IA Scholar Query: The Boolean Hierarchy over Level 1/2 of the Straubing-Therien Hierarchy
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Internet Archive Scholar query results feedeninfo@archive.orgMon, 03 Oct 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440A generic polynomial time approach to separation by first-order logic without quantifier alternation
https://scholar.archive.org/work/uks65dtuxng3rhtr24ckozdilu
We look at classes of languages associated to the fragment of first-order logic BΣ1 which disallows quantifier alternations. Each class is defined by choosing the set of predicates on positions that may be used. Two key such fragments are those equipped with the linear ordering and possibly the successor relation. It is known that these two variants have decidable membership: "does an input regular language belong to the class ?". We rely on a characterization of BΣ1 by the operator BPol: given an input class C, it outputs a class BPol(C) that corresponds to a variant of BΣ1 equipped with special predicates associated to C. We extend these results in two orthogonal directions. First, we use two kinds of inputs: classes G of group languages (i.e., recognized by a DFA in which each letter induces a permutation of the states) and extensions thereof, written G+. The classes BPol(G) and BPol(G+) capture many variants of BΣ1 which use predicates such as the linear ordering, the successor, the modular predicates or the alphabetic modular predicates. Second, instead of membership, we explore the more general separation problem: decide if two regular languages can be separated by a language from the class under study. We show it is decidable for BPol(G) and BPol(G+) when this is the case for G. This was known for BPol(G) and for two particular classes BPol(G+). Yet, the algorithms were indirect and relied on involved frameworks, yielding poor upper complexity bounds. Our approach is direct. We work with elementary concepts (mainly, finite automata). Our main contribution consists in polynomial time Turing reductions from both BPol(G)- and BPol(G+)-separation to G-separation. This yields polynomial algorithms for key variants of BΣ1, including those equipped with the linear ordering and possibly the successor and/or the modular predicates.Thomas Place, Marc Zeitounwork_uks65dtuxng3rhtr24ckozdiluMon, 03 Oct 2022 00:00:00 GMTAll about unambiguous polynomial closure
https://scholar.archive.org/work/umdpibpkrvddpeculi22og5kre
We investigate a standard operator on classes of languages: unambiguous polynomial closure. We prove that for every class C of regular languages satisfying mild properties, the membership problem for its unambiguous polynomial closure UPol(C) reduces to the same problem for C. We also show that unambiguous polynomial closure coincides with alternating left and right deterministic closure. Moreover, we prove that if additionally C is finite, the separation and covering problems are decidable for UPol(C). Finally, we present an overview of the generic logical characterizations of the classes built using unambiguous polynomial closure.Thomas Place, Marc Zeitounwork_umdpibpkrvddpeculi22og5kreSun, 14 Aug 2022 00:00:00 GMTThe Regular Languages of First-Order Logic with One Alternation
https://scholar.archive.org/work/u7fttq35lnhirbjja2yravt4dm
The regular languages with a neutral letter expressible in firstorder logic with one alternation are characterized. Specifically, it is shown that if an arbitrary Σ 2 formula defines a regular language with a neutral letter, then there is an equivalent Σ 2 formula that only uses the order predicate. This shows that the so-called Central Conjecture of Straubing holds for Σ 2 over languages with a neutral letter, the first progress on the Conjecture in more than 20 years. To show the characterization, lower bounds against polynomial-size depth-3 Boolean circuits with constant top fan-in are developed. The heart of the combinatorial argument resides in studying how positions within a language are determined from one another, a technique of independent interest.Corentin Barloy, Michael Cadilhac, Charles Paperman, Thomas Zeumework_u7fttq35lnhirbjja2yravt4dmTue, 02 Aug 2022 00:00:00 GMTThe Regular Languages of First-Order Logic with One Alternation
https://scholar.archive.org/work/nfhru457l5exhacq7yesu2b66e
The regular languages with a neutral letter expressible in first-order logic with one alternation are characterized. Specifically, it is shown that if an arbitrary Σ_2 formula defines a regular language with a neutral letter, then there is an equivalent Σ_2 formula that only uses the order predicate. This shows that the so-called Central Conjecture of Straubing holds for Σ_2 over languages with a neutral letter, the first progress on the Conjecture in more than 20 years. To show the characterization, lower bounds against polynomial-size depth-3 Boolean circuits with constant top fan-in are developed. The heart of the combinatorial argument resides in studying how positions within a language are determined from one another, a technique of independent interest.Corentin Barloy and Michaël Cadilhac and Charles Paperman and Thomas Zeumework_nfhru457l5exhacq7yesu2b66eFri, 11 Mar 2022 00:00:00 GMTMembership and separation problems inside two-variable first order logic
https://scholar.archive.org/work/cv3jfs6mdncsrloghnfugdzsbm
This thesis studies the expressive power of restricted fragments of first order logic on words with the order predicate. In particular, we consider particular instances of the following two questions.Given a regular language L, and a fragment of first order logic, can L be described by formulae in this fragment?Given two regular languages L1 and L2, and a fragment of first order logic, can we find a language L which can be described by formulae in this fragment such that L1 is a subset of L and the intersection of L2 and L is empty.The former is known as the membership problem, and the latter the separation problem. These problems are well studied in the theory of regular languages, and their solution has often relied on tools from algebra, such as finite monoids and pointlikes.We consider formulae using only two variables, a fragment known as FO2[For languages of finite words, membership for these fragments has already been solved. We generalise these results in multiple directions.First, we solve the membership problem for the aforementioned fragments for inf-regular languages, i.e. languages consisting of both infinitely and finitely long words. This in particular solves the membership problem for omega-regular languages, i.e. those containing only infinitely long words. Our main tool is to impose algebraic conditions on the syntactic monoid: a monoid which can be computed from any major presentation of regular languages. For the lower levels, we also need to consider topological properties of the languages.The above shows that membership for inf- and omega-regular languages is decidable. However, given a language presented by an automata, calculating the syntactic monoid is generally not efficient. Our second main contribution is therefore an efficient way to decide membership for languages presented by deterministic finite automata (for regular languages) and Carton--Michel automata (for omega-regular languages). We give forbidden patterns for these fragments; that is, we specify [...]Viktor Henrikssonwork_cv3jfs6mdncsrloghnfugdzsbmThu, 03 Feb 2022 00:00:00 GMTSmaller ACC0 Circuits for Symmetric Functions
https://scholar.archive.org/work/dgstpfee7ve7peos3dxpbkyqua
What is the power of constant-depth circuits with MOD_m gates, that can count modulo m? Can they efficiently compute MAJORITY and other symmetric functions? When m is a constant prime power, the answer is well understood. In this regime, Razborov and Smolensky proved in the 1980s that MAJORITY and MOD_m require super-polynomial-size MOD_q circuits, where q is any prime power not dividing m. However, relatively little is known about the power of MOD_m gates when m is not a prime power. For example, it is still open whether every problem decidable in exponential time can be computed by depth-3 circuits of polynomial-size and only MOD_6 gates. In this paper, we shed some light on the difficulty of proving lower bounds for MOD_m circuits, by giving new upper bounds. We show how to construct MOD_m circuits computing symmetric functions with non-prime power m, with size-depth tradeoffs that beat the longstanding lower bounds for AC^0[m] circuits when m is a prime power. Furthermore, we observe that our size-depth tradeoff circuits have essentially optimal dependence on m and d in the exponent, under a natural circuit complexity hypothesis. For example, we show that for every ε > 0, every symmetric function can be computed using MOD_m circuits of depth 3 and 2^{n^ε} size, for a constant m depending only on ε > 0. In other words, depth-3 CC^0 circuits can compute any symmetric function in subexponential size. This demonstrates a significant difference in the power of depth-3 CC^0 circuits, compared to other models: for certain symmetric functions, depth-3 AC^0 circuits require 2^{Ω(√n)} size [Håstad 1986], and depth-3 AC^0[p^k] circuits (for fixed prime power p^k) require 2^{Ω(n^{1/6})} size [Smolensky 1987]. Even for depth-2 MOD_p ∘ MOD_m circuits, 2^{Ω(n)} lower bounds were known [Barrington Straubing Thérien 1990].Brynmor Chapman, R. Ryan Williams, Mark Bravermanwork_dgstpfee7ve7peos3dxpbkyquaTue, 25 Jan 2022 00:00:00 GMTCharacterizing level one in group-based concatenation hierarchies
https://scholar.archive.org/work/kelwluwwbzg2dludjahrtnvdyi
We investigate two operators on classes of regular languages: polynomial closure (Pol) and Boolean closure (Bool). We apply these operators to classes of group languages G and to their well-suited extensions G+, which is the least Boolean algebra containing G and the singleton language containing the empty word. This yields the classes Bool(Pol(G)) and Bool(Pol(G+)). These classes form the first level in important classifications of classes of regular languages, called concatenation hierarchies, which admit natural logical characterizations. We present generic algebraic characterizations of these classes. They imply that one may decide whether a regular language belongs to such a class, provided that a more general problem called separation is decidable for the input class G. The proofs are constructive and rely exclusively on notions from language and automata theory.Thomas Place, Marc Zeitounwork_kelwluwwbzg2dludjahrtnvdyiTue, 18 Jan 2022 00:00:00 GMTOn the Complexity of Algebraic Numbers, and the Bit-Complexity of Straight-Line Programs
https://scholar.archive.org/work/s4hbrcbtyrarxgfvp3o7v3ngny
We investigate the complexity of languages that correspond to algebraic real numbers, and we present improved upper bounds on the complexity of these languages. Our key technical contribution is the presentation of improved uniform TC 0 circuits for division, matrix powering, and related problems, where the improvement is in terms of "majority depth" (initially studied by Maciel and Thérien). As a corollary, we obtain improved bounds on the complexity of certain problems involving arithmetic circuits, which are known to lie in the counting hierarchy, and we answer a question posed by Yap.Eric Allender, Nikhil Balaji, Samir Datta, Rameshwar Pratapwork_s4hbrcbtyrarxgfvp3o7v3ngnyEvaluation and Enumeration of Regular Simple Path and Trail Queries
https://scholar.archive.org/work/s65qs34zhrg45fns44745oshpi
Regular path queries (RPQs) are an essential component of graph query languages. Such queries consider a regular expression r and a directed edge-labeled graph G and search for paths in G for which the sequence of labels is in the language of r. In order to avoid having to consider infinitely many paths, some database engines restrict such paths to paths without repeated nodes or edges which are called simple paths or trails, respectively. Whereas arbitrary paths can be dealt with efficiently, simple paths and trails become computationally difficult already for very small RPQs. In this dissertation we investigate decision and enumeration problems concerning simple path and trail semantics. Evaluation Problem on Directed Graphs: Bagan, Bonifati, and Groz gave a trichotomy for the evaluation problem for simple paths when the RPQ is fixed. We complement their work by giving a similar trichotomy for the evaluation problem for trails and studying various characteristics of this class. We also study RPQs used in query logs and define a class of simple transitive expressions that is prominent in practice and for which we can prove dichotomies for the evaluation problem when the input language is not fixed, but used as a parameter. We observe that, even though simple path and trail semantics are intractable for RPQs in general, they are feasible for the vast majority of RPQs that are used in practice. At the heart of this study is a result of independent interest: the two disjoint paths problem in directed graphs is W[1]-hard if parameterized by the length of one of the two paths. Evaluation Problem on Undirected Graphs: While graph databases focus on directed graphs, there are edges which are naturally bidirectional, such as "sibling" or "married". Furthermore, database systems often allow to navigate an edge in its inverse direction (2RPQ), thus the study of the undirected setting gives us a better idea of what is possible. We are able to identify several tractable and intractable subclasses of regular languages when t [...]Tina Poppwork_s65qs34zhrg45fns44745oshpi