Descriptive complexity theory aims at inferring a problem's computational complexity from the syntactic complexity of its description. A cornerstone of this theory is Fagin's Theorem, by which a graph property is expressible in existential second-order logic (ESO logic) if, and only if, it is in NP. A natural question, from the theory's point of view, is which syntactic fragments of ESO logic also still characterize NP. Research on this question has culminated in a dichotomy result by Gottlob, Kolatis, and Schwentick: for each possible quantifier prefix of an ESO formula, the resulting prefix class either contains an NP-complete problem or is contained in P. However, the exact complexity of the prefix classes inside P remained elusive. In the present paper, we clear up the picture by showing that for each prefix class of ESO logic, its reduction closure under first-order reductions is either FO, L, NL, or NP. For undirected, self-loop-free graphs two containment results are especially challenging to prove: containment in L for the prefix @math and containment in FO for the prefix @math for monadic @math . The complex argument by Gottlob, Kolatis, and Schwentick concerning polynomial time needs to be carefully reexamined and either combined with the logspace version of Courcelle's Theorem or directly improved to first-order computations. A different challenge is posed by formulas with the prefix @math : We show that they express special constraint satisfaction problems that lie in L.
Concerning syntactic fragments of logic, the two papers most closely related to the present paper are @cite_12 by Eiter, Gottlob, and Gurevich and @cite_0 by Gottlob, Kolaitis, and Schwentick. In the first paper, a similar kind of classification is presented as in the present paper, only over rather than . It is shown there that for all prefix patterns @math the class @math is either equal to @math ; is not equal to @math but contains an @math -complete problem; is equal to @math ; or is a subclass of @math . Interestingly, two classes of special interest are @math and @math , both of which are the minimal classes equal to @math (by the results of Büchi @cite_7 ). In comparison, by the results of the present paper @math , while @math , and @math , while @math .
@cite_0 @cite_7 @cite_12
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Abstract Given two infinite binary sequences A , B we say that B can compress at least as well as A if the prefix-free Kolmogorov complexity relative to B of any binary string is at most as much as the prefix-free Kolmogorov complexity relative to A , modulo a constant. This relation, introduced in Nies (2005) [14] and denoted by A ≤ L K B , is a measure of relative compressing power of oracles, in the same way that Turing reducibility is a measure of relative information. The equivalence classes induced by ≤ L K are called L K degrees (or degrees of compressibility) and there is a least degree containing the oracles which can only compress as much as a computable oracle, also called the ‘low for K ’ sets. A well-known result from Nies (2005) [14] states that these coincide with the K-trivial sets, which are the ones whose initial segments have minimal prefix-free Kolmogorov complexity. We show that with respect to ≤ L K , given any non-trivial Δ 2 0 sets X , Y there is a computably enumerable set A which is not K-trivial and it is below X , Y . This shows that the local structures of Σ 1 0 and Δ 2 0 Turing degrees are not elementarily equivalent to the corresponding local structures in the LK degrees. It also shows that there is no pair of sets computable from the halting problem which forms a minimal pair in the LK degrees; this is sharp in terms of the jump, as it is known that there are sets computable from 0 ″ which form a minimal pair in the LK degrees. We also show that the structure of LK degrees below the LK degree of the halting problem is not elementarily equivalent to the Δ 2 0 or Σ 1 0 structures of LK degrees. The proofs introduce a new technique of permitting below a Δ 2 0 set that is not K -trivial, which is likely to have wider applications. Fagin's theorem, the first important result of descriptive complexity, asserts that a property of graphs is in NP if and only if it is definable by an existential second-order formula. In this article, we study the complexity of evaluating existential second-order formulas that belong to prefix classses of existential second-order logic, where a prefix class is the collection of all existential second-order formulas in prenex normal form such that the second-order and the first-order quantifiers obey a certain quantifier pattern. We completely characterize the computational complexity of prefix classes of existential second-order logic in three different contexts: (1) over directed graphs, (2) over undirected graphs with self-loops and (3) over undirected graphs without self-loops. Our main result is that in each of these three contexts a dichotomy holds, that is to say, each prefix class of existential second-order logic either contains sentences that can express NP-complete problems, or each of its sentences expresses a polynomial-time solvable problem. Although the boundary of the dichotomy coincides for the first two cases, it changes, as one moves to undirected graphs without self-loops. The key difference is that a certain prefix class, based on the well-known Ackermann class of first-order logic, contains sentences that can express NP-complete problems over graphs of the first two types, but becomes tractable over undirected graphs without self-loops. Moreover, establishing the dichotomy over undirected graphs without self-loops turns out to be a technically challenging problem that requires the use of sophisticated machinery from graph theory and combinatorics, including results about graphs of bounded tree-width and Ramsey's theorem. In this paper we compare the expressive power of elementary representation formats for vague, incomplete or conflicting information. These include Boolean valuation pairs introduced by Lawry and Gonzalez-Rodriguez, orthopairs of sets of variables, Boolean possibility and necessity measures, three-valued valuations, supervaluations. We make explicit their connections with strong Kleene logic and with Belnap logic of conflicting information. The formal similarities between 3-valued approaches to vagueness and formalisms that handle incomplete information often lead to a confusion between degrees of truth and degrees of uncertainty. Yet there are important differences that appear at the interpretive level: while truth-functional logics of vagueness are accepted by a part of the scientific community (even if questioned by supervaluationists), the truth-functionality assumption of three-valued calculi for handling incomplete information looks questionable, compared to the non-truth-functional approaches based on Boolean possibility-necessity pairs. This paper aims to clarify the similarities and differences between the two situations. We also study to what extent operations for comparing and merging information items in the form of orthopairs can be expressed by means of operations on valuation pairs, three-valued valuations and underlying possibility distributions. We explore the connections between several representations of imperfect information.In each case we compare the expressive power of these formalisms.In each case we study how to express aggregation operations.We demonstrate the formal similarities among these approaches.We point out the differences in interpretations between these approaches. The following paper is reproduced from a Russian journal of the character of our own Proceedings of the National Academy of Sciences, Comptes Rendus (Doklady) de I'Academie des Sciences de I'URSS, 1942, Volume XXXVII, No. 7-8. The author is one of the most distinguished of Russian mathematicians. He has made very important contributions in pure mathematics in the theory of functional analysis, and has made equally important contributions to applied mathematics in numerical analysis and the theory and practice of computation. Although his exposition in this paper is quite terse and couched in mathematical language which may be difficult for some readers of Management Science to follow, it is thought that this presentation will: (1) make available to American readers generally an important work in the field of linear programming, (2) provide an indication of the type of analytic work which has been done and is being done in connection with rational planning in Russia, (3) through the specific examples mentioned indicate the types of interpretation which the Russians have made of the abstract mathematics (for example, the potential and field interpretations adduced in this country recently by W. Prager were anticipated in this paper). It is to be noted, however, that the problem of determining an effective method of actually acquiring the solution to a specific problem is not solved in this paper. In the category of development of such methods we seem to be, currently, ahead of the Russians.--A. Charnes, Northwestern Technological Institute and The Transportation Center.