/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^2)) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). (0) CpxTRS (1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, 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, 270 ms] (10) BOUNDS(1, n^2) (11) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (12) TRS for Loop Detection (13) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (14) BEST (15) proven lower bound (16) LowerBoundPropagationProof [FINISHED, 0 ms] (17) BOUNDS(n^1, INF) (18) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: sub(0, 0) -> 0 sub(s(x), 0) -> s(x) sub(0, s(x)) -> 0 sub(s(x), s(y)) -> sub(x, y) zero(nil) -> zero2(0, nil) zero(cons(x, xs)) -> zero2(sub(x, x), cons(x, xs)) zero2(0, nil) -> nil zero2(0, cons(x, xs)) -> cons(sub(x, x), zero(xs)) zero2(s(y), nil) -> zero(nil) zero2(s(y), cons(x, xs)) -> zero(cons(x, xs)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, 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^2). The TRS R consists of the following rules: sub(0, 0) -> 0 [1] sub(s(x), 0) -> s(x) [1] sub(0, s(x)) -> 0 [1] sub(s(x), s(y)) -> sub(x, y) [1] zero(nil) -> zero2(0, nil) [1] zero(cons(x, xs)) -> zero2(sub(x, x), cons(x, xs)) [1] zero2(0, nil) -> nil [1] zero2(0, cons(x, xs)) -> cons(sub(x, x), zero(xs)) [1] zero2(s(y), nil) -> zero(nil) [1] zero2(s(y), cons(x, xs)) -> zero(cons(x, xs)) [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: sub(0, 0) -> 0 [1] sub(s(x), 0) -> s(x) [1] sub(0, s(x)) -> 0 [1] sub(s(x), s(y)) -> sub(x, y) [1] zero(nil) -> zero2(0, nil) [1] zero(cons(x, xs)) -> zero2(sub(x, x), cons(x, xs)) [1] zero2(0, nil) -> nil [1] zero2(0, cons(x, xs)) -> cons(sub(x, x), zero(xs)) [1] zero2(s(y), nil) -> zero(nil) [1] zero2(s(y), cons(x, xs)) -> zero(cons(x, xs)) [1] The TRS has the following type information: sub :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s zero :: nil:cons -> nil:cons nil :: nil:cons zero2 :: 0:s -> nil:cons -> nil:cons cons :: 0:s -> nil:cons -> nil:cons 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: none And the following fresh constants: none ---------------------------------------- (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: sub(0, 0) -> 0 [1] sub(s(x), 0) -> s(x) [1] sub(0, s(x)) -> 0 [1] sub(s(x), s(y)) -> sub(x, y) [1] zero(nil) -> zero2(0, nil) [1] zero(cons(x, xs)) -> zero2(sub(x, x), cons(x, xs)) [1] zero2(0, nil) -> nil [1] zero2(0, cons(x, xs)) -> cons(sub(x, x), zero(xs)) [1] zero2(s(y), nil) -> zero(nil) [1] zero2(s(y), cons(x, xs)) -> zero(cons(x, xs)) [1] The TRS has the following type information: sub :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s zero :: nil:cons -> nil:cons nil :: nil:cons zero2 :: 0:s -> nil:cons -> nil:cons cons :: 0:s -> nil:cons -> nil:cons 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: 0 => 0 nil => 0 ---------------------------------------- (8) Obligation: Complexity RNTS consisting of the following rules: sub(z, z') -{ 1 }-> sub(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x sub(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 sub(z, z') -{ 1 }-> 0 :|: z' = 1 + x, x >= 0, z = 0 sub(z, z') -{ 1 }-> 1 + x :|: x >= 0, z = 1 + x, z' = 0 zero(z) -{ 1 }-> zero2(sub(x, x), 1 + x + xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 zero(z) -{ 1 }-> zero2(0, 0) :|: z = 0 zero2(z, z') -{ 1 }-> zero(0) :|: y >= 0, z = 1 + y, z' = 0 zero2(z, z') -{ 1 }-> zero(1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, y >= 0, x >= 0, z = 1 + y zero2(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0 zero2(z, z') -{ 1 }-> 1 + sub(x, x) + zero(xs) :|: xs >= 0, z' = 1 + x + xs, x >= 0, z = 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,[sub(V1, V, Out)],[V1 >= 0,V >= 0]). eq(start(V1, V),0,[zero(V1, Out)],[V1 >= 0]). eq(start(V1, V),0,[zero2(V1, V, Out)],[V1 >= 0,V >= 0]). eq(sub(V1, V, Out),1,[],[Out = 0,V1 = 0,V = 0]). eq(sub(V1, V, Out),1,[],[Out = 1 + V2,V2 >= 0,V1 = 1 + V2,V = 0]). eq(sub(V1, V, Out),1,[],[Out = 0,V = 1 + V3,V3 >= 0,V1 = 0]). eq(sub(V1, V, Out),1,[sub(V4, V5, Ret)],[Out = Ret,V = 1 + V5,V4 >= 0,V5 >= 0,V1 = 1 + V4]). eq(zero(V1, Out),1,[zero2(0, 0, Ret1)],[Out = Ret1,V1 = 0]). eq(zero(V1, Out),1,[sub(V6, V6, Ret0),zero2(Ret0, 1 + V6 + V7, Ret2)],[Out = Ret2,V1 = 1 + V6 + V7,V7 >= 0,V6 >= 0]). eq(zero2(V1, V, Out),1,[],[Out = 0,V1 = 0,V = 0]). eq(zero2(V1, V, Out),1,[sub(V8, V8, Ret01),zero(V9, Ret11)],[Out = 1 + Ret01 + Ret11,V9 >= 0,V = 1 + V8 + V9,V8 >= 0,V1 = 0]). eq(zero2(V1, V, Out),1,[zero(0, Ret3)],[Out = Ret3,V10 >= 0,V1 = 1 + V10,V = 0]). eq(zero2(V1, V, Out),1,[zero(1 + V12 + V13, Ret4)],[Out = Ret4,V13 >= 0,V = 1 + V12 + V13,V11 >= 0,V12 >= 0,V1 = 1 + V11]). input_output_vars(sub(V1,V,Out),[V1,V],[Out]). input_output_vars(zero(V1,Out),[V1],[Out]). input_output_vars(zero2(V1,V,Out),[V1,V],[Out]). CoFloCo proof output: Preprocessing Cost Relations ===================================== #### Computed strongly connected components 0. recursive : [sub/3] 1. recursive : [zero/2,zero2/3] 2. non_recursive : [start/2] #### Obtained direct recursion through partial evaluation 0. SCC is partially evaluated into sub/3 1. SCC is partially evaluated into zero/2 2. SCC is partially evaluated into start/2 Control-Flow Refinement of Cost Relations ===================================== ### Specialization of cost equations sub/3 * CE 13 is refined into CE [14] * CE 11 is refined into CE [15] * CE 12 is refined into CE [16] * CE 10 is refined into CE [17] ### Cost equations --> "Loop" of sub/3 * CEs [15] --> Loop 11 * CEs [16] --> Loop 12 * CEs [17] --> Loop 13 * CEs [14] --> Loop 14 ### Ranking functions of CR sub(V1,V,Out) * RF of phase [14]: [V,V1] #### Partial ranking functions of CR sub(V1,V,Out) * Partial RF of phase [14]: - RF of loop [14:1]: V V1 ### Specialization of cost equations zero/2 * CE 9 is refined into CE [18] * CE 8 is refined into CE [19,20,21,22] * CE 7 is discarded (unfeasible) ### Cost equations --> "Loop" of zero/2 * CEs [20,22] --> Loop 15 * CEs [19,21] --> Loop 16 * CEs [18] --> Loop 17 ### Ranking functions of CR zero(V1,Out) * RF of phase [15,16]: [V1] #### Partial ranking functions of CR zero(V1,Out) * Partial RF of phase [15,16]: - RF of loop [15:1]: V1-1 - RF of loop [16:1]: V1 ### Specialization of cost equations start/2 * CE 2 is refined into CE [23] * CE 3 is refined into CE [24,25,26,27] * CE 1 is refined into CE [28] * CE 4 is refined into CE [29] * CE 5 is refined into CE [30,31,32,33,34,35] * CE 6 is refined into CE [36,37] ### Cost equations --> "Loop" of start/2 * CEs [33] --> Loop 18 * CEs [23,28,32,34,35,37] --> Loop 19 * CEs [24,25,26,27,29,30,31,36] --> Loop 20 ### Ranking functions of CR start(V1,V) #### Partial ranking functions of CR start(V1,V) Computing Bounds ===================================== #### Cost of chains of sub(V1,V,Out): * Chain [[14],13]: 1*it(14)+1 Such that:it(14) =< V1 with precondition: [Out=0,V1=V,V1>=1] * Chain [[14],12]: 1*it(14)+1 Such that:it(14) =< V1 with precondition: [Out=0,V1>=1,V>=V1+1] * Chain [[14],11]: 1*it(14)+1 Such that:it(14) =< V with precondition: [V1=Out+V,V>=1,V1>=V+1] * Chain [13]: 1 with precondition: [V1=0,V=0,Out=0] * Chain [12]: 1 with precondition: [V1=0,Out=0,V>=1] * Chain [11]: 1 with precondition: [V=0,V1=Out,V1>=1] #### Cost of chains of zero(V1,Out): * Chain [[15,16],17]: 10*it(15)+1*s(10)+1*s(12)+2 Such that:aux(6) =< V1 it(15) =< aux(6) aux(3) =< aux(6)-1 s(10) =< it(15)*aux(6) s(12) =< it(15)*aux(3) with precondition: [Out>=1,V1>=Out] * Chain [17]: 2 with precondition: [V1=0,Out=0] #### Cost of chains of start(V1,V): * Chain [20]: 22*s(14)+2*s(16)+2*s(17)+4 Such that:aux(8) =< V s(14) =< aux(8) s(15) =< aux(8)-1 s(16) =< s(14)*aux(8) s(17) =< s(14)*s(15) with precondition: [V1=0] * Chain [19]: 11*s(26)+1*s(28)+1*s(29)+11*s(30)+1*s(35)+1*s(36)+3 Such that:aux(9) =< V1 aux(10) =< V s(30) =< aux(9) s(26) =< aux(10) s(34) =< aux(9)-1 s(35) =< s(30)*aux(9) s(36) =< s(30)*s(34) s(27) =< aux(10)-1 s(28) =< s(26)*aux(10) s(29) =< s(26)*s(27) with precondition: [V1>=1] * Chain [18]: 1*s(37)+1 Such that:s(37) =< V with precondition: [V1=V,V1>=1] Closed-form bounds of start(V1,V): ------------------------------------- * Chain [20] with precondition: [V1=0] - Upper bound: nat(V)*22+4+nat(V)*2*nat(V)+nat(V)*2*nat(nat(V)+ -1) - Complexity: n^2 * Chain [19] with precondition: [V1>=1] - Upper bound: 11*V1+3+V1*V1+(V1-1)*V1+nat(V)*11+nat(V)*nat(V)+nat(nat(V)+ -1)*nat(V) - Complexity: n^2 * Chain [18] with precondition: [V1=V,V1>=1] - Upper bound: V+1 - Complexity: n ### Maximum cost of start(V1,V): nat(V)*10+2+nat(V)*nat(V)+nat(nat(V)+ -1)*nat(V)+max([11*V1+V1*V1+nat(V1-1)*V1,nat(V)*11+1+nat(V)*nat(V)+nat(nat(V)+ -1)*nat(V)])+(nat(V)+1) Asymptotic class: n^2 * Total analysis performed in 192 ms. ---------------------------------------- (10) BOUNDS(1, n^2) ---------------------------------------- (11) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (12) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: sub(0, 0) -> 0 sub(s(x), 0) -> s(x) sub(0, s(x)) -> 0 sub(s(x), s(y)) -> sub(x, y) zero(nil) -> zero2(0, nil) zero(cons(x, xs)) -> zero2(sub(x, x), cons(x, xs)) zero2(0, nil) -> nil zero2(0, cons(x, xs)) -> cons(sub(x, x), zero(xs)) zero2(s(y), nil) -> zero(nil) zero2(s(y), cons(x, xs)) -> zero(cons(x, xs)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (13) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence sub(s(x), s(y)) ->^+ sub(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 [ ]. ---------------------------------------- (14) Complex Obligation (BEST) ---------------------------------------- (15) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: sub(0, 0) -> 0 sub(s(x), 0) -> s(x) sub(0, s(x)) -> 0 sub(s(x), s(y)) -> sub(x, y) zero(nil) -> zero2(0, nil) zero(cons(x, xs)) -> zero2(sub(x, x), cons(x, xs)) zero2(0, nil) -> nil zero2(0, cons(x, xs)) -> cons(sub(x, x), zero(xs)) zero2(s(y), nil) -> zero(nil) zero2(s(y), cons(x, xs)) -> zero(cons(x, xs)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (16) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (17) BOUNDS(n^1, INF) ---------------------------------------- (18) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: sub(0, 0) -> 0 sub(s(x), 0) -> s(x) sub(0, s(x)) -> 0 sub(s(x), s(y)) -> sub(x, y) zero(nil) -> zero2(0, nil) zero(cons(x, xs)) -> zero2(sub(x, x), cons(x, xs)) zero2(0, nil) -> nil zero2(0, cons(x, xs)) -> cons(sub(x, x), zero(xs)) zero2(s(y), nil) -> zero(nil) zero2(s(y), cons(x, xs)) -> zero(cons(x, xs)) S is empty. Rewrite Strategy: INNERMOST