/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).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, 220 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: minus(x, 0) -> x minus(s(x), s(y)) -> minus(x, y) quot(0, s(y)) -> 0 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) RcToIrcProof (BOTH BOUNDS(ID, ID)) Converted rc-obligation to irc-obligation. The duplicating contexts are: quot(s(x), s([])) The defined contexts are: quot([], s(x1)) minus([], x1) [] just represents basic- or constructor-terms in the following defined contexts: quot([], s(x1)) 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^1). The TRS R consists of the following rules: minus(x, 0) -> x minus(s(x), s(y)) -> minus(x, y) quot(0, s(y)) -> 0 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 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: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] quot(0, s(y)) -> 0 [1] quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [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: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] quot(0, s(y)) -> 0 [1] quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1] The TRS has the following type information: minus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s quot :: 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: minus(v0, v1) -> null_minus [0] quot(v0, v1) -> null_quot [0] And the following fresh constants: null_minus, null_quot ---------------------------------------- (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: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] quot(0, s(y)) -> 0 [1] quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1] minus(v0, v1) -> null_minus [0] quot(v0, v1) -> null_quot [0] The TRS has the following type information: minus :: 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot 0 :: 0:s:null_minus:null_quot s :: 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot quot :: 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot -> 0:s:null_minus:null_quot null_minus :: 0:s:null_minus:null_quot null_quot :: 0:s:null_minus:null_quot 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 null_minus => 0 null_quot => 0 ---------------------------------------- (10) Obligation: Complexity RNTS consisting of the following rules: minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 quot(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0 quot(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 quot(z, z') -{ 1 }-> 1 + quot(minus(x, y), 1 + y) :|: z' = 1 + y, x >= 0, y >= 0, z = 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(V1, V),0,[minus(V1, V, Out)],[V1 >= 0,V >= 0]). eq(start(V1, V),0,[quot(V1, V, Out)],[V1 >= 0,V >= 0]). eq(minus(V1, V, Out),1,[],[Out = V2,V2 >= 0,V1 = V2,V = 0]). eq(minus(V1, V, Out),1,[minus(V3, V4, Ret)],[Out = Ret,V = 1 + V4,V3 >= 0,V4 >= 0,V1 = 1 + V3]). eq(quot(V1, V, Out),1,[],[Out = 0,V = 1 + V5,V5 >= 0,V1 = 0]). eq(quot(V1, V, Out),1,[minus(V7, V6, Ret10),quot(Ret10, 1 + V6, Ret1)],[Out = 1 + Ret1,V = 1 + V6,V7 >= 0,V6 >= 0,V1 = 1 + V7]). eq(minus(V1, V, Out),0,[],[Out = 0,V9 >= 0,V8 >= 0,V1 = V9,V = V8]). eq(quot(V1, V, Out),0,[],[Out = 0,V11 >= 0,V10 >= 0,V1 = V11,V = V10]). input_output_vars(minus(V1,V,Out),[V1,V],[Out]). input_output_vars(quot(V1,V,Out),[V1,V],[Out]). CoFloCo proof output: Preprocessing Cost Relations ===================================== #### Computed strongly connected components 0. recursive : [minus/3] 1. recursive : [quot/3] 2. non_recursive : [start/2] #### Obtained direct recursion through partial evaluation 0. SCC is partially evaluated into minus/3 1. SCC is partially evaluated into quot/3 2. SCC is partially evaluated into start/2 Control-Flow Refinement of Cost Relations ===================================== ### Specialization of cost equations minus/3 * CE 5 is refined into CE [9] * CE 3 is refined into CE [10] * CE 4 is refined into CE [11] ### Cost equations --> "Loop" of minus/3 * CEs [11] --> Loop 7 * CEs [9] --> Loop 8 * CEs [10] --> Loop 9 ### Ranking functions of CR minus(V1,V,Out) * RF of phase [7]: [V,V1] #### Partial ranking functions of CR minus(V1,V,Out) * Partial RF of phase [7]: - RF of loop [7:1]: V V1 ### Specialization of cost equations quot/3 * CE 6 is refined into CE [12] * CE 8 is refined into CE [13] * CE 7 is refined into CE [14,15,16] ### Cost equations --> "Loop" of quot/3 * CEs [16] --> Loop 10 * CEs [15] --> Loop 11 * CEs [14] --> Loop 12 * CEs [12,13] --> Loop 13 ### Ranking functions of CR quot(V1,V,Out) * RF of phase [10]: [V1-1,V1-V+1] * RF of phase [12]: [V1] #### Partial ranking functions of CR quot(V1,V,Out) * Partial RF of phase [10]: - RF of loop [10:1]: V1-1 V1-V+1 * Partial RF of phase [12]: - RF of loop [12:1]: V1 ### Specialization of cost equations start/2 * CE 1 is refined into CE [17,18,19] * CE 2 is refined into CE [20,21,22,23,24] ### Cost equations --> "Loop" of start/2 * CEs [20] --> Loop 14 * CEs [17,18,19,21,22,23,24] --> Loop 15 ### Ranking functions of CR start(V1,V) #### Partial ranking functions of CR start(V1,V) Computing Bounds ===================================== #### Cost of chains of minus(V1,V,Out): * Chain [[7],9]: 1*it(7)+1 Such that:it(7) =< V with precondition: [V1=Out+V,V>=1,V1>=V] * Chain [[7],8]: 1*it(7)+0 Such that:it(7) =< V with precondition: [Out=0,V1>=1,V>=1] * Chain [9]: 1 with precondition: [V=0,V1=Out,V1>=0] * Chain [8]: 0 with precondition: [Out=0,V1>=0,V>=0] #### Cost of chains of quot(V1,V,Out): * Chain [[12],13]: 2*it(12)+1 Such that:it(12) =< Out with precondition: [V=1,Out>=1,V1>=Out] * Chain [[12],11,13]: 2*it(12)+1*s(2)+2 Such that:s(2) =< 1 it(12) =< Out with precondition: [V=1,Out>=2,V1>=Out] * Chain [[10],13]: 2*it(10)+1*s(5)+1 Such that:it(10) =< V1-V+1 aux(3) =< V1 it(10) =< aux(3) s(5) =< aux(3) with precondition: [V>=2,Out>=1,V1+2>=2*Out+V] * Chain [[10],11,13]: 2*it(10)+1*s(2)+1*s(5)+2 Such that:it(10) =< V1-V+1 s(2) =< V aux(4) =< V1 it(10) =< aux(4) s(5) =< aux(4) with precondition: [V>=2,Out>=2,V1+3>=2*Out+V] * Chain [13]: 1 with precondition: [Out=0,V1>=0,V>=0] * Chain [11,13]: 1*s(2)+2 Such that:s(2) =< V with precondition: [Out=1,V1>=1,V>=1] #### Cost of chains of start(V1,V): * Chain [15]: 4*s(9)+4*s(12)+2*s(14)+2 Such that:aux(6) =< V1 aux(7) =< V1-V+1 aux(8) =< V s(12) =< aux(7) s(9) =< aux(8) s(12) =< aux(6) s(14) =< aux(6) with precondition: [V1>=0,V>=0] * Chain [14]: 1*s(19)+4*s(21)+2 Such that:s(19) =< 1 s(20) =< V1 s(21) =< s(20) with precondition: [V=1,V1>=1] Closed-form bounds of start(V1,V): ------------------------------------- * Chain [15] with precondition: [V1>=0,V>=0] - Upper bound: 2*V1+4*V+2+nat(V1-V+1)*4 - Complexity: n * Chain [14] with precondition: [V=1,V1>=1] - Upper bound: 4*V1+3 - Complexity: n ### Maximum cost of start(V1,V): 2*V1+2+max([2*V1+1,nat(V1-V+1)*4+4*V]) Asymptotic class: n * Total analysis performed in 164 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: minus(x, 0) -> x minus(s(x), s(y)) -> minus(x, y) quot(0, s(y)) -> 0 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 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 minus(s(x), s(y)) ->^+ minus(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^1). The TRS R consists of the following rules: minus(x, 0) -> x minus(s(x), s(y)) -> minus(x, y) quot(0, s(y)) -> 0 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) 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: minus(x, 0) -> x minus(s(x), s(y)) -> minus(x, y) quot(0, s(y)) -> 0 quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) S is empty. Rewrite Strategy: FULL