/export/starexec/sandbox2/solver/bin/starexec_run_tct_dc /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(?,O(n^2))
* Step 1: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            a(b(x)) -> b(a(x))
            a(c(x)) -> x
        - Signature:
            {a/1} / {b/1,c/1}
        - Obligation:
             derivational complexity wrt. signature {a,b,c}
    + 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 + [64] 
          p(b) = [1] x1 + [144]
          p(c) = [1] x1 + [148]
        
        Following rules are strictly oriented:
        a(c(x)) = [1] x + [212]
                > [1] x + [0]  
                = x            
        
        
        Following rules are (at-least) weakly oriented:
        a(b(x)) =  [1] x + [208]
                >= [1] x + [208]
                =  b(a(x))      
        
* Step 2: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            a(b(x)) -> b(a(x))
        - Weak TRS:
            a(c(x)) -> x
        - Signature:
            {a/1} / {b/1,c/1}
        - Obligation:
             derivational complexity wrt. signature {a,b,c}
    + Applied Processor:
        NaturalMI {miDimension = 2, miDegree = 2, 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 4] x1 + [0]    
                 [0 1]      [1]    
          p(b) = [1 32] x1 + [132] 
                 [0  1]      [32]  
          p(c) = [1 252] x1 + [176]
                 [0   1]      [106]
        
        Following rules are strictly oriented:
        a(b(x)) = [1 36] x + [260]
                  [0  1]     [33] 
                > [1 36] x + [164]
                  [0  1]     [33] 
                = b(a(x))         
        
        
        Following rules are (at-least) weakly oriented:
        a(c(x)) =  [1 256] x + [600]
                   [0   1]     [107]
                >= [1 0] x + [0]    
                   [0 1]     [0]    
                =  x                
        
* Step 3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a(b(x)) -> b(a(x))
            a(c(x)) -> x
        - Signature:
            {a/1} / {b/1,c/1}
        - Obligation:
             derivational complexity wrt. signature {a,b,c}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))