1,925 research outputs found
The Brownian continuum random tree as the unique solution to a fixed point equation
In this note, we provide a new characterization of Aldous' Brownian continuum
random tree as the unique fixed point of a certain natural operation on
continuum trees (which gives rise to a recursive distributional equation). We
also show that this fixed point is attractive.Comment: 15 pages, 3 figure
MALL proof equivalence is Logspace-complete, via binary decision diagrams
Proof equivalence in a logic is the problem of deciding whether two proofs
are equivalent modulo a set of permutation of rules that reflects the
commutative conversions of its cut-elimination procedure. As such, it is
related to the question of proofnets: finding canonical representatives of
equivalence classes of proofs that have good computational properties. It can
also be seen as the word problem for the notion of free category corresponding
to the logic.
It has been recently shown that proof equivalence in MLL (the multiplicative
with units fragment of linear logic) is PSPACE-complete, which rules out any
low-complexity notion of proofnet for this particular logic.
Since it is another fragment of linear logic for which attempts to define a
fully satisfactory low-complexity notion of proofnet have not been successful
so far, we study proof equivalence in MALL- (multiplicative-additive without
units fragment of linear logic) and discover a situation that is totally
different from the MLL case. Indeed, we show that proof equivalence in MALL-
corresponds (under AC0 reductions) to equivalence of binary decision diagrams,
a data structure widely used to represent and analyze Boolean functions
efficiently.
We show these two equivalent problems to be LOGSPACE-complete. If this
technically leaves open the possibility for a complete solution to the question
of proofnets for MALL-, the established relation with binary decision diagrams
actually suggests a negative solution to this problem.Comment: in TLCA 201
Decision diagrams in machine learning: an empirical study on real-life credit-risk data.
Decision trees are a widely used knowledge representation in machine learning. However, one of their main drawbacks is the inherent replication of isomorphic subtrees, as a result of which the produced classifiers might become too large to be comprehensible by the human experts that have to validate them. Alternatively, decision diagrams, a generalization of decision trees taking on the form of a rooted, acyclic digraph instead of a tree, have occasionally been suggested as a potentially more compact representation. Their application in machine learning has nonetheless been criticized, because the theoretical size advantages of subgraph sharing did not always directly materialize in the relatively scarce reported experiments on real-world data. Therefore, in this paper, starting from a series of rule sets extracted from three real-life credit-scoring data sets, we will empirically assess to what extent decision diagrams are able to provide a compact visual description. Furthermore, we will investigate the practical impact of finding a good attribute ordering on the achieved size savings.Advantages; Classifiers; Credit scoring; Data; Decision; Decision diagrams; Decision trees; Empirical study; Knowledge; Learning; Real life; Representation; Size; Studies;
Multiplicative-Additive Proof Equivalence is Logspace-complete, via Binary Decision Trees
Given a logic presented in a sequent calculus, a natural question is that of
equivalence of proofs: to determine whether two given proofs are equated by any
denotational semantics, ie any categorical interpretation of the logic
compatible with its cut-elimination procedure. This notion can usually be
captured syntactically by a set of rule permutations.
Very generally, proofnets can be defined as combinatorial objects which
provide canonical representatives of equivalence classes of proofs. In
particular, the existence of proof nets for a logic provides a solution to the
equivalence problem of this logic. In certain fragments of linear logic, it is
possible to give a notion of proofnet with good computational properties,
making it a suitable representation of proofs for studying the cut-elimination
procedure, among other things.
It has recently been proved that there cannot be such a notion of proofnets
for the multiplicative (with units) fragment of linear logic, due to the
equivalence problem for this logic being Pspace-complete.
We investigate the multiplicative-additive (without unit) fragment of linear
logic and show it is closely related to binary decision trees: we build a
representation of proofs based on binary decision trees, reducing proof
equivalence to decision tree equivalence, and give a converse encoding of
binary decision trees as proofs. We get as our main result that the complexity
of the proof equivalence problem of the studied fragment is Logspace-complete.Comment: arXiv admin note: text overlap with arXiv:1502.0199
- …