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Electronic Colloquium on Computational Complexity

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Reports tagged with quantified Boolean formulas:
TR08-076 | 17th June 2008
Ryan Williams

Non-Linear Time Lower Bound for (Succinct) Quantified Boolean Formulas

We prove a model-independent non-linear time lower bound for a slight generalization of the quantified Boolean formula problem (QBF). In particular, we give a reduction from arbitrary languages in alternating time t(n) to QBFs describable in O(t(n)) bits by a reasonable (polynomially) succinct encoding. The reduction works for many reasonable ... more >>>

TR13-097 | 25th June 2013
Mikolas Janota, Joao Marques-Silva

On Propositional QBF Expansions and Q-Resolution

Over the years, proof systems for propositional satisfiability (SAT)
have been extensively studied. Recently, proof systems for
quantified Boolean formulas (QBFs) have also been gaining attention.
Q-resolution is a calculus enabling producing proofs from
DPLL-based QBF solvers. While DPLL has become a dominating technique
for SAT, QBF has been tackled ... more >>>

TR13-108 | 9th August 2013
Rahul Santhanam, Ryan Williams

New Algorithms for QBF Satisfiability and Implications for Circuit Complexity

Revisions: 1

We revisit the complexity of the satisfiability problem for quantified Boolean formulas. We show that satisfiability
of quantified CNFs of size $\poly(n)$ on $n$ variables with $O(1)$
quantifier blocks can be solved in time $2^{n-n^{\Omega(1)}}$ by zero-error
randomized algorithms. This is the first known improvement over brute force search in ... more >>>

TR15-059 | 10th April 2015
Olaf Beyersdorff, Leroy Chew, Meena Mahajan, Anil Shukla

Feasible Interpolation for QBF Resolution Calculi

In sharp contrast to classical proof complexity we are currently short of lower bound techniques for QBF proof systems. In this paper we establish the feasible interpolation technique for all resolution-based QBF systems, whether modelling CDCL or expansion-based solving. This both provides the first general lower bound method for QBF ... more >>>

TR16-002 | 18th January 2016
Ryan Williams

Strong ETH Breaks With Merlin and Arthur: Short Non-Interactive Proofs of Batch Evaluation

We present an efficient proof system for Multipoint Arithmetic Circuit Evaluation: for every arithmetic circuit $C(x_1,\ldots,x_n)$ of size $s$ and degree $d$ over a field ${\mathbb F}$, and any inputs $a_1,\ldots,a_K \in {\mathbb F}^n$,
$\bullet$ the Prover sends the Verifier the values $C(a_1), \ldots, C(a_K) \in {\mathbb F}$ and ... more >>>

TR17-032 | 17th February 2017
Olaf Beyersdorff, Joshua Blinkhorn

Formulas with Large Weight: a New Technique for Genuine QBF Lower Bounds

We devise a new technique to prove lower bounds for the proof size in resolution-type calculi for quantified Boolean formulas (QBF). The new technique applies to the strong expansion system IR-calc and thereby also to the most studied QBF system Q-Resolution.

Our technique exploits a clear semantic paradigm, showing the ... more >>>

TR17-037 | 25th February 2017
Olaf Beyersdorff, Leroy Chew, Meena Mahajan, Anil Shukla

Understanding Cutting Planes for QBFs

We define a cutting planes system CP+$\forall$red for quantified Boolean formulas (QBF) and analyse the proof-theoretic strength of this new calculus. While in the propositional case, Cutting Planes is of intermediate strength between resolution and Frege, our findings here show that the situation in QBF is slightly more complex: while ... more >>>

TR17-044 | 21st February 2017
Olaf Beyersdorff, Luke Hinde, Ján Pich

Reasons for Hardness in QBF Proof Systems

Revisions: 1

We aim to understand inherent reasons for lower bounds for QBF proof systems and revisit and compare two previous approaches in this direction.

The first of these relates size lower bounds for strong QBF Frege systems to circuit lower bounds via strategy extraction (Beyersdorff & Pich, LICS'16). Here we ... more >>>

TR17-137 | 11th September 2017
Olaf Beyersdorff, Joshua Blinkhorn, Luke Hinde

Size, Cost, and Capacity: A Semantic Technique for Hard Random QBFs

As a natural extension of the SAT problem, different proof systems for quantified Boolean formulas (QBF) have been proposed. Many of these extend a propositional system to handle universal quantifiers.

By formalising the construction of the QBF proof system from a propositional proof system, by the addition of the ... more >>>

TR18-024 | 1st February 2018
Olaf Beyersdorff, Judith Clymo, Stefan Dantchev, Barnaby Martin

The Riis Complexity Gap for QBF Resolution

We give an analogue of the Riis Complexity Gap Theorem for Quanti fied Boolean Formulas (QBFs). Every fi rst-order sentence $\phi$ without finite models gives rise to a sequence of QBFs whose minimal refutations in tree-like Q-Resolution are either of polynomial size (if $\phi$ has no models) or at least ... more >>>

TR18-025 | 1st February 2018
Olaf Beyersdorff, Judith Clymo

More on Size and Width in QBF Resolution

In their influential paper `Short proofs are narrow -- resolution made simple', Ben-Sasson and Wigderson introduced a crucial tool for proving lower bounds on the lengths of proofs in the resolution calculus. Over a decade later their technique for showing lower bounds on the size of proofs, by examining the ... more >>>

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