1,082 research outputs found
An O(T log T) reduction from RAM computations to satisfiability
AbstractA new method is given for obtaining a boolean expression whose satisfiability is equivalent to the existence of an accepting computation of some nondeterministic machine. Although starting from random access machines, this method gives an expression of the same O(T log T) length as the best reduction from general Turing machines
Alternation-Trading Proofs, Linear Programming, and Lower Bounds
A fertile area of recent research has demonstrated concrete polynomial time
lower bounds for solving natural hard problems on restricted computational
models. Among these problems are Satisfiability, Vertex Cover, Hamilton Path,
Mod6-SAT, Majority-of-Majority-SAT, and Tautologies, to name a few. The proofs
of these lower bounds follow a certain proof-by-contradiction strategy that we
call alternation-trading. An important open problem is to determine how
powerful such proofs can possibly be.
We propose a methodology for studying these proofs that makes them amenable
to both formal analysis and automated theorem proving. We prove that the search
for better lower bounds can often be turned into a problem of solving a large
series of linear programming instances. Implementing a small-scale theorem
prover based on this result, we extract new human-readable time lower bounds
for several problems. This framework can also be used to prove concrete
limitations on the current techniques.Comment: To appear in STACS 2010, 12 page
A Casual Tour Around a Circuit Complexity Bound
I will discuss the recent proof that the complexity class NEXP
(nondeterministic exponential time) lacks nonuniform ACC circuits of polynomial
size. The proof will be described from the perspective of someone trying to
discover it.Comment: 21 pages, 2 figures. An earlier version appeared in SIGACT News,
September 201
Complexity Hierarchies Beyond Elementary
We introduce a hierarchy of fast-growing complexity classes and show its
suitability for completeness statements of many non elementary problems. This
hierarchy allows the classification of many decision problems with a
non-elementary complexity, which occur naturally in logic, combinatorics,
formal languages, verification, etc., with complexities ranging from simple
towers of exponentials to Ackermannian and beyond.Comment: Version 3 is the published version in TOCT 8(1:3), 2016. I will keep
updating the catalogue of problems from Section 6 in future revision
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 of size and
degree over a field , and any inputs ,
the Prover sends the Verifier the values and a proof of length, and
the Verifier tosses coins and can check the proof in about time, with probability of error less than .
For small degree , this "Merlin-Arthur" proof system (a.k.a. MA-proof
system) runs in nearly-linear time, and has many applications. For example, we
obtain MA-proof systems that run in time (for various ) for the
Permanent, Circuit-SAT for all sublinear-depth circuits, counting
Hamiltonian cycles, and infeasibility of - linear programs. In general,
the value of any polynomial in Valiant's class can be certified
faster than "exhaustive summation" over all possible assignments. These results
strongly refute a Merlin-Arthur Strong ETH and Arthur-Merlin Strong ETH posed
by Russell Impagliazzo and others.
We also give a three-round (AMA) proof system for quantified Boolean formulas
running in time, nearly-linear time MA-proof systems for
counting orthogonal vectors in a collection and finding Closest Pairs in the
Hamming metric, and a MA-proof system running in -time for
counting -cliques in graphs.
We point to some potential future directions for refuting the
Nondeterministic Strong ETH.Comment: 17 page
Elements of computability, decidability, and complexity (Third edition)
These lecture notes are intended to introduce the reader to the
basic notions of computability theory, decidability, and complexity. More
information on these subjects can be found in classical books such as [Cut80,Dav58,Her69,HoU79,Rog67].
The results reported in these notes are taken from those books and in various
parts we closely follow their style of presentation. The reader is encouraged
to look at those books for improving his/her knowledge on these topics. Some
parts of the chapter on complexity are taken from the lecture notes of a
beautiful course given by Prof. Leslie Valiant at Edinburgh University,
Scotland, in 1979. It was, indeed, a very stimulating and enjoyable course.
For the notions of Predicate Calculus we have used in this book the reader
may refer to [Men87].
I would like to thank Dr. Maurizio Proietti at IASI-CNR (Roma, Italy),
my colleagues, and my students at the University of Roma Tor Vergata and,
in particular, Michele Martone. They have been for me a source of continuous
inspiration and enthusiasm.
Finally, I would like to thank Dr. Gioacchino Onorati and Lorenzo Costantini
of the Aracne Publishing Company for their helpful cooperation
Elements of computability, decidability, and complexity (Third edition)
These lecture notes are intended to introduce the reader to the
basic notions of computability theory, decidability, and complexity. More
information on these subjects can be found in classical books such as [Cut80,Dav58,Her69,HoU79,Rog67].
The results reported in these notes are taken from those books and in various
parts we closely follow their style of presentation. The reader is encouraged
to look at those books for improving his/her knowledge on these topics. Some
parts of the chapter on complexity are taken from the lecture notes of a
beautiful course given by Prof. Leslie Valiant at Edinburgh University,
Scotland, in 1979. It was, indeed, a very stimulating and enjoyable course.
For the notions of Predicate Calculus we have used in this book the reader
may refer to [Men87].
I would like to thank Dr. Maurizio Proietti at IASI-CNR (Roma, Italy),
my colleagues, and my students at the University of Roma Tor Vergata and,
in particular, Michele Martone. They have been for me a source of continuous
inspiration and enthusiasm.
Finally, I would like to thank Dr. Gioacchino Onorati and Lorenzo Costantini
of the Aracne Publishing Company for their helpful cooperation
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