7 research outputs found
The Complexity of Reasoning for Fragments of Autoepistemic Logic
Autoepistemic logic extends propositional logic by the modal operator L. A
formula that is preceded by an L is said to be "believed". The logic was
introduced by Moore 1985 for modeling an ideally rational agent's behavior and
reasoning about his own beliefs. In this paper we analyze all Boolean fragments
of autoepistemic logic with respect to the computational complexity of the
three most common decision problems expansion existence, brave reasoning and
cautious reasoning. As a second contribution we classify the computational
complexity of counting the number of stable expansions of a given knowledge
base. To the best of our knowledge this is the first paper analyzing the
counting problem for autoepistemic logic
Complexity of Non-Monotonic Logics
Over the past few decades, non-monotonic reasoning has developed to be one of
the most important topics in computational logic and artificial intelligence.
Different ways to introduce non-monotonic aspects to classical logic have been
considered, e.g., extension with default rules, extension with modal belief
operators, or modification of the semantics. In this survey we consider a
logical formalism from each of the above possibilities, namely Reiter's default
logic, Moore's autoepistemic logic and McCarthy's circumscription.
Additionally, we consider abduction, where one is not interested in inferences
from a given knowledge base but in computing possible explanations for an
observation with respect to a given knowledge base.
Complexity results for different reasoning tasks for propositional variants
of these logics have been studied already in the nineties. In recent years,
however, a renewed interest in complexity issues can be observed. One current
focal approach is to consider parameterized problems and identify reasonable
parameters that allow for FPT algorithms. In another approach, the emphasis
lies on identifying fragments, i.e., restriction of the logical language, that
allow more efficient algorithms for the most important reasoning tasks. In this
survey we focus on this second aspect. We describe complexity results for
fragments of logical languages obtained by either restricting the allowed set
of operators (e.g., forbidding negations one might consider only monotone
formulae) or by considering only formulae in conjunctive normal form but with
generalized clause types.
The algorithmic problems we consider are suitable variants of satisfiability
and implication in each of the logics, but also counting problems, where one is
not only interested in the existence of certain objects (e.g., models of a
formula) but asks for their number.Comment: To appear in Bulletin of the EATC
Do Hard SAT-Related Reasoning Tasks Become Easier in the Krom Fragment?
Many reasoning problems are based on the problem of satisfiability (SAT).
While SAT itself becomes easy when restricting the structure of the formulas in
a certain way, the situation is more opaque for more involved decision
problems. We consider here the CardMinSat problem which asks, given a
propositional formula and an atom , whether is true in some
cardinality-minimal model of . This problem is easy for the Horn
fragment, but, as we will show in this paper, remains -complete (and
thus -hard) for the Krom fragment (which is given by formulas in
CNF where clauses have at most two literals). We will make use of this fact to
study the complexity of reasoning tasks in belief revision and logic-based
abduction and show that, while in some cases the restriction to Krom formulas
leads to a decrease of complexity, in others it does not. We thus also consider
the CardMinSat problem with respect to additional restrictions to Krom formulas
towards a better understanding of the tractability frontier of such problems
Moons Are Planets: Scientific Usefulness Versus Cultural Teleology in the Taxonomy of Planetary Science
We argue that taxonomical concept development is vital for planetary science
as in all branches of science, but its importance has been obscured by unique
historical developments. The literature shows that the concept of planet
developed by scientists during the Copernican Revolution was theory-laden and
pragmatic for science. It included both primaries and satellites as planets due
to their common intrinsic, geological characteristics. About two centuries
later the non-scientific public had just adopted heliocentrism and was
motivated to preserve elements of geocentrism including teleology and the
assumptions of astrology. This motivated development of a folk concept of
planet that contradicted the scientific view. The folk taxonomy was based on
what an object orbits, making satellites out to be non-planets and ignoring
most asteroids. Astronomers continued to keep primaries and moons classed
together as planets and continued teaching that taxonomy until the 1920s. The
astronomical community lost interest in planets ca. 1910 to 1955 and during
that period complacently accepted the folk concept. Enough time has now elapsed
so that modern astronomers forgot this history and rewrote it to claim that the
folk taxonomy is the one that was created by the Copernican scientists.
Starting ca. 1960 when spacecraft missions were developed to send back detailed
new data, there was an explosion of publishing about planets including the
satellites, leading to revival of the Copernican planet concept. We present
evidence that taxonomical alignment with geological complexity is the most
useful scientific taxonomy for planets. It is this complexity of both primary
and secondary planets that is a key part of the chain of origins for life in
the cosmos.Comment: 68 pages, 16 figures. For supplemental data files, see
https://www.philipmetzger.com/moons_are_planets
Counting complexity of propositional abduction
Abduction is an important method of non-monotonic reasoning with many applications in AI and related topics. In this paper, we concentrate on propositional abduction, where the background knowledge is given by a propositional formula. Decision problems of great interest are the existence and the relevance problems. The complexity of these decision problems has been systematically studied while the counting complexity of propositional abduction has remained obscure. The goal of this work is to provide a comprehensive analysis of the counting complexity of propositional abduction in various classes of theories.
Counting complexity of minimal cardinality and minimal weight abduction
Abstract. Abduction is an important method of non-monotonic reasoning with many applications in artificial intelligence and related topics. In this paper, we concentrate on propositional abduction, where the background knowledge is given by a propositional formula. We have recently started to study the counting complexity of propositional abduction. However, several important cases have been left open, namely, the cases when we restrict ourselves to solutions with minimal cardinality or with minimal weight. These cases – possibly combined with priorities – are now settled in this paper. We thus arrive at a complete picture of the counting complexity of propositional abduction.