/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: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a(X) -> e()
            b(X) -> e()
            c(X) -> e()
            c(b(a(X))) -> a(a(b(b(c(c(X))))))
        - Signature:
            {a/1,b/1,c/1} / {e/0}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,c,e}
    + 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 + [0]
            p(b) = [1] x1 + [3]
            p(c) = [1] x1 + [0]
            p(e) = [1]         
          
          Following rules are strictly oriented:
          b(X) = [1] X + [3]
               > [1]        
               = e()        
          
          
          Following rules are (at-least) weakly oriented:
                a(X) =  [1] X + [0]        
                     >= [1]                
                     =  e()                
          
                c(X) =  [1] X + [0]        
                     >= [1]                
                     =  e()                
          
          c(b(a(X))) =  [1] X + [3]        
                     >= [1] X + [6]        
                     =  a(a(b(b(c(c(X))))))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a(X) -> e()
            c(X) -> e()
            c(b(a(X))) -> a(a(b(b(c(c(X))))))
        - Weak TRS:
            b(X) -> e()
        - Signature:
            {a/1,b/1,c/1} / {e/0}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,c,e}
    + 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 + [1]
            p(b) = [1] x1 + [0]
            p(c) = [1] x1 + [2]
            p(e) = [0]         
          
          Following rules are strictly oriented:
          a(X) = [1] X + [1]
               > [0]        
               = e()        
          
          c(X) = [1] X + [2]
               > [0]        
               = e()        
          
          
          Following rules are (at-least) weakly oriented:
                b(X) =  [1] X + [0]        
                     >= [0]                
                     =  e()                
          
          c(b(a(X))) =  [1] X + [3]        
                     >= [1] X + [6]        
                     =  a(a(b(b(c(c(X))))))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            c(b(a(X))) -> a(a(b(b(c(c(X))))))
        - Weak TRS:
            a(X) -> e()
            b(X) -> e()
            c(X) -> e()
        - Signature:
            {a/1,b/1,c/1} / {e/0}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,c,e}
    + Applied Processor:
        NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind triangular matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
          p(a) = [1 0 0]      [0]
                 [0 0 1] x1 + [0]
                 [0 0 0]      [0]
          p(b) = [1 1 0]      [0]
                 [0 0 0] x1 + [0]
                 [0 0 0]      [1]
          p(c) = [1 0 1]      [0]
                 [0 0 0] x1 + [0]
                 [0 0 0]      [0]
          p(e) = [0]             
                 [0]             
                 [0]             
        
        Following rules are strictly oriented:
        c(b(a(X))) = [1 0 1]     [1]    
                     [0 0 0] X + [0]    
                     [0 0 0]     [0]    
                   > [1 0 1]     [0]    
                     [0 0 0] X + [0]    
                     [0 0 0]     [0]    
                   = a(a(b(b(c(c(X))))))
        
        
        Following rules are (at-least) weakly oriented:
        a(X) =  [1 0 0]     [0]
                [0 0 1] X + [0]
                [0 0 0]     [0]
             >= [0]            
                [0]            
                [0]            
             =  e()            
        
        b(X) =  [1 1 0]     [0]
                [0 0 0] X + [0]
                [0 0 0]     [1]
             >= [0]            
                [0]            
                [0]            
             =  e()            
        
        c(X) =  [1 0 1]     [0]
                [0 0 0] X + [0]
                [0 0 0]     [0]
             >= [0]            
                [0]            
                [0]            
             =  e()            
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a(X) -> e()
            b(X) -> e()
            c(X) -> e()
            c(b(a(X))) -> a(a(b(b(c(c(X))))))
        - Signature:
            {a/1,b/1,c/1} / {e/0}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,c,e}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))