282 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
On the relative asymptotic expressivity of inference frameworks
Let be a first-order signature and let be the set of
all -structures with domain . By an inference
framework we mean a class of pairs , where
and is a
probability distribution on , and is a logic with truth
values in the unit interval . An inference framework is
asymptotically at least as expressive as another inference framework
if for every there is
such that is asymptotically
total-variation-equivalent to and for every there is such that is
asymptotically equivalent to with respect to .
This relation is a preorder and we describe a partial order on the equivalence
classes of some inference frameworks that seem natural in the context of
machine learning and artificial intelligence. Several previous results about
asymptotic (or almost sure) equivalence of formulas or convergence in
probability can be formulated in terms of relative asymptotic strength of
inference frameworks. We incorporate these results in our classification of
inference frameworks and prove two new results. Both concern sequences of
probability distributions defined by directed graphical models that use
``continuous'' aggregation functions. The first considers queries expressed by
a logic with truth values in which employs continuous aggregation
functions. The second considers queries expressed by a two-valued conditional
logic that can express statements about relative frequencies.Comment: 52 page
Bounded Relativization
Relativization is one of the most fundamental concepts in complexity theory, which explains the difficulty of resolving major open problems. In this paper, we propose a weaker notion of relativization called bounded relativization. For a complexity class ?, we say that a statement is ?-relativizing if the statement holds relative to every oracle ? ? ?. It is easy to see that every result that relativizes also ?-relativizes for every complexity class ?. On the other hand, we observe that many non-relativizing results, such as IP = PSPACE, are in fact PSPACE-relativizing.
First, we use the idea of bounded relativization to obtain new lower bound results, including the following nearly maximum circuit lower bound: for every constant ? > 0, BPE^{MCSP}/2^{?n} ? SIZE[2?/n].
We prove this by PSPACE-relativizing the recent pseudodeterministic pseudorandom generator by Lu, Oliveira, and Santhanam (STOC 2021).
Next, we study the limitations of PSPACE-relativizing proof techniques, and show that a seemingly minor improvement over the known results using PSPACE-relativizing techniques would imply a breakthrough separation NP ? L. For example:
- Impagliazzo and Wigderson (JCSS 2001) proved that if EXP ? BPP, then BPP admits infinitely-often subexponential-time heuristic derandomization. We show that their result is PSPACE-relativizing, and that improving it to worst-case derandomization using PSPACE-relativizing techniques implies NP ? L.
- Oliveira and Santhanam (STOC 2017) recently proved that every dense subset in P admits an infinitely-often subexponential-time pseudodeterministic construction, which we observe is PSPACE-relativizing. Improving this to almost-everywhere (pseudodeterministic) or (infinitely-often) deterministic constructions by PSPACE-relativizing techniques implies NP ? L.
- Santhanam (SICOMP 2009) proved that pr-MA does not have fixed polynomial-size circuits. This lower bound can be shown PSPACE-relativizing, and we show that improving it to an almost-everywhere lower bound using PSPACE-relativizing techniques implies NP ? L.
In fact, we show that if we can use PSPACE-relativizing techniques to obtain the above-mentioned improvements, then PSPACE ? EXPH. We obtain our barrier results by constructing suitable oracles computable in EXPH relative to which these improvements are impossible
Search versus Search for Collapsing Electoral Control Types
Electoral control types are ways of trying to change the outcome of elections
by altering aspects of their composition and structure [BTT92]. We say two
compatible (i.e., having the same input types) control types that are about the
same election system E form a collapsing pair if for every possible input
(which typically consists of a candidate set, a vote set, a focus candidate,
and sometimes other parameters related to the nature of the attempted
alteration), either both or neither of the attempted attacks can be
successfully carried out [HHM20]. For each of the seven general (i.e., holding
for all election systems) electoral control type collapsing pairs found by
Hemaspaandra, Hemaspaandra, and Menton [HHM20] and for each of the additional
electoral control type collapsing pairs of Carleton et al. [CCH+ 22] for veto
and approval (and many other election systems in light of that paper's Theorems
3.6 and 3.9), both members of the collapsing pair have the same complexity
since as sets they are the same set. However, having the same complexity (as
sets) is not enough to guarantee that as search problems they have the same
complexity. In this paper, we explore the relationships between the search
versions of collapsing pairs. For each of the collapsing pairs of Hemaspaandra,
Hemaspaandra, and Menton [HHM20] and Carleton et al. [CCH+ 22], we prove that
the pair's members' search-version complexities are polynomially related (given
access, for cases when the winner problem itself is not in polynomial time, to
an oracle for the winner problem). Beyond that, we give efficient reductions
that from a solution to one compute a solution to the other. For the concrete
systems plurality, veto, and approval, we completely determine which of their
(due to our results) polynomially-related collapsing search-problem pairs are
polynomial-time computable and which are NP-hard.Comment: The metadata's abstract is abridged due to arXiv.org's
abstract-length limit. The paper itself has the unabridged (i.e., full)
abstrac
Recent Advances in Research on Island Phenomena
In natural languages, filler-gap dependencies can straddle across an unbounded distance. Since the 1960s, the term “island” has been used to describe syntactic structures from which extraction is impossible or impeded. While examples from English are ubiquitous, attested counterexamples in the Mainland Scandinavian languages have continuously been dismissed as illusory and alternative accounts for the underlying structure of such cases have been proposed. However, since such extractions are pervasive in spoken Mainland Scandinavian, these languages may not have been given the attention that they deserve in the syntax literature. In addition, recent research suggests that extraction from certain types of island structures in English might not be as unacceptable as previously assumed either. These findings break new empirical ground, question perceived knowledge, and may indeed have substantial ramifications for syntactic theory. This volume provides an overview of state-of-the-art research on island phenomena primarily in English and the Scandinavian languages, focusing on how languages compare to English, with the aim to shed new light on the nature of island constraints from different theoretical perspectives
Power of Counting by Nonuniform Families of Polynomial-Size Finite Automata
Lately, there have been intensive studies on strengths and limitations of
nonuniform families of promise decision problems solvable by various types of
polynomial-size finite automata families, where "polynomial-size" refers to the
polynomially-bounded state complexity of a finite automata family. In this line
of study, we further expand the scope of these studies to families of partial
counting and gap functions, defined in terms of nonuniform families of
polynomial-size nondeterministic finite automata, and their relevant families
of promise decision problems. Counting functions have an ability of counting
the number of accepting computation paths produced by nondeterministic finite
automata. With no unproven hardness assumption, we show numerous separations
and collapses of complexity classes of those partial counting and gap function
families and their induced promise decision problem families. We also
investigate their relationships to pushdown automata families of polynomial
stack-state complexity.Comment: (A4, 10pt, 21 pages) This paper corrects and extends a preliminary
report published in the Proceedings of the 24th International Symposium on
Fundamentals of Computation Theory (FCT 2023), Trier, Germany, September
18-24, 2023, Lecture Notes in Computer Science, vol. 14292, pp. 421-435,
Springer Cham, 202
Upward Translation of Optimal and P-Optimal Proof Systems in the Boolean Hierarchy over NP
We study the existence of optimal and p-optimal proof systems for classes in
the Boolean hierarchy over . Our main results concern
, i.e., the second level of this hierarchy:
If all sets in have p-optimal proof systems, then all sets in
have p-optimal proof systems. The analogous implication for
optimal proof systems fails relative to an oracle.
As a consequence, we clarify such implications for all classes
and in the Boolean hierarchy over : either we can
prove the implication or show that it fails relative to an oracle. Furthermore,
we show that the sets and have p-optimal proof
systems, if and only if all sets in the Boolean hierarchy over
have p-optimal proof systems which is a new characterization of a conjecture
studied by Pudl\'ak
Recommended from our members
Associative Plurals
The goal of this dissertation is to present an analysis of associative plurals in Japanese, Turkish, and Armenian that captures their associative interpretation along with a series of cross-linguistically consistent behaviours that do not seem to stem directly from these special meanings. For associative plurals, group affiliation is established through spatio-temporal or conceptual contiguity rather than a shared description (Moravcsik 2003). Approaches to English-like additive plurality are unable to capture associative plurals because they predict a plurality based on similarity, where every element of a plural noun is either an element of the corresponding singular or a concatenation of those elements. I propose that unlike additives, associative plurals are formed from a contextually specified individual concept that behaves like a group noun. This accounts for data which suggests associative plurals are inherently intensional, with a life that exists across indices. I will suggest that this individual concept is introduced as the plural marker. The noun being pluralized is actually part of a complex determiner that introduces a possessive like R relation that establishes the relationship between the group and the named individual
- …