V1,V,V10): max([max([V1+1+max([V,2*V1+3+V1*V1]),nat(V1-V)*V1+V+nat(V1-V)*3])+V1,V+1+nat(V-V10)*V+nat(V-V10)*3])+3 Asymptotic class: n^2 * Total analysis performed in 168 ms. ---------------------------------------- (12) BOUNDS(1, n^2) ---------------------------------------- (13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (14) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: lt(0, s(x)) -> true lt(x, 0) -> false lt(s(x), s(y)) -> lt(x, y) minus(x, y) -> help(lt(y, x), x, y) help(true, x, y) -> s(minus(x, s(y))) help(false, x, y) -> 0 S is empty. Rewrite Strategy: FULL ---------------------------------------- (15) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence lt(s(x), s(y)) ->^+ lt(x, y) gives rise to a decreasing loop by considering the right hand sides subterm at position []. The pumping substitution is [x / s(x), y / s(y)]. The result substitution is [ ]. ---------------------------------------- (16) Complex Obligation (BEST) ---------------------------------------- (17) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: lt(0, s(x)) -> true lt(x, 0) -> false lt(s(x), s(y)) -> lt(x, y) minus(x, y) -> help(lt(y, x), x, y) help(true, x, y) -> s(minus(x, s(y))) help(false, x, y) -> 0 S is empty. Rewrite Strategy: FULL ---------------------------------------- (18) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (19) BOUNDS(n^1, INF) ---------------------------------------- (20) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: lt(0, s(x)) -> true lt(x, 0) -> false lt(s(x), s(y)) -> lt(x, y) minus(x, y) -> help(lt(y, x), x, y) help(true, x, y) -> s(minus(x, s(y))) help(false, x, y) -> 0 S is empty. Rewrite Strategy: FULL