Finite and infinite basis in P and NP

1. Overview This paper talks about monotonicity of P and NP by using monotone circuit and Log Space Transducer (LST). We can reduce Deterministic Turing Machine (DTM) to monotone circuit value problem by using LST. Therefore DTM has nonmonotonicity of the degree that LST can compute it. But Nondeterministic Turing Machine (NTM) have nonmono-tonicity that LST cannot compute it. Therefore We cannot reduce NP-Complete problem to P-Complete problem by using LST. The NP-Complete problem is different from P-Complete problem in non-monotonicity. And NP-Complete is not in P-Complete. 2. Nonmonotonicity of LST First, I show nonmonotonicity of LST. LST have read only input tape, write only output tape, read and write working tape that is logarithm size, head, and state register that is constant size. The information of working tape, head position, and state registers is replaceable and become non-monotonicity as information. Definition 1. I will use the term " Nonmonotonicity information " as the total information of working tape, head position, and state registers. I will use the term " Monotonic information " as the total information of input tape and output tape. Theorem 2. Nonmonotonic information size of LST is at most logarithm size. Proof. We can record working tape, head position and state register in logarithm size, logarithm size pointer, constant size of record. Therefore, we can record all of nonmonotonic information in logarithm size. P-Complete problem have some nonmonotonic information. But We can reduce P-Complete problem to monotone information by using LST. Definition 3. I will use the term " Flatten input " as changing each value of an input (Target input) of problem (Target problem) to 0 → 10 1 → 01 And I will use the term " Flatten problem " as that problem that reduce object input to flatten input, " Flatten transducer " as transducer that change target input to flatten input. LST can change target input to flatten input. Theorem 4. We can make Flatten transducer with LST.