30,561 research outputs found
Packed rules for automatic transfer-rule induction
We present a method of encoding transfer rules in a highly efficient packed structure using contextualized constraints (Maxwell and Kaplan, 1991), an existing method of encoding
adopted from LFG parsing (Kaplan and Bresnan, 1982; Bresnan, 2001; Dalrymple, 2001). The packed representation allows us to encode O(2n) transfer rules in a single packed
representation only requiring O(n) storage space. Besides reducing space requirements, the representation also has a high impact on the amount of time taken to load large numbers of transfer rules to memory with very little trade-off in time needed to unpack the rules. We include an experimental evaluation which shows a considerable reduction in space and time requirements for a large set of automatically induced transfer rules by storing the rules in the packed representation
Kripke Models for Classical Logic
We introduce a notion of Kripke model for classical logic for which we
constructively prove soundness and cut-free completeness. We discuss the
novelty of the notion and its potential applications
Computational reverse mathematics and foundational analysis
Reverse mathematics studies which subsystems of second order arithmetic are
equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main
philosophical application of reverse mathematics proposed thus far is
foundational analysis, which explores the limits of different foundations for
mathematics in a formally precise manner. This paper gives a detailed account
of the motivations and methodology of foundational analysis, which have
heretofore been largely left implicit in the practice. It then shows how this
account can be fruitfully applied in the evaluation of major foundational
approaches by a careful examination of two case studies: a partial realization
of Hilbert's program due to Simpson [1988], and predicativism in the extended
form due to Feferman and Sch\"{u}tte.
Shore [2010, 2013] proposes that equivalences in reverse mathematics be
proved in the same way as inequivalences, namely by considering only
-models of the systems in question. Shore refers to this approach as
computational reverse mathematics. This paper shows that despite some
attractive features, computational reverse mathematics is inappropriate for
foundational analysis, for two major reasons. Firstly, the computable
entailment relation employed in computational reverse mathematics does not
preserve justification for the foundational programs above. Secondly,
computable entailment is a complete relation, and hence employing it
commits one to theoretical resources which outstrip those available within any
foundational approach that is proof-theoretically weaker than
.Comment: Submitted. 41 page
Some observations on the logical foundations of inductive theorem proving
In this paper we study the logical foundations of automated inductive theorem
proving. To that aim we first develop a theoretical model that is centered
around the difficulty of finding induction axioms which are sufficient for
proving a goal.
Based on this model, we then analyze the following aspects: the choice of a
proof shape, the choice of an induction rule and the language of the induction
formula. In particular, using model-theoretic techniques, we clarify the
relationship between notions of inductiveness that have been considered in the
literature on automated inductive theorem proving. This is a corrected version
of the paper arXiv:1704.01930v5 published originally on Nov.~16, 2017
Local Induction and Provably Total Computable Functions: A Case Study
Let IÎ â2 denote the fragment of Peano Arithmetic obtained by restricting the induction scheme to parameter free Î 2 formulas. Answering a question of R. Kaye, L. Beklemishev showed that the provably total computable functions (p.t.c.f.) of IÎ â2 are, precisely, the primitive recursive ones. In this work we give a new proof of this fact through an analysis of the p.t.c.f. of certain local versions of induction principles closely related to IÎ â2 . This analysis is essentially based on the equivalence between local induction rules and restricted forms of iteration. In this way, we obtain a more direct answer to Kayeâs question, avoiding the metamathematical machinery (reflection principles, provability logic,...) needed for Beklemishevâs original proof.Ministerio de Ciencia e InnovaciĂłn MTM2008â0643
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