WORST_CASE(?,O(n^1)) proof of input_qTCkWAAziP.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). (0) CpxTRS (1) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (2) CpxWeightedTrs (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxTypedWeightedTrs (5) CompletionProof [UPPER BOUND(ID), 0 ms] (6) CpxTypedWeightedCompleteTrs (7) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (8) CpxRNTS (9) CompleteCoflocoProof [FINISHED, 156 ms] (10) BOUNDS(1, n^1) ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1). The TRS R consists of the following rules: ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X) u21(ackout(X), Y) -> u22(ackin(Y, X)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (2) 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: ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X) [1] u21(ackout(X), Y) -> u22(ackin(Y, X)) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (4) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X) [1] u21(ackout(X), Y) -> u22(ackin(Y, X)) [1] The TRS has the following type information: ackin :: s -> s -> ackout:u22 s :: s -> s u21 :: ackout:u22 -> s -> ackout:u22 ackout :: s -> ackout:u22 u22 :: ackout:u22 -> ackout:u22 Rewrite Strategy: INNERMOST ---------------------------------------- (5) 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: ackin(v0, v1) -> null_ackin [0] u21(v0, v1) -> null_u21 [0] And the following fresh constants: null_ackin, null_u21, const ---------------------------------------- (6) 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: ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X) [1] u21(ackout(X), Y) -> u22(ackin(Y, X)) [1] ackin(v0, v1) -> null_ackin [0] u21(v0, v1) -> null_u21 [0] The TRS has the following type information: ackin :: s -> s -> ackout:u22:null_ackin:null_u21 s :: s -> s u21 :: ackout:u22:null_ackin:null_u21 -> s -> ackout:u22:null_ackin:null_u21 ackout :: s -> ackout:u22:null_ackin:null_u21 u22 :: ackout:u22:null_ackin:null_u21 -> ackout:u22:null_ackin:null_u21 null_ackin :: ackout:u22:null_ackin:null_u21 null_u21 :: ackout:u22:null_ackin:null_u21 const :: s Rewrite Strategy: INNERMOST ---------------------------------------- (7) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: null_ackin => 0 null_u21 => 0 const => 0 ---------------------------------------- (8) Obligation: Complexity RNTS consisting of the following rules: ackin(z, z') -{ 1 }-> u21(ackin(1 + X, Y), X) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0 ackin(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 u21(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 u21(z, z') -{ 1 }-> 1 + ackin(Y, X) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (9) CompleteCoflocoProof (FINISHED) Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo: eq(start(V1, V),0,[ackin(V1, V, Out)],[V1 >= 0,V >= 0]). eq(start(V1, V),0,[u21(V1, V, Out)],[V1 >= 0,V >= 0]). eq(ackin(V1, V, Out),1,[ackin(1 + X1, Y1, Ret0),u21(Ret0, X1, Ret)],[Out = Ret,V1 = 1 + X1,Y1 >= 0,V = 1 + Y1,X1 >= 0]). eq(u21(V1, V, Out),1,[ackin(Y2, X2, Ret1)],[Out = 1 + Ret1,V1 = 1 + X2,V = Y2,Y2 >= 0,X2 >= 0]). eq(ackin(V1, V, Out),0,[],[Out = 0,V3 >= 0,V2 >= 0,V1 = V3,V = V2]). eq(u21(V1, V, Out),0,[],[Out = 0,V5 >= 0,V4 >= 0,V1 = V5,V = V4]). input_output_vars(ackin(V1,V,Out),[V1,V],[Out]). input_output_vars(u21(V1,V,Out),[V1,V],[Out]). CoFloCo proof output: Preprocessing Cost Relations ===================================== #### Computed strongly connected components 0. recursive [multiple] : [ackin/3,u21/3] 1. non_recursive : [start/2] #### Obtained direct recursion through partial evaluation 0. SCC is partially evaluated into ackin/3 1. SCC is partially evaluated into start/2 Control-Flow Refinement of Cost Relations ===================================== ### Specialization of cost equations ackin/3 * CE 6 is refined into CE [7] * CE 5 is refined into CE [8] * CE 4 is refined into CE [9] ### Cost equations --> "Loop" of ackin/3 * CEs [9] --> Loop 5 * CEs [8] --> Loop 6 * CEs [7] --> Loop 7 ### Ranking functions of CR ackin(V1,V,Out) #### Partial ranking functions of CR ackin(V1,V,Out) * Partial RF of phase [5,6]: - RF of loop [5:1,6:1]: V depends on loops [6:2] - RF of loop [6:2]: V1 ### Specialization of cost equations start/2 * CE 1 is refined into CE [10] * CE 2 is refined into CE [11] * CE 3 is refined into CE [12] ### Cost equations --> "Loop" of start/2 * CEs [10,11,12] --> Loop 8 ### Ranking functions of CR start(V1,V) #### Partial ranking functions of CR start(V1,V) Computing Bounds ===================================== #### Cost of chains of ackin(V1,V,Out): * Chain [7]: 0 with precondition: [Out=0,V1>=0,V>=0] * Chain [multiple([5,6],[[7]])]: 1*it(5)+0 Such that:it(5) =< V with precondition: [Out=0,V1>=1,V>=1] #### Cost of chains of start(V1,V): * Chain [8]: 1*s(2)+1*s(3)+1 Such that:s(2) =< V1 s(3) =< V with precondition: [V1>=0,V>=0] Closed-form bounds of start(V1,V): ------------------------------------- * Chain [8] with precondition: [V1>=0,V>=0] - Upper bound: V1+V+1 - Complexity: n ### Maximum cost of start(V1,V): V1+V+1 Asymptotic class: n * Total analysis performed in 75 ms. ---------------------------------------- (10) BOUNDS(1, n^1)