88 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
1991 July, Memphis State University bulletin
Vol. 80, No. 4 of the Memphis State University bulletin containing the graduate catalog for 1991-92, 1991 July.https://digitalcommons.memphis.edu/speccoll-ua-pub-bulletins/1173/thumbnail.jp
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Symmetric Circuits and Model-Theoretic Logics
The question of whether there is a logic that characterises polynomial-time is arguably the
most important open question in finite model theory. The study of extensions of fixed-point
logic are of central importance to this question. It was shown by Anderson and Dawar that
fixed-point logic with counting (FPC) has the same expressive power as uniform families of
symmetric circuits over a basis with threshold functions.
In this thesis we prove a far-reaching generalisation of their result and establish an
analogous circuit characterisation for each from a broad range of extensions of fixed-point
logic. In order to do so we fist develop a very general framework for defining and studying
extensions of fixed-point logics, which we call generalised operators. These operators generalise
Lindström quantifiers as well as the counting and rank operators used to define FPC and
fixed-point logic with rank (FPR).
We also show that in order to define a symmetric circuit model that goes beyond FPC
we need to consider circuits with gates that are allowed to compute non-symmetric functions.
In order to do so we develop a far more general framework for studying circuits. We also
show that key notions, such as the notion of a symmetric circuit, can be analogously defined
in this more general framework. The characterisation of FPC in terms of symmetric circuits,
and the treatment of circuits generally, relies heavily on the assumption that the gates in
the circuit compute symmetric functions. We develop a broad range of new techniques and
approaches in order to study these more general symmetric circuit models.
As a corollary of our main result we establish a circuit characterisation of FPR. We also
show that the question of whether there is a logic that characterises polynomial-time can
be understood as a question about the symmetry property of circuits. We lastly propose
a number of new approaches that might exploit this new-found connection between circuit
complexity and descriptive complexity.Gates Cambridge Scholarship
Large-alphabet sequence modelling - a comparative study
Most raw data is not binary, but over some often large and structured alphabet. Sometimes it is convenient to deal with binarised data sequence, but typically exploiting the original structure of the data significantly improves performance in many practical applications. In this thesis, we study Martin-Lof random sequences that are maximally incompressible and provide a topological view on the size of the set of random sequences. We also investigate the relationship between binary data compression techniques and modelling natural language text with the latter using raw unbinarised data sequence from a large alphabet. We perform an experimental comparative study for them, including an empirical comparison between Kneser-Ney (KN) variants with regular Context Tree Weighting algorithm (CTW) and phase CTW, and with large-alphabet CTW with different estimators. We also apply the idea of Hutter's adaptive sparse Dirichlet-multinomial coding to the KN method and provide a heuristic to make the discounting parameter adaptive. The KN with this adaptive discounting parameter outperforms the traditional KN method on the Large Calgary corpus
Election-Attack Complexity for More Natural Models
Elections are arguably the best way that a group of agents with preferences over a set of choices can reach a decision. This can include political domains, as well as multiagent systems in artificial-intelligence settings. It is well-known that every reasonable election system is manipulable, but determining whether such a manipulation exists may be computationally infeasible. We build on an exciting line of research that considers the complexity of election-attack problems, which include voters misrepresenting their preferences (manipulation) and attacks on the structure of the election itself (control). We must properly model such attacks and the preferences of the electorate to give us insight into the difficulty of election attacks in natural settings. This includes models for how the voters can state their preferences, their structure, and new models for the election attack itself.
We study several different natural models on the structure of the voters. In the computational study of election attacks it is generally assumed that voters strictly rank all of the candidates from most to least preferred. We consider the very natural model where voters are able to cast votes with ties, and the model where they additionally have a single-peaked structure. Specifically, we explore how voters with varying amounts of ties and structure in their preferences affect the computational complexity of different election attacks and the complexity of determining whether a given electorate is single-peaked.
For the representation of the voters, we consider how representing the voters succinctly affects the complexity of election attacks and discuss how approaches for the nonsuccinct case can be adapted.
Control and manipulation are two of the most commonly studied election-attack problems. We introduce a model of electoral control in the setting where some of the voters act strategically (i.e., are manipulators), and consider both the case where the agent controlling the election and the manipulators share a goal, and the case where they have competing goals.
The computational study of election-attack problems allows us to better understand how different election systems compare to one another, and it is important to study these problems for natural settings, as this thesis does
Query Answering in Probabilistic Data and Knowledge Bases
Probabilistic data and knowledge bases are becoming increasingly important in academia and industry. They are continuously extended with new data, powered by modern information extraction tools that associate probabilities with knowledge base facts. The state of the art to store and process such data is founded on probabilistic database systems, which are widely and successfully employed. Beyond all the success stories, however, such systems still lack the fundamental machinery to convey some of the valuable knowledge hidden in them to the end user, which limits their potential applications in practice. In particular, in their classical form, such systems are typically based on strong, unrealistic limitations, such as the closed-world assumption, the closed-domain assumption, the tuple-independence assumption, and the lack of commonsense knowledge. These limitations do not only lead to unwanted consequences, but also put such systems on weak footing in important tasks, querying answering being a very central one. In this thesis, we enhance probabilistic data and knowledge bases with more realistic data models, thereby allowing for better means for querying them. Building on the long endeavor of unifying logic and probability, we develop different rigorous semantics for probabilistic data and knowledge bases, analyze their computational properties and identify sources of (in)tractability and design practical scalable query answering algorithms whenever possible. To achieve this, the current work brings together some recent paradigms from logics, probabilistic inference, and database theory
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