The polynomial-time hierarchy

AbstractThe polynomial-time hierarchy is that subrecursive analog of the Kleene arithmetical hierarchy in which deterministic (nondeterministic) polynomial time plays the role of recursive (recursively enumerable) time. Known properties of the polynomial-time hierarchy are summarized. A word problem which is complete in the second stage of the hierarchy is exhibited. In the analogy between the polynomial-time hierarchy and the arithmetical hierarchy, the first order theory of equality plays the role of elementary arithmetic (as the ω-jump of the hierarchy). The problem of deciding validity in the theory of equality is shown to be complete in polynomial-space, and close upper and lower bounds on the space complexity of this problem are established

Independence-friendly logic without Henkin quantification

We analyze the expressive resources of IF logic that do not stem from Henkin (partially-ordered) quantification. When one restricts attention to regular IF sentences, this amounts to the study of the fragment of IF logic which is individuated by the game-theoretical property of action recall (AR). We prove that the fragment of prenex AR sentences can express all existential second-order properties. We then show that the same can be achieved in the non-prenex fragment of AR, by using “signalling by disjunction” instead of Henkin or signalling patterns. We also study irregular IF logic (in which requantification of variables is allowed) and analyze its correspondence to regular IF logic. By using new methods, we prove that the game-theoretical property of knowledge memory is a first-order syntactical constraint also for irregular sentences, and we identify another new first-order fragment. Finally we discover that irregular prefixes behave quite differently in finite and infinite models. In particular, we show that, over infinite structures, every irregular prefix is equivalent to a regular one; and we present an irregular prefix which is second order on finite models but collapses to a first-order prefix on infinite models.Peer reviewe

Mathematical Logic: Proof Theory, Constructive Mathematics

[no abstract available

Positive Versions of Polynomial Time

AbstractWe show that restricting a number of characterizations of the complexity classPto be positive (in natural ways) results in the same class of (monotone) problems, which we denote byposP. By a well-known result of Razborov,posPis a proper subclass of the class of monotone problems inP. We exhibit complete problems forposPvia weak logical reductions, as we do for other logically defined classes of problems. Our work is a continuation of research undertaken by Grigni and Sipser, and subsequently Stewart; indeed, we introduce the notion of a positive deterministic Turing machine and consequently solve a problem posed by Grigni and Sipser

Expressiveness and complexity of graph logic

We investigate the complexity and expressive power of the spatial logic for querying graphs introduced by Cardelli, Gardner and Ghelli (ICALP 2002).We show that the model-checking complexity of versions of this logic with and without recursion is PSPACE-complete. In terms of expressive power, the version without recursion is a fragment of the monadic second-order logic of graphs and we show that it can express complete problems at every level of the polynomial hierarchy. We also show that it can define all regular languages, when interpretation is restricted to strings. The expressive power of the logic with recursion is much greater as it can express properties that are PSPACE-complete and therefore unlikely to be definable in second-order logic

Regular Representations of Uniform TC^0

The circuit complexity class DLOGTIME-uniform AC^0 is known to be a modest subclass of DLOGTIME-uniform TC^0. The weakness of AC^0 is caused by the fact that AC^0 is not closed under restricting AC^0-computable queries into simple subsequences of the input. Analogously, in descriptive complexity, the logics corresponding to DLOGTIME-uniform AC^0 do not have the relativization property and hence they are not regular. This weakness of DLOGTIME-uniform AC^0 has been elaborated in the line of research on the Crane Beach Conjecture. The conjecture (which was refuted by Barrington, Immerman, Lautemann, Schweikardt and Th{\'e}rien) was that if a language L has a neutral letter, then L can be defined in first-order logic with the collection of all numerical built-in relations, if and only if L can be already defined in FO with order. In the first part of this article we consider logics in the range of AC^0 and TC^0. First we formulate a combinatorial criterion for a cardinality quantifier C_S implying that all languages in DLOGTIME-uniform TC^0 can be defined in FO(C_S). For instance, this criterion is satisfied by C_S if S is the range of some polynomial with positive integer coefficients of degree at least two. In the second part of the paper we first adapt the key properties of abstract logics to accommodate built-in relations. Then we define the regular interior R-int(L) and regular closure R-cl(L), of a logic L, and show that the Crane Beach Conjecture can be interpreted as a statement concerning the regular interior of first-order logic with built-in relations B. We show that if B={+}, or B contains only unary relations besides the order, then R-int(FO_B) collapses to FO with order. In contrast, our results imply that if B contains the order and the range of a polynomial of degree at least two, then R-cl(FO_B) includes all languages in DLOGTIME-uniform TC^0