1,030 research outputs found
On the complexity of solving linear congruences and computing nullspaces modulo a constant
We consider the problems of determining the feasibility of a linear
congruence, producing a solution to a linear congruence, and finding a spanning
set for the nullspace of an integer matrix, where each problem is considered
modulo an arbitrary constant k>1. These problems are known to be complete for
the logspace modular counting classes {Mod_k L} = {coMod_k L} in special case
that k is prime (Buntrock et al, 1992). By considering variants of standard
logspace function classes --- related to #L and functions computable by UL
machines, but which only characterize the number of accepting paths modulo k
--- we show that these problems of linear algebra are also complete for
{coMod_k L} for any constant k>1.
Our results are obtained by defining a class of functions FUL_k which are low
for {Mod_k L} and {coMod_k L} for k>1, using ideas similar to those used in the
case of k prime in (Buntrock et al, 1992) to show closure of Mod_k L under NC^1
reductions (including {Mod_k L} oracle reductions). In addition to the results
above, we briefly consider the relationship of the class FUL_k for arbitrary
moduli k to the class {F.coMod_k L} of functions whose output symbols are
verifiable by {coMod_k L} algorithms; and consider what consequences such a
comparison may have for oracle closure results of the form {Mod_k L}^{Mod_k L}
= {Mod_k L} for composite k.Comment: 17 pages, one Appendix; minor corrections and revisions to
presentation, new observations regarding the prospect of oracle closures.
Comments welcom
The complexity and distribution of computationally useful problems
The solutions of certain natural decision problems such as the halting problem and the boolean satisfiability problem contain large amounts of useful information about computation that is highly organized and readily available to efficient computational processes. Such problems are computationally useful. This dissertation investigates the complexity and distribution of these computationally useful problems. The main results of this dissertation are of the following three general types. (1) Useful problems contain highly organized information. (2) Very useful problems are so highly organized that they are unusually simple and hence rare. (3) Useful problems are, as a whole, not rare and thus are not necessarily simple;A result of type (1) is proven in Chapter 3. Bennett recently extended algorithmic information theory to include a notion of computational depth that appears to quantify the level of organization in binary strings and sequences. The main result of Chapter 3 states that every weakly useful sequence is strongly deep. (A sequence x is weakly useful if a non-negligible set of recursive problems are decidable within a fixed recursive time bound when given access to x.);Results of type (2) are presented in Chapters 4 and 5. These results say that the ≤[subscript]sp m P-complete problems for E = DTIME(2[superscript] linear) and the ≤[subscript]sp m p/poly-complete problems for ESPACE = DSPACE(2[superscript] linear) are unusually simple and hence rare. Complete problems are very useful because every problem in E or ESPACE is efficiently decidable when given access to one of these problems;Chapter 6 develops a result of type (3). This result says that the weakly ≤[subscript]sp m P-complete problems for E and ESPACE are not rare and hence are not necessarily simple. Weakly complete problems are useful because every problem in a non-negligible subset of E or ESPACE is efficiently decidable when given access to one of these problems;The above results (and others along the way) are obtained through a systematic investigation of the measure-theoretic structure of complexity classes
Characterization and Lower Bounds for Branching Program Size using Projective Dimension
We study projective dimension, a graph parameter (denoted by pd for a
graph ), introduced by (Pudl\'ak, R\"odl 1992), who showed that proving
lower bounds for pd for bipartite graphs associated with a Boolean
function imply size lower bounds for branching programs computing .
Despite several attempts (Pudl\'ak, R\"odl 1992 ; Babai, R\'{o}nyai, Ganapathy
2000), proving super-linear lower bounds for projective dimension of explicit
families of graphs has remained elusive.
We show that there exist a Boolean function (on bits) for which the
gap between the projective dimension and size of the optimal branching program
computing (denoted by bpsize), is . Motivated by the
argument in (Pudl\'ak, R\"odl 1992), we define two variants of projective
dimension - projective dimension with intersection dimension 1 (denoted by
upd) and bitwise decomposable projective dimension (denoted by
bitpdim).
As our main result, we show that there is an explicit family of graphs on vertices such that the projective dimension is , the
projective dimension with intersection dimension is and the
bitwise decomposable projective dimension is .
We also show that there exist a Boolean function (on bits) for which
the gap between upd and bpsize is . In contrast, we
also show that the bitwise decomposable projective dimension characterizes size
of the branching program up to a polynomial factor. That is, there exists a
constant and for any function , . We also study two other
variants of projective dimension and show that they are exactly equal to
well-studied graph parameters - bipartite clique cover number and bipartite
partition number respectively.Comment: 24 pages, 3 figure
PPP-Completeness with Connections to Cryptography
Polynomial Pigeonhole Principle (PPP) is an important subclass of TFNP with
profound connections to the complexity of the fundamental cryptographic
primitives: collision-resistant hash functions and one-way permutations. In
contrast to most of the other subclasses of TFNP, no complete problem is known
for PPP. Our work identifies the first PPP-complete problem without any circuit
or Turing Machine given explicitly in the input, and thus we answer a
longstanding open question from [Papadimitriou1994]. Specifically, we show that
constrained-SIS (cSIS), a generalized version of the well-known Short Integer
Solution problem (SIS) from lattice-based cryptography, is PPP-complete.
In order to give intuition behind our reduction for constrained-SIS, we
identify another PPP-complete problem with a circuit in the input but closely
related to lattice problems. We call this problem BLICHFELDT and it is the
computational problem associated with Blichfeldt's fundamental theorem in the
theory of lattices.
Building on the inherent connection of PPP with collision-resistant hash
functions, we use our completeness result to construct the first natural hash
function family that captures the hardness of all collision-resistant hash
functions in a worst-case sense, i.e. it is natural and universal in the
worst-case. The close resemblance of our hash function family with SIS, leads
us to the first candidate collision-resistant hash function that is both
natural and universal in an average-case sense.
Finally, our results enrich our understanding of the connections between PPP,
lattice problems and other concrete cryptographic assumptions, such as the
discrete logarithm problem over general groups
The Quantitative Structure of Exponential Time
Recent results on the internal, measure-theoretic structure of the exponential time complexity classes E = DTIME(2^linear) and E2 = DTIME(2^polynomial) are surveyed. The measure structure of these classes is seen to interact in informative ways with bi-immunity, complexity cores, polynomial-time many-one reducibility, circuit-size complexity, Kolmogorov complexity, and the density of hard languages. Possible implications for the structure of NP are also discussed
The Quantitative Structure of Exponential Time
Department of Computer Science Iowa State University Ames, Iowa 50010 Recent results on the internal, measure-theoretic structure of the exponential time complexity classes linear polynomial E = DTIME(2 ) and E = DTIME(2 ) 2 are surveyed. The measure structure of these classes is seen to interact in informative ways with bi-immunity, complexity cores, polynomial-time many-one reducibility, circuit-size complexity, Kolmogorov complexity, and the density of hard languages. Possible implications for the structure of NP are also discussed
Computational Difficulty of Computing the Density of States
We study the computational difficulty of computing the ground state
degeneracy and the density of states for local Hamiltonians. We show that the
difficulty of both problems is exactly captured by a class which we call #BQP,
which is the counting version of the quantum complexity class QMA. We show that
#BQP is not harder than its classical counting counterpart #P, which in turn
implies that computing the ground state degeneracy or the density of states for
classical Hamiltonians is just as hard as it is for quantum Hamiltonians.Comment: v2: Accepted version. 9 pages, 1 figur
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