YES

Problem:
 strict:
  half(0()) -> 0()
  half(s(s(x))) -> s(half(x))
  log(s(0())) -> 0()
  log(s(s(x))) -> s(log(s(half(x))))
 weak:
  rand(x) -> x
  rand(x) -> rand(s(x))

Proof:
 Arctic Interpretation Processor:
  dimension: 1
  interpretation:
   [rand](x0) = 2x0,
   
   [half](x0) = x0,
   
   [log](x0) = x0,
   
   [0] = 1,
   
   [s](x0) = x0
  orientation:
   half(0()) = 1 >= 1 = 0()
   
   half(s(s(x))) = x >= x = s(half(x))
   
   log(s(0())) = 1 >= 1 = 0()
   
   log(s(s(x))) = x >= x = s(log(s(half(x))))
   
   rand(x) = 2x >= x = x
   
   rand(x) = 2x >= 2x = rand(s(x))
  problem:
   strict:
    half(0()) -> 0()
    half(s(s(x))) -> s(half(x))
    log(s(0())) -> 0()
    log(s(s(x))) -> s(log(s(half(x))))
   weak:
    rand(x) -> rand(s(x))
  Arctic Interpretation Processor:
   dimension: 1
   interpretation:
    [rand](x0) = 8x0,
    
    [half](x0) = x0,
    
    [log](x0) = 4x0,
    
    [0] = 4,
    
    [s](x0) = x0
   orientation:
    half(0()) = 4 >= 4 = 0()
    
    half(s(s(x))) = x >= x = s(half(x))
    
    log(s(0())) = 8 >= 4 = 0()
    
    log(s(s(x))) = 4x >= 4x = s(log(s(half(x))))
    
    rand(x) = 8x >= 8x = rand(s(x))
   problem:
    strict:
     half(0()) -> 0()
     half(s(s(x))) -> s(half(x))
     log(s(s(x))) -> s(log(s(half(x))))
    weak:
     rand(x) -> rand(s(x))
   Matrix Interpretation Processor: dim=2
    
    interpretation:
                  [2 0]     [0]
     [rand](x0) = [0 0]x0 + [1],
     
                    
     [half](x0) = x0,
     
                 [2 1]     [0]
     [log](x0) = [0 1]x0 + [1],
     
           [0]
     [0] = [0],
     
                    [0]
     [s](x0) = x0 + [1]
    orientation:
                 [0]    [0]      
     half(0()) = [0] >= [0] = 0()
     
                         [0]        [0]             
     half(s(s(x))) = x + [2] >= x + [1] = s(half(x))
     
                    [2 1]    [2]    [2 1]    [1]                     
     log(s(s(x))) = [0 1]x + [3] >= [0 1]x + [3] = s(log(s(half(x))))
     
               [2 0]    [0]    [2 0]    [0]             
     rand(x) = [0 0]x + [1] >= [0 0]x + [1] = rand(s(x))
    problem:
     strict:
      half(0()) -> 0()
      half(s(s(x))) -> s(half(x))
     weak:
      rand(x) -> rand(s(x))
    Arctic Interpretation Processor:
     dimension: 1
     interpretation:
      [rand](x0) = x0,
      
      [half](x0) = 12x0,
      
      [0] = 4,
      
      [s](x0) = x0
     orientation:
      half(0()) = 16 >= 4 = 0()
      
      half(s(s(x))) = 12x >= 12x = s(half(x))
      
      rand(x) = x >= x = rand(s(x))
     problem:
      strict:
       half(s(s(x))) -> s(half(x))
      weak:
       rand(x) -> rand(s(x))
     String Reversal Processor:
      strict:
       s(s(half(x))) -> half(s(x))
      weak:
       rand(x) -> s(rand(x))
      Matrix Interpretation Processor: dim=2
       
       interpretation:
                     [1 0]     [2]
        [rand](x0) = [0 0]x0 + [0],
        
                          [0]
        [half](x0) = x0 + [1],
        
                  [1 1]  
        [s](x0) = [0 1]x0
       orientation:
                        [1 2]    [2]    [1 1]    [0]             
        s(s(half(x))) = [0 1]x + [1] >= [0 1]x + [1] = half(s(x))
        
                  [1 0]    [2]    [1 0]    [2]             
        rand(x) = [0 0]x + [0] >= [0 0]x + [0] = s(rand(x))
       problem:
        strict:
         
        weak:
         rand(x) -> s(rand(x))
       Qed