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  • 1.
    Lagerkvist, Victor
    Linköping University, Department of Computer and Information Science. Linköping University, The Institute of Technology.
    Restricted Constraint Satisfaction Problems and the Exponential-time Hypothesis2012Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    A constraint satisfaction problem (CSP) can be represented as two structures: the structure induced by the variables and the structure induced by the constraint language. Both these parameters are amenable to restrictions which affects the complexity of the resulting problems. In this thesis we shall use both constraint language restrictions and structural restrictions in order to create problems that can be solved as efficiently as possible. The language restrictions are based on creating a language that in terms of frozen partial clone theory has the largest number of polymorphic functions. Such a language can according to the Galois connection between functions and relations be implemented by as many languages as possible and is therefore the Boolean language with the lowest complexity. The structural restrictions are mainly based on limiting the number of times a variable is allowed to occur in an instance. We shall prove that the easiest language does not contain a Delta-matroid relation and is NP-complete even with the very restricted structure where no variable can occur in more than two constraints. We also give a branch-and-reduce algorithm for this problem with time complexity O(1.0493^n). This problem is then related to the exponential-time hypothesis, which postulates that k-SAT is not sub-exponential for k >= 3. We show that the exponential-time hypothesis holds if and only if this restricted problem is not sub-exponential, if and only if all NP-complete Boolean languages are not sub-exponential. In the process we also prove a stronger version of Impagliazzo's sparsification lemma for k-SAT; namely that all finite, NP-complete Boolean languages can be sparsified into each other. This should be contrasted with Santhanam's negative result which states that the same does not hold for all infinite Boolean languages.

  • 2.
    Lagerqvist, Victor
    Linköping University, Department of Computer and Information Science, TCSLAB - Theoretical Computer Science Laboratory.
    A comparison of SL- and unit-resolution search rules for stratified logic programs2010Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    There are two symmetrical resolution rules applicable to logic programs - SL-resolution which yields a top-down refutation and unit-resolution which yields a bottom-up refutation. Both resolution principles need to be coupled with a search rule before they can be used in practice. The search rule determines in which order program clauses are used in the refutation and affects both performance, completeness and quality of solutions. The thesis surveys exhaustive and heuristic search rules for SL-resolution and transformation techniques for (general) logic programs that makes unit-resolution goal oriented.

    The search rules were implemented as meta-interpreters for Prolog and were benchmarked on a suite of programs incorporating both deterministic and nondeterministic code. Whenever deemed applicable benchmark programs were permuted with respect to clause and goal ordering to see if it affected the interpreters performance and termination.

    With the help of the evaluation the conclusion was that alternative search rules for SL-resolution should not be used for performance gains but can in some cases greatly improve the quality of solutions, e.g. in planning or other applications where the quality of an answer correlates with the length of the refutation. It was also established that A* is more flexible than exhaustive search rules since its behavior can be fine-tuned with weighting, and can in some cases be more efficient than both iterative deepening and breadth-first search. The bottom-up interpreter based on unit-resolution and magic transformation had several advantages over the top-down interpreters. Notably for programs where subgoals are recomputed many times. The great disparity in implementation techniques made direct performance comparisons hard however, and it is not clear if even an optimized bottom-up interpreter is competitive against a top-down interpreter with tabling of answers.

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