3,635 research outputs found
The Complexity of Reasoning with FODD and GFODD
Recent work introduced Generalized First Order Decision Diagrams (GFODD) as a
knowledge representation that is useful in mechanizing decision theoretic
planning in relational domains. GFODDs generalize function-free first order
logic and include numerical values and numerical generalizations of existential
and universal quantification. Previous work presented heuristic inference
algorithms for GFODDs and implemented these heuristics in systems for decision
theoretic planning. In this paper, we study the complexity of the computational
problems addressed by such implementations. In particular, we study the
evaluation problem, the satisfiability problem, and the equivalence problem for
GFODDs under the assumption that the size of the intended model is given with
the problem, a restriction that guarantees decidability. Our results provide a
complete characterization placing these problems within the polynomial
hierarchy. The same characterization applies to the corresponding restriction
of problems in first order logic, giving an interesting new avenue for
efficient inference when the number of objects is bounded. Our results show
that for formulas, and for corresponding GFODDs, evaluation and
satisfiability are complete, and equivalence is
complete. For formulas evaluation is complete, satisfiability
is one level higher and is complete, and equivalence is
complete.Comment: A short version of this paper appears in AAAI 2014. Version 2
includes a reorganization and some expanded proof
Advances in Functional Decomposition: Theory and Applications
Functional decomposition aims at finding efficient representations for Boolean functions. It is used in many applications, including multi-level logic synthesis, formal verification, and testing.
This dissertation presents novel heuristic algorithms for functional decomposition. These algorithms take advantage of suitable representations of the Boolean functions in order to be efficient.
The first two algorithms compute simple-disjoint and disjoint-support decompositions. They are based on representing the target function by a Reduced Ordered Binary Decision Diagram (BDD). Unlike other BDD-based algorithms, the presented ones can deal with larger target functions and produce more decompositions without requiring expensive manipulations of the representation, particularly BDD reordering.
The third algorithm also finds disjoint-support decompositions, but it is based on a technique which integrates circuit graph analysis and BDD-based decomposition. The combination of the two approaches results in an algorithm which is more robust than a purely BDD-based one, and that improves both the quality of the results and the running time.
The fourth algorithm uses circuit graph analysis to obtain non-disjoint decompositions. We show that the problem of computing non-disjoint decompositions can be reduced to the problem of computing multiple-vertex dominators. We also prove that multiple-vertex dominators can be found in polynomial time. This result is important because there is no known polynomial time algorithm for computing all non-disjoint decompositions of a Boolean function.
The fifth algorithm provides an efficient means to decompose a function at the circuit graph level, by using information derived from a BDD representation. This is done without the expensive circuit re-synthesis normally associated with BDD-based decomposition approaches.
Finally we present two publications that resulted from the many detours we have taken along the winding path of our research
Phase Transitions of the Typical Algorithmic Complexity of the Random Satisfiability Problem Studied with Linear Programming
Here we study the NP-complete -SAT problem. Although the worst-case
complexity of NP-complete problems is conjectured to be exponential, there
exist parametrized random ensembles of problems where solutions can typically
be found in polynomial time for suitable ranges of the parameter. In fact,
random -SAT, with as control parameter, can be solved quickly
for small enough values of . It shows a phase transition between a
satisfiable phase and an unsatisfiable phase. For branch and bound algorithms,
which operate in the space of feasible Boolean configurations, the empirically
hardest problems are located only close to this phase transition. Here we study
-SAT () and the related optimization problem MAX-SAT by a linear
programming approach, which is widely used for practical problems and allows
for polynomial run time. In contrast to branch and bound it operates outside
the space of feasible configurations. On the other hand, finding a solution
within polynomial time is not guaranteed. We investigated several variants like
including artificial objective functions, so called cutting-plane approaches,
and a mapping to the NP-complete vertex-cover problem. We observed several
easy-hard transitions, from where the problems are typically solvable (in
polynomial time) using the given algorithms, respectively, to where they are
not solvable in polynomial time. For the related vertex-cover problem on random
graphs these easy-hard transitions can be identified with structural properties
of the graphs, like percolation transitions. For the present random -SAT
problem we have investigated numerous structural properties also exhibiting
clear transitions, but they appear not be correlated to the here observed
easy-hard transitions. This renders the behaviour of random -SAT more
complex than, e.g., the vertex-cover problem.Comment: 11 pages, 5 figure
Exact Algorithms for Mixed-Integer Multilevel Programming Problems
We examine multistage optimization problems, in which one or more decision makers solve a sequence of interdependent optimization problems. In each stage the corresponding decision maker determines values for a set of variables, which in turn parameterizes the subsequent problem by modifying its constraints and objective function. The optimization literature has covered multistage optimization problems in the form of bilevel programs, interdiction problems, robust optimization, and two-stage stochastic programming. One of the main differences among these research areas lies in the relationship between the decision makers. We analyze the case in which the decision makers are self-interested agents seeking to optimize their own objective function (bilevel programming), the case in which the decision makers are opponents working against each other, playing a zero-sum game (interdiction), and the case in which the decision makers are cooperative agents working towards a common goal (two-stage stochastic programming). Traditional exact approaches for solving multistage optimization problems often rely on strong duality either for the purpose of achieving single-level reformulations of the original multistage problems, or for the development of cutting-plane approaches similar to Benders\u27 decomposition. As a result, existing solution approaches usually assume that the last-stage problems are linear or convex, and fail to solve problems for which the last-stage is nonconvex (e.g., because of the presence of discrete variables). We contribute exact finite algorithms for bilevel mixed-integer programs, three-stage defender-attacker-defender problems, and two-stage stochastic programs. Moreover, we do not assume linearity or convexity for the last-stage problem and allow the existence of discrete variables. We demonstrate how our proposed algorithms significantly outperform existing state-of-the-art algorithms. Additionally, we solve for the first time a class of interdiction and fortification problems in which the third-stage problem is NP-hard, opening a venue for new research and applications in the field of (network) interdiction
Solving Limited Memory Influence Diagrams
We present a new algorithm for exactly solving decision making problems
represented as influence diagrams. We do not require the usual assumptions of
no forgetting and regularity; this allows us to solve problems with
simultaneous decisions and limited information. The algorithm is empirically
shown to outperform a state-of-the-art algorithm on randomly generated problems
of up to 150 variables and solutions. We show that the problem is
NP-hard even if the underlying graph structure of the problem has small
treewidth and the variables take on a bounded number of states, but that a
fully polynomial time approximation scheme exists for these cases. Moreover, we
show that the bound on the number of states is a necessary condition for any
efficient approximation scheme.Comment: 43 pages, 8 figure
Semantic Similarity of Spatial Scenes
The formalization of similarity in spatial information systems can unleash their functionality and contribute technology not only useful, but also desirable by broad groups of users. As a paradigm for information retrieval, similarity supersedes tedious querying techniques and unveils novel ways for user-system interaction by naturally supporting modalities such as speech and sketching. As a tool within the scope of a broader objective, it can facilitate such diverse tasks as data integration, landmark determination, and prediction making. This potential motivated the development of several similarity models within the geospatial and computer science communities. Despite the merit of these studies, their cognitive plausibility can be limited due to neglect of well-established psychological principles about properties and behaviors of similarity. Moreover, such approaches are typically guided by experience, intuition, and observation, thereby often relying on more narrow perspectives or restrictive assumptions that produce inflexible and incompatible measures. This thesis consolidates such fragmentary efforts and integrates them along with novel formalisms into a scalable, comprehensive, and cognitively-sensitive framework for similarity queries in spatial information systems. Three conceptually different similarity queries at the levels of attributes, objects, and scenes are distinguished. An analysis of the relationship between similarity and change provides a unifying basis for the approach and a theoretical foundation for measures satisfying important similarity properties such as asymmetry and context dependence. The classification of attributes into categories with common structural and cognitive characteristics drives the implementation of a small core of generic functions, able to perform any type of attribute value assessment. Appropriate techniques combine such atomic assessments to compute similarities at the object level and to handle more complex inquiries with multiple constraints. These techniques, along with a solid graph-theoretical methodology adapted to the particularities of the geospatial domain, provide the foundation for reasoning about scene similarity queries. Provisions are made so that all methods comply with major psychological findings about people’s perceptions of similarity. An experimental evaluation supplies the main result of this thesis, which separates psychological findings with a major impact on the results from those that can be safely incorporated into the framework through computationally simpler alternatives
Programming Quantum Computers Using Design Automation
Recent developments in quantum hardware indicate that systems featuring more
than 50 physical qubits are within reach. At this scale, classical simulation
will no longer be feasible and there is a possibility that such quantum devices
may outperform even classical supercomputers at certain tasks. With the rapid
growth of qubit numbers and coherence times comes the increasingly difficult
challenge of quantum program compilation. This entails the translation of a
high-level description of a quantum algorithm to hardware-specific low-level
operations which can be carried out by the quantum device. Some parts of the
calculation may still be performed manually due to the lack of efficient
methods. This, in turn, may lead to a design gap, which will prevent the
programming of a quantum computer. In this paper, we discuss the challenges in
fully-automatic quantum compilation. We motivate directions for future research
to tackle these challenges. Yet, with the algorithms and approaches that exist
today, we demonstrate how to automatically perform the quantum programming flow
from algorithm to a physical quantum computer for a simple algorithmic
benchmark, namely the hidden shift problem. We present and use two tool flows
which invoke RevKit. One which is based on ProjectQ and which targets the IBM
Quantum Experience or a local simulator, and one which is based on Microsoft's
quantum programming language Q.Comment: 10 pages, 10 figures. To appear in: Proceedings of Design, Automation
and Test in Europe (DATE 2018
Descriptional complexity of cellular automata and decidability questions
We study the descriptional complexity of cellular automata (CA), a parallel model of computation. We show that between one of the simplest cellular models, the realtime-OCA. and "classical" models like deterministic finite automata (DFA) or pushdown automata (PDA), there will be savings concerning the size of description not bounded by any recursive function, a so-called nonrecursive trade-off. Furthermore, nonrecursive trade-offs are shown between some restricted classes of cellular automata. The set of valid computations of a Turing machine can be recognized by a realtime-OCA. This implies that many decidability questions are not even semi decidable for cellular automata. There is no pumping lemma and no minimization algorithm for cellular automata
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