ABSTRACTS
A3.4 Computational Logic and Applications of Logic
Hybrid Logic for Qualitative Reasoning about Location
Michal Zawidzki, Institute of Philosophy, University of Lodz/University of Warsaw, Warszawa, POLAND
Tomasz Lechowski, Institute of Philosophy, University of Warsaw, Warszawa, POLAND
Przemyslaw Walega,University of Warsaw, Warszawa, POLAND
Spatial reasoning is one of the most interesting abilities that humans posses
but can hardly reproduce by means of artificial intelligence algorithms. Humanlike methods are usually less precise than artificial ones but, on the other hand,
their practical results are far better and more universal than of any artificial system.
At the intersection of logic and computer science there emerged a whole
field of research called qualitative spatial reasoning (QSR in short) investigating
methods that try to imitate or model human-like reasoning.
In this talk, we present a hybrid modal framework for qualitative reasoning
about location of objects in a flat 2-dimensional, subject-centered environment.
In contrast to many existing qualitative approaches for reasoning about location,
our system is devised to simultaneously capture intuitive notions expressing
relative directions such as: to the right, to the left, behind, before etc., and
qualitative distance relations like far or close. Furtheremore, since our approach
is subject-oriented, it captures spatial representation in a human-like manner.
The language of our Hybrid Logic for Qualitative Location (HLQL) is the
basic hybrid multi-modal language (involving nominals and satisfaction operators) augmented with appropriately tailored accessibility relations and the constant symbol s (for the subject). The semantics for the logic is Kripke-structure
based. A frame for HLQL is a plane, either finite or infinite, with polar coordinates, divided into cells of arbitrary length and angle-width. The central locus
is occupied by the subject.
In the talk, we provide an axiomatization for HLQL and claim its soundness. We also show that, notwithstanding the PSpace -completeness of the
basic hybrid multi-modal logic, the consequence of certain accessibility relations
(like right and behind) causes the exponential blow-up of the computational
complexity of HLQL raising it to NExpTime-hardness.
Contrary-to-Duty Imperatives: A Paraconsistent Deontic Approach
Can Baskent, Semagramme, INRIA, Vandoeuvre-lès-Nancy Cedex, FRANCE
Contrary-to-duty imperatives present an interesting take on deontology, and describes what a moral agent must do
when she neglects her duties. In short, contrary-to-duty (CtD, henceforth) obligations are of the following form:
“You ought to do a, but if you do not do a, then you must do b”.
CtD imperatives turn out to be inconsistent if the actions a and b contradict each other. Modal deontic logic can
express this situation. Yet, this contradiction trivializes the classical modal deontic logic. For that reason, we need
to adopt a logical formalism that does not collapse under inconsistencies.
Paraconsistent logic suggests sound alternatives when moral obligations contradict each other. In this paper, we
suggest a paraconsistent framework to express CtD imperatives. Our approach builds on some previously
suggested paraconsistent deontic logics of da Costa and Carnielli.
The central contribution of this paper is to suggest a dynamic paraconsistent deontic logic to express CtD
imperatives. The dynamic take helps us analyze how the model changes in a paraconsistent way when the moral
agent neglects her initial obligation, and how the negligence of duties can be read as a dynamic update.
In order to achieve this, we syntactically define CtD as follows: “C(p, q) ≡ Op ∧ ¬p → Oq”, meaning that the agent
is obliged to p, yet p is not the case, then she is obliged to q. If we assume the classical deontic axiom that ¬(Op ∧
O¬p) then C(p,q) produces a contradiction if q is taken as ¬p. This justifies the use of paraconsistent logic for CtD
imperatives.
Based on this definition, we define updated models that satisfies the second obligation (here, q), and observe how
CtD imperatives may benefit from a dynamic approach. Furthermore, we consider some frame properties if the
underlying logic enjoys various additional frame properties.
Generalized Quantifiers and Higher-order Logic Programming
Peter Gabrovsky, Computer Science, California State University, Northridge, USA
We study the effects of incorporating generalized quantifiers in the bodies of program clauses in a higher-order
logic programming language, such as the one introduced in [1]. We observe that there are some quantifiers
(e.g., the existential) that keep the enriched language within the realm of computability, which is to say that the
interpreter of that language is computable in the sense of Church-Turing thesis (i.e., it is a partial recursive
function). However, there are quantifiers (e.g., the universal) such that the enriched language does not have a
computable interpreter. Furthermore, among these quantifiers there are some that when used in the bodies of
program clauses, define a language whose interpreter is not even representable (programmable in the
generalized sense) in that language. A logic programming language that does not have an interpreter
programmable in that language fails the so-called self-reflection property, which is one of the two fundamental
properties of universal programming languages - the other being composition. We are concerned here with the
effects of certain kinds of quantifiers, the so-called monotonic quantifiers, and we show that if we restrict the
use to only those quantifiers, the enriched logic programming language retains not only the self-reflection
property, but also that of composition. We also show that the interpreter of such a language is sound and
complete with respect to the declarative semantics of the language. The results here are essentially a
generalization of the results in [2] and [3], where we dealt only with the traditional first and higher-order logic
programming languages.
[1] Miller and Nadathur, Higher-order logic programming, Third International Conference on Logic
Programming, 1986.
[2] Peter Gabrovsky, Logic programming with generalized quantifiers, Journal of Computing Sciences in
Colleges, 2010.
[3] Peter Gabrovsky, A recursion-theoretic semantics for higher-order logic programming, Fifth Southeastern
Logic Symposium 1989.