13,192 research outputs found
Classifying the Arithmetical Complexity of Teaching Models
This paper classifies the complexity of various teaching models by their
position in the arithmetical hierarchy. In particular, we determine the
arithmetical complexity of the index sets of the following classes: (1) the
class of uniformly r.e. families with finite teaching dimension, and (2) the
class of uniformly r.e. families with finite positive recursive teaching
dimension witnessed by a uniformly r.e. teaching sequence. We also derive the
arithmetical complexity of several other decision problems in teaching, such as
the problem of deciding, given an effective coding of all uniformly r.e. families, any such that
, any and , whether or not the
teaching dimension of with respect to is upper bounded
by .Comment: 15 pages in International Conference on Algorithmic Learning Theory,
201
A Theory of Formal Synthesis via Inductive Learning
Formal synthesis is the process of generating a program satisfying a
high-level formal specification. In recent times, effective formal synthesis
methods have been proposed based on the use of inductive learning. We refer to
this class of methods that learn programs from examples as formal inductive
synthesis. In this paper, we present a theoretical framework for formal
inductive synthesis. We discuss how formal inductive synthesis differs from
traditional machine learning. We then describe oracle-guided inductive
synthesis (OGIS), a framework that captures a family of synthesizers that
operate by iteratively querying an oracle. An instance of OGIS that has had
much practical impact is counterexample-guided inductive synthesis (CEGIS). We
present a theoretical characterization of CEGIS for learning any program that
computes a recursive language. In particular, we analyze the relative power of
CEGIS variants where the types of counterexamples generated by the oracle
varies. We also consider the impact of bounded versus unbounded memory
available to the learning algorithm. In the special case where the universe of
candidate programs is finite, we relate the speed of convergence to the notion
of teaching dimension studied in machine learning theory. Altogether, the
results of the paper take a first step towards a theoretical foundation for the
emerging field of formal inductive synthesis
Bounding Embeddings of VC Classes into Maximum Classes
One of the earliest conjectures in computational learning theory-the Sample
Compression conjecture-asserts that concept classes (equivalently set systems)
admit compression schemes of size linear in their VC dimension. To-date this
statement is known to be true for maximum classes---those that possess maximum
cardinality for their VC dimension. The most promising approach to positively
resolving the conjecture is by embedding general VC classes into maximum
classes without super-linear increase to their VC dimensions, as such
embeddings would extend the known compression schemes to all VC classes. We
show that maximum classes can be characterised by a local-connectivity property
of the graph obtained by viewing the class as a cubical complex. This geometric
characterisation of maximum VC classes is applied to prove a negative embedding
result which demonstrates VC-d classes that cannot be embedded in any maximum
class of VC dimension lower than 2d. On the other hand, we show that every VC-d
class C embeds in a VC-(d+D) maximum class where D is the deficiency of C,
i.e., the difference between the cardinalities of a maximum VC-d class and of
C. For VC-2 classes in binary n-cubes for 4 <= n <= 6, we give best possible
results on embedding into maximum classes. For some special classes of Boolean
functions, relationships with maximum classes are investigated. Finally we give
a general recursive procedure for embedding VC-d classes into VC-(d+k) maximum
classes for smallest k.Comment: 22 pages, 2 figure
Optimal Collusion-Free Teaching
Formal models of learning from teachers need to respect certain criteria toavoid collusion. The most commonly accepted notion of collusion-freeness wasproposed by Goldman and Mathias (1996), and various teaching models obeyingtheir criterion have been studied. For each model and each concept class, a parameter - refers to theteaching dimension of concept class in model ---defined to bethe number of examples required for teaching a concept, in the worst case overall concepts in . This paper introduces a new model of teaching, called no-clash teaching,together with the corresponding parameter .No-clash teaching is provably optimal in the strong sense that, given anyconcept class and any model obeying Goldman and Mathias'scollusion-freeness criterion, one obtains \mathrm{NCTD}(\mathcal{C})\leM-. We also study a corresponding notion for the case of learning from positive data only, establishuseful bounds on and , and discuss relationsof these parameters to the VC-dimension and to sample compression. In addition to formulating an optimal model of collusion-free teaching, ourmain results are on the computational complexity of deciding whether (or ) for given and . We show some such decision problems to be equivalent tothe existence question for certain constrained matchings in bipartite graphs.Our NP-hardness results for the latter are of independent interest in the studyof constrained graph matchings.<br
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