66,452 research outputs found
Syntactic Complexity of Finite/Cofinite, Definite, and Reverse Definite Languages
We study the syntactic complexity of finite/cofinite, definite and reverse
definite languages. The syntactic complexity of a class of languages is defined
as the maximal size of syntactic semigroups of languages from the class, taken
as a function of the state complexity n of the languages. We prove that (n-1)!
is a tight upper bound for finite/cofinite languages and that it can be reached
only if the alphabet size is greater than or equal to (n-1)!-(n-2)!. We prove
that the bound is also (n-1)! for reverse definite languages, but the minimal
alphabet size is (n-1)!-2(n-2)!. We show that \lfloor e\cdot (n-1)!\rfloor is a
lower bound on the syntactic complexity of definite languages, and conjecture
that this is also an upper bound, and that the alphabet size required to meet
this bound is \floor{e \cdot (n-1)!} - \floor{e \cdot (n-2)!}. We prove the
conjecture for n\le 4.Comment: 10 pages. An error concerning the size of the alphabet has been
corrected in Theorem
Quantum Proofs
Quantum information and computation provide a fascinating twist on the notion
of proofs in computational complexity theory. For instance, one may consider a
quantum computational analogue of the complexity class \class{NP}, known as
QMA, in which a quantum state plays the role of a proof (also called a
certificate or witness), and is checked by a polynomial-time quantum
computation. For some problems, the fact that a quantum proof state could be a
superposition over exponentially many classical states appears to offer
computational advantages over classical proof strings. In the interactive proof
system setting, one may consider a verifier and one or more provers that
exchange and process quantum information rather than classical information
during an interaction for a given input string, giving rise to quantum
complexity classes such as QIP, QSZK, and QMIP* that represent natural quantum
analogues of IP, SZK, and MIP. While quantum interactive proof systems inherit
some properties from their classical counterparts, they also possess distinct
and uniquely quantum features that lead to an interesting landscape of
complexity classes based on variants of this model.
In this survey we provide an overview of many of the known results concerning
quantum proofs, computational models based on this concept, and properties of
the complexity classes they define. In particular, we discuss non-interactive
proofs and the complexity class QMA, single-prover quantum interactive proof
systems and the complexity class QIP, statistical zero-knowledge quantum
interactive proof systems and the complexity class \class{QSZK}, and
multiprover interactive proof systems and the complexity classes QMIP, QMIP*,
and MIP*.Comment: Survey published by NOW publisher
The PITA System: Tabling and Answer Subsumption for Reasoning under Uncertainty
Many real world domains require the representation of a measure of
uncertainty. The most common such representation is probability, and the
combination of probability with logic programs has given rise to the field of
Probabilistic Logic Programming (PLP), leading to languages such as the
Independent Choice Logic, Logic Programs with Annotated Disjunctions (LPADs),
Problog, PRISM and others. These languages share a similar distribution
semantics, and methods have been devised to translate programs between these
languages. The complexity of computing the probability of queries to these
general PLP programs is very high due to the need to combine the probabilities
of explanations that may not be exclusive. As one alternative, the PRISM system
reduces the complexity of query answering by restricting the form of programs
it can evaluate. As an entirely different alternative, Possibilistic Logic
Programs adopt a simpler metric of uncertainty than probability. Each of these
approaches -- general PLP, restricted PLP, and Possibilistic Logic Programming
-- can be useful in different domains depending on the form of uncertainty to
be represented, on the form of programs needed to model problems, and on the
scale of the problems to be solved. In this paper, we show how the PITA system,
which originally supported the general PLP language of LPADs, can also
efficiently support restricted PLP and Possibilistic Logic Programs. PITA
relies on tabling with answer subsumption and consists of a transformation
along with an API for library functions that interface with answer subsumption
Formalized proof, computation, and the construction problem in algebraic geometry
An informal discussion of how the construction problem in algebraic geometry
motivates the search for formal proof methods. Also includes a brief discussion
of my own progress up to now, which concerns the formalization of category
theory within a ZFC-like environment
Symbolic Algorithms for Language Equivalence and Kleene Algebra with Tests
We first propose algorithms for checking language equivalence of finite
automata over a large alphabet. We use symbolic automata, where the transition
function is compactly represented using a (multi-terminal) binary decision
diagrams (BDD). The key idea consists in computing a bisimulation by exploring
reachable pairs symbolically, so as to avoid redundancies. This idea can be
combined with already existing optimisations, and we show in particular a nice
integration with the disjoint sets forest data-structure from Hopcroft and
Karp's standard algorithm. Then we consider Kleene algebra with tests (KAT), an
algebraic theory that can be used for verification in various domains ranging
from compiler optimisation to network programming analysis. This theory is
decidable by reduction to language equivalence of automata on guarded strings,
a particular kind of automata that have exponentially large alphabets. We
propose several methods allowing to construct symbolic automata out of KAT
expressions, based either on Brzozowski's derivatives or standard automata
constructions. All in all, this results in efficient algorithms for deciding
equivalence of KAT expressions
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