Prospettive logiche sul ragionamento incerto in intelligenza artificiale

Prospettive logiche sul ragionamento incerto in
intelligenza artificiale
(lavori in corso!!!)
H. Hosni
hykel@maths.man.ac.uk,
12 novembre 2003
1 Un po’ di coordinate
1.1 Intelligenza artificiale
Per cominciare: [McC96]
In via del tutto generale si possono individuare due scopi principali dell’indagine
in IA: la spiegazione del “comportamento intelligente” e la sua
implementazione in sistemi computazionali.
Sebbene non sia ragionevolmente possibile fornire una caratterizzazione
univoca del dominio di indagine dell’IA, può essere utile, in via del tutto
approssimativa, elencare alcuni dei suoi ambiti principali di ricerca.
IA logica Caratterizzata dall’uso della logica matematica come strumento
metodologico e al tempo stesso strumento di indagine principale.
Uno degli obiettivi di questo filone di ricerca consiste nel fornire una
caratterizzazione formale (logica) del “comportamento intelligente”.
IA cognitiva Rivolta principalmente alla spiegazione del comportamento
intelligente osservato negli esseri umani e negli animali. La spiegazione
fornita è quindi algoritmica.
IA robotica Principalmente incentrata nella realizzazione fisica di agenti
intelligenti costituisce l’ambito meno omogeneo, e quindi immediatamente
caratterizzabile, sia per strumenti, sia per metodologia.
Come si ha modo di notare, ogni ambito è connesso all’ambito immediatamente
adiacente ma, nello schema precedente, l’IA logica non non è
necessariamente connessa a quella robotica né per scopi, né per metodologia.
1

1.1.1 Indicazioni bibliografiche generali
Tre manuali classici: [RN95, Nil98, PMG98]. Con la possibile eccezione
del primo, in questi manuali vengono trattati tutti gli aspetti indicati nello
schema precedente.
2 Ragionamento incerto in IA
La centralità del ragionamento incerto in IA è diretta conseguenza delle limitazioni
che caratterizzano tutti gli agenti finiti in termini di capacità computazionale
(e quindi memorizzazione e estrazione di informazione) e capacità
di disporre di “informazione perfetta”, cioè completa, precisa, consistente.
Alcuni aspetti della seconda limitazione dipendono in modo significativo dall’ambiente
in cui operano gli agenti. Nel caso di ambienti realistici, si può
ragionevolmente assumere che l’informazione a disposizione degli agenti sia
inevitabilmente imperfetta.
Esistono molti approcci logico-matematici al problema, ognuno dei quali
tende alla formalizzazione di una delle caratteristiche che pregiudicano la
perfezione dell’informazione e, di conseguenza, la certezza –intesa come, p.e.
grado “razionale” di convinzione– delle inferenze. Per facilità di esposizione
si possono distinguere (in modo assolutamente non esclusivo né esaustivo)
impostazioni non-monotoniche e probabilistiche da un lato e imposazioni
basate su sematiche a “mondi possibili” e polivalenti dall’altro. Le ultime
si distinguono dai sistemi logici non-monotonici e probabilistici nella misura
in cui, pur generalizzando il ragionamento classico -bivalente e monotoniconon
sono direttamente incentrate sulla caratterizzazione formale del comportamento/ragionamento
intelligente. Questa nota è fortemente sbilanciata
verso la mia area di ricerca, ovvero la caratterizzazione matematica del
“comportamento intelligente”.
Nei termini della suddivisione qui proposta, è interessante notare come il
fenomeno di imperfezione dell’informazione che viene attaccato dalle logiche
nonmonotoniche e probabilistiche (e il plurale è irriducibile in entrambe
le aree) consiste nell’incompletezza. Si assume, cioè che gli agenti si trovino
costantemente nell’impossibilità di possedere informazioni sufficienti
per la determinazione univoca e “certa” della chiusura logica (nel senso
Tarskiano, quindi, classica) della propria “base di conoscenza”. Nel caso
delle logiche polivalenti, per esempio, questa non è un’assunzione necessaria,
poichè il fenomeno largamente preso in considerazione è, per quanto
riguarda logiche con infiniti valori di verità, quello dell’intrinseca granularità
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dell’informazione. Ma logica di Lukasiewicz, per esempio, è classicamente
monotonica.
2.0.2 Indicazioni bibliografiche generali
Una raccolta, datata ma abbastanza generale è [SP90]. Più recente e di
livello più avanzato sono i manuali [GS98a, GS98b, GS98c]. Recentissimo
e generale, soprattutto per quanto riguarda la trattazione di impostazioni
logiche probabilistiche e non è il manuale [Hal03].
(Sui vari tipi di imperfezione nell’informazione e la loro relazione con
i rispettivi sistemi logici vedi la sezione 4 di [Hos03] e i riferimenti bibliografici
lì contenuti mentre per un’analisi più approfondita sulla relazione
tra “razionalità”, “incertezza” in intelligenza artificiale si veda la sezione 2
di [Hos02].)
2.1 Ragionamento non-monotonico
Per cominciare: [LM92]
Some say [GW03] that nonmonotonic logic begun with Aristotle. There
is little doubt, however, that nowadays underpinning the interest in this subject
is logic-based artificial intelligence. The special issue of Artificial Intelligence,
edited by D.G. Bobrow [Bob80] in 1980, constitutes for many respects
the starting point of a new logico-mathematical field of research. The main
motivation for such an undertaking can be summed up very roughly as the
intention to characterize logical (not necessarily probabilistic) systems capable
of representing flexible as well as plausible forms of reasoning. Reiter’s
Default Logic, [Rei80, RC81], McCarthy’s Circumscription [McC80, McC90]
and McDermott and Doyle’s Non-monotonic Logic [MD80] were but a few
initial attempts to bring about such an ambitious aim. Among them, particular
attention was received by Reiter’s proposal which provided formal
constructions for extending agents’ knowledge bases beyond the rigidity of
classical monotonic logic, hence allowing flexible, exception tolerant, patterns
of reasoning to be represented logically. 1
Those initial proposals focussed essentially on the problem of characterizing
flexibility in reasoning. The seminal papers by Gabbay [Gab85], Makin-
1 Indeed Reiter’s logic was developed and then widely appreciated in the theory of
databases, especially in connection with the so-called Closed World assumption that allowed
agents to “jump to conclusions” that nonetheless could have been subsequently
withdrawn on the addition of new information. On the “success of default logic” see
[Lif99].
3

son [Mak89] and Kraus, Lehmann and Magidor [KLM90] put the investigations
on nonmonotonic logics into a more abstract framework –developed at
the traditional levels of proof-theory and model-theory– with the intention of
providing a clear account of the logical properties of nonmonotonic inference.
From this new, abstract, perspective, the question of plausibility of reasoning
could be tackled in a more direct way. By studying, in fact, consequence
relations it becomes possible to devise appropriate rules which are meant
to capture the conditions that sensible patterns reasoning under imperfect
information should reasonably satisfy.
2.2 Basic concepts and notation
Let SL be the set of sentences built up from a classical propositional language
L in the usual way. We use small Greek letters for elements of SL and block
capital Greek letters for subsets of SL. We denote by 2 L the set of classical
valuations on L, that is the set of all maps from L to 2 = {0, 1}. Valuations
extend uniquely to SL in the usual, recursive way. We write, as usual, Cn
for the classical consequence operation (sometimes called Tarskian), that is
to say an operation which is reflexive, transitive and monotonic, the corresponding
notation for the classical consequence relation being ⊢. As we will
only consider finitary consequence (i.e. with premises being finite sets of SL)
we’ll adopt the usual convention in the area of freely swapping the notions
of consequence operation and consequence relations. 2
Consequence relations on 2 SL × SL other than the classical one will be
denoted by |∼, possibly with decorations. Note that all of the consequence
relations that we’ll be considering rest on the same underlying language L.
With this little background we are now in a position to discuss the role
of monotonicity as far as the representation of AI agents’ reasoning is concerned.
To do that we’ll have to make the crucial idealization that (some
essential aspects of) agents’ reasoning can be captured by some appropriate
consequence relation. Recall that, in the present notation, monotonicity says
that if θ ∈ Cn(Γ) then θ ∈ Cn(Γ ∪ ∆). If we assume that subsets of SL represent
agents’ information, then monotonicity just says that the introduction
of additional information to the premises of a derivation cannot cause the
rejection of any of the conclusions previously drawn from the initial set of
premises, no matter what this new information turns out to be. This lack
of qualification is an immediate consequence of the fact that no formal con-
2 Indeed, given Γ ⊆ SL, θ ∈ SL and the operation Cn, we can define a relation
⊢⊆ 2 SL × SL as the set of ordered pairs 〈Γ, θ〉 such that θ ∈ Cn(Γ), and conversely
Cn(Γ) = {θ | Γ ⊢ θ}.
4

straint is imposed on the additional set ∆. We call this the unconstrained
form of monotonicity.
A very immediate reflection on the import of unconstrained monotonicity
is sufficient to appreciate its inadequacy to represent flexible, adaptive, plausible
reasoning. What unconstrained monotonicity sanctions, so to speak, is
the sufficiency of a given set of premises (Γ in our case) with respect to a
certain conclusion (θ). This is a consequence of the fact that as far as (the
derivation of) θ is concerned (the addition of) ∆ is completely irrelevant.
But since monotonicity does not impose any constraint on the enlargement
of the set of premises, ∆ can indeed be any subset of SL. Hence, any possible
addition to Γ will be a priori irrelevant, which is to say that Γ was indeed
sufficient to conclude θ. Put in more emphatic terms, once a monotonic agent
draws a conclusion from a given set of premises, nothing it might possibly
come to learn will cause it changing its mind. In realistic (i.e. “world-like”)
environments, however, this cannot reasonably happen: agents will hardly
enjoy the priviledge of having complete, i.e. sufficient, evidence on which
their rasoning, decisions, choices, etc. can be grounded. Hence, in its unconstrained
form, monotonicity cannot underlie adequate pattern of reasoning
for realistic agents, those of interesest to us.
Still, as far as the characterization of intelligent reasoning is concerned,
monotonic patterns of reasoning have a number of desirable properties, so
that throwing away monotonicity as a whole will not do.
Therefore monotonicity embeds –blindly, in its unconstrained form– a
general principle of informational economy which can loosely be formulated
by saying that agents should not change their mind unless they have good reasons
to do so. 3 It cannot be surpising then, that a significant part of research
into nonmonotonic reasoning focuses on the characterization of suitable constraints
to be imposed on monotonicity so as to retain the (economically)
desirable aspects of its conservativeness, while disposing of the inadequate
conclusions arising from its unconstrained application. The result of this enquiry
is a very well-behaved class of logical systems which do indeed capture
this aspect of rational or common-sensical reasoning, yet being liable, as we
will shortly see, of a purely deductive characterization.
2.3 Constrained monotonicity
An important difference between monotonic and nonmonotonic logic lies in
the fact that for the latter no smallest or otherwise “best” consequence re-
3 In the literature on Belief Revision the principle of informational economy (see e.g.
[Gär88]) requires that the revision of a belief set should preserve old beliefs “as much as
possible”.
5

lation can be characterized. Indeed various notions of nonmonotonic consequence
arise as soon as certain conditions have been fixed with the remarkable
result that the intersection of those various relations just brings us back to
classical monotonic logic [Mak94, Mak03], the logic we use at the metalevel.
There is then a diversity which is peculiar to nonmonotonic logics. This
doesn’t mean, however, that nonmonotonic consequences are chosen arbitrarily,
or that there is a lack of shared intuitions underlying the various formal
developments. During the last decade or so, in fact, scholars in the subject
have shown significant signs of agreement towards the so-called Preferential
Inference (System P) [KLM90] or, as it is sometimes called after the authors’
names, System KLM. The reasons for such an appreciation mainly consist
in the following. 4 Firstly, the formal constraint to be discussed in the last
part of this section are intuitively justified on the grounds of “rationality”,
“minimality” and “informational economy”. This relates the investigation
on constrained monotonicity to the commonsense-based approach to uncertain
reasoning (see, e.g. [Par94, Par99]), where rational reasoning under
conditions of imperfect information is mathematically characterized as the
obedience to certain “common-sense” principles, some of which turn out to
be remarkably close the KLM rules/conditions. 5 The second reason for the
convergence on the KLM system is that it can be proved to be sound and
complete with respect to various semantic interpretations of reasoning based
on usualness, typicality, normality, very high probability, and so on. 6 Finally
there is a very tight connection, indeed a translation, between preferential inference
and the classic postulates for Belief Revision introduced in [AGM85]. 7
Of course, the fact that certain nonmonotonic consequence relations can be
put in formal correspondence with the classical paradigm for belief revision
gives us further reasons to consider nonmonotonic logics as capturing crucial
features of the kind of flexibility and sensibility that the pervasiveness of
4Precise details can be found in recent comprehensive monographs like, e.g., [Boc01,
Rot01, KI01].
5The first formal aspects of this connection have been put forward by relating
nonmonotonic logics to maximum entropy reasoning. See e.g. [KI01, HP02].
6Friedman and Halpern have recently suggested a formal explanation of this deep connection
between the proof-theoretic characterization and a wide class of semantics for
which nonmonotonic consequence is complete, based on the general framework of plausibility
measures [FH01]. Particularly relevant to us is the semantical interpretation according
to which the relation θ |∼P φ holds if φ is (classically) satisfied in all the most preferred
worlds in which θ is (classically) satisfied.
7The main idea providing this translation is the so-called Makinson-Gärdenfors Identity,
according to which inferring nonmonotonically a conclusion from a sentence θ amounts
to revising a certain set of background assumptions by θ. For precise details see [MG91,
GM94, Rot01, KI01, Boc01].
6

imperfection in information demands from intelligent agents.
Let’s now turn to the formal rules/conditions characterizing preferential
consequence relations. 8 The inference system is presented here, as usual, as
a Gentzen-style system with individual sentences as arguments of the consequence
relations. 9 Relations satisfying the following conditions are called
rational consequence relations, whereas relations satisfying all but RMO are
called preferential consequence relations.
Reflexivity is the only axiom scheme:
θ |∼ θ (REF).
Then three sorts of rules (or conditions) are imposed on the relation
|∼. Firstly what are normally referred as “pure conditions”: Left Logical
Equivalence and Right Weakening: 10
⊢ θ ↔ φ, θ |∼ ψ
φ |∼ ψ
θ |∼ φ, φ ⊢ ψ
θ |∼ ψ
(LLE)
(RWE)
Secondly the usual Conjunction in the conclusions and Disjunction in the
premises are introduced:
θ |∼ φ, θ |∼ ψ
θ |∼ φ ∧ ψ
θ |∼ ψ, φ |∼ ψ
θ ∨ φ |∼ ψ
(AND)
(OR)
Finally, the formal constraints on monotonicity. Cautious Monotonicity
and Rational Monotonicity:
8 Note that the following rules of inference are generally understood also as constraints
that consequence relations should satisfy in order to account for sensible patterns of
reasoning. The latter being their intended interpretation as far as their justification is
concerned.
9 The first generalization to the infinitary case was studied in [FL93].
10 Note that any consequence relation |∼ ∗ such that ⊢ ⊆ |∼ ∗ is called supraclassical, and
Reflexivity and Right Weakening entail that. Intuitively then, a supraclassical consequence
relation extends the deductive power of classical reasoning. Very roughly, the desirability
of this property is a consequence of the intuition that reasoning under perfect information
is just a limiting case of reasoning under uncertainty. This extension, however, is not
priceless. Supraclassical relations, in fact, generally have a “less regular behaviour”, in
Makinson’s phraseology. See [Mak03, Rot01] for more on this.
7

θ |∼ φ, θ |∼ ψ
θ ∧ φ |∼ ψ
θ |∼ φ, θ |∼¬ψ
θ ∧ ψ |∼ φ
(CMO)
(RMO)
Cautious monotonicity, the first form of constrained monotonicity that
has been discussed in the characterization of nonmonotonic consequence
[Gab85], puts a very natural condition on the application of monotonicity:
the addition of a sentence that has been previously concluded from a given
premise should not cause its consequence to be abandoned. This is clearly an
important aspect of the conservativeness of monotonicity that need not be
rejected in the characterization of sensible reasoning. 11 Rational Monotonicity,
on the other hand, puts much a stronger constraint on the applicability
of monotonicity. As for CMO, the underlying idea is that of filtering away
“irrelevant information”, where relevance is mainly understood as the capability
of invalidating the conclusion drawn from given premises. Thus, for
CMO, the addition of φ is irrelevant as far as the derivation of ψ from θ
is concerned. Instead of characterizing as irrelevant what has already been
concluded, RMO requires, in order to determine the irrelevance of a certain
sentence, that the possibility for the added hypothesis (ψ) to clash with the
conclusion (φ) should be ruled out. This can clearly be accomplished by
requiring that ¬ψ cannot be (nonmonotonically) proved from θ. There is a
general feeling, however, that weaker conditions might capture this side of
relevance more adequately. 12
11 Due to space constraints, we cannot delve into the discussion on Cautions Monotonicity,
or the following Rational Monotonicity, referring the interested reader to
[KLM90, LM92] for detailed explanations and justifications, and to [Rot01, KI01, Boc01]
for general surveys on the principles of nonmonotonic inference. [GW03] stresses (see
especially chapter 3) the benefits of conservative reasoning for realistic agents.
12 Rational Monotonicity is often criticised (see, e.g. [Mak94, Rot01]) on the grounds
that it fails to be a Horn-condition, thus preventing the resulting consequence relation
from being closed under intersection. There might also be concerns on the semantics of
rational consequence relations which require a total ordering (or equivalent conditions)
among preferences in order to yield soundness and completeness results for the axiom system.
(See the seminal [KLM90, LM92] for a fully detailed account on (the semantics of)
rational consequence relations). As far as “positive” consequence of “positive” knowledge
is concerned, however, the significance of the latter concern is undermined by the characterization
results by Lehmann and Magidor [LM92] according to which preferential and
rational consequence relations are in fact equivalent. See [BP98] for more on this.
8

2.3.1 Indicazioni bibliografiche
La prima raccolta è il numero speciale di Artificial Intelligence [Bob80] da
cui si sono sviluppati linee distinte di ricerca nell’area. Un’altra raccolta fondamentale
(con parziali sovrapposizioni rispetto alla precedente) è [Gin87].
Più avanzata è la raccolta nel volume [GHR94] (che contiene uno degli studi
fondamentali sulle relazioni di conseguenza nonmonotoniche [Mak94]).
[Rot01, KI01, Boc01] sono tre recenti monografie di livello avanzato.
2.4 Ragionamento probabilistico
Per cominciare: [Par99]
Esistono varie interpretazioni della probabilità e di conseguenza varie sono
le nozioni di ragionamento probabilistico che si ritrovano in IA. Si distinguono,
essenzialmente a seconda che venga sviluppata una prospettiva soggettivistica
o oggettivistica della probabilità. 13 Nel primo caso, la probabilità
è intesa come grado di convinzione di un agente nei confronti di un certo
evento (ovvero della proposizione che lo descrive. La prospettiva soggettivistica,
principalmente sviluppata nella prima metà del Novecento da de
Finetti[dF31, dF37, dF70], Ramsey [Ram64] e altri, viene spesso etichettata
come “Bayesiana”. In IA questa terminologia è particolarmente utilizzata
per indicare le cosiddette reti bayesiane, menzionate sotto. Nella prospettiva
oggettivistica, invece, la probabilità viene interpretata usualmente in termini
di una relazione logica (prospettiva “logicista”, essenzialmente introdotta
da Keynes in [Key21]) o nei termini tradizionali di limite della frequenza di
casi positivi rispetto a una certa classe di riferimento. Nononstante alcuni
studiosi sostengano un parere opposto, il paradigma soggettivista fornisce la
prospettiva più feconda in cui inquadrare il problema dell’incertezza in IA.
2.4.1 Indicazioni bibliografiche
Formalizzazione del buon senso o razionalità e Massima Entropia. Un’ottima
lettura introduttiva è [Par99]. [Par94] fornisce una prospettiva matematica
del problema (contiene la derivazione del principio di massima entropia). In
[PV97] vengono discusse e rigettate alcune critiche comuni al principio di
massima entropia. In [PV01] viene discussa la generalizzazione dei principi
di buon senso a basi di conoscenza non lineari.
13 La raccolta di lezioni di de Finetti [Mur95] fornisce un resoconto molto dettagliato e
accessibile delle varie interpretazioni della probabilità e delle conseguenze che ne derivano.
9

Reti bayesiane: [Pea88] è considerato un classico. [Wil01] è un recente studio
in cui viene esposta la stretta correlazione che sussiste tra reti bayesiane
e sistemi inferenziali intesi tradizionalmente.
3 Logiche epistemiche (modali)
Per cominciare: [Hal98]
3.0.2 Indicazioni bibliografiche
[Háj98]
4 Logiche polivalenti
Per cominciare: [Háj00]
4.0.3 Indicazioni bibliografiche
[HP97]
Riferimenti bibliografici
[AGM85] C. E. Alchourrón, P. Gärdenfors, and D. Makinson. On the logic of
theory change: partial meet functions for contraction and revision.
Journal of Symbolic Logic, 50:510–530, 1985.
[Bob80] D.G. Bobrow. Special issue on non-monotonic logics. Artificial
Intelligence, 13(1-2), 1980.
[Boc01] A. Bochman. A Logical Theory of Nonmonotonic Inference and
Belief Change. Springer, 2001.
[BP98] R. Booth and J. B. Paris. A note on the rational closure of knowledge
bases with both positive and negative knowledge. Journal of
Logic, Language, and Information, 7(2):165–190, 1998.
[dF31] B. de Finetti. Sul significato soggettivo della probabilità.
Fundamenta Mathematicae, 17:289–329, 1931.
[dF37] B. de Finetti. La prévision: Ses lois logiques, ses sources
subjectives. Annales de l’Institut H. Poincaré, 7:1–68, 1937.
10

[dF70] B. de Finetti. Teoria delle probabilità. Einaudi, Torino, 1970.
[FH01] N. Friedman and J. Halpern. Plausibility measures and default
reasoning. Journal of the ACM, 48(4):648–685, 2001.
[FL93] M. Freund and D. Lehmann. Nonmonotonic inference operations.
Bullettin of the Interest Group in Pure and Applied Logics, 1(1):23–
68, 1993.
[Gab85] D.M. Gabbay. Theoretical foundations for non-monotonic reasoning
in expert systems, pages 439–457. Proceedings NATO Advanced
Institute on Logics and Models of Cuncurrent Systems.
Springer, Berlin, 1985.
[Gär88] P. Gärdenfors. Knowledge in Flux: Modeling the Dynamics of
Epistemic States. MIT Press, Bradford Books, 1988.
[GHR94] D.M. Gabbay, C. Hogger, and J. Robinson, editors. Handbook of
Logic in Artificial Intelligence and Logic Programming, volume 3.
Oxford University Press, Oxford UK, 1994.
[Gin87] M.L. Ginsberg, editor. Readings in Nonmonotonic Reasoning.
Morgan Kaufmann, San Mateo, Ca, 1987.
[GM94] P. Gärdenfors and D. Makinson. Nonmonotonic inferences based
on expectations. Artificial Intelligence, 55:1–60, 1994.
[GS98a] D.M. Gabbay and P. Smets, editors. Handbook of Defeasible
Reasoning and Uncertainty Management System. Vol. 1. Kluwer,
1998.
[GS98b] D.M. Gabbay and P. Smets, editors. Handbook of Defeasible
Reasoning and Uncertainty Management System. Vol. 2. Kluwer,
1998.
[GS98c] D.M. Gabbay and P. Smets, editors. Handbook of Defeasible
Reasoning and Uncertainty Management System. Vol. 3. Kluwer,
1998.
[GW03] D.M. Gabbay and J. Woods. Agenda Relevance: A Study in
Formal Pragmatics. North-Holland, 2003.
[Háj98] P. Hájek. Logics of knowing and believing. In U. Ratsch, M.M.
Richter, and I-O. Stamatescu, editors, Intelligence and Artificial
Intelligence, pages 96–108. Springer, 1998.
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[Háj00] P. Hájek. Mathematical fuzzy logic—state of art. In Sam Buss,
Petr Hájek, and Pavel Pudlák, editors, Proc. Logic Colloquium ’98,
Prague, Czech Republic, volume 13 of Lecture Notes in Logic, pages
197–205. Association for Symbolic Logic with A K Peters, 2000.
[Hal98] J. Halpern. A logical approach to reasoning about uncertainty: a
tutorial. In K. Korta X. Arrazola and F.J. Pelletier, editors, Discourse,
Interaction, and Communication, pages 141–155. Kluwer,
1998.
[Hal03] J. Halpern. Reasoning About Uncertainty. MIT Press, 2003.
[Hos02] H. Hosni. Rationality constraints on BDI logics. In
M. Wooldridge and M. Pauli, editors, Symposium on Logic in
Games and Multiagent Systems, University of Liverpool, UK, 2002.
http://www.maths.man.ac.uk/˜hykel/work/radtbdi.pdf.
[Hos03] H. Hosni. Imperfect information management
in marine GIS. Work in progress, 2003.
http://www.maths.man.ac.uk/˜hykel/work/cosit03.pdf.
[HP97] P. Hájek and J.B. Paris. A dialogue on fuzzy logic.
Soft Computing—A Fusion of Foundations, Methodologies and
Applications, 1(1):3–5, 1997.
[HP02] L.C. Hill and J.B. Paris. When maximising entropy gives the
rational closure. Journal of Logic and Computation, 12(1):1–19,
2002.
[Key21] J.M. Keynes. A Treatise on Probability. Macmillan, London, 1921.
[KI01] G. Kern-Isberner. Conditionals in Nonmonotonic Reasoning and
Belief Revision - Considering Conditionals as Agents, volume 2087
of Lecture Notes in Computer Science. Springer, 2001.
[KLM90] S. Kraus, D Lehmann, and M. Magidor. Nonmonotonic reasoning,
preferential models and cumulative inference. Artificial
Intelligence, 44(1):167–207, 1990.
[Lif99] V. Lifschitz. Success of Default Logic. In H. Levesque and F Pirri,
editors, Logical Foundations for Cognitive Agents, pages 208–212.
Springer-Verlag, 1999.
[LM92] D. Lehmann and M Magidor. What does a conditional knowledge
base entail? Artificial Intelligence, 55:1–60, 1992.
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[Mak89] D. Makinson. General theory of cumulative inference, volume n.346
of Lecture Notes in Artificial Intelligence, pages 1–18. Springer,
1989.
[Mak94] D. Makinson. General patterns in nonmonotonic reasoning. In
D.M. Gabbay, C. Hogger, and J. Robinson, editors, Handbook of
Logic in Artificial Intelligence and Logic Programming, pages 35–
110, 1994.
[Mak03] D. Makinson. Bridges between classical and nonmonotonic logic.
Logic Journal of the IGPL, 11(1):69–96, 2003.
[McC80] J. McCarthy. Circumscription: a form of non-monotonic logic.
Artificial Intelligence, (13):27–39, 1980.
[McC90] J. McCarthy. Applications of circumscription to formalizing common
sense knowledge. In Vladimir Lifschitz, editor, Formalizing
Common Sense: Papers by John McCarthy, pages 198–225,
Norwood, New Jersey, 1990. Ablex Publishing Corporation.
[McC96] J. McCarthy. From here to human-level AI. In L. Carlucci Aiello,
J. Doyle, and S Shapiro, editors, Proceedings of the Fifth International
Conference on Principles of Knowledge Representation and
Reasoning, pages 640–646, San Francisc, 1996. Morgan Kaufmann.
[MD80] D. McDermott and J. Doyle. Non-monotonic logic 1. Artificial
Intelligence, 13(1–2):41–72, 1980.
[MG91] D. Makinson and P. Gärdenfors. Relations between the logic of theory
change and nonmonotonic logic. In A. Fuhrmann and M. Morreau,
editors, Proceedings of the Workshop on The Logic of Theory
Change, volume 465 of Lecture Notes in Computer Science, pages
185–205, Konstanz, FRG, 1991. Springer-Verlag.
[Mur95] A. Mura, editor. Filosofia della probabilità. Il Saggiatore, Milano,
1995.
[Nil98] N. Nilsson. Artificial Intelligence: A New Syntesis. Morgan
Kaufmann Publishers, San Francisco, 1998.
[Par94] J.B. Paris. The Uncertain Reasoner’s Companion: A Mathematical
Perspective. Cambridge University Press, Cambridge, England,
1994.
13

[Par99] J.B Paris. Comon sense and maximum entropy. Synthese,
117(1):73–93, 1999.
[Pea88] J. Pearl. Probabilistic Reasoning in Intelligent Systems. Morgan
Kaufmann, Los Altos, California, 1988.
[PMG98] D. Poole, A. Mackworth, and R. Goebel. Computational
Intelligence. Oxford University Press, Oxford, 1998.
[PV97] J.B. Paris and A. Vencovská. In defence of the maximum entropy
inference process. International Journal of Approximate
Reasoning, 17:77–103, 1997.
[PV01] J.B. Paris and A. Vencovská. Common sense and stochastic independence.
In D. Corfield and J. Williamson, editors, Foundations
of Bayesianism, pages 203–240. Kluwer Academic Press, 2001.
[Ram64] F.P. Ramsey. Truth and probability (1931). In Jr. H. E. Kyburg
and H. E. Smokler, editors, Studies in Subjective Probability, pages
61–92. Wiley, New York, 1964.
[RC81] R. Reiter and G. Criscuolo. On interacting defaults. In Proceedings
of IJCAI-81, 7th International Joint Conference on Artificial
Intelligence, pages 270–276, Vancouver, BC, 1981.
[Rei80] R. Reiter. A logic for default reasoning. Artificial Intelligence,
13(1–2):81–132, 1980.
[RN95] S. Russell and P. Norvig. Artificial Intelligence: A modern
approach. Prentice-Hall, 1995.
[Rot01] H. Rott. Change, Choice and Inference : A Study of Belief Revision
and Nonmonotonic Reasoning. Oxford University Press,
2001.
[SP90] G. Shafer and J. Pearl, editors. Readings in Uncertain Reasoning.
Morgan Kaufmann, San Mateo, California, 1990.
[Wil01] J. Williamson. Foundations of bayesian networks. In D. Corfield
and J. Williamson, editors, Foundations of Bayesianism. Kluwer
Academic Press, 2001.
14