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


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WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X)
            u21(ackout(X),Y) -> u22(ackin(Y,X))
        - Signature:
            {ackin/2,u21/2} / {ackout/1,s/1,u22/1}
        - Obligation:
             runtime complexity wrt. defined symbols {ackin,u21} and constructors {ackout,s,u22}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X)
            u21(ackout(X),Y) -> u22(ackin(Y,X))
        - Signature:
            {ackin/2,u21/2} / {ackout/1,s/1,u22/1}
        - Obligation:
             runtime complexity wrt. defined symbols {ackin,u21} and constructors {ackout,s,u22}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          ackin(s(x),y){y -> s(y)} =
            ackin(s(x),s(y)) ->^+ u21(ackin(s(x),y),x)
              = C[ackin(s(x),y) = ackin(s(x),y){}]

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X)
            u21(ackout(X),Y) -> u22(ackin(Y,X))
        - Signature:
            {ackin/2,u21/2} / {ackout/1,s/1,u22/1}
        - Obligation:
             runtime complexity wrt. defined symbols {ackin,u21} and constructors {ackout,s,u22}
    + Applied Processor:
        NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(linear):
        The following argument positions are considered usable:
          uargs(u21) = {1},
          uargs(u22) = {1}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
           p(ackin) = 0     
          p(ackout) = 2     
               p(s) = 0     
             p(u21) = 2*x1  
             p(u22) = 3 + x1
        
        Following rules are strictly oriented:
        u21(ackout(X),Y) = 4              
                         > 3              
                         = u22(ackin(Y,X))
        
        
        Following rules are (at-least) weakly oriented:
        ackin(s(X),s(Y)) =  0                   
                         >= 0                   
                         =  u21(ackin(s(X),Y),X)
        
** Step 1.b:2: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X)
        - Weak TRS:
            u21(ackout(X),Y) -> u22(ackin(Y,X))
        - Signature:
            {ackin/2,u21/2} / {ackout/1,s/1,u22/1}
        - Obligation:
             runtime complexity wrt. defined symbols {ackin,u21} and constructors {ackout,s,u22}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(u21) = {1},
          uargs(u22) = {1}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
           p(ackin) = [1] x2 + [8]
          p(ackout) = [1] x1 + [9]
               p(s) = [1] x1 + [8]
             p(u21) = [1] x1 + [0]
             p(u22) = [1] x1 + [1]
        
        Following rules are strictly oriented:
        ackin(s(X),s(Y)) = [1] Y + [16]        
                         > [1] Y + [8]         
                         = u21(ackin(s(X),Y),X)
        
        
        Following rules are (at-least) weakly oriented:
        u21(ackout(X),Y) =  [1] X + [9]    
                         >= [1] X + [9]    
                         =  u22(ackin(Y,X))
        
** Step 1.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            ackin(s(X),s(Y)) -> u21(ackin(s(X),Y),X)
            u21(ackout(X),Y) -> u22(ackin(Y,X))
        - Signature:
            {ackin/2,u21/2} / {ackout/1,s/1,u22/1}
        - Obligation:
             runtime complexity wrt. defined symbols {ackin,u21} and constructors {ackout,s,u22}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^1))