/export/starexec/sandbox2/solver/bin/starexec_run_tct_dci /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:
            and(x,x) -> x
            not(and(x,y)) -> or(not(x),not(y))
            not(not(x)) -> x
            not(or(x,y)) -> and(not(x),not(y))
            or(x,x) -> x
        - Signature:
            {and/2,not/1,or/2} / {}
        - Obligation:
            innermost derivational complexity wrt. signature {and,not,or}
    + 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(and) = [1] x1 + [1] x2 + [2]
          p(not) = [1] x1 + [0]         
           p(or) = [1] x1 + [1] x2 + [2]
        
        Following rules are strictly oriented:
        and(x,x) = [2] x + [2]
                 > [1] x + [0]
                 = x          
        
         or(x,x) = [2] x + [2]
                 > [1] x + [0]
                 = x          
        
        
        Following rules are (at-least) weakly oriented:
        not(and(x,y)) =  [1] x + [1] y + [2]
                      >= [1] x + [1] y + [2]
                      =  or(not(x),not(y))  
        
          not(not(x)) =  [1] x + [0]        
                      >= [1] x + [0]        
                      =  x                  
        
         not(or(x,y)) =  [1] x + [1] y + [2]
                      >= [1] x + [1] y + [2]
                      =  and(not(x),not(y)) 
        
* Step 2: WeightGap.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            not(and(x,y)) -> or(not(x),not(y))
            not(not(x)) -> x
            not(or(x,y)) -> and(not(x),not(y))
        - Weak TRS:
            and(x,x) -> x
            or(x,x) -> x
        - Signature:
            {and/2,not/1,or/2} / {}
        - Obligation:
            innermost derivational complexity wrt. signature {and,not,or}
    + 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(and) = [1] x1 + [1] x2 + [0]
            p(not) = [1] x1 + [2]         
             p(or) = [1] x1 + [1] x2 + [8]
          
          Following rules are strictly oriented:
           not(not(x)) = [1] x + [4]         
                       > [1] x + [0]         
                       = x                   
          
          not(or(x,y)) = [1] x + [1] y + [10]
                       > [1] x + [1] y + [4] 
                       = and(not(x),not(y))  
          
          
          Following rules are (at-least) weakly oriented:
               and(x,x) =  [2] x + [0]         
                        >= [1] x + [0]         
                        =  x                   
          
          not(and(x,y)) =  [1] x + [1] y + [2] 
                        >= [1] x + [1] y + [12]
                        =  or(not(x),not(y))   
          
                or(x,x) =  [2] x + [8]         
                        >= [1] x + [0]         
                        =  x                   
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            not(and(x,y)) -> or(not(x),not(y))
        - Weak TRS:
            and(x,x) -> x
            not(not(x)) -> x
            not(or(x,y)) -> and(not(x),not(y))
            or(x,x) -> x
        - Signature:
            {and/2,not/1,or/2} / {}
        - Obligation:
            innermost derivational complexity wrt. signature {and,not,or}
    + 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(and) = [1 0] x1 + [1 0] x2 + [0]
                   [0 1]      [0 1]      [1]
          p(not) = [1 1] x1 + [0]           
                   [0 1]      [0]           
           p(or) = [1 0] x1 + [1 0] x2 + [0]
                   [0 1]      [0 1]      [1]
        
        Following rules are strictly oriented:
        not(and(x,y)) = [1 1] x + [1 1] y + [1]
                        [0 1]     [0 1]     [1]
                      > [1 1] x + [1 1] y + [0]
                        [0 1]     [0 1]     [1]
                      = or(not(x),not(y))      
        
        
        Following rules are (at-least) weakly oriented:
            and(x,x) =  [2 0] x + [0]          
                        [0 2]     [1]          
                     >= [1 0] x + [0]          
                        [0 1]     [0]          
                     =  x                      
        
         not(not(x)) =  [1 2] x + [0]          
                        [0 1]     [0]          
                     >= [1 0] x + [0]          
                        [0 1]     [0]          
                     =  x                      
        
        not(or(x,y)) =  [1 1] x + [1 1] y + [1]
                        [0 1]     [0 1]     [1]
                     >= [1 1] x + [1 1] y + [0]
                        [0 1]     [0 1]     [1]
                     =  and(not(x),not(y))     
        
             or(x,x) =  [2 0] x + [0]          
                        [0 2]     [1]          
                     >= [1 0] x + [0]          
                        [0 1]     [0]          
                     =  x                      
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            and(x,x) -> x
            not(and(x,y)) -> or(not(x),not(y))
            not(not(x)) -> x
            not(or(x,y)) -> and(not(x),not(y))
            or(x,x) -> x
        - Signature:
            {and/2,not/1,or/2} / {}
        - Obligation:
            innermost derivational complexity wrt. signature {and,not,or}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))