Remarks on Inheritance Systems
ABSTRACT We try a conceptual analysis of inheritance diagrams, first in abstract terms, and then compare to "normality" and the "small/big sets" of preferential and related reasoning. The main ideas are about nodes as truth values and information sources, truth comparison by paths, accessibility or relevance of information by paths, relative normality, and prototypical reasoning.
arXiv:math/0611937v4 [math.LO] 19 Aug 2007
AN ANALYSIS OF DEFEASIBLE INHERITANCE
Laboratoire d’Informatique Fondamentale de Marseille†
August 18, 2007
We give a conceptual analysis of (defeasible or nonmonotonic) inheritance dia-
grams, and compare our analysis to the ”small/big sets” of preferential and related
In our analysis, we consider nodes as information sources and truth values, direct
links as information, and valid paths as information channels and comparisons of
truth values. This results in an upward chaining, split validity, off-path preclusion
We show that the small/big sets of preferential reasoning have to be relativized
if we want them to conform to inheritance theory, resulting in a more cautious
approach, perhaps closer to actual human reasoning.
Finally, we interpret inheritance diagrams as theories of prototypical reasoning,
based on two distances: set difference, and information difference.
We will also see that some of the major distinctions between inheritance for-
malisms are consequences of deeper and more general problems of treating conflict-
AMS Classification: 68T27, 68T30
∗email@example.com, firstname.lastname@example.org, http://www.cmi.univ-mrs.fr/ ∼ ks
†UMR 6166, CNRS and Universit´ e de Provence, Address: CMI, 39, rue Joliot-Curie, F-13453 Marseille
Cedex 13, France
2Introduction to nonmonotonic inheritance4
2.1Basic discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4
2.2Directly sceptical split validity upward chaining off-path inheritance. . .8
2.3Review of other approaches and problems . . . . . . . . . . . . . . . . . .17
3Introduction to small and big sets and the logical systems P and R19
4.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24
4.2 Informal comparison of inheritance with the systems P and R . . . . . . .24
4.3Inheritance as information transfer. . . . . . . . . . . . . . . . . . . . .25
4.4 Inheritance as reasoning with prototypes . . . . . . . . . . . . . . . . . .30
5 Detailed translation of inheritance to modified systems of small sets34
5.1Normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34
5.2Small sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34
Throughout this paper, ”inheritance” will stand for ”nonmonotonic or defeasible inher-
itance”. We will use indiscriminately ”inheritance system”, ”inheritance diagram”, ”in-
heritance network”, ”inheritance net”.
In this introduction, we first give the motivation for this article, then describe in very
brief terms the basic components of inheritance diagrams, mention the basic ideas of our
analysis, as well as some more general decisions about treating contradictory information.
Inheritance sytems or diagrams have an intuitive appeal. They seem close to human rea-
soning, natural, and are also implemented (see [Mor98]). Yet, they are a more procedural
approach to nonmonotonic reasoning, and, to the author’s knowledge, a conceptual analy-
sis, leading to a formal semantics, as well as a comparison to more logic based formalisms
like the systems P and R of preferential systems are lacking. We attempt to reduce the
gap between the more procedural and the more analytical approaches in this particular
case. This will also give indications how to modify systems P and R to approach them
more to actual human reasoning. Moreover, we establish a link to multi-valued logics and
the logics of information sources (see e.g. [ABK07] and forthcoming work of the same
authors, and also [BGH95]).
Inheritance diagrams are deceptively simple. Their conceptually complicated nature is
seen by e.g. the fundamental difference between direct links and valid paths, and the
multitude of existing formalisms, upward vs. downward chaining, intersection of exten-
sions vs. direct scepticism, on-path vs. off-path preclusion (or pre-emption), split validity
vs. total validity preclusion etc., to name a few. Such a proliferation of formalisms usually
hints at deeper problems on the conceptual side, i.e. that the underlying ideas are am-
bigous, and not sufficiently analysed. Therefore, any clarification and resulting reduction
of possible formalisms seems a priori to make progress. Such clarification will involve
conceptual decisions, which need not be shared by all, they can only be suggestions. Of
course, a proof that such decisions are correct is impossible, and so is its contrary.
We will introduce into the analysis of inheritance systems a number of concepts not
usually found in the field, like multiple truth values, access to information, comparison of
truth values, etc. We think that this additional conceptual burden pays off by a better
comprehension and analysis of the problems behind the surface of inheritance.
We will also see that some distinctions between inheritance formalisms go far beyond ques-
tions of inheritance, and concern general problems of treating contradictory information
- isolating some of these is another objective of this article.
The text is essentially self-contained, still some familiarity with the basic concepts of
inheritance systems and nonmonotonic logics in general is helpful. For a presentation, the
reader might look into [Sch97-2] and [Sch04].
A.Bochman, has pointed out to author work by J.Barwise, D.Gabbay, C.Hartonas,
[BGH95], on information flow. This has a superficial resemblance with the present pages.
But, first, the BGH work is much deeper into logic, presenting sequent calculi, complete-
ness results, etc. Second, our work is on non-monotonic logics, which BGH is not, and our
main thrust is a conceptual analysis of inheritance networks, also working with multiple
truth values. But the basic ideas are about similar situations.
The text is organized as follows. After an introduction to inheritance theory and big/small
subsets and the systems P and R in Section 2 and Section 3, we turn to an informal
description of the fundamental differences between inheritance and the systems P and
R in Section 4.2, give an analysis of inheritance systems in terms of information and
information flow in Section 4.3, then in terms of reasoning with prototypes in Section
4.4, and conclude in Section 5 with a translation of inheritance into (necessarily deeply
modified) coherent systems of big/small sets, respectively logical systems P and R.
One of the main modifications will be to relativize the notions of small/big, which thus
become less ”mathematically pure” but perhaps closer to actual use in ”dirty” common
2Introduction to nonmonotonic inheritance
We give here an informal discussion. The reader unfamiliar with inheritance systems
should consult in parallel Definition 2.3 and Definition 2.4. As there are many variants of
the definitions, it seems reasonable to discuss them before a formal introduction, which,
otherwise, would pretend to be definite without being so.
(Defeasible or nonmonotonic) inheritance networks or diagrams
Nonmonotonic inheritance systems describe situations like ”normally, birds fly”, written
birds → fly. Exceptions are permitted, ”normally penguins don′t fly”, penguins ?→ fly.
(+++*** Orig. No.: Definition 2.1 )
A nonmonotonic inheritance net is a finite DAG, directed, acyclic graph, with two types
of arrows or links, → and ?→, and labelled nodes. We will use Γ etc. for such graphs, and
σ etc. for paths - the latter to be defined below.
Roughly (and to be made precise and modified below, we try to give here just a first
intuition), X → Y means that ”normal” elements of X are in Y, and X ?→ Y means that
”normal” elements of X are not in Y. In a semi-quantitative set interpretation, we will
read ”most elements of X are in Y′′, ”most elements of X are not in Y′′, etc. This is by
no means the only interpretation, as we will see.
According to the set intrepretation, we will also use informally expressions like X ∩ Y,
X-Y, CX, etc. But we will also use nodes informally as formulas, like X ∧ Y, X ∧ ¬Y,
¬X, etc. All this will only be used as an appeal to intuition.
Nodes at the beginning of an arrow can also stand for individuals, so Tweety ?→ fly
means something like: ”normally, Tweety will not fly”.
systems, exceptions are permitted, so the soft rules ”birds fly”, ”penguins don′t fly”,
As always in nonmonotonic
and (the hard rule) ”penguins are birds” can coexist in one diagram, penguins are then
abnormal birds (with respect to flying). The direct link penguins ?→ fly will thus be
accepted, or considered valid, but not the composite path penguins → birds → fly, by
specificity - see below. This is illustrated by Diagram 2.1, where a stands for Tweety, c
for penguins, b for birds, d for flying animals or objects.
(Remark: The arrows a → c, a → b, and c → b can also be composite paths - see below
for the details.)
The Tweety diagram
(Of course, there is an analogous case for the opposite polarity, i.e. when the arrow from
b to d is negative, and the one from c to d is positive.)
The main problem is to define in an intuitively acceptable way a notion of valid path, i.e.
concatenations of arrows satisfying certain properties.
We will write Γ |= σ, if σ is a valid path in the network Γ, and if x is the origin, and
y the endpoint of σ, and σ is positive, we will write Γ |= xy, i.e. we will accept the
conclusion that x’s are y′s, and analogously Γ |= xy for negative paths. Note that we
will not accept any other conclusions, only those established by a valid path, so many
questions about conclusions have a trivial negative answer: there is obviously no path
from x to y. E.g., there is no path from b to c in Diagram 2.1. Likewise, there are no