/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^1)) proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). (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, 175 ms] (12) BOUNDS(1, n^1) (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^1). The TRS R consists of the following rules: half(0) -> 0 half(s(0)) -> 0 half(s(s(x))) -> s(half(x)) bits(0) -> 0 bits(s(x)) -> s(bits(half(s(x)))) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) RcToIrcProof (BOTH BOUNDS(ID, ID)) Converted rc-obligation to irc-obligation. As the TRS is a non-duplicating overlay system, we have rc = irc. ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: half(0) -> 0 half(s(0)) -> 0 half(s(s(x))) -> s(half(x)) bits(0) -> 0 bits(s(x)) -> s(bits(half(s(x)))) 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^1). The TRS R consists of the following rules: half(0) -> 0 [1] half(s(0)) -> 0 [1] half(s(s(x))) -> s(half(x)) [1] bits(0) -> 0 [1] bits(s(x)) -> s(bits(half(s(x)))) [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: half(0) -> 0 [1] half(s(0)) -> 0 [1] half(s(s(x))) -> s(half(x)) [1] bits(0) -> 0 [1] bits(s(x)) -> s(bits(half(s(x)))) [1] The TRS has the following type information: half :: 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s bits :: 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: half(0) -> 0 [1] half(s(0)) -> 0 [1] half(s(s(x))) -> s(half(x)) [1] bits(0) -> 0 [1] bits(s(x)) -> s(bits(half(s(x)))) [1] The TRS has the following type information: half :: 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s bits :: 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 ---------------------------------------- (10) Obligation: Complexity RNTS consisting of the following rules: bits(z) -{ 1 }-> 0 :|: z = 0 bits(z) -{ 1 }-> 1 + bits(half(1 + x)) :|: x >= 0, z = 1 + x half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 1 + half(x) :|: x >= 0, z = 1 + (1 + x) 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(V),0,[half(V, Out)],[V >= 0]). eq(start(V),0,[bits(V, Out)],[V >= 0]). eq(half(V, Out),1,[],[Out = 0,V = 0]). eq(half(V, Out),1,[],[Out = 0,V = 1]). eq(half(V, Out),1,[half(V1, Ret1)],[Out = 1 + Ret1,V1 >= 0,V = 2 + V1]). eq(bits(V, Out),1,[],[Out = 0,V = 0]). eq(bits(V, Out),1,[half(1 + V2, Ret10),bits(Ret10, Ret11)],[Out = 1 + Ret11,V2 >= 0,V = 1 + V2]). input_output_vars(half(V,Out),[V],[Out]). input_output_vars(bits(V,Out),[V],[Out]). CoFloCo proof output: Preprocessing Cost Relations ===================================== #### Computed strongly connected components 0. recursive : [half/2] 1. recursive : [bits/2] 2. non_recursive : [start/1] #### Obtained direct recursion through partial evaluation 0. SCC is partially evaluated into half/2 1. SCC is partially evaluated into bits/2 2. SCC is partially evaluated into start/1 Control-Flow Refinement of Cost Relations ===================================== ### Specialization of cost equations half/2 * CE 5 is refined into CE [8] * CE 4 is refined into CE [9] * CE 3 is refined into CE [10] ### Cost equations --> "Loop" of half/2 * CEs [9] --> Loop 7 * CEs [10] --> Loop 8 * CEs [8] --> Loop 9 ### Ranking functions of CR half(V,Out) * RF of phase [9]: [V-1] #### Partial ranking functions of CR half(V,Out) * Partial RF of phase [9]: - RF of loop [9:1]: V-1 ### Specialization of cost equations bits/2 * CE 7 is refined into CE [11,12,13] * CE 6 is refined into CE [14] ### Cost equations --> "Loop" of bits/2 * CEs [14] --> Loop 10 * CEs [13] --> Loop 11 * CEs [12] --> Loop 12 * CEs [11] --> Loop 13 ### Ranking functions of CR bits(V,Out) * RF of phase [11,12]: [V-1] #### Partial ranking functions of CR bits(V,Out) * Partial RF of phase [11,12]: - RF of loop [11:1]: V/2-1 - RF of loop [12:1]: V-1 ### Specialization of cost equations start/1 * CE 1 is refined into CE [15,16,17,18] * CE 2 is refined into CE [19,20,21] ### Cost equations --> "Loop" of start/1 * CEs [17,18,21] --> Loop 14 * CEs [16,20] --> Loop 15 * CEs [15,19] --> Loop 16 ### Ranking functions of CR start(V) #### Partial ranking functions of CR start(V) Computing Bounds ===================================== #### Cost of chains of half(V,Out): * Chain [[9],8]: 1*it(9)+1 Such that:it(9) =< 2*Out with precondition: [V=2*Out,V>=2] * Chain [[9],7]: 1*it(9)+1 Such that:it(9) =< 2*Out with precondition: [V=2*Out+1,V>=3] * Chain [8]: 1 with precondition: [V=0,Out=0] * Chain [7]: 1 with precondition: [V=1,Out=0] #### Cost of chains of bits(V,Out): * Chain [[11,12],13,10]: 2*it(11)+2*it(12)+2*s(5)+3 Such that:it(11) =< V/2 aux(5) =< V aux(6) =< 2*V it(11) =< aux(5) it(12) =< aux(5) it(12) =< aux(6) s(5) =< aux(6) with precondition: [Out>=2,V+2>=2*Out] * Chain [13,10]: 3 with precondition: [V=1,Out=1] * Chain [10]: 1 with precondition: [V=0,Out=0] #### Cost of chains of start(V): * Chain [16]: 1 with precondition: [V=0] * Chain [15]: 3 with precondition: [V=1] * Chain [14]: 2*s(7)+2*s(9)+2*s(12)+2*s(13)+3 Such that:s(11) =< 2*V s(9) =< V/2 aux(7) =< V s(7) =< aux(7) s(9) =< aux(7) s(12) =< aux(7) s(12) =< s(11) s(13) =< s(11) with precondition: [V>=2] Closed-form bounds of start(V): ------------------------------------- * Chain [16] with precondition: [V=0] - Upper bound: 1 - Complexity: constant * Chain [15] with precondition: [V=1] - Upper bound: 3 - Complexity: constant * Chain [14] with precondition: [V>=2] - Upper bound: 9*V+3 - Complexity: n ### Maximum cost of start(V): 9*V+3 Asymptotic class: n * Total analysis performed in 109 ms. ---------------------------------------- (12) BOUNDS(1, n^1) ---------------------------------------- (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^1). The TRS R consists of the following rules: half(0) -> 0 half(s(0)) -> 0 half(s(s(x))) -> s(half(x)) bits(0) -> 0 bits(s(x)) -> s(bits(half(s(x)))) 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 half(s(s(x))) ->^+ s(half(x)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [x / s(s(x))]. 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^1). The TRS R consists of the following rules: half(0) -> 0 half(s(0)) -> 0 half(s(s(x))) -> s(half(x)) bits(0) -> 0 bits(s(x)) -> s(bits(half(s(x)))) 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^1). The TRS R consists of the following rules: half(0) -> 0 half(s(0)) -> 0 half(s(s(x))) -> s(half(x)) bits(0) -> 0 bits(s(x)) -> s(bits(half(s(x)))) S is empty. Rewrite Strategy: FULL