/export/starexec/sandbox/solver/bin/starexec_run_tct_dci /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(?,O(n^1))
* Step 1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a(a(x1)) -> b(c(x1))
            b(b(x1)) -> c(d(x1))
            c(c(x1)) -> d(d(d(x1)))
            d(d(d(x1))) -> a(c(x1))
        - Signature:
            {a/1,b/1,c/1,d/1} / {}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,c,d}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind triangular matrix interpretation:
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
          p(a) = [1] x1 + [12]
          p(b) = [1] x1 + [10]
          p(c) = [1] x1 + [12]
          p(d) = [1] x1 + [8] 
        
        Following rules are strictly oriented:
        a(a(x1)) = [1] x1 + [24]
                 > [1] x1 + [22]
                 = b(c(x1))     
        
        
        Following rules are (at-least) weakly oriented:
           b(b(x1)) =  [1] x1 + [20]
                    >= [1] x1 + [20]
                    =  c(d(x1))     
        
           c(c(x1)) =  [1] x1 + [24]
                    >= [1] x1 + [24]
                    =  d(d(d(x1)))  
        
        d(d(d(x1))) =  [1] x1 + [24]
                    >= [1] x1 + [24]
                    =  a(c(x1))     
        
* Step 2: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            b(b(x1)) -> c(d(x1))
            c(c(x1)) -> d(d(d(x1)))
            d(d(d(x1))) -> a(c(x1))
        - Weak TRS:
            a(a(x1)) -> b(c(x1))
        - Signature:
            {a/1,b/1,c/1,d/1} / {}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,c,d}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind triangular matrix interpretation:
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
          p(a) = [1] x1 + [11]
          p(b) = [1] x1 + [10]
          p(c) = [1] x1 + [12]
          p(d) = [1] x1 + [8] 
        
        Following rules are strictly oriented:
        d(d(d(x1))) = [1] x1 + [24]
                    > [1] x1 + [23]
                    = a(c(x1))     
        
        
        Following rules are (at-least) weakly oriented:
        a(a(x1)) =  [1] x1 + [22]
                 >= [1] x1 + [22]
                 =  b(c(x1))     
        
        b(b(x1)) =  [1] x1 + [20]
                 >= [1] x1 + [20]
                 =  c(d(x1))     
        
        c(c(x1)) =  [1] x1 + [24]
                 >= [1] x1 + [24]
                 =  d(d(d(x1)))  
        
* Step 3: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            b(b(x1)) -> c(d(x1))
            c(c(x1)) -> d(d(d(x1)))
        - Weak TRS:
            a(a(x1)) -> b(c(x1))
            d(d(d(x1))) -> a(c(x1))
        - Signature:
            {a/1,b/1,c/1,d/1} / {}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,c,d}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind triangular matrix interpretation:
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
          p(a) = [1] x1 + [12]
          p(b) = [1] x1 + [12]
          p(c) = [1] x1 + [12]
          p(d) = [1] x1 + [8] 
        
        Following rules are strictly oriented:
        b(b(x1)) = [1] x1 + [24]
                 > [1] x1 + [20]
                 = c(d(x1))     
        
        
        Following rules are (at-least) weakly oriented:
           a(a(x1)) =  [1] x1 + [24]
                    >= [1] x1 + [24]
                    =  b(c(x1))     
        
           c(c(x1)) =  [1] x1 + [24]
                    >= [1] x1 + [24]
                    =  d(d(d(x1)))  
        
        d(d(d(x1))) =  [1] x1 + [24]
                    >= [1] x1 + [24]
                    =  a(c(x1))     
        
* Step 4: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            c(c(x1)) -> d(d(d(x1)))
        - Weak TRS:
            a(a(x1)) -> b(c(x1))
            b(b(x1)) -> c(d(x1))
            d(d(d(x1))) -> a(c(x1))
        - Signature:
            {a/1,b/1,c/1,d/1} / {}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,c,d}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind triangular matrix interpretation:
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
            p(a) = [1] x1 + [10]
            p(b) = [1] x1 + [9] 
            p(c) = [1] x1 + [11]
            p(d) = [1] x1 + [7] 
          
          Following rules are strictly oriented:
          c(c(x1)) = [1] x1 + [22]
                   > [1] x1 + [21]
                   = d(d(d(x1)))  
          
          
          Following rules are (at-least) weakly oriented:
             a(a(x1)) =  [1] x1 + [20]
                      >= [1] x1 + [20]
                      =  b(c(x1))     
          
             b(b(x1)) =  [1] x1 + [18]
                      >= [1] x1 + [18]
                      =  c(d(x1))     
          
          d(d(d(x1))) =  [1] x1 + [21]
                      >= [1] x1 + [21]
                      =  a(c(x1))     
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a(a(x1)) -> b(c(x1))
            b(b(x1)) -> c(d(x1))
            c(c(x1)) -> d(d(d(x1)))
            d(d(d(x1))) -> a(c(x1))
        - Signature:
            {a/1,b/1,c/1,d/1} / {}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,c,d}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))