16 research outputs found
Quasipolynomial Hitting Sets for Circuits with Restricted Parse Trees
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
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
such that 1. can be computed by an abecedarian ABP of size O(nd);
2. any abecedarian formula computing must have size that is
super-polynomial in n.
We also show that a super-polynomial lower bound against abecedarian formulas
for would separate the powers of formulas and ABPs in the
non-commutative setting
Classifying Problems into Complexity Classes
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 âx â A? â 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
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 â 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
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum