Abstract. In questo lavoro si vedranno nuove
motivazioni a favore di uno studio della contrazione su basi
finite (non chiuse), oltre a quelle già note in
letteratura, e si mostrerà un algoritmo efficiente su base
finita.
Abstract. Il concetto che sta alla base della
assiomatizzazione della revisione è quello di cambiamento
minimo (minimal change), ossia che alfa*beta deve venire ottenuto da alfa
operando una modifica che rende vero beta, in modo tale che questo
cambiamento sia il più piccolo possibile. Ci possono poi
essere diversi criteri per valutare la minimalità di un
cambiamento: in ogni caso se alfa¨beta è consistente allora
alfa*beta = alfa¨beta, perché in questo modo sto solo aggiungendo beta
alla vecchia base senza fare altre modifiche. Nell'arbitraggio
il concetto guida è quello del massimo mantenimento della
conoscenza, ossia quello di fare in modo che il risultato
dell'arbitraggio di due basi mantenga quanta più
informazione possibile delle due. Inoltre, è possibile
pensare di fare in modo da escludere che le informazioni di una
base che sono molto in contrasto con quelle dell'altra, vengano
escluse. In questo modo, si suppone che ognuna delle due formule
di partenza abbia in sé una ''parte di verità'', e
che questo sia verosimilmente l'informazione su cui esse si
''somigliano''.
Abstract. The ability of Database Systems to cope with
changing situations and information coming from different sources
is crucial for their applicability in real-world scenarios.
Recent work in the field of belief revision and update has
provided us tecniques to handle change. In this paper we
introduce a different form of revision aiming at capturing the
process of ''merging'' possibly inconsistent pieces of
information. We call this process arbitration. Along the lines
of Gardenfors' work, we propose a set of postulates for this
operator and prove a representation theorem.
Abstract. Knowledge-based systems must be able to
''intelligently'' manage a large amount of information coming
from different sources and at different moments in time.
Intelligent systems must be able to cope with a changing world by
adopting a ''principled'' strategy. Many formalisms have been
put forward in the AI and DB literature to address this problem.
Among them, belief revision is one of the most successful
frameworks to deal with dynamically changing worlds. Formal
properties of belief revision have been investigated by
Alchourron, Gärdenfors and Makinson, who put forward a set
of postulates stating the properties that a belief revision
operator should satisfy. Among these properties, a basic
assumption of revision is that the new piece of information is
totally reliable and, therefore, must be in the revised knowledge
base. Different principles must be applied when there are two
different sources of information and each one has a different
view of the situation, the two views contradicting each other.
If we do not have any reason to consider any of the sources
completely unreliable, the best we can do is to ''merge'' the two
views in a new and consistent one, trying to preserve as much
information as possible. We call this merging process arbitration.
In this paper we investigate the properties that any arbitration
operator should satisfy. In the style of Alchourron,
Gärdenfors and Makinson we propose a set of postulates,
analyze their properties and propose actual operators for
arbitration.
Abstract. The process of integrating knowledge coming
from different sources has been widely investigated in the
literature. Three distinct conceptual approaches to this problem
have been most succesful: belief revision, merging and update.
In this paper we present a framework that integrates these three
approaches. In the proposed framework all three operations can
be performed. We provide an example that can only be solved by
applying more than one single style of knowledge integration and,
therefore, cannot be addressed by anyone of the approaches alone.
The framework has been implemented, and the examples shown in
this paper (as well as other examples from the belief revision
literature) have been successfully tested.
Abstract. In this paper we show how several different
semantics for belief update can be expressed in a framework for
reasoning about actions. This framework can therefore be
considered as a common core of all these update formalisms, thus
making it clear what they have in common. This framework also
allows expressing scenarios that are problematic for the
classical formalization of belief update.
Abstract. Belief integration methods are often aimed
at deriving a single and consistent knowledge base that retains
as much as possible of the knowledge bases to integrate. The
rationale behind this approach is the minimal change principle: the
result of the integration process should differ as less as
possible from the knowledge bases to integrate. We show that
this principle can be reformulated in terms of a more general
model of belief revision, based on the assumption that
inconsistency is due to the mistakes the knowledge bases contain.
Current belief revision strategies are based on a specific kind
of mistakes, which however does not include all possible ones.
Some alternative possibilities are discussed.
Abstract. This article proposes a solution to the
problem of obtaining plausibility information, which is necessary
to perform belief revision: given a sequence of revisions,
together with their results, derive a possible initial order that
has generated them; this is different from the usual assumption
of starting from an all-equal initial order and modifying it by a
sequence of revisions. Four semantics for iterated revision are
considered: natural, restrained, lexicographic and
reinforcement. For each, a necessary and sufficient condition to
the existence of an order generating a given history of revisions
and results is proved. Complexity is proved coNP complete in all
cases but one (reinforcement revision with unbounded sequence
length).
Abstract. One of the main challenges in the formal
modeling of common-sense reasoning is the ability to cope with
the dynamic nature of the world. Among the approaches put
forward to address this problem are belief revision and update. Given
a knowledge base T, representing our knowledge of the ''state of
affairs'' of the world of interest, it is possible that we are
lead to trust another piece of information P, possibly
inconsistent with the old one T. The aim of revision and update
operators is to characterize the revised knowledge base T' that
incorporates the new formula P into the old one T while
preserving consistency and, at the same time, avoiding the loss
of too much information in this process. In this paper we study
the computational complexity of one of the main computational
problems of belief revision and update: deciding if an
interpretation M is a model of the revised knowledge base.
Abstract. In this paper we analyze the complexity of
revising a knowledge base when an iteration of this process is
necessary. The analysis concerns both the classical problems of
belief revision (inference, model checking, computation of the
new base) and new issues, related to the problem of committing
the changes.
Abstract. Belief revision and belief update are two
different forms of belief change, and they serve different
purposes. In this paper we focus on belief update, the
formalization of change in beliefs due to changes in the world.
The complexity of the basic update (introduced by Winslett
[1990]) has been proved by Eiter and Gottlob (1991). Since then,
many other formalizations have been proposed to overcome the
limitations and drawbacks of Winslett's update. In this paper we
analyze the complexity of the proposals presented in the
literature, and relate some of them to previous work on closed
world reasoning.
Abstract. One of the main challenges in the formal
modeling of common-sense reasoning is the ability to cope with
the dynamic nature of the world. Among the approaches put
forward to address this problem are belief revision and update. Given
a knowledge base T, representing our knowledge of the ''state of
affairs'' of the world of interest, it is possible that we are
lead to trust another piece of information P, possibly
inconsistent with the old one T. The aim of revision and update
operators is to characterize the revised knowledge base T' that
incorporates the new formula P into the old one T while
preserving consistency and, at the same time, avoiding the loss
of too much information in this process. In this paper we study
the computational complexity, in the propositional case, of one
of the main computational problems of belief revision and update:
deciding if an interpretation M is a model of the revised
knowledge base.
Abstract. Belief revision and belief update are two
different forms of belief change, and they serve different
purposes. In this paper we focus on belief update, the
formalization of change in beliefs due to changes in the world.
The complexity of the basic update (introduced by Winslett) has
been determined by Eiter and Gottlob. Since then, many other
formalizations have been proposed to overcome the limitations and
drawbacks of Winslett's update. In this paper we analyze the
complexity of the proposals presented in the literature: the
standard semantics by Winslett, the minimal change with exception
and the minimal change with maximal disjunctive information by
Zhang and Foo, the update with disjunctive function by Herzig,
the abduction-based update and the generalized update by
Boutilier. We relate some of these approaches to belief update
to previous work on closed world reasoning.
Abstract. In this paper we address a specific
computational aspect of belief revision: The size of the
propositional formula obtained by means of the revision of a
formula with a new one. In particular, we focus on the size of
the smallest formula equivalent to the revised knowledge base.
The main result of this paper is that not all formalizations of
belief revision are equal from this point of view. For some of
them we show that the revised knowledge base can be expressed
with a formula admitting a polynomial-space representation (we
call these results ''compactability'' results). On the other
hand we are able to prove that for other ones the revised
knowledge base does not always admit a polynomial-space
representation, unless the polynomial hierarchy collapses at a
sufficiently low level (''non-compactability'' results). The
time complexity of query answering for the revised knowledge base
has definitely an impact on being able to represent the result of
the revision compactly. Nevertheless formalisms with the same
complexity may have different compactability properties.
Abstract. Several methods are proposed, in the past
years, to attempt to define how to deal with a dynamic knowledge.
From a computational point of view, different definitions lead to
different different computational property, namely to a
complexity between [Delta]p2 to [Delta]p3 in the polynomial hierarchy.
Restricting the analysis to cases in which some operations
becomes tractable (Horn case), other problems arises. We show
that, providing some assumption on the polynomial hierarchy, many
update operators are uncompactable, i.e. it is impossible to find
a representation of the result without an exponential-size
explosion. We supply also the explicit representations of the
updates for which it is possible.
Abstract. In this paper we address a specific
computational aspect of belief revision: the size of the
propositional formula obtained by means of the revision of a
formula with a new one. In particular, we focus on the size of
the smallest formula which is logically equivalent to the revised
knowledge base. The main result of this paper is that not all
formalizations of belief revision are equal from this point of
view. For some of them we show that the revised knowledge base
can be represented by a polynomial-size formula (we call these
results ''compactability'' results). On the other hand, for
other ones the revised knowledge base does not always admit a
polynomial-space representation, unless the polynomial hierarchy
collapses at a sufficiently low level (''non-compactability''
results). We also show that the time complexity of query
answering for the revised knowledge base has definitely an impact
on being able to represent the result of the revision compactly.
Nevertheless, formalisms with the same complexity may have
different compactability properties. We also study
compactability properties for a weaker form of equivalence,
called query equivalence, which allows to introduce new
propositional symbols. Moreover, we extend our analysis to the
special case in which the new formula has constant size and to
the case of sequences of revisions (i.e., iterated belief
revision). A complete analysis along these four coordinates is
shown.
Abstract. Several methods have been proposed as an
attempt to deal with dynamically-changing scenarios. From a
computational point of view, different formalisms have different
computational properties. In this paper we consider knowledge
bases represented as sets of Horn clauses. The importance of
this case is twofold: first, inference is polynomial, thus
tractable; second, Horn clauses represent causal relations
between facts, thus they are of great practical importance,
although not all propositional knowledge bases can be represented
in Horn form. The complexity of Horn revision is still high, and
in some cases coincides with the complexity of the general
(non-Horn) case. We analyze the complexity of belief revision
from the point of view of the compilation: we study the
possibility of reducing the complexity by allowing a (possibly
expensive) preprocessing of part of the input of the problem.
Extending the work by Cadoli et al. [1999], we consider the
problem of compact representation of revision in the Horn case,
that is, given a knowledge base T and an update P (both
represented by Horn clauses) decide whether T *P, the result of
the revision, can be represented with a propositional formula
whose size is polynomial in the size of T and P. We give this
representation for all formalisms for which it exists, and we
show that the existence of a compact representation is related to
the possibility of decreasing the complexity of a formalism via a
preprocessing.
Abstract. A propositional knowledge base can be seen
as a compact representation of a set of models. When a knowledge
base T is updated with a formula P, the resulting set of models
can be represented in two ways: either by a theory T' that is
equivalent to T*P or by the pair (T,P). The second representation
can be super-polinomially more compact than the first. In this
paper, we prove that the compactness of this representation
depends on the specific semantics of *, e.g., Winslett's
semantics is more compact than Ginsberg's.
Abstract. Nonmonotonic formalisms and belief revision
operators have been introduced as useful tools to describe and
reason about evolving scenarios. Both approaches have been
proven effective in a number of different situations. However,
little is known about their relationship. Previous work by
Winslett has shown some correlations between a specific operator
and circumscription. In this paper we greatly extend Winslett's
work by establishing new relations between circumscription and a
large number of belief revision operators. This highlights
similarities and differences between these formalisms.
Furthermore, these connections provide us with the possibility of
importing results in one field into the other one.
Abstract. Nonmonotonic formalisms and belief revision
operators have been introduced as useful tools to describe and
reason about evolving scenarios. Both approaches have been
proven effective in a number of different situations. However,
little is known about their relationship. Previous work by
Winslett has shown some correlations between a specific operator
and circumscription. In this paper we greatly extend Winslett's
work by establishing new relations between circumscription and a
large number of belief revision operators. This highlights
similarities and differences between these formalisms.
Furthermore, these connections provide us with the possibility of
importing results in one field into the other one.
Abstract. Default logic is one of the most widely used
formalisms to formalize commonsense reasoning. In this paper we
analyze the complexity of deciding whether a propositional
interpretation is a model of a default theory for some of the
variants of default logic presented in the literature. We prove
that all the analyzed variants have the same complexity and that
this problem is in general Sigma2 complete, while it is CoNP
complete under some restrictions on the form of the defaults.
Abstract. Many variants of default logics exist. Two
of the main differences among them are the choice between local
or global consistency and the choice of accepting maximally
successful sets of defaults. Proving a result that is valid in
all variants amounts to showing either a proof that holds for all
semantics or a different proof for each semantics. In this
paper, we characterize theories that do not depend at all on what
makes the semantics different. A result that is proved on such
theories holds not only for all considered semantics, but also on
any other semantics that differs on the classical one because of
the two choices. These theories are also of interest for
practical applications of default logic, as an implemented system
should be able to detect (for example, to warn the user) any
theory whose semantics is debatable.
Abstract. Many variants of default logics exist. Two
of the main differences among them arise from the choice between
local and global consistency, and the choice of whether or not to
accept maximally successful sets of defaults. In this paper, we
characterize theories that do not depend at all on what makes the
semantics different, that is, theories for which these two
choices do not matter. A result that is proved for such theories
holds not only for all the considered semantics, but also for
every other semantics that differs from them on the two choices.
Abstract. Most of the work in default logic is about
default theories that are completely specified. In this category
are the proposals of appropriate semantics for default logic, the
characterizations of the complexity of reasoning with a default
theory, the algorithms for finding consequences of default
theories, etc. Relatively little attention has been paid to the
process of building a default theory, and most of the work on
this topic is about translating knowledge bases from other
formalisms (such as circumscription, autoepistemic logic, and
action description languages) into default logic. This paper is
about expressing knowledge in default logic. In particular, we
assume that defaults are initially formulated as normal, and are
then corrected using specific inference examples.
Abstract. A default theory can be seen as a way for
representing a set of formulae, i.e., its extensions. In this
paper, we characterize the sets of formulae that can be expressed
by a default theory according to various semantics: justified,
constrained, rational, cumulative, QDL, CADL, and two semantics
with priorities. These characterizations imply some
non-translatability results between semantics.
Abstract. We analyze the complexity of deciding
whether a propositional interpretation is a model of a default
theory for some of the variants of default logic presented in the
literature: Reiter's, justified, constrained, rational, and
cumulative. We prove that all the analyzed variants are
Sigma2-complete in the general case, and remains so even if all
defaults are prerequisite-free and seminormal. Cumulative
default logic is also Sigma2-complete even if all defaults are
normal and prerequisite-free. The other semantics are
Delta-log2-complete and coNP-complete if all defaults are normal
and prerequisites are allowed or not, respectively.
Abstract. In this paper we analyze the problem of
checking whether a default theory has a single extension. This
problem is important for at least three reasons. First, if a
theory has a single extension, nonmonotonic inference can be
reduced to entailment in propositional logic (which is
computationally easier) using the set of consequences of the
generating defaults. Second, a theory with many extensions is
typically weak i.e., it has few consequences; this indicates that
the theory is of little use, and that new information has to be
added to it, either as new formulae, or as preferences over
defaults. Third, some applications require as few extensions as
possible (e.g. diagnosis). We study the complexity of checking
whether a default theory has a single extension. We consider the
combination of several restrictions of default logics:
seminormal, normal, disjunction-free, unary, ordered. Complexity
varies from the first to the third level of the polynomial
hierarchy. The problem of checking whether a theory has a given
number of extensions is also discussed.
Abstract. The complexity of deciding equivalence of a
formula to an extension of a default theory is investigated for
the Reiter, justified, constrained, and rational semantics.
Abstract. A consistency default is a propositional
inference rule that asserts the consistency of a formula in its
consequence. Consistency defaults allow for a straightforward
encoding of domains in which it is explicitely known when
something is possible. The logic of consistency defaults can be
seen as a variant of cumulative default logic or as a
generalization of justified default logic; it is also able to
simulate Reiter default logic in the seminormal case. A
semantical characterization of consistency defaults in terms of
processes and in terms of a fixpoint equation is given, as well
as a normal form.
Abstract. In this article, we report results about
translations between variants of defaults logics such that the
extensions of the theories that are the input and the output of
the translation are in a bijective correspondence. We assume
that a translation can introduce new variables and that the
result of translating a theory can either be produced in time
polynomial in the size of the theory or its output is of size
polynomial in the size of the theory; we restrict to the case in
which the original theory has extensions. This study fills a gap
between two previous works, one studying bijective translations
among restrictions of default logics and one studying
non-bijective translations between default variants of default
logic.
Abstract. Abduction is one of the most important forms
of reasoning and it has been successfully applied to several
practical problems such as diagnosis. In this paper we
investigate whether the computational complexity of abduction can
be reduced by an appropriate use of preprocessing or compilation.
This is motivated by the fact that part of the data of the
problem (namely, the set of all possible assumptions and the
theory relating assumptions and manifestations) are often known
before the rest of the problem. We present a detailed analysis
of the computational complexity of abduction when compilation is
allowed.
Abstract. We call verification the process of finding
the actual explanation of a given set of manifestations. We
consider an abductive setting, in which explanations are sets of
assumptions. To filter out erroneous explanations, a
verification program should propose which assumptions to check.
Given the abductive setting of manifestations, assumptions, and a
theory relating them, we study the complexity of providing a
minimal set of assumptions to be checked in order to identify the
actual explanation. We study also the case in which assumptions
to be checked are given in a tree-like order.
Abstract. Abduction is one of the most important forms
of reasoning; it has been successfully applied to several
practical problems such as diagnosis. In this paper we
investigate whether the computational complexity of abduction can
be reduced by an appropriate use of preprocessing. This is
motivated by the fact that part of the data of the problem
(namely, the set of all possible assumptions and the theory
relating assumptions and manifestations) are often known before
the rest of the problem. In this paper, we show some complexity
results about abduction when compilation is allowed.
Abstract. When looking for a propositional abductive
explanation of a given set of manifestations, an ordering between
possible solutions is often assumed. While the complexity of
computing optimal solutions is already known, in this paper we
consider second-best solutions with respect to different
orderings, and different definitions of what a second-best
solution is: an optimal solution not already found, or a
solution that is optimal among the ones not previously found.
Abstract. In this paper we anliyze the complexity of
the language A, proposed by Gelfond and Lifschitz (1993) to
formalize properties of actions. We prove that the general
language is NP complete, thus intractable, and show some
tractable (polynomial) subclasses of it. We also show how states
that are unreachable affect the semantics of the language.
Abstract. The problem of minimally modifying a plan in
response to changes in the specification of the planning problem
has already been investigated in the literature. In this paper
we consider the problem of non-minimal modification of plans from a
computational point of view. We prove that while the general
problem is intractable (that is, replanning from scratch is
equally hard), there are scenarios in which the new plan can be
built more efficiently (in polynomial time).
Abstract. Policies of Markov Decision Processes (MDPs)
tell the next action to execute, given the current state and
(possibly) the history of actions executed so far. Factorization
is used when the number of states is exponentially large: both
the MDP and the policy can be then represented using a compact
form, for example employing circuits. We prove that there are
MDPs whose optimal policies require exponential space even in
factored form.
Abstract. Policies of Markov Decision Processes (MDPs)
determine the next action to execute from the current state and,
possibly, the history (the past states). When the number of
states is large, succinct representations are often used to
compactly represent both the MDPs and the policies in a reduced
amount of space. In this paper, some problems related to the
size of succinctly represented policies are analyzed. Namely, it
is shown that some MDPs have policies that can only be
represented in space super-polynomial in the size of the MDP,
unless the polynomial hierarchy collapses. This fact motivates
the study of the problem of deciding whether a given MDP has a
policy of a given size and reward. Since some algorithms for
MDPs work by finding a succinct representation of the value
function, the problem of deciding the existence of a succinct
representation of a value function of a given size and reward is
also considered.
Abstract. This paper analyses the computational
complexity of problems related to case-based planning: planning
when a plan for a similar instance is known, and planning from a
library of plans. It is proven that planning from a single case
has the same complexity than generative planning (i.e. planning
from scratch); using an extended definition of cases, complexity
is reduced if the domain stored in the case is similar to the one
to search plans for. Planning from a library of cases is shown
to have the same complexity. In both cases, the complexity of
planning remains, in the worst case, PSPACE-complete.
Abstract. In this paper we consider the problem of
finding minimal models of a set of propositional clauses. This
problem has many applications: for instance, finding
minimal-length plans, minimal diagnosis, and solving many other
minimization and maximization problems. We have adapted the
Davis-Putnam algorithm (that finds generic models of a set of
clauses) in order to find minimal models. Experiments have been
performed to determine which heuristics can be useful, and to
decide in which cases the problem is hard and in which cases it
is easy. A comparison with a local search algorithm is also
shown.
Abstract. We study the computational complexity of an
optimization version of the constraint satisfaction problem:
given a set F of constraint functions, an instance consists of a
set of variables V related by constraints chosen from F and a
natural number k. The problem is to decide whether there exists
a subset V' of V such that the size of V' is greater or equal
than k and the subinstance induced by V' has a solution. For all
possible choices of F, we show that this problem is either
NP-hard or trivial. This hardness result makes it interesting to
study relaxations of the problem which may have better
computational properties. Thus, we study the approximability of
the problem and we consider certain compilation techniques. In
both cases, the results are not encouraging.
Abstract. In this paper we analyze a computational
problem raising from the well-known DPLL algorithm for checking
satisfiability of propositional formulae. The DPLL algorithm is
a backtracking algorithm enhanced with three rules to improve
efficency. We study the computational complexity of determining
whether a variable leads to an optimal search tree, when it is
chosen as the branching literal.
Abstract. The possibility of solving the Quantified
Boolean Formulae (QBF) problems using the SMV system is a
consequence of two well-known theoretical results: the
membership of QBF to PSPACE, and the PSPACE-hardness of LTL (and
therefore, of SMV). Nevertheless, such results do not imply the
existence of a reduction that is also of practical utility. In
this paper, we show a reduction from QBF to SMV that is linear
(instead of cubic), and uses a constant-size specification. This
new reduction has three applications the previous one has not:
first, it allows for solving QBF problems using SMV-like systems,
which are now more developed than direct QBF solvers; second, we
can use it to verify whether the performance behavior of direct
QBF solvers is intrinsic of the problem, or rather an effect of
the solving algorithm; third, random hard SMV instances can be
easily generated by reduction from QBF hard instances (whose
generation method is now established).
Abstract. DPLL and resolution are two popular methods
for solving the problem of propositional satisfiability. Rather
than algorithms, they are families of algorithms, as their
behavior depend on some choices they face during execution: DPLL
depends on the choice of the literal to branch on; resolution
depends on the choice of the pair of clauses to resolve at each
step. The complexity of making the optimal choice is analyzed in
this paper. Extending previous results, we prove that choosing
the optimal literal to branch on in DPLL is Delta-log2-hard, and
becomes NPˆPP-hard if branching is only allowed on a subset
of variables. Optimal choice in regular resolution is both
NP-hard and CoNP-hard. The problem of determining the size of
the optimal proofs is also analyzed: it is CoNP-hard for DPLL,
and Delta-log2-hard if a conjecture we make is true. This
problem is CoNP-hard for regular resolution.
Abstract. A knowledge base is redundant if it contains
parts that can be inferred from the rest of it. We study the
problem of checking whether a CNF formula (a set of clauses) is
redundant, that is, it contains clauses that can be derived from
the other ones. Any CNF formula can be made irredundant by
deleting some of its clauses: what results is an irredundant
equivalent subset (I.E.S.) We study the complexity of some
related problems: verification, checking existence of a I.E.S.
with a given size, checking necessary and possible presence of
clauses in I.E.S.'s, and uniqueness.
Abstract. Independence--the study of what is relevant
to a given problem of reasoning--is an important AI topic. In
this paper, we investigate several notions of conditional
independence in propositional logic: Darwiche and Pearl's
conditional independence, and some more restricted forms of it.
Many characterizations and properties of these independence
relations are provided. We show them related to many other
notions of independence pointed out so far in the literature
(mainly formula-variable independence, irrelevance and novelty
under various forms, separability, interactivity). We identify
the computational complexity of conditional independence and of
all these related independence relations.
Abstract. In this paper we study the problem of
approximating a boolean function using the Hamming distance as
the approximation measure. Namely, given a boolean function f,
its k-approximation is the function fk returning true on the same
points in which f does, plus all points whose Hamming distance
from the previous set is at most k. We investigate whether
k-approximation generates an exponential increase in size or not,
when functions are represented as circuits. We also briefly
investigate the increase in the size of the circuit for other
forms of approximation.
Abstract. Independence - the study of what is
relevant to a given problem of reasoning - has received an
increasing attention from the AI community. In this paper, we
consider two basic forms of independence, namely, a syntactic one
and a semantic one. We show features and drawbacks of them. In
particular, while the syntactic form of independence is
computationally easy to check, there are cases in which things
that intuitively are not relevant are not recognized as such. We
also consider the problem of forgetting, i.e., distilling from a
knowledge base only the part that is relevant to the set of
queries constructed from a subset of the alphabet. While such
process is computationally hard, it allows for a simplification
of subsequent reasoning, and can thus be viewed as a form of
compilation: once the relevant part of a knowledge base has been
extracted, all reasoning tasks to be performed can be simplified.
Abstract. A knowledge base is redundant if it contains
parts that can be inferred from the rest of it. We study some
problems related to the redundancy of a CNF formula. In
particular, any CNF formula can be made irredundant by deleting
some of its clauses: what results is an irredundant equivalent
subset. We study the complexity of problems related to
irredundant equivalent subsets: verification, checking existence
of an irredundant equivalent subset with a given size, checking
necessary and possible presence of clauses in irredundant
equivalent subsets, and uniqueness. We also consider the problem
of redundancy with different definitions of equivalence.
Abstract. We report results about the redundancy of
formulae in 2CNF form. In particular, we give a slight
improvement over the trivial redundancy algorithm and give some
complexity results about some problems related to finding
Irredundant Equivalent Subsets (I.E.S.) of 2CNF formulae. The
problems of checking whether a 2CNF formula has a unique I.E.S.
and checking whether a clause in is all its I.E.S.'s are
polynomial. Checking whether a 2CNF formula has an I.E.S. of a
given size and checking whether a clause is in some I.E.S.'s of a
2CNF formula are polynomial or NP-complete depending on whether
the formula is cyclic. Some results about Horn formulae are also
reported.
Abstract. Results about the redundancy of certain
versions of circumscription and default logic are presented. In
particular, propositional circumscription where all variables are
minimized and skeptical default logics are considered. This
restricted version of circumscription is shown to have the
unitary redundancy property: a CNF formula is redundant (it is
equivalent to one of its proper subsets) if and only if it
contains a redundant clause (it is equivalent to itself minus one
clause); default logic does not have this property in general.
We also give the complexity of checking redundancy in the
considered formalisms.
Abstract. We formally investigate the idea of
processing off-line part of the input data in order to speed up
on-line computing. In particular, we focus on off-line
processing for intractable decision problems. To this end, we
define new complexity classes and reductions, and find complete
problems.
Abstract. We investigate the space efficiency of a
Propositional Knowledge Representation (PKR) formalism.
Informally, the space efficiency of a formalism F in representing
a certain piece of knowledge alfa, is the size of the shortest
formula of F that represents alfa. In this paper we assume that
knowledge is either a set of propositional interpretations or a
set of formulae (theorems). We provide a formal way of talking
about the relative ability of PKR formalisms to compactly
represent a set of models or a set of theorems. We introduce two
new compactness measures, the corresponding classes, and show
that the relative space efficiency of a PKR formalism in
representing models/theorems is directly related to such classes.
In particular, we consider formalisms for nonmonotonic reasoning,
such as circumscription and default logic, as well as belief
revision operators.
Abstract. In this paper we investigate the usefulness
of preprocessing part of the input of a given problem to improve
the efficiency. We extend the results of [Cadoli et al. ISTCS,
1996] by giving sufficient conditions to prove the unfeasibility
of reducing the on-line complexity via an off-line preprocessing.
We analyze the problems of diagnosis, planning, reasoning about
actions, and belief revision. We analyze other problems from
various fields.
Abstract. Intractable problems are hard to solve.
Since the work ''has to be done'', various solutions have been
developed. In this dissertation we concentrate on compilation.
We have an intractable problem, such that each instance is
composed of two parts, one (called the fixed part) is known in
advance, while the other (called the varying part) only comes at
execution time. Our idea is to solve the problem in two steps:
1. Take the fixed part and compile it into a new data structure;
2. Take the new data structure and the varying part, and produce
the output. If the second step can be accomplished in polynomial
time, the problem is said to be compilable to polynomial time.
The dissertation contains a formal definition of compilation and
compilabil- ity of hard problems. Namely, two hierarchies of
classes of problems (similar to the polynomial hierarchy) are
defined. One of them is used to prove that problems are
compilable, while the other one is mainly used to prove that some
problems are not compilable. Applications of this framework
includes: compilability of AI problems, problems on graphs,
compact representation of AI operators. A comparison with the
non-uniform polynomial hierarchy and with the FPT classes are
given.
Abstract. Some computationally hard problems -e.g.,
deduction in logical knowledge bases- are such that part of an
instance is known well before the rest of it, and remains the
same for several subsequent instances of the problem. In these
cases, it is useful to preprocess off-line this known part so as to
simplify the remaining on-line problem. In this paper we
investigate such a technique in the context of intractable, i.e.,
NP-hard, problems. Recent results in the literature show that
not all NP-hard problems behave in the same way: for some of
them preprocessing yields polynomial-time on-line simplified
problems (we call them compilable), while for other ones their
compilability imply some consequences that are considered
unlikely. Our primary goal is to provide a sound methodology
that can be used to either prove or disprove that a problem is
compilable. To this end, we define new models of computation,
complexity classes, and reductions. We find complete problems
for such classes, ''completeness'' meaning they are ''the less
likely to be compilable''. We also investigate preprocessing
that does not yield polynomial-time on-line algorithms, but
generically ''decreases'' complexity. This leads us to define
''hierarchies of compilability'', that are the analog of the
polynomial hierarchy. A detailed comparison of our framework to
the idea of ''parameterized tractability'' shows the differences
between the two approaches.
Abstract. The idea of preprocessing part of the input
of a problem in order to improve efficiency has been employed by
several researchers in several areas of computer science. In
this paper we show sufficient conditions to prove that an
intractable problem cannot be efficiently solved even allowing an
exponentially long preprocessing phase. The generality of such
conditions is shown by applying them to various problems coming
from different fields. While the results may seem to discourage
the use of compilation, we present some evidence that such
negative results are useful in practice.
Abstract. A knowledge base is redundant if it contains
parts that can be inferred from the rest of it. We study some
problems related to the redundancy of a CNF formula. In
particular, any CNF formula can be made irredundant by deleting
some of its clauses: what results is an irredundant equivalent
subset. We study the complexity of problems related to
irredundant equivalent subsets: verification, checking existence
of an irredundant equivalent subset with a given size, checking
necessary and possible presence of clauses in irredundant
equivalent subsets, and uniqueness. We also consider the problem
of redundancy with different definitions of equivalence.
Abstract. Temporal Logic Model Checking is a
verification method in which we describe a system, the model, and
then we verify whether important properties, expressed in a
temporal logic formula, hold in the system. Many Model Checking
tools employ BDDs or some other data structure to represent sets
of states. It has been empirically observed that the BDDs used
in these algorithms may grow exponentially as the model and
formula increase in size. We formally prove that no kind of data
structure of polynomial size can represent the set of valid
initial states for all models and all formulae. This result
holds for all data structures where a state can be checked in
polynomial time. Therefore, it holds not only for all types of
BDDs regardless of variable ordering, but also for more powerful
data structures, such as RBCs, MTBDDs, ADDs and SDDs. Thus, the
size explosion of BDDs is not a limit of these specific data
representation structures, but is unavoidable: every formalism
used in the same way would lead to an exponential size blow up.