721 research outputs found
Sub-computable Boundedness Randomness
This paper defines a new notion of bounded computable randomness for certain
classes of sub-computable functions which lack a universal machine. In
particular, we define such versions of randomness for primitive recursive
functions and for PSPACE functions. These new notions are robust in that there
are equivalent formulations in terms of (1) Martin-L\"of tests, (2) Kolmogorov
complexity, and (3) martingales. We show these notions can be equivalently
defined with prefix-free Kolmogorov complexity. We prove that one direction of
van Lambalgen's theorem holds for relative computability, but the other
direction fails. We discuss statistical properties of these notions of
randomness
Stretching demi-bits and nondeterministic-secure pseudorandomness
We develop the theory of cryptographic nondeterministic-secure pseudorandomness beyond the point reached by Rudich's original work [25], and apply it to draw new consequences in average-case complexity and proof complexity. Specifically, we show the following: Demi-bit stretch: Super-bits and demi-bits are variants of cryptographic pseudorandom generators which are secure against nondeterministic statistical tests [25]. They were introduced to rule out certain approaches to proving strong complexity lower bounds beyond the limitations set out by the Natural Proofs barrier of Razborov and Rudich [23]. Whether demi-bits are stretchable at all had been an open problem since their introduction. We answer this question affirmatively by showing that: every demi-bit b : {0, 1}n → {0, 1}n+1 can be stretched into sublinear many demi-bits b′: {0, 1}n → {0, 1}n+nc , for every constant 0 < c < 1. Average-case hardness: Using work by Santhanam [26], we apply our results to obtain new averagecase Kolmogorov complexity results: we show that Kpoly[n-O(1)] is zero-error average-case hard against NP/poly machines iff Kpoly[n-o(n)] is, where for a function s(n) : N → N, Kpoly[s(n)] denotes the languages of all strings x ∈ {0, 1}n for which there are (fixed) polytime Turing machines of description-length at most s(n) that output x. Characterising super-bits by nondeterministic unpredictability: In the deterministic setting, Yao [31] proved that super-polynomial hardness of pseudorandom generators is equivalent to ("nextbit") unpredictability. Unpredictability roughly means that given any strict prefix of a random string, it is infeasible to predict the next bit. We initiate the study of unpredictability beyond the deterministic setting (in the cryptographic regime), and characterise the nondeterministic hardness of generators from an unpredictability perspective. Specifically, we propose four stronger notions of unpredictability: NP/poly-unpredictability, coNP/poly-unpredictability, ∩-unpredictability and ∪unpredictability, and show that super-polynomial nondeterministic hardness of generators lies between ∩-unpredictability and ∪unpredictability. Characterising super-bits by nondeterministic hard-core predicates: We introduce a nondeterministic variant of hard-core predicates, called super-core predicates. We show that the existence of a super-bit is equivalent to the existence of a super-core of some non-shrinking function. This serves as an analogue of the equivalence between the existence of a strong pseudorandom generator and the existence of a hard-core of some one-way function [8, 12], and provides a first alternative characterisation of super-bits. We also prove that a certain class of functions, which may have hard-cores, cannot possess any super-core
Conspiracies between learning algorithms, circuit lower bounds, and pseudorandomness
We prove several results giving new and stronger connections between learning theory, circuit
complexity and pseudorandomness. Let C be any typical class of Boolean circuits, and C[s(n)]
denote n-variable C-circuits of size ≤ s(n). We show:
Learning Speedups. If C[poly(n)] admits a randomized weak learning algorithm under the
uniform distribution with membership queries that runs in time 2n/nω(1), then for every k ≥ 1
and ε > 0 the class C[n
k
] can be learned to high accuracy in time O(2n
ε
). There is ε > 0 such that
C[2n
ε
] can be learned in time 2n/nω(1) if and only if C[poly(n)] can be learned in time 2(log n)
O(1)
.
Equivalences between Learning Models. We use learning speedups to obtain equivalences
between various randomized learning and compression models, including sub-exponential
time learning with membership queries, sub-exponential time learning with membership and
equivalence queries, probabilistic function compression and probabilistic average-case function
compression.
A Dichotomy between Learnability and Pseudorandomness. In the non-uniform setting,
there is non-trivial learning for C[poly(n)] if and only if there are no exponentially secure
pseudorandom functions computable in C[poly(n)].
Lower Bounds from Nontrivial Learning. If for each k ≥ 1, (depth-d)-C[n
k
] admits a
randomized weak learning algorithm with membership queries under the uniform distribution
that runs in time 2n/nω(1), then for each k ≥ 1, BPE * (depth-d)-C[n
k
]. If for some ε > 0 there
are P-natural proofs useful against C[2n
ε
], then ZPEXP * C[poly(n)].
Karp-Lipton Theorems for Probabilistic Classes. If there is a k > 0 such that BPE ⊆
i.o.Circuit[n
k
], then BPEXP ⊆ i.o.EXP/O(log n). If ZPEXP ⊆ i.o.Circuit[2n/3
], then ZPEXP ⊆
i.o.ESUBEXP.
Hardness Results for MCSP. All functions in non-uniform NC1
reduce to the Minimum
Circuit Size Problem via truth-table reductions computable by TC0
circuits. In particular, if
MCSP ∈ TC0
then NC1 = TC0
Conspiracies Between Learning Algorithms, Circuit Lower Bounds, and Pseudorandomness
We prove several results giving new and stronger connections between learning theory, circuit complexity and pseudorandomness. Let C be any typical class of Boolean circuits, and C[s(n)] denote n-variable C-circuits of size <= s(n). We show:
Learning Speedups: If C[s(n)] admits a randomized weak learning algorithm under the uniform distribution with membership queries that runs in time 2^n/n^{omega(1)}, then for every k >= 1 and epsilon > 0 the class C[n^k] can be learned to high accuracy in time O(2^{n^epsilon}). There is epsilon > 0 such that C[2^{n^{epsilon}}] can be learned in time 2^n/n^{omega(1)} if and only if C[poly(n)] can be learned in time 2^{(log(n))^{O(1)}}.
Equivalences between Learning Models: We use learning speedups to obtain equivalences between various randomized learning and compression models, including sub-exponential time learning with membership queries, sub-exponential time learning with membership and equivalence queries, probabilistic function compression and probabilistic average-case function compression.
A Dichotomy between Learnability and Pseudorandomness: In the non-uniform setting, there is non-trivial learning for C[poly(n)] if and only if there are no exponentially secure pseudorandom functions computable in C[poly(n)].
Lower Bounds from Nontrivial Learning: If for each k >= 1, (depth-d)-C[n^k] admits a randomized weak learning algorithm with membership queries under the uniform distribution that runs in time 2^n/n^{omega(1)}, then for each k >= 1, BPE is not contained in (depth-d)-C[n^k]. If for some epsilon > 0 there are P-natural proofs useful against C[2^{n^{epsilon}}], then ZPEXP is not contained in C[poly(n)].
Karp-Lipton Theorems for Probabilistic Classes: If there is a k > 0 such that BPE is contained in i.o.Circuit[n^k], then BPEXP is contained in i.o.EXP/O(log(n)). If ZPEXP is contained in i.o.Circuit[2^{n/3}], then ZPEXP is contained in i.o.ESUBEXP.
Hardness Results for MCSP: All functions in non-uniform NC^1 reduce to the Minimum Circuit Size Problem via truth-table reductions computable by TC^0 circuits. In particular, if MCSP is in TC^0 then NC^1 = TC^0
Algorithms and lower bounds for de Morgan formulas of low-communication leaf gates
The class consists of Boolean functions
computable by size- de Morgan formulas whose leaves are any Boolean
functions from a class . We give lower bounds and (SAT, Learning,
and PRG) algorithms for , for classes
of functions with low communication complexity. Let
be the maximum -party NOF randomized communication
complexity of . We show:
(1) The Generalized Inner Product function cannot be computed in
on more than fraction of inputs
for As a corollary, we get an average-case lower bound for
against .
(2) There is a PRG of seed length that -fools . For
, we get the better seed length . This gives the first
non-trivial PRG (with seed length ) for intersections of half-spaces
in the regime where .
(3) There is a randomized -time SAT algorithm for , where In particular, this implies a nontrivial
#SAT algorithm for .
(4) The Minimum Circuit Size Problem is not in .
On the algorithmic side, we show that can be
PAC-learned in time
- …