Listings and Logics

Abstract

There are standard logics DTC, TC, and LFP capturing the complexity classes L, NL, and P on ordered structures, respectively. In [4] we have shown that LFP inv , the "order-invariant least fixed-point logic LFP," captures P (on all finite structures) if and only if there is a listing of the Psubsets of the set TAUT of propositional tautologies. We are able to extend the result to listings of the L-subsets (NL-subsets) of TAUT and the logic DTC inv (TC inv ). As a byproduct we get that LFP inv captures P if DTC inv captures L.Furthermore, we show that the existence of a listing of the L-subsets of TAUT is equivalent to the existence of an almost space optimal algorithm for TAUT. To obtain this result we have to derive a space version of a theorem of Levin on optimal inverters. I . INTRODUCTIONIt is well-known that for standard complexity classes C (as L, NL and P) the existence of a logic capturing C is equivalent to the existence of a listing (or effective enumeration) of the classes of structures closed under isomorphism in C by means of Turing machines of type C that decide them (or equivalently, to such a listing of the classes of graphs closed under isomorphism). Recently [4] we have shown for C = P that such a listing exists if there is a listing of the P-subsets of the set TAUT of propositional tautologies; more explicitly, a listing of the subsets in P of TAUT by means of polynomial time Turing machines deciding them. Even more, it is shown in [4] that the following two statements are equivalent:(i) There is a listing of the P-subsets of TAUT.(ii) The logic LFP inv , the "order-invariant least fixed-point logic LFP," 1 captures P. The starting point for the investigations which led to this paper was the observation (cf. Proposition II.3) that one gets a listing of the P-subsets of TAUT if one assumes that there is a listing of its L-subsets. In particular, then LFP inv captures P. Thus, we asked ourselves whether then we even get that DTC inv , the "order-invariant deterministic transitive closure logic DTC," 1 captures L.At the end we realized that the equivalence of (i) and (ii) lifts to their L-analogues and to their NL-analogues. In 1 It is well-known that LFP captures P on ordered structures and that deterministic transitive closure logic DTC captures L on ordered structures. particular, LFP inv captures P if DTC inv captures L. Note that it is not known whether the existence of a logic capturing P is implied by the existence of a logic capturing L.A more general notion of listing turned out to be helpful. For complexity classes C and C we consider listings of the C-subsets of TAUT by means of Turing machines of type C ; we write LIST(C, TAUT, C ) if such a listing exists. For the classes P and NP such listings were already considered and put to good use by Sadowski in [16]. This more general notion is also meaningful in the context of logics (for P and NP already remarked in [3]); loosely speaking, if LIST(C, TAUT, C ), then in the order-invariant logic corresponding to C we can axiomatize the classe...

Hard Instances of Algorithms and Proof Systems

“…We show for arbitrary Q that the existence of hard sets for all algorithms is equivalent to the existence of an effective enumeration of all polynomial time decidable subsets of Q, a property which has turned out to be useful in various contexts (cf. [12,2,3]). We analyze what Messner's result means for proof systems.…”