3,459 research outputs found
On the pigeonhole and related principles in deep inference and monotone systems
International audienceWe construct quasipolynomial-size proofs of the propositional pigeonhole principle in the deep inference system KS, addressing an open problem raised in previous works and matching the best known upper bound for the more general class of monotone proofs. We make significant use of monotone formulae computing boolean threshold functions, an idea previously considered in works of Atserias et al. The main construction, monotone proofs witnessing the symmetry of such functions, involves an implementation of merge-sort in the design of proofs in order to tame the structural behaviour of atoms, and so the complexity of normalization. Proof transformations from previous work on atomic flows are then employed to yield appropriate KS proofs. As further results we show that our constructions can be applied to provide quasipolynomial-size KS proofs of the parity principle and the generalized pigeonhole principle. These bounds are inherited for the class of monotone proofs, and we are further able to construct n^O(log log n) -size monotone proofs of the weak pigeonhole principle with (1 + ε)n pigeons and n holes for ε = 1/ polylog n, thereby also improving the best known bounds for monotone proofs
Online Learning of k-CNF Boolean Functions
This paper revisits the problem of learning a k-CNF Boolean function from
examples in the context of online learning under the logarithmic loss. In doing
so, we give a Bayesian interpretation to one of Valiant's celebrated PAC
learning algorithms, which we then build upon to derive two efficient, online,
probabilistic, supervised learning algorithms for predicting the output of an
unknown k-CNF Boolean function. We analyze the loss of our methods, and show
that the cumulative log-loss can be upper bounded, ignoring logarithmic
factors, by a polynomial function of the size of each example.Comment: 20 LaTeX pages. 2 Algorithms. Some Theorem
Resolution over Linear Equations and Multilinear Proofs
We develop and study the complexity of propositional proof systems of varying
strength extending resolution by allowing it to operate with disjunctions of
linear equations instead of clauses. We demonstrate polynomial-size refutations
for hard tautologies like the pigeonhole principle, Tseitin graph tautologies
and the clique-coloring tautologies in these proof systems. Using the
(monotone) interpolation by a communication game technique we establish an
exponential-size lower bound on refutations in a certain, considerably strong,
fragment of resolution over linear equations, as well as a general polynomial
upper bound on (non-monotone) interpolants in this fragment.
We then apply these results to extend and improve previous results on
multilinear proofs (over fields of characteristic 0), as studied in
[RazTzameret06]. Specifically, we show the following:
1. Proofs operating with depth-3 multilinear formulas polynomially simulate a
certain, considerably strong, fragment of resolution over linear equations.
2. Proofs operating with depth-3 multilinear formulas admit polynomial-size
refutations of the pigeonhole principle and Tseitin graph tautologies. The
former improve over a previous result that established small multilinear proofs
only for the \emph{functional} pigeonhole principle. The latter are different
than previous proofs, and apply to multilinear proofs of Tseitin mod p graph
tautologies over any field of characteristic 0.
We conclude by connecting resolution over linear equations with extensions of
the cutting planes proof system.Comment: 44 page
Complexity of Non-Monotonic Logics
Over the past few decades, non-monotonic reasoning has developed to be one of
the most important topics in computational logic and artificial intelligence.
Different ways to introduce non-monotonic aspects to classical logic have been
considered, e.g., extension with default rules, extension with modal belief
operators, or modification of the semantics. In this survey we consider a
logical formalism from each of the above possibilities, namely Reiter's default
logic, Moore's autoepistemic logic and McCarthy's circumscription.
Additionally, we consider abduction, where one is not interested in inferences
from a given knowledge base but in computing possible explanations for an
observation with respect to a given knowledge base.
Complexity results for different reasoning tasks for propositional variants
of these logics have been studied already in the nineties. In recent years,
however, a renewed interest in complexity issues can be observed. One current
focal approach is to consider parameterized problems and identify reasonable
parameters that allow for FPT algorithms. In another approach, the emphasis
lies on identifying fragments, i.e., restriction of the logical language, that
allow more efficient algorithms for the most important reasoning tasks. In this
survey we focus on this second aspect. We describe complexity results for
fragments of logical languages obtained by either restricting the allowed set
of operators (e.g., forbidding negations one might consider only monotone
formulae) or by considering only formulae in conjunctive normal form but with
generalized clause types.
The algorithmic problems we consider are suitable variants of satisfiability
and implication in each of the logics, but also counting problems, where one is
not only interested in the existence of certain objects (e.g., models of a
formula) but asks for their number.Comment: To appear in Bulletin of the EATC
Model Counting of Query Expressions: Limitations of Propositional Methods
Query evaluation in tuple-independent probabilistic databases is the problem
of computing the probability of an answer to a query given independent
probabilities of the individual tuples in a database instance. There are two
main approaches to this problem: (1) in `grounded inference' one first obtains
the lineage for the query and database instance as a Boolean formula, then
performs weighted model counting on the lineage (i.e., computes the probability
of the lineage given probabilities of its independent Boolean variables); (2)
in methods known as `lifted inference' or `extensional query evaluation', one
exploits the high-level structure of the query as a first-order formula.
Although it is widely believed that lifted inference is strictly more powerful
than grounded inference on the lineage alone, no formal separation has
previously been shown for query evaluation. In this paper we show such a formal
separation for the first time.
We exhibit a class of queries for which model counting can be done in
polynomial time using extensional query evaluation, whereas the algorithms used
in state-of-the-art exact model counters on their lineages provably require
exponential time. Our lower bounds on the running times of these exact model
counters follow from new exponential size lower bounds on the kinds of d-DNNF
representations of the lineages that these model counters (either explicitly or
implicitly) produce. Though some of these queries have been studied before, no
non-trivial lower bounds on the sizes of these representations for these
queries were previously known.Comment: To appear in International Conference on Database Theory (ICDT) 201
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