Learning algebraic structures from text

AbstractThe present work investigates the learnability of classes of substructures of some algebraic structures: submonoids and subgroups of given groups, ideals of given commutative rings, subfields of given vector spaces. The learner sees all positive data but no negative one and converges to a program enumerating or computing the set to be learned. Besides semantical (BC) and syntactical (Ex) convergence also the more restrictive ordinal bounds on the number of mind changes are considered. The following is shown: (a) Learnability depends much on the amount of semantic knowledge given at the synthesis of the learner where this knowledge is represented by programs for the algebraic operations, codes for prominent elements of the algebraic structure (like 0 and 1 fields) and certain parameters (like the dimension of finite-dimensional vector spaces). For several natural examples, good knowledge of the semantics may enable to keep ordinal mind change bounds while restricted knowledge may either allow only BC-convergence or even not permit learnability at all.(b) The class of all ideals of a recursive ring is BC-learnable iff the ring is Noetherian. Furthermore, one has either only a BC-learner outputting enumerable indices or one can already get an Ex-learner converging to decision procedures and respecting an ordinal bound on the number of mind changes. The ring is Artinian iff the ideals can be Ex-learned with a constant bound on the number of mind changes, this constant is the length of the ring. Ex-learnability depends not only on the ring but also on the representation of the ring. Polynomial rings over the field of rationals with n variables have exactly the ordinal mind change bound ωn in the standard representation. Similar results can be established for unars. Noetherian unars with one function can be learned with an ordinal mind change bound aω for some a

A Decision Procedure for XPath Containment

XPath is the standard language for addressing parts of an XML document. We present a sound and complete decision procedure for containment of XPath queries. The considered XPath fragment covers most of the language features used in practice. Specifically, we show how XPath queries can be translated into equivalent formulas in monadic second-order logic. Using this translation, we construct an optimized logical formulation of the containment problem, which is decided using tree automata. When the containment relation does not hold between two XPath expressions, a counter-example XML tree is generated. We provide a complexity analysis together with practical experiments that illustrate the efficiency of the decision procedure for realistic scenarios

The Derivational Complexity Induced by the Dependency Pair Method

We study the derivational complexity induced by the dependency pair method, enhanced with standard refinements. We obtain upper bounds on the derivational complexity induced by the dependency pair method in terms of the derivational complexity of the base techniques employed. In particular we show that the derivational complexity induced by the dependency pair method based on some direct technique, possibly refined by argument filtering, the usable rules criterion, or dependency graphs, is primitive recursive in the derivational complexity induced by the direct method. This implies that the derivational complexity induced by a standard application of the dependency pair method based on traditional termination orders like KBO, LPO, and MPO is exactly the same as if those orders were applied as the only termination technique

Algorithms and the mathematical foundations of computer science

The goal of this chapter is to bring to the attention of philosophers of mathematics the concept of algorithm as it is studied incontemporary theoretical computer science, and at the same time address several foundational questions about the role this notion plays in our practices. A view known as algorithmic realism will be described which maintains that individual algorithms are identical to mathematical objects. Upon considering several ways in which the details of algorithmic realism might be formulated, it will be argued (pace Moschovakis and Gurevich) that there are principled reasons to think that this view cannot be systematically developed in a manner which is compatible with the practice of computational complexity theory and algorithmic analysis

The use of proof plans in tactic synthesis

We undertake a programme of tactic synthesis. We first formalize the notion of a tactic as a rewrite rule, then give a correctness criterion for this by means of a reflection mechanism in the constructive type theory OYSTER. We further formalize the notion of a tactic specification, given as a synthesis goal and a decidability goal. We use a proof planner. CIAM. to guide the search for inductive proofs of these, and are able to successfully synthesize several tactics in this fashion. This involves two extensions to existing methods: context-sensitive rewriting and higher-order wave rules. Further, we show that from a proof of the decidability goal one may compile to a Prolog program a pseudo- tactic which may be run to efficiently simulate the input/output behaviour of the synthetic tacti