Small Jump with Negation-UTM Trampoline

1. Overview This paper talk about that NP is not AL=P by using circular founded set. I make set structure of CHAOS and computation history of Turing Machine. CHAOS is circular founded set and computation history of Turing Machine is well founded set. And Log space ATM can not compute CHAOS. 2. Computational problem as set In this paper, we think computational problem as set of related problems each other, and we analyze the structure of the set. First, we define Turing Machine (TM) as set, and extend the structure to the whole computational problems. Definition 1. I will use the term " DTM " , " NTM " , " ATM " as Deterministic TM, Nondeterministic TM, Alternating TM. And I will use the term " LSDTM " , " LSNTM " , " LSATM " to the TM limited log space. And I will use the term " PTDTM " , " PT-NTM " , " PTATM " to the TM limited polynomial time. And in this paper, TM is decider that does not have circular computation history. Theorem 2. We think the set which minimum elements is singleton include accepting configuration and empty set. Computation history of DTM is the singleton (or singleton of singleton), Computation history of NTM is the singleton of the set, Computation history of ATM is the set of the set. But depth of inclusion decide either universal state or existential state of the set. Universal state set include only existential state set, and Existential state set include only universal state set. And all computation history start universal state. All universal state set include no empty set. Proof. To think about the relation TM's computation history and result. Because Computation history have no cyclic path in this paper, computation history is directed acyclic graph (DAG). This DAG have root as starting configuration, trunk as computing configuration, leaf as halting configuration. We can associate each TM's result to the DAG of the computation history. Therefore, we can associate TM with set that correspond with the DAG of TM's computation history. And the set is well-founded set which minimal elements are target configurations because DAG have no cyclic part. For simplicity, all computation history of this set are accepting computation history. Computation history of DTM is only one path and have only one accepting configuration. Therefore, computation history correspond with singleton of the minimum elements.