1,528 research outputs found
Compaction of Church Numerals for Higher-Order Compression
In this study, we address the problem of compacting Church numerals. Church
numerals appear as a representation of the repetitive part of data in
higher-order compression. We propose a novel decomposition scheme for a natural
number using tetration, which leads to a compact representation of
-terms equivalent to the original Church numerals. For natural number
, we prove that the size of the -term obtained by the proposed
method is . Moreover, we
quantitatively confirmed experimentally that the proposed method outperforms a
binary expression of Church numerals when is less than approximately 10000
Linear numeral systems
We investigate numeral systems in the lambda calculus; specifically in the linear lambda calculus where terms cannot be copied or erased. Our interest is threefold: representing numbers in the linear calculus, finding constant time arithmetic operations when possible for successor, addition and predecessor, and finally, efficiently encoding subtraction—an operation that is problematic in many numeral systems. This paper defines systems that address these points, and in addition provides a characterisation of linear numeral systems
Advanced Proof Viewing in ProofTool
Sequent calculus is widely used for formalizing proofs. However, due to the
proliferation of data, understanding the proofs of even simple mathematical
arguments soon becomes impossible. Graphical user interfaces help in this
matter, but since they normally utilize Gentzen's original notation, some of
the problems persist. In this paper, we introduce a number of criteria for
proof visualization which we have found out to be crucial for analyzing proofs.
We then evaluate recent developments in tree visualization with regard to these
criteria and propose the Sunburst Tree layout as a complement to the
traditional tree structure. This layout constructs inferences as concentric
circle arcs around the root inference, allowing the user to focus on the
proof's structural content. Finally, we describe its integration into ProofTool
and explain how it interacts with the Gentzen layout.Comment: In Proceedings UITP 2014, arXiv:1410.785
Exhaustible sets in higher-type computation
We say that a set is exhaustible if it admits algorithmic universal
quantification for continuous predicates in finite time, and searchable if
there is an algorithm that, given any continuous predicate, either selects an
element for which the predicate holds or else tells there is no example. The
Cantor space of infinite sequences of binary digits is known to be searchable.
Searchable sets are exhaustible, and we show that the converse also holds for
sets of hereditarily total elements in the hierarchy of continuous functionals;
moreover, a selection functional can be constructed uniformly from a
quantification functional. We prove that searchable sets are closed under
intersections with decidable sets, and under the formation of computable images
and of finite and countably infinite products. This is related to the fact,
established here, that exhaustible sets are topologically compact. We obtain a
complete description of exhaustible total sets by developing a computational
version of a topological Arzela--Ascoli type characterization of compact
subsets of function spaces. We also show that, in the non-empty case, they are
precisely the computable images of the Cantor space. The emphasis of this paper
is on the theory of exhaustible and searchable sets, but we also briefly sketch
applications
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