110,441 research outputs found
Derived rules for predicative set theory: an application of sheaves
We show how one may establish proof-theoretic results for constructive
Zermelo-Fraenkel set theory, such as the compactness rule for Cantor space and
the Bar Induction rule for Baire space, by constructing sheaf models and using
their preservation properties
Rerepresenting and Restructuring Domain Theories: A Constructive Induction Approach
Theory revision integrates inductive learning and background knowledge by
combining training examples with a coarse domain theory to produce a more
accurate theory. There are two challenges that theory revision and other
theory-guided systems face. First, a representation language appropriate for
the initial theory may be inappropriate for an improved theory. While the
original representation may concisely express the initial theory, a more
accurate theory forced to use that same representation may be bulky,
cumbersome, and difficult to reach. Second, a theory structure suitable for a
coarse domain theory may be insufficient for a fine-tuned theory. Systems that
produce only small, local changes to a theory have limited value for
accomplishing complex structural alterations that may be required.
Consequently, advanced theory-guided learning systems require flexible
representation and flexible structure. An analysis of various theory revision
systems and theory-guided learning systems reveals specific strengths and
weaknesses in terms of these two desired properties. Designed to capture the
underlying qualities of each system, a new system uses theory-guided
constructive induction. Experiments in three domains show improvement over
previous theory-guided systems. This leads to a study of the behavior,
limitations, and potential of theory-guided constructive induction.Comment: See http://www.jair.org/ for an online appendix and other files
accompanying this articl
Constructive set theory and Brouwerian principles
The paper furnishes realizability models of constructive Zermelo-Fraenkel set theory, CZF, which also validate Brouwerian principles such as the axiom of continuous choice (CC), the fan theorem (FT), and monotone bar induction (BIM), and thereby determines the proof-theoretic strength of CZF augmented by these principles. The upshot is that CZF+CC+FT possesses the same strength as CZF, or more precisely, that CZF+CC+FTis conservative over CZF for 02 statements of arithmetic, whereas the addition of a restricted version of bar induction to CZF (called decidable bar induction, BID) leads to greater proof-theoretic strength in that CZF+BID proves the consistency of CZF
Constructive approaches to Program Induction
Search is a key technique in artificial intelligence, machine learning and Program Induction. No
matter how efficient a search procedure, there exist spaces that are too large to search effectively
and they include the search space of programs. In this dissertation we show that in the context
of logic-program induction (Inductive Logic Programming, or ILP) it is not necessary to search
for a correct program, because if one exists, there also exists a unique object that is the most
general correct program, and that can be constructed directly, without a search, in polynomial
time and from a polynomial number of examples. The existence of this unique object, that we
term the Top Program because of its maximal generality, does not so much solve the problem
of searching a large program search space, as it completely sidesteps it, thus improving the
efficiency of the learning task by orders of magnitude commensurate with the complexity of a
program space search.
The existence of a unique Top Program and the ability to construct it given finite resources
relies on the imposition, on the language of hypotheses, from which programs are constructed,
of a strong inductive bias with relevance to the learning task. In common practice, in machine
learning, Program Induction and ILP, such relevant inductive bias is selected, or created,
manually, by the human user of a learning system, with intuition or knowledge of the problem
domain, and in the form of various kinds of program templates. In this dissertation we show
that by abandoning the reliance on such extra-logical devices as program templates, and instead
defining inductive bias exclusively as First- and Higher-Order Logic formulae, it is possible to
learn inductive bias itself from examples, automatically, and efficiently, by Higher-Order Top
Program construction.
In Chapter 4 we describe the Top Program in the context of the Meta-Interpretive Learning
approach to ILP (MIL) and describe an algorithm for its construction, the Top Program
Construction algorithm (TPC). We prove the efficiency and accuracy of TPC and describe
its implementation in a new MIL system called Louise. We support theoretical results with
experiments comparing Louise to the state-of-the-art, search-based MIL system, Metagol, and
find that Louise improves Metagol’s efficiency and accuracy. In Chapter 5 we re-frame MIL as
specialisation of metarules, Second-Order clauses used as inductive bias in MIL, and prove that
problem-specific metarules can be derived by specialisation of maximally general metarules, by
MIL. We describe a sub-system of Louise, called TOIL, that learns new metarules by MIL and
demonstrate empirically that the metarules learned by TOIL match those selected manually,
while maintaining the accuracy and efficiency of learning.
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A Direct Version of Veldman's Proof of Open Induction on Cantor Space via Delimited Control Operators
First, we reconstruct Wim Veldman's result that Open Induction on Cantor
space can be derived from Double-negation Shift and Markov's Principle. In
doing this, we notice that one has to use a countable choice axiom in the proof
and that Markov's Principle is replaceable by slightly strengthening the
Double-negation Shift schema. We show that this strengthened version of
Double-negation Shift can nonetheless be derived in a constructive intermediate
logic based on delimited control operators, extended with axioms for
higher-type Heyting Arithmetic. We formalize the argument and thus obtain a
proof term that directly derives Open Induction on Cantor space by the shift
and reset delimited control operators of Danvy and Filinski
Discrete Jordan Curve Theorem: A proof formalized in Coq with hypermaps
This paper presents a formalized proof of a discrete form of the Jordan Curve
Theorem. It is based on a hypermap model of planar subdivisions, formal
specifications and proofs assisted by the Coq system. Fundamental properties
are proven by structural or noetherian induction: Genus Theorem, Euler's
Formula, constructive planarity criteria. A notion of ring of faces is
inductively defined and a Jordan Curve Theorem is stated and proven for any
planar hypermap
Right at the start : a research and development agenda for teacher induction
Recent developments in teacher induction in both England and Scotland are bringing long overdue improvements, but there is a range of issues in need of further exploration if policy is to be developed. Current evaluations have begun to reveal the absence of some important conceptual aspects of induction in the somewhat hasty implementation. Some of these have been well rehearsed in the literature over the years but have generally failed to make any impact hitherto in induction policy. This paper picks up and discusses some of the conceptual tensions and weaknesses that have, or are likely to, become practical issues of quality, in both Scottish and English induction policies. These include the use of competence-based descriptions, the non-formal dimension of learning to teach, open narrative and focused approaches to classroom observation and feedback, individualism and a pupil perspective. The array of concepts is organised into a constructive, topical agenda which, it is argued, bring a much needed formative dimension to research and development in this crucial area of professional learning
The principle of pointfree continuity
In the setting of constructive pointfree topology, we introduce a notion of
continuous operation between pointfree topologies and the corresponding
principle of pointfree continuity. An operation between points of pointfree
topologies is continuous if it is induced by a relation between the bases of
the topologies; this gives a rigorous condition for Brouwer's continuity
principle to hold. The principle of pointfree continuity for pointfree
topologies and says that any relation which induces
a continuous operation between points is a morphism from to
. The principle holds under the assumption of bi-spatiality of
. When is the formal Baire space or the formal unit
interval and is the formal topology of natural numbers, the
principle is equivalent to spatiality of the formal Baire space and formal unit
interval, respectively. Some of the well-known connections between spatiality,
bar induction, and compactness of the unit interval are recast in terms of our
principle of continuity.
We adopt the Minimalist Foundation as our constructive foundation, and
positive topology as the notion of pointfree topology. This allows us to
distinguish ideal objects from constructive ones, and in particular, to
interpret choice sequences as points of the formal Baire space
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