Inherent logic and complexity

The old intuitive question ”what does the machine think” at different stages of its computation is examined. Our paper is based on the formal definitions and results which are collected in the branching program theory around the intuitive question ”what does the program know about the contents of the input bits” [1],[2],[3],[4],[5]. We further develop these results above and we present a formal counterpart of the intuitive notion ”what does the program think ” at different stages of its computation on the processed input word. The definition is constructed as the logical consequences of the definition of the knowledge about the contents of input bits above. Our formal definition is in a good relation to the world of intuitive ideas. We prove the theorem saying that the programs which are allowed to compute (think) in a more sophisticated way can compute more effectively. We also demonstrate an example that for some programs a small enrichment of their inherent logical possibilities implies a dramatic complexity drop. So, our definition lives up to the expectations inspired by intuition. The present paper opens a large field of possible investigations of relations between logic on one hand and complexity on the other hand.