393 research outputs found
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
Small Normalized Boolean Circuits for Semi-disjoint Bilinear Forms Require Logarithmic Conjunction-depth
We consider normalized Boolean circuits that use binary operations of disjunction and conjunction, and unary negation, with the restriction that negation can be only applied to input variables. We derive a lower bound trade-off between the size of normalized Boolean circuits computing Boolean semi-disjoint bilinear forms and their conjunction-depth (i.e., the maximum number of and-gates on a directed path to an output gate). In particular, we show that any normalized Boolean circuit of at most epsilon log n conjunction-depth computing the n-dimensional Boolean vector convolution has Omega(n^{2-4 epsilon}) and-gates. Analogously, any normalized Boolean circuit of at most epsilon log n conjunction-depth computing the n x n Boolean matrix product has Omega(n^{3-4 epsilon}) and-gates. We complete our lower-bound trade-offs with upper-bound trade-offs of similar form yielded by the known fast algebraic algorithms
On generating the irredundant conjunctive and disjunctive normal forms of monotone Boolean functions
AbstractLet f:{0,1}nâ{0,1} be a monotone Boolean function whose value at any point xâ{0,1}n can be determined in time t. Denote by c=âIâCâiâIxi the irredundant CNF of f, where C is the set of the prime implicates of f. Similarly, let d=âJâDâjâJxj be the irredundant DNF of the same function, where D is the set of the prime implicants of f. We show that given subsets Câ˛âC and Dâ˛âD such that (Câ˛,Dâ˛)â (C,D), a new term in (C⧚Câ˛)âŞ(D⧚Dâ˛) can be found in time O(n(t+n))+mo(logm), where m=|Câ˛|+|Dâ˛|. In particular, if f(x) can be evaluated for every xâ{0,1}n in polynomial time, then the forms c and d can be jointly generated in incremental quasi-polynomial time. On the other hand, even for the class of â§,â¨-formulae f of depth 2, i.e., for CNFs or DNFs, it is unlikely that uniform sampling from within the set of the prime implicates and implicants of f can be carried out in time bounded by a quasi-polynomial 2polylog(¡) in the input size of f. We also show that for some classes of polynomial-time computable monotone Boolean functions it is NP-hard to test either of the conditions Dâ˛=D or Câ˛=C. This provides evidence that for each of these classes neither conjunctive nor disjunctive irredundant normal forms can be generated in total (or incremental) quasi-polynomial time. Such classes of monotone Boolean functions naturally arise in game theory, networks and relay contact circuits, convex programming, and include a subset of â§,â¨-formulae of depth 3
A visual interactive method for prime implicants identification
We propose a visual interactive method for the identification of the Prime Implicants (PIs) of dynamic non-coherent systems. Visual interactive methods integrate mathematical and symbolic models with runtime interaction and real-time graphic display, which allow visualizing the underlying physical relationships among process parameters. The proposed method is based on a parallel coordinates data mining tool that relies on an innovative pruning procedure which, on the basis of a proper selection of characteristic features of the accident sequences, retrieves the PIs among the whole set of Implicants in terms of process parameters values and/or components failure states. The method is exemplified on an artificial case study and, then, applied for the dynamic reliability analysis of the Airlock System (AS) of a CANDU reactor
Incremental polynomial time dualization of quadratic functions and a subclass of degree-k functions
Cataloged from PDF version of article.We consider the problem of dualizing a Boolean function f represented by
a DNF. In its most general form, this problem is commonly believed not to be solvable
by a quasi-polynomial total time algorithm.We show that if the input DNF is quadratic or is
a special degree-k DNF, then dualization turns out to be equivalent to hypergraph dualization
in hypergraphs of bounded degree and hence it can be achieved in incremental polynomial
time
Karchmer-Wigderson Games for Hazard-free Computation
We present a Karchmer-Wigderson game to study the complexity of hazard-free
formulas. This new game is both a generalization of the monotone
Karchmer-Wigderson game and an analog of the classical Boolean
Karchmer-Wigderson game. Therefore, it acts as a bridge between the existing
monotone and general games.
Using this game, we prove hazard-free formula size and depth lower bounds
that are provably stronger than those possible by the standard technique of
transferring results from monotone complexity in a black-box fashion. For the
multiplexer function we give (1) a hazard-free formula of optimal size and (2)
an improved low-depth hazard-free formula of almost optimal size and (3) a
hazard-free formula with alternation depth that has optimal depth. We then
use our optimal constructions to obtain an improved universal worst-case
hazard-free formula size upper bound. We see our results as a significant step
towards establishing hazard-free computation as an independent missing link
between Boolean complexity and monotone complexity.Comment: 34 pages, To appear in ITCS 202
Efficient state reduction methods for PLA-based sequential circuits
Experiences with heuristics for the state reduction of finite-state machines are presented and two new heuristic algorithms described in detail. Results on machines from the literature and from the MCNC benchmark set are shown. The area of the PLA implementation of the combinational component and the design time are used as figures of merit. The comparison of such parameters, when the state reduction step is included in the design process and when it is not, suggests that fast state-reduction heuristics should be implemented within FSM automatic synthesis systems
Karchmer-Wigderson Games for Hazard-Free Computation
We present a Karchmer-Wigderson game to study the complexity of hazard-free formulas. This new game is both a generalization of the monotone Karchmer-Wigderson game and an analog of the classical Boolean Karchmer-Wigderson game. Therefore, it acts as a bridge between the existing monotone and general games.
Using this game, we prove hazard-free formula size and depth lower bounds that are provably stronger than those possible by the standard technique of transferring results from monotone complexity in a black-box fashion. For the multiplexer function we give (1) a hazard-free formula of optimal size and (2) an improved low-depth hazard-free formula of almost optimal size and (3) a hazard-free formula with alternation depth 2 that has optimal depth. We then use our optimal constructions to obtain an improved universal worst-case hazard-free formula size upper bound. We see our results as a step towards establishing hazard-free computation as an independent missing link between Boolean complexity and monotone complexity
Lower Bounds for DeMorgan Circuits of Bounded Negation Width
We consider Boolean circuits over {or, and, neg} with negations applied only to input variables. To measure the "amount of negation" in such circuits, we introduce the concept of their "negation width". In particular, a circuit computing a monotone Boolean function f(x_1,...,x_n) has negation width w if no nonzero term produced (purely syntactically) by the circuit contains more than w distinct negated variables. Circuits of negation width w=0 are equivalent to monotone Boolean circuits, while those of negation width w=n have no restrictions. Our motivation is that already circuits of moderate negation width w=n^{epsilon} for an arbitrarily small constant epsilon>0 can be even exponentially stronger than monotone circuits.
We show that the size of any circuit of negation width w computing f is roughly at least the minimum size of a monotone circuit computing f divided by K=min{w^m,m^w}, where m is the maximum length of a prime implicant of f. We also show that the depth of any circuit of negation width w computing f is roughly at least the minimum depth of a monotone circuit computing f minus log K. Finally, we show that formulas of bounded negation width can be balanced to achieve a logarithmic (in their size) depth without increasing their negation width
Prime implicants for modularised non-coherent fault trees using binary decision diagrams
This paper presents an extended strategy for the analysis of complex fault trees. The method utilises simplification rules, which are applied to the fault tree to reduce it to a series of smaller subtrees, whose solution is equivalent to the original fault tree. The smaller subtree units are less sensitive to the basic event ordering during BDD conversion. BDDs are constructed for every subtree. Qualitative analysis is performed on the set of BDDs to obtain the prime implicant sets for the original top event. It is shown how to extract the prime implicant sets from complex and modular events in order to obtain the prime implicant sets of the original fault tree in terms of basic events
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