/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^2))
* Step 1: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            half(0()) -> 0()
            half(s(0())) -> 0()
            half(s(s(x))) -> s(half(x))
            log(s(x)) -> s(log(half(s(x))))
            s(log(0())) -> s(0())
        - Signature:
            {half/1,log/1,s/1} / {0/0}
        - Obligation:
            innermost derivational complexity wrt. signature {0,half,log,s}
    + 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(0) = [0]         
          p(half) = [1] x1 + [0]
           p(log) = [1] x1 + [1]
             p(s) = [1] x1 + [0]
        
        Following rules are strictly oriented:
        s(log(0())) = [1]   
                    > [0]   
                    = s(0())
        
        
        Following rules are (at-least) weakly oriented:
            half(0()) =  [0]               
                      >= [0]               
                      =  0()               
        
         half(s(0())) =  [0]               
                      >= [0]               
                      =  0()               
        
        half(s(s(x))) =  [1] x + [0]       
                      >= [1] x + [0]       
                      =  s(half(x))        
        
            log(s(x)) =  [1] x + [1]       
                      >= [1] x + [1]       
                      =  s(log(half(s(x))))
        
* Step 2: WeightGap.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            half(0()) -> 0()
            half(s(0())) -> 0()
            half(s(s(x))) -> s(half(x))
            log(s(x)) -> s(log(half(s(x))))
        - Weak TRS:
            s(log(0())) -> s(0())
        - Signature:
            {half/1,log/1,s/1} / {0/0}
        - Obligation:
            innermost derivational complexity wrt. signature {0,half,log,s}
    + 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(0) = [0]         
            p(half) = [1] x1 + [5]
             p(log) = [1] x1 + [2]
               p(s) = [1] x1 + [0]
          
          Following rules are strictly oriented:
             half(0()) = [5]
                       > [0]
                       = 0()
          
          half(s(0())) = [5]
                       > [0]
                       = 0()
          
          
          Following rules are (at-least) weakly oriented:
          half(s(s(x))) =  [1] x + [5]       
                        >= [1] x + [5]       
                        =  s(half(x))        
          
              log(s(x)) =  [1] x + [2]       
                        >= [1] x + [7]       
                        =  s(log(half(s(x))))
          
            s(log(0())) =  [2]               
                        >= [0]               
                        =  s(0())            
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            half(s(s(x))) -> s(half(x))
            log(s(x)) -> s(log(half(s(x))))
        - Weak TRS:
            half(0()) -> 0()
            half(s(0())) -> 0()
            s(log(0())) -> s(0())
        - Signature:
            {half/1,log/1,s/1} / {0/0}
        - Obligation:
            innermost derivational complexity wrt. signature {0,half,log,s}
    + 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(0) = [14]        
            p(half) = [1] x1 + [0]
             p(log) = [1] x1 + [1]
               p(s) = [1] x1 + [1]
          
          Following rules are strictly oriented:
          half(s(s(x))) = [1] x + [2]
                        > [1] x + [1]
                        = s(half(x)) 
          
          
          Following rules are (at-least) weakly oriented:
             half(0()) =  [14]              
                       >= [14]              
                       =  0()               
          
          half(s(0())) =  [15]              
                       >= [14]              
                       =  0()               
          
             log(s(x)) =  [1] x + [2]       
                       >= [1] x + [3]       
                       =  s(log(half(s(x))))
          
           s(log(0())) =  [16]              
                       >= [15]              
                       =  s(0())            
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: MI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            log(s(x)) -> s(log(half(s(x))))
        - Weak TRS:
            half(0()) -> 0()
            half(s(0())) -> 0()
            half(s(s(x))) -> s(half(x))
            s(log(0())) -> s(0())
        - Signature:
            {half/1,log/1,s/1} / {0/0}
        - Obligation:
            innermost derivational complexity wrt. signature {0,half,log,s}
    + Applied Processor:
        MI {miKind = Automaton (Just 2), miDimension = 3, miUArgs = NoUArgs, miURules = NoURules, miSelector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind Automaton (Just 2):
        
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
             p(0) = [2]              
                    [0]              
                    [0]              
          p(half) = [1 0 0]       [1]
                    [0 1 0] x_1 + [0]
                    [0 1 0]       [0]
           p(log) = [1 0 2]       [3]
                    [0 0 1] x_1 + [0]
                    [0 0 1]       [3]
             p(s) = [1 0 0]       [0]
                    [0 1 0] x_1 + [2]
                    [0 1 0]       [4]
        
        Following rules are strictly oriented:
        log(s(x)) = [1 2 0]     [11]  
                    [0 1 0] x + [4]   
                    [0 1 0]     [7]   
                  > [1 2 0]     [8]   
                    [0 1 0] x + [4]   
                    [0 1 0]     [6]   
                  = s(log(half(s(x))))
        
        
        Following rules are (at-least) weakly oriented:
            half(0()) =  [3]            
                         [0]            
                         [0]            
                      >= [2]            
                         [0]            
                         [0]            
                      =  0()            
        
         half(s(0())) =  [3]            
                         [2]            
                         [2]            
                      >= [2]            
                         [0]            
                         [0]            
                      =  0()            
        
        half(s(s(x))) =  [1 0 0]     [1]
                         [0 1 0] x + [4]
                         [0 1 0]     [4]
                      >= [1 0 0]     [1]
                         [0 1 0] x + [2]
                         [0 1 0]     [4]
                      =  s(half(x))     
        
          s(log(0())) =  [5]            
                         [2]            
                         [4]            
                      >= [2]            
                         [2]            
                         [4]            
                      =  s(0())         
        
* Step 5: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            half(0()) -> 0()
            half(s(0())) -> 0()
            half(s(s(x))) -> s(half(x))
            log(s(x)) -> s(log(half(s(x))))
            s(log(0())) -> s(0())
        - Signature:
            {half/1,log/1,s/1} / {0/0}
        - Obligation:
            innermost derivational complexity wrt. signature {0,half,log,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))