7 research outputs found
A Circuit-Based Approach to Efficient Enumeration
We study the problem of enumerating the satisfying valuations of a circuit while bounding the delay, i.e., the time needed to compute each successive valuation. We focus on the class of structured d-DNNF circuits originally introduced in knowledge compilation, a sub-area of artificial intelligence. We propose an algorithm for these circuits that enumerates valuations with linear preprocessing and delay linear in the Hamming weight of each valuation. Moreover, valuations of constant Hamming weight can be enumerated with linear preprocessing and constant delay.
Our results yield a framework for efficient enumeration that applies to all problems whose solutions can be compiled to structured d-DNNFs. In particular, we use it to recapture classical results in database theory, for factorized database representations and for MSO evaluation. This gives an independent proof of constant-delay enumeration for MSO formulae with first-order free variables on bounded-treewidth structures
Descriptive Complexity for Counting Complexity Classes
Descriptive Complexity has been very successful in characterizing complexity
classes of decision problems in terms of the properties definable in some
logics. However, descriptive complexity for counting complexity classes, such
as FP and #P, has not been systematically studied, and it is not as developed
as its decision counterpart. In this paper, we propose a framework based on
Weighted Logics to address this issue. Specifically, by focusing on the natural
numbers we obtain a logic called Quantitative Second Order Logics (QSO), and
show how some of its fragments can be used to capture fundamental counting
complexity classes such as FP, #P and FPSPACE, among others. We also use QSO to
define a hierarchy inside #P, identifying counting complexity classes with good
closure and approximation properties, and which admit natural complete
problems. Finally, we add recursion to QSO, and show how this extension
naturally captures lower counting complexity classes such as #L