Expressibility in Σ11

2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), 2021

We study multi-structural games, played on two sets ${\mathcal{A}}$ and ${\mathcal{B}}$ of structures. These games generalize Ehrenfeucht-Fraïssé games. Whereas Ehrenfeucht-Fraïssé games capture the quantifier rank of a first-order sentence, multi-structural games capture the number of quantifiers, in the sense that Spoiler wins the r-round game if and only if there is a first-order sentence ϕ with at most r quantifiers, where every structure in ${\mathcal{A}}$ satisfies ϕ and no structure in ${\mathcal{B}}$ satisfies ϕ. We use these games to give a complete characterization of the number of quantifiers required to distinguish linear orders of different sizes, and develop machinery for analyzing structures beyond linear orders.

Citeseer

The evaluation of a logical formula can be viewed as a game played by two opponents, one trying to show that the formula is true and the other trying to prove it false. This correspondence is exploited algorithmically to evaluate formulas of first and second-order logic on finite structures. We extend the game-based algorithmic approach to first-order logic on infinite structures that arise in computer science. Such structures are stored and manipulated by a computer, thus elements and relations must be represented in a finite way. We study a prominent class of finitely presentable structures, namely automatic structures. Automatic structures consist of elements represented by words over a finite alphabet. Relations within these structures are represented by synchronous automata that perform step-by-step transitions on tuples of symbols from the alphabet. A prominent example of an automatic structure is Presburger arithmetic, for which the natural way of writing numbers as sequences of digits and the standard column addition method constitute an automatic presentation. An important property of automatic structures is that first-order logic is decidable on these structures. To develop the correspondence between games and logic on automatic structures, we first look for suitable extensions of first-order logic that remain decidable. We study the notion of game quantification and extend the notions of open and closed game quantifiers, introduced in the model theory of infinitary logics, to a regular game quantifier. We show that this game quantifier preserves regularity on automatic presentations and investigate its expressive power. We identify the classes of structures on which first-order logic extended with this quantifier collapses to pure first-order logic. For better understanding of the classes where this is not the case, we introduce the notion of inductive automorphisms and show that they preserve relations defined using the game quantifier. Model-checking games for the above extension of first-order logic can be defined in a more natural way than for pure first-order logic. Towards this, we extend the classical two-player parity games to a multiplayer setting where two coalitions play against each other with a particular kind of hierarchical imperfect information about actions of the players. In contrast to the classical case, in hierarchical games it is essential that players alternate. We show that otherwise the problem which coalition wins a hierarchical game is undecidable. Nevertheless, there is an important aspect in which they are similar to classical games: hierarchical games are robust with respect to changes of the representation of winning condition. One reason for this robustness of hierarchical games is that only a finite memory is needed to reduce games with complex winning conditions to games with simpler ones. While this technique has been well-known for games with finitely many priorities, our approach is the first to extend it to games with infinitely many priorities on infinite arenas. We generalize the classical notion of latest appearance record to a new memory structure, which we show to be sufficient for winning Muller games with a finite or co-finite number of sets in the Muller condition, and additionally for a few other classes of games with infinitely many priorities. Certain classes of winning conditions on infinitely many priorities admit descriptions in terms of generalized Zielonka trees. We investigate such conditions and show that the correspondence between the number of branches of a Zielonka tree and the memory needed for strategies generalizes to the case of infinitely many priorities. A basic way of extending first-order logic is by adding generalized Lindström quantifiers. We address the following question: which generalized unary quantifiers can be added to first-order logic without introducing non-regular relations on automatic structures. We answer this question by giving a complete characterization of such quantifiers in terms of definability using cardinality and modulo counting quantifiers. We show that these quantifiers indeed preserve regularity on all automatic structures, including the non-injective omega-automatic ones. As a corollary we answer a question of Blumensath and prove that all countable omega-automatic structures are in fact finite-word automatic. Further, we study cardinality quantifiers on a large class of generalized-automatic structures. We use the composition method for monadic second-order logic to show that on such structures cardinality quantifiers collapse to first-order logic.

2005

Annals of Pure and Applied Logic, 1993

J. and J. Valnanen, Game-theoretic inductive definability, Annals of Pure and Applied Logic 65 (1993) 265-306.

Journal of Computer and System Sciences, 2005

A language L over an alphabet A is said to have a neutral letter if there is a letter e ∈ A such that inserting or deleting e's from any word in A * does not change its membership or non-membership in L. The presence of a neutral letter affects the definability of a language in firstorder logic. It was conjectured that it renders all numerical predicates apart from the order predicate useless, i.e., that if a language L with a neutral letter is not definable in first-order logic with linear order, then it is not definable in first-order logic with any set N of numerical predicates. Named after the location of its first, flawed, proof this conjecture is called the Crane Beach conjecture (CBC, for short). The CBC is closely related to uniformity conditions in circuit complexity theory and to collapse results in database theory. We investigate the CBC in detail, showing that it fails for N = {+, ×}, or, possibly stronger, for any set N that allows counting up to the m times iterated logarithm, for any constant m. On the positive side, we prove the conjecture for the case of all monadic numerical predicates, for the addition predicate +, for the fragment BC(Σ 1) of first-order logic, for regular languages, and for languages over a binary alphabet. We explain the precise relation between the CBC and so-called natural-generic collapse results in database theory. Furthermore, we introduce a framework that gives better understanding of what exactly may cause a failure of the conjecture.

Theoretical Computer Science, 2004

In this paper, we introduce and study some syntactical fragments of monadic second-order and ÿrst-order (PEANO) arithmetic which we will prove the connection to famous complexity classes. Starting from descriptive complexity results, and giving an e ective method for translating formulas between di erent logical structures representing encodings of integers, we give some new arithmetical characterizations of NP, PH, NL, and P.

PhD thesis, 2007

Descriptive complexity theory is a branch of complexity theory that views the hardness of a problem in terms of the complexity of expressing it in some logical formalism; among the resources considered are the number of object variables, quantifier depth, type, and alternation, sentences length (finite/infinite), etc. In this field we have studied two problems: (i) expressibility in ∃SO and (ii) the descriptive complexity of finite abelian groups. Inspired by Fagin’s result that NP = ∃SO, we have developed a partial framework to investigate expressibility inside ∃SO so as to have a finer look into NP. The framework uses combinatorics derived from second-order Ehrenfeucht-Fraısse games and the notion of game types. Among the results obtained is that for any k, divisibility by k is not expressible by an ∃SO sentence where (1) each second-order variable has arity at most 2, (2) the first-order part has at most 2 first-order variables, and (3) the first-order part has quantifier depth at most 3. In the second project we have investigated the descriptive complexity of finite abelian groups. Using Ehrenfeucht-Fraısse games we find upper and lower bounds on quantifier depth, quantifier alternations, and number of variables of a first-order sentence that distinguishes two finite abelian groups. Our main results are the following. Let G1 and G2 be a pair of non-isomorphic finite abelian groups, and let m be a number that divides one of the two groups’ orders. Then the following hold: (1) there exists a first-order sentence φ that distinguishes G1 and G2 such that φ is existential, has quantifier depth O(log m), and has at most 5 variables and (2) if φ is a sentence that distinguishes G1 and G2 then φ must have quantifier depth Ω(log m). In infinitary model theory we have studied abstract elementary classes. We have defined Galois types over arbitrary subsets of the monster (large enough homogeneous model), have defined a simple notion of splitting, and have proved some properties of this notion such as invariance under isomorphism, monotonicity, reflexivity, existence of non-splitting extensions.

DIMACS Series in Discrete Mathematics and Theoretical Computer Science

The classes W [t] of the Downey-Fellows W hierarchy are defined, for each t, by fixed-parameter reductions to the weighted-assignment satisfiability problem for weft-t circuits. This paper proves that for each t ≥ 1, W [t] equals the closure under fixed-parameter reductions of the class of languages L definable by formulas of the form φ = (∃U)ψ, where U is a set variable and ψ is a first-order formula in t prenex form. This is a fixed-parameter analogue of Fagin's well-known characterization of NP by second-order existential formulas. An equivalent form of this result states that the fixed-parameter "slices" L k of L are definable by a family { φ k } of first-order formulas in t prenex form, subject to the restriction that the quantifier blocks in φ k after the leading existential block are independent of k. Whether this restriction can be removed is connected to open problems in other recent papers on the W hierarchy.

Proceedings of the 41st annual ACM symposium on Symposium on theory of computing - STOC '09, 2009

The well known dichotomy conjecture of Feder and Vardi states that for every family Γ of constraints CSP(Γ) is either polynomially solvable or NP-hard. Bulatov and Jeavons reformulated this conjecture in terms of the properties of the algebra P ol(Γ), where the latter is the collection of those m-ary operations (m = 1, 2, . . .) that keep all constraints in Γ invariant. We show that the algebraic condition boils down to whether there are arbitrarily resilient functions in P ol(Γ). Equivalently, we can express this in the terms of the PCP theory: CSP(Γ) is NP-hard iff all long code tests created from Γ that passes with zero error admits only juntas 1 . Then, using this characterization and a result of Dinur, Friedgut and Regev, we give an entirely new and transparent proof to the Hell-Nešetřil theorem, which states that for a simple connected undirected graph H,

Studia Logica, 1981