1,293 research outputs found
On the complexity of automatic complexity
Generalizing the notion of automatic complexity of individual strings due to
Shallit and Wang, we define the automatic complexity of an equivalence
relation on a finite set of strings.
We prove that the problem of determining whether equals the number
of equivalence classes of is -complete. The problem of
determining whether for a fixed is complete for the
second level of the Boolean hierarchy for , i.e.,
-complete.
Let be the language consisting of all strings of maximal nondeterministic
automatic complexity. We characterize the complexity of infinite subsets of
by showing that they can be co-context-free but not context-free, i.e., is
-immune, but not -immune.
We show that for each , , where
is the set of all strings whose deterministic automatic complexity
satisfies
Incompleteness in the finite domain
Motivated by the problem of finding finite versions of classical
incompleteness theorems, we present some conjectures that go beyond . These conjectures formally connect computational complexity
with the difficulty of proving some sentences, which means that high
computational complexity of a problem associated with a sentence implies that
the sentence is not provable in a weak theory, or requires a long proof.
Another reason for putting forward these conjectures is that some results in
proof complexity seem to be special cases of such general statements and we
want to formalize and fully understand these statements. In this paper we
review some conjectures that we have presented earlier, introduce new
conjectures, systematize them and prove new connections between them and some
other statements studied before
On the Cauchy Completeness of the Constructive Cauchy Reals
It is consistent with constructive set theory (without Countable Choice,
clearly) that the Cauchy reals (equivalence classes of Cauchy sequences of
rationals) are not Cauchy complete. Related results are also shown, such as
that a Cauchy sequence of rationals may not have a modulus of convergence, and
that a Cauchy sequence of Cauchy sequences may not converge to a Cauchy
sequence, among others
Expansions of pseudofinite structures and circuit and proof complexity
I shall describe a general model-theoretic task to construct expansions of
pseudofinite structures and discuss several examples of particular relevance to
computational complexity. Then I will present one specific situation where
finding a suitable expansion would imply that, assuming a one-way permutation
exists, the computational class NP is not closed under complementation.Comment: Preliminary version May 201
Flexible constraint satisfiability and a problem in semigroup theory
We examine some flexible notions of constraint satisfaction, observing some
relationships between model theoretic notions of universal Horn class
membership and robust satisfiability. We show the \texttt{NP}-completeness of
-robust monotone 1-in-3 3SAT in order to give very small examples of finite
algebras with \texttt{NP}-hard variety membership problem. In particular we
give a -element algebra with this property, and solve a widely stated
problem by showing that the -element Brandt monoid has \texttt{NP}-hard
variety membership problem. These are the smallest possible sizes for a general
algebra and a semigroup to exhibit \texttt{NP}-hardness for the membership
problem of finite algebras in finitely generated varieties
Algebraic foundations for qualitative calculi and networks
A qualitative representation is like an ordinary representation of a
relation algebra, but instead of requiring , as
we do for ordinary representations, we only require that , for each in the algebra. A constraint
network is qualitatively satisfiable if its nodes can be mapped to elements of
a qualitative representation, preserving the constraints. If a constraint
network is satisfiable then it is clearly qualitatively satisfiable, but the
converse can fail. However, for a wide range of relation algebras including the
point algebra, the Allen Interval Algebra, RCC8 and many others, a network is
satisfiable if and only if it is qualitatively satisfiable.
Unlike ordinary composition, the weak composition arising from qualitative
representations need not be associative, so we can generalise by considering
network satisfaction problems over non-associative algebras. We prove that
computationally, qualitative representations have many advantages over ordinary
representations: whereas many finite relation algebras have only infinite
representations, every finite qualitatively representable algebra has a finite
qualitative representation; the representability problem for (the atom
structures of) finite non-associative algebras is NP-complete; the network
satisfaction problem over a finite qualitatively representable algebra is
always in NP; the validity of equations over qualitative representations is
co-NP-complete. On the other hand we prove that there is no finite
axiomatisation of the class of qualitatively representable algebras.Comment: 22 page
An Effective Dichotomy for the Counting Constraint Satisfaction Problem
Bulatov (2008) gave a dichotomy for the counting constraint satisfaction
problem #CSP. A problem from #CSP is characterised by a constraint language,
which is a fixed, finite set of relations over a finite domain D. An instance
of the problem uses these relations to constrain the variables in a larger set.
Bulatov showed that the problem of counting the satisfying assignments of
instances of any problem from #CSP is either in polynomial time (FP) or is
#P-complete. His proof draws heavily on techniques from universal algebra and
cannot be understood without a secure grasp of that field. We give an
elementary proof of Bulatov's dichotomy, based on succinct representations,
which we call frames, of a class of highly structured relations, which we call
strongly rectangular. We show that these are precisely the relations which are
invariant under a Mal'tsev polymorphism. En route, we give a simplification of
a decision algorithm for strongly rectangular constraint languages, due to
Bulatov and Dalmau (2006). We establish a new criterion for the #CSP dichotomy,
which we call strong balance, and we prove that this property is decidable. In
fact, we establish membership in NP. Thus, we show that the dichotomy is
effective, resolving the most important open question concerning the #CSP
dichotomy.Comment: 31 pages. Corrected some errors from previous version
Logic Blog 2015f
The 2015 Logic Blog contains a large variety of results connected to logic,
some of them unlikely to be submitted to a journal. For the first time there is
a group theory part. There are results in higher randomness, and in computable
ergodic theory
Algebraic approach to promise constraint satisfaction
The complexity and approximability of the constraint satisfaction problem
(CSP) has been actively studied over the last 20 years. A new version of the
CSP, the promise CSP (PCSP) has recently been proposed, motivated by open
questions about the approximability of variants of satisfiability and graph
colouring. The PCSP significantly extends the standard decision CSP. The
complexity of CSPs with a fixed constraint language on a finite domain has
recently been fully classified, greatly guided by the algebraic approach, which
uses polymorphisms --- high-dimensional symmetries of solution spaces --- to
analyse the complexity of problems. The corresponding classification for PCSPs
is wide open and includes some long-standing open questions, such as the
complexity of approximate graph colouring, as special cases.
The basic algebraic approach to PCSP was initiated by Brakensiek and
Guruswami, and in this paper we significantly extend it and lift it from
concrete properties of polymorphisms to their abstract properties. We introduce
a new class of problems that can be viewed as algebraic versions of the (Gap)
Label Cover problem, and show that every PCSP with a fixed constraint language
is equivalent to a problem of this form. This allows us to identify a "measure
of symmetry" that is well suited for comparing and relating the complexity of
different PCSPs via the algebraic approach. We demonstrate how our theory can
be applied by improving the state-of-the-art in approximate graph colouring: we
show that, for any , it is NP-hard to find a -colouring of a
given -colourable graph.Comment: Extended version (73 pages). Preliminary versions of parts of this
paper were published in the proceedings of STOC 2019 and LICS 201
Learning implicitly in reasoning in PAC-Semantics
We consider the problem of answering queries about formulas of propositional
logic based on background knowledge partially represented explicitly as other
formulas, and partially represented as partially obscured examples
independently drawn from a fixed probability distribution, where the queries
are answered with respect to a weaker semantics than usual -- PAC-Semantics,
introduced by Valiant (2000) -- that is defined using the distribution of
examples. We describe a fairly general, efficient reduction to limited versions
of the decision problem for a proof system (e.g., bounded space treelike
resolution, bounded degree polynomial calculus, etc.) from corresponding
versions of the reasoning problem where some of the background knowledge is not
explicitly given as formulas, only learnable from the examples. Crucially, we
do not generate an explicit representation of the knowledge extracted from the
examples, and so the "learning" of the background knowledge is only done
implicitly. As a consequence, this approach can utilize formulas as background
knowledge that are not perfectly valid over the distribution---essentially the
analogue of agnostic learning here
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