Insignificant Choice Polynomial Time

In the late 1980s Gurevich conjectured that there is no logic capturing PTIME, where logic has to be understood in a very general way comprising computation models over structures. In this article we first refute Gurevich's conjecture. For this we extend the seminal research of Blass, Gurevich and Shelah on {\em choiceless polynomial time} (CPT), which exploits deterministic Abstract State Machines (ASMs) supporting unbounded parallelism to capture the choiceless fragment of PTIME. CPT is strictly included in PTIME. We observe that choice is unavoidable, but that a restricted version suffices, which guarantees that the final result is independent from the choice. Such a version of polynomially bounded ASMs, which we call {\em insignificant choice polynomial time} (ICPT) will indeed capture PTIME. Even more, insignificant choice can be captured by ASMs with choice restricted to atoms such that a {\em local insignificance condition} is satisfied. As this condition can be expressed in the logic of non-deterministic ASMs, we obtain a logic capturing PTIME. Furthermore, using inflationary fixed-points we can capture problems in PTIME by fixed-point formulae in a fragment of the logic of non-deterministic ASMs plus inflationary fixed-points. We use this result for our second contribution showing that PTIME differs from NP. For the proof we build again on the research on CPT first establishing a limitation on permutation classes of the sets that can be activated by an ICPT computation. We then prove an inseparability theorem, which characterises classes of structures that cannot be separated by the logic. In particular, this implies that SAT cannot be decided by an ICPT computation.Comment: 69 page

The linear time hierarchy theorems for abstract state machines and rams

We prove the Linear Time Hierarchy Theorems for random access machines and Gurevich abstract state machines. One long-term goal of this line or research isto prove lower bounds for natural linear time problems.