72 research outputs found
Transitive Closure Logic and Multihead Automata with Nested Pebbles
Several extensions of first-order logic are studied in descriptive complexity theory. These extensions include transitive closure logic and deterministic transitive closure logic, which extend first-order logic with transitive closure operators. It is known that deterministic transitive closure logic captures the complexity class of the languages that are decidable by some deterministic Turing machine using a logarithmic amount of memory space. An analogous result holds for transitive closure logic and nondeterministic Turing machines.
This thesis concerns the k-ary fragments of these two logics. In each k-ary fragment, the arities of transitive closure operators appearing in formulas are restricted to a nonzero natural number k. The expressivity of these fragments can be studied in terms of multihead finite automata. The type of automaton that we consider in this thesis is a two-way multihead automaton with nested pebbles.
We look at the expressive power of multihead automata and the k-ary fragments of transitive closure logics in the class of finite structures called word models. We show that deterministic twoway k-head automata with nested pebbles have the same expressive power as first-order logic with k-ary deterministic transitive closure. For a corresponding result in the case of nondeterministic automata, we restrict to the positive fragment of k-ary transitive closure logic. The two theorems and their proofs are based on the article âAutomata with nested pebbles capture first-order logic with transitive closureâ by Joost Engelfriet and Hendrik Jan Hoogeboom. In the article, the results are proved in the case of trees. Since word models can be viewed as a special type of trees, the theorems considered in this thesis are a special case of a more general result
An exponential lower bound for Individualization-Refinement algorithms for Graph Isomorphism
The individualization-refinement paradigm provides a strong toolbox for
testing isomorphism of two graphs and indeed, the currently fastest
implementations of isomorphism solvers all follow this approach. While these
solvers are fast in practice, from a theoretical point of view, no general
lower bounds concerning the worst case complexity of these tools are known. In
fact, it is an open question whether individualization-refinement algorithms
can achieve upper bounds on the running time similar to the more theoretical
techniques based on a group theoretic approach.
In this work we give a negative answer to this question and construct a
family of graphs on which algorithms based on the individualization-refinement
paradigm require exponential time. Contrary to a previous construction of
Miyazaki, that only applies to a specific implementation within the
individualization-refinement framework, our construction is immune to changing
the cell selector, or adding various heuristic invariants to the algorithm.
Furthermore, our graphs also provide exponential lower bounds in the case when
the -dimensional Weisfeiler-Leman algorithm is used to replace the standard
color refinement operator and the arguments even work when the entire
automorphism group of the inputs is initially provided to the algorithm.Comment: 21 page
Polar Varieties and Efficient Real Elimination
Let be a smooth and compact real variety given by a reduced regular
sequence of polynomials . This paper is devoted to the
algorithmic problem of finding {\em efficiently} a representative point for
each connected component of . For this purpose we exhibit explicit
polynomial equations that describe the generic polar varieties of . This
leads to a procedure which solves our algorithmic problem in time that is
polynomial in the (extrinsic) description length of the input equations and in a suitably introduced, intrinsic geometric parameter, called
the {\em degree} of the real interpretation of the given equation system .Comment: 32 page
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Solving linear programs without breaking abstractions
We show that the ellipsoid method for solving linear programs can be implemented in a way that respects the symmetry of the program being solved. That is to say, there is an algorithmic implementation of the method that does not distinguish, or make choices, between variables or constraints in the program unless they are distinguished by properties definable from the program. In particular, we demonstrate that the solvability of linear programs can be expressed in fixed-point logic with counting (FPC) as long as the program is given by a separation oracle that is itself definable in FPC. We use this to show that the size of a maximum matching in a graph is definable in FPC. This settles an open problem first posed by Blass, Gurevich and Shelah [Blass et al. 1999]. On the way to defining a suitable separation oracle for the maximum matching program, we provide FPC formulas defining canonical maximum flows and minimum cuts in undirected capacitated graphs.Research supported by EPSRC grant EP/H026835.This is the author accepted manuscript. The final version is available from ACM via http://dx.doi.org/10.1145/282289
Machines at play: The attraction of automation
Taking as its starting point the ubiquitous nature of automated technology, this research asks how play may be used in an antagonistic form against the regimentation of machines but, conversely, may also be employed to instrumentalise them. The work undertaken specifically focuses on how play (a quality considered here as intrinsic to human culture and nature following Johan Huizingaâs Homo Ludens) can expose issues of control, agency and authority within a technological context.
While automated machines have become increasingly complex over time (synchronous to the trickle-down availability of computing devices to the everyday consumer), the understanding of their function and the means through which they produce, represent or declare forms of âknowledgeâ are today even more opaque. An automated machineâthought of here as being any set of infinitely repeatable, programmed proceduresâraises anxiety as to the human condition. Machina ludens, the figure of the playing machine that I propose, takes this model a step further and uses âattractiveâ effects to produce (what Huizinga terms) âfalse playâ so as to hide the ramification of any social or political design by its engineer. Following VilĂ©m Flusser and Bruno Latourâs notion of the âblack boxâ, how then can an artist open up an automated machine and its script in order to declare this?
The research is undertaken through an interlinked practical and written component. These components use a methodology that undertakes an analysis of the play-element, alongside a technological/engineering analysis of the machine-element in culture. In practice, following a lineage of artists who have similarly made use of technology in the production of machines in their artwork, from Jean Tinguely and the E.A.T. group to Harold Cohenâs AARON, the research examines various forms of the âart machineâ. Both the written and practical works use the tension (or contention) between disciplines, the researcher overtly taking the position of being simultaneously engineer and artist. As such, this research is a re-reading of Huizingaâs understanding of the play-element of culture through a contemporary, technological lens that bridges the gap between a humanities/philosophical approach and an engineering approach, applying this to contemporary issues surrounding automated âart machinesâ
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