16 research outputs found

    Quasipolynomial Hitting Sets for Circuits with Restricted Parse Trees

    Get PDF
    We study the class of non-commutative Unambiguous circuits or Unique-Parse-Tree (UPT) circuits, and a related model of Few-Parse-Trees (FewPT) circuits (which were recently introduced by Lagarde, Malod and Perifel [Guillaume Lagarde et al., 2016] and Lagarde, Limaye and Srinivasan [Guillaume Lagarde et al., 2017]) and give the following constructions: - An explicit hitting set of quasipolynomial size for UPT circuits, - An explicit hitting set of quasipolynomial size for FewPT circuits (circuits with constantly many parse tree shapes), - An explicit hitting set of polynomial size for UPT circuits (of known parse tree shape), when a parameter of preimage-width is bounded by a constant. The above three results are extensions of the results of [Manindra Agrawal et al., 2015], [Rohit Gurjar et al., 2015] and [Rohit Gurjar et al., 2016] to the setting of UPT circuits, and hence also generalize their results in the commutative world from read-once oblivious algebraic branching programs (ROABPs) to UPT-set-multilinear circuits. The main idea is to study shufflings of non-commutative polynomials, which can then be used to prove suitable depth reduction results for UPT circuits and thereby allow a careful translation of the ideas in [Manindra Agrawal et al., 2015], [Rohit Gurjar et al., 2015] and [Rohit Gurjar et al., 2016]

    Separating ABPs and Some Structured Formulas in the Non-Commutative Setting

    Get PDF
    The motivating question for this work is a long standing open problem, posed by Nisan (1991), regarding the relative powers of algebraic branching programs (ABPs) and formulas in the non-commutative setting. Even though the general question continues to remain open, we make some progress towards its resolution. To that effect, we generalise the notion of ordered polynomials in the non-commutative setting (defined by \Hrubes, Wigderson and Yehudayoff (2011)) to define abecedarian polynomials and models that naturally compute them. Our main contribution is a possible new approach towards separating formulas and ABPs in the non-commutative setting, via lower bounds against abecedarian formulas. In particular, we show the following. There is an explicit n-variate degree d abecedarian polynomial fn,d(x)f_{n,d}(x) such that 1. fn,d(x)f_{n, d}(x) can be computed by an abecedarian ABP of size O(nd); 2. any abecedarian formula computing fn,log⁥n(x)f_{n, \log n}(x) must have size that is super-polynomial in n. We also show that a super-polynomial lower bound against abecedarian formulas for flog⁥n,n(x)f_{\log n, n}(x) would separate the powers of formulas and ABPs in the non-commutative setting

    Classifying Problems into Complexity Classes

    Full text link
    A fundamental problem in computer science is, stated informally: Given a problem, how hard is it?. We measure hardness by looking at the following question: Given a set A whats is the fastest algorithm to determine if “x ∈ A? ” We measure the speed of an algorithm by how long it takes to run on inputs of length n, as a function of n. For example, sorting a list of length n can be done in roughly n log n steps. Obtaining a fast algorithm is only half of the problem. Can you prove that there is no better algorithm? This is notoriously difficult; however, we can classify problems into complexity classes where those in the same class are roughly equally hard. In this chapter we define many complexity classes and describing natural problems that are in them. Our classes go all the way from regular languages to various shades of undecidable. We then summarize all that is known about these classes.

    Tools and Algorithms for the Construction and Analysis of Systems

    Get PDF
    This open access two-volume set constitutes the proceedings of the 27th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS 2021, which was held during March 27 – April 1, 2021, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2021. The conference was planned to take place in Luxembourg and changed to an online format due to the COVID-19 pandemic. The total of 41 full papers presented in the proceedings was carefully reviewed and selected from 141 submissions. The volume also contains 7 tool papers; 6 Tool Demo papers, 9 SV-Comp Competition Papers. The papers are organized in topical sections as follows: Part I: Game Theory; SMT Verification; Probabilities; Timed Systems; Neural Networks; Analysis of Network Communication. Part II: Verification Techniques (not SMT); Case Studies; Proof Generation/Validation; Tool Papers; Tool Demo Papers; SV-Comp Tool Competition Papers

    35th Symposium on Theoretical Aspects of Computer Science: STACS 2018, February 28-March 3, 2018, Caen, France

    Get PDF

    LIPIcs, Volume 274, ESA 2023, Complete Volume

    Get PDF
    LIPIcs, Volume 274, ESA 2023, Complete Volum

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

    Get PDF
    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

    Get PDF
    LIPIcs, Volume 261, ICALP 2023, Complete Volum
    corecore