41,986 research outputs found
NP-hardness of circuit minimization for multi-output functions
Can we design efficient algorithms for finding fast algorithms? This question is captured by various circuit minimization problems, and algorithms for the corresponding tasks have significant practical applications. Following the work of Cook and Levin in the early 1970s, a central question is whether minimizing the circuit size of an explicitly given function is NP-complete. While this is known to hold in restricted models such as DNFs, making progress with respect to more expressive classes of circuits has been elusive.
In this work, we establish the first NP-hardness result for circuit minimization of total functions in the setting of general (unrestricted) Boolean circuits. More precisely, we show that computing the minimum circuit size of a given multi-output Boolean function f : {0,1}^n ? {0,1}^m is NP-hard under many-one polynomial-time randomized reductions. Our argument builds on a simpler NP-hardness proof for the circuit minimization problem for (single-output) Boolean functions under an extended set of generators.
Complementing these results, we investigate the computational hardness of minimizing communication. We establish that several variants of this problem are NP-hard under deterministic reductions. In particular, unless ? = ??, no polynomial-time computable function can approximate the deterministic two-party communication complexity of a partial Boolean function up to a polynomial. This has consequences for the class of structural results that one might hope to show about the communication complexity of partial functions
New bounds on classical and quantum one-way communication complexity
In this paper we provide new bounds on classical and quantum distributional
communication complexity in the two-party, one-way model of communication. In
the classical model, our bound extends the well known upper bound of Kremer,
Nisan and Ron to include non-product distributions. We show that for a boolean
function f:X x Y -> {0,1} and a non-product distribution mu on X x Y and
epsilon in (0,1/2) constant: D_{epsilon}^{1, mu}(f)= O((I(X:Y)+1) vc(f)), where
D_{epsilon}^{1, mu}(f) represents the one-way distributional communication
complexity of f with error at most epsilon under mu; vc(f) represents the
Vapnik-Chervonenkis dimension of f and I(X:Y) represents the mutual
information, under mu, between the random inputs of the two parties. For a
non-boolean function f:X x Y ->[k], we show a similar upper bound on
D_{epsilon}^{1, mu}(f) in terms of k, I(X:Y) and the pseudo-dimension of f' =
f/k. In the quantum one-way model we provide a lower bound on the
distributional communication complexity, under product distributions, of a
function f, in terms the well studied complexity measure of f referred to as
the rectangle bound or the corruption bound of f . We show for a non-boolean
total function f : X x Y -> Z and a product distribution mu on XxY,
Q_{epsilon^3/8}^{1, mu}(f) = Omega(rec_ epsilon^{1, mu}(f)), where
Q_{epsilon^3/8}^{1, mu}(f) represents the quantum one-way distributional
communication complexity of f with error at most epsilon^3/8 under mu and rec_
epsilon^{1, mu}(f) represents the one-way rectangle bound of f with error at
most epsilon under mu . Similarly for a non-boolean partial function f:XxY -> Z
U {*} and a product distribution mu on X x Y, we show, Q_{epsilon^6/(2 x
15^4)}^{1, mu}(f) = Omega(rec_ epsilon^{1, mu}(f)).Comment: ver 1, 19 page
Low-Sensitivity Functions from Unambiguous Certificates
We provide new query complexity separations against sensitivity for total
Boolean functions: a power separation between deterministic (and even
randomized or quantum) query complexity and sensitivity, and a power
separation between certificate complexity and sensitivity. We get these
separations by using a new connection between sensitivity and a seemingly
unrelated measure called one-sided unambiguous certificate complexity
(). We also show that is lower-bounded by fractional block
sensitivity, which means we cannot use these techniques to get a
super-quadratic separation between and . We also provide a
quadratic separation between the tree-sensitivity and decision tree complexity
of Boolean functions, disproving a conjecture of Gopalan, Servedio, Tal, and
Wigderson (CCC 2016).
Along the way, we give a power separation between certificate
complexity and one-sided unambiguous certificate complexity, improving the
power separation due to G\"o\"os (FOCS 2015). As a consequence, we
obtain an improved lower-bound on the
co-nondeterministic communication complexity of the Clique vs. Independent Set
problem.Comment: 25 pages. This version expands the results and adds Pooya Hatami and
Avishay Tal as author
Boolean Delay Equations: A simple way of looking at complex systems
Boolean Delay Equations (BDEs) are semi-discrete dynamical models with
Boolean-valued variables that evolve in continuous time. Systems of BDEs can be
classified into conservative or dissipative, in a manner that parallels the
classification of ordinary or partial differential equations. Solutions to
certain conservative BDEs exhibit growth of complexity in time. They represent
therewith metaphors for biological evolution or human history. Dissipative BDEs
are structurally stable and exhibit multiple equilibria and limit cycles, as
well as more complex, fractal solution sets, such as Devil's staircases and
``fractal sunbursts``. All known solutions of dissipative BDEs have stationary
variance. BDE systems of this type, both free and forced, have been used as
highly idealized models of climate change on interannual, interdecadal and
paleoclimatic time scales. BDEs are also being used as flexible, highly
efficient models of colliding cascades in earthquake modeling and prediction,
as well as in genetics. In this paper we review the theory of systems of BDEs
and illustrate their applications to climatic and solid earth problems. The
former have used small systems of BDEs, while the latter have used large
networks of BDEs. We moreover introduce BDEs with an infinite number of
variables distributed in space (``partial BDEs``) and discuss connections with
other types of dynamical systems, including cellular automata and Boolean
networks. This research-and-review paper concludes with a set of open
questions.Comment: Latex, 67 pages with 15 eps figures. Revised version, in particular
the discussion on partial BDEs is updated and enlarge
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