/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1). (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, 186 ms] (10) BOUNDS(1, n^1) (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^1). The TRS R consists of the following rules: digits -> d(0) d(x) -> if(le(x, s(s(s(s(s(s(s(s(s(0)))))))))), x) if(true, x) -> cons(x, d(s(x))) if(false, x) -> nil le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) 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^1). The TRS R consists of the following rules: digits -> d(0) [1] d(x) -> if(le(x, s(s(s(s(s(s(s(s(s(0)))))))))), x) [1] if(true, x) -> cons(x, d(s(x))) [1] if(false, x) -> nil [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [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: digits -> d(0) [1] d(x) -> if(le(x, s(s(s(s(s(s(s(s(s(0)))))))))), x) [1] if(true, x) -> cons(x, d(s(x))) [1] if(false, x) -> nil [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] The TRS has the following type information: digits :: cons:nil d :: 0:s -> cons:nil 0 :: 0:s if :: true:false -> 0:s -> cons:nil le :: 0:s -> 0:s -> true:false s :: 0:s -> 0:s true :: true:false cons :: 0:s -> cons:nil -> cons:nil false :: true:false nil :: cons:nil 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: digits -> d(0) [1] d(x) -> if(le(x, s(s(s(s(s(s(s(s(s(0)))))))))), x) [1] if(true, x) -> cons(x, d(s(x))) [1] if(false, x) -> nil [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] The TRS has the following type information: digits :: cons:nil d :: 0:s -> cons:nil 0 :: 0:s if :: true:false -> 0:s -> cons:nil le :: 0:s -> 0:s -> true:false s :: 0:s -> 0:s true :: true:false cons :: 0:s -> cons:nil -> cons:nil false :: true:false nil :: cons:nil 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 true => 1 false => 0 nil => 0 ---------------------------------------- (8) Obligation: Complexity RNTS consisting of the following rules: d(z) -{ 1 }-> if(le(x, 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), x) :|: x >= 0, z = x digits -{ 1 }-> d(0) :|: if(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0 if(z, z') -{ 1 }-> 1 + x + d(1 + x) :|: z' = x, z = 1, x >= 0 le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, 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(V, V2),0,[digits(Out)],[]). eq(start(V, V2),0,[d(V, Out)],[V >= 0]). eq(start(V, V2),0,[if(V, V2, Out)],[V >= 0,V2 >= 0]). eq(start(V, V2),0,[le(V, V2, Out)],[V >= 0,V2 >= 0]). eq(digits(Out),1,[d(0, Ret)],[Out = Ret]). eq(d(V, Out),1,[le(V1, 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), Ret0),if(Ret0, V1, Ret1)],[Out = Ret1,V1 >= 0,V = V1]). eq(if(V, V2, Out),1,[d(1 + V3, Ret11)],[Out = 1 + Ret11 + V3,V2 = V3,V = 1,V3 >= 0]). eq(if(V, V2, Out),1,[],[Out = 0,V2 = V4,V4 >= 0,V = 0]). eq(le(V, V2, Out),1,[],[Out = 1,V5 >= 0,V = 0,V2 = V5]). eq(le(V, V2, Out),1,[],[Out = 0,V6 >= 0,V = 1 + V6,V2 = 0]). eq(le(V, V2, Out),1,[le(V7, V8, Ret2)],[Out = Ret2,V2 = 1 + V8,V7 >= 0,V8 >= 0,V = 1 + V7]). input_output_vars(digits(Out),[],[Out]). input_output_vars(d(V,Out),[V],[Out]). input_output_vars(if(V,V2,Out),[V,V2],[Out]). input_output_vars(le(V,V2,Out),[V,V2],[Out]). CoFloCo proof output: Preprocessing Cost Relations ===================================== #### Computed strongly connected components 0. recursive : [le/3] 1. recursive : [d/2,if/3] 2. non_recursive : [digits/1] 3. non_recursive : [start/2] #### Obtained direct recursion through partial evaluation 0. SCC is partially evaluated into le/3 1. SCC is partially evaluated into d/2 2. SCC is completely evaluated into other SCCs 3. SCC is partially evaluated into start/2 Control-Flow Refinement of Cost Relations ===================================== ### Specialization of cost equations le/3 * CE 10 is refined into CE [11] * CE 9 is refined into CE [12] * CE 8 is refined into CE [13] ### Cost equations --> "Loop" of le/3 * CEs [12] --> Loop 7 * CEs [13] --> Loop 8 * CEs [11] --> Loop 9 ### Ranking functions of CR le(V,V2,Out) * RF of phase [9]: [V,V2] #### Partial ranking functions of CR le(V,V2,Out) * Partial RF of phase [9]: - RF of loop [9:1]: V V2 ### Specialization of cost equations d/2 * CE 7 is refined into CE [14,15] * CE 6 is refined into CE [16] ### Cost equations --> "Loop" of d/2 * CEs [16] --> Loop 10 * CEs [15] --> Loop 11 * CEs [14] --> Loop 12 ### Ranking functions of CR d(V,Out) * RF of phase [11]: [-V+10] #### Partial ranking functions of CR d(V,Out) * Partial RF of phase [11]: - RF of loop [11:1]: -V+10 ### Specialization of cost equations start/2 * CE 1 is refined into CE [17] * CE 2 is refined into CE [18,19] * CE 3 is refined into CE [20] * CE 4 is refined into CE [21,22,23] * CE 5 is refined into CE [24,25,26,27] ### Cost equations --> "Loop" of start/2 * CEs [17,18,19,20,21,22,23,24,25,26,27] --> Loop 13 ### Ranking functions of CR start(V,V2) #### Partial ranking functions of CR start(V,V2) Computing Bounds ===================================== #### Cost of chains of le(V,V2,Out): * Chain [[9],8]: 1*it(9)+1 Such that:it(9) =< V with precondition: [Out=1,V>=1,V2>=V] * Chain [[9],7]: 1*it(9)+1 Such that:it(9) =< V2 with precondition: [Out=0,V2>=1,V>=V2+1] * Chain [8]: 1 with precondition: [V=0,Out=1,V2>=0] * Chain [7]: 1 with precondition: [V2=0,Out=0,V>=1] #### Cost of chains of d(V,Out): * Chain [[11],10]: 3*it(11)+1*s(1)+1*s(4)+3 Such that:s(1) =< 9 s(4) =< -9*V+90 it(11) =< -V+10 with precondition: [9>=V,V>=1] * Chain [12,[11],10]: 4*it(11)+1*s(4)+6 Such that:s(4) =< 81 aux(1) =< 9 it(11) =< aux(1) with precondition: [V=0] * Chain [10]: 1*s(1)+3 Such that:s(1) =< 9 with precondition: [Out=0,V>=10] #### Cost of chains of start(V,V2): * Chain [13]: 12*s(5)+1*s(7)+3*s(8)+2*s(9)+1*s(17)+3*s(18)+1*s(19)+1*s(20)+7 Such that:s(17) =< -9*V+90 s(18) =< -V+10 s(20) =< V s(7) =< -9*V2+81 s(8) =< -V2+9 s(19) =< V2 aux(2) =< 9 aux(3) =< 81 s(5) =< aux(2) s(9) =< aux(3) with precondition: [] Closed-form bounds of start(V,V2): ------------------------------------- * Chain [13] with precondition: [] - Upper bound: nat(V)+277+nat(V2)+nat(-V+10)*3+nat(-V2+9)*3+nat(-9*V+90)+nat(-9*V2+81) - Complexity: n ### Maximum cost of start(V,V2): nat(V)+277+nat(V2)+nat(-V+10)*3+nat(-V2+9)*3+nat(-9*V+90)+nat(-9*V2+81) Asymptotic class: n * Total analysis performed in 121 ms. ---------------------------------------- (10) BOUNDS(1, n^1) ---------------------------------------- (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^1). The TRS R consists of the following rules: digits -> d(0) d(x) -> if(le(x, s(s(s(s(s(s(s(s(s(0)))))))))), x) if(true, x) -> cons(x, d(s(x))) if(false, x) -> nil le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) 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 le(s(x), s(y)) ->^+ le(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^1). The TRS R consists of the following rules: digits -> d(0) d(x) -> if(le(x, s(s(s(s(s(s(s(s(s(0)))))))))), x) if(true, x) -> cons(x, d(s(x))) if(false, x) -> nil le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) 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^1). The TRS R consists of the following rules: digits -> d(0) d(x) -> if(le(x, s(s(s(s(s(s(s(s(s(0)))))))))), x) if(true, x) -> cons(x, d(s(x))) if(false, x) -> nil le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) S is empty. Rewrite Strategy: INNERMOST