A simplified proof of a fixpoint theorem for functions over stratified complete lattices

Παρουσιάζω μια απλουστευμένη και πιο άμεση απόδειξη του θεωρήματος ‘A fixed point theorem for non-monotonic functions’ ([Z. Ésik and P. Rondogiannis. Theor. Comput. Sci., 574:18–38, 2015]). Το Θεώρημα αυτό αποδεικνύει την ύπαρξη ενός ελάχιστου σταθερού σημείου (πιθανώς) μη-μονοτονικών συναρτήσεων σε ειδικά δομημένα πλήρη πλέγματα, τα οποία θα ορίσω ως στρωματοποιημένα πλήρη πλέγματα. Όταν το θεώρημα εφαρμοστεί σε μονοτονικές συναρτήσεις είναι ισοδύναμο του θεωρήματος Knaster-Tarski. Το θεώρημα έχει άμεσες εφαρμογές στην σημασιολογία της άρνησης μέσω αποτυχίας του λογικού προγραμματισμού. Για παράδειγμα με την χρήση του θα γινόταν πιο άμεση η απόδειξη της ύπαρξης ελάχιστου σταθερού σημείου στο [P. Rondogiannis and W.W. Wadge. ACM Transactions on Computational Logic, 6(2):441–467, 2005].I present a simplified and more direct proof of the fixed point theorem for non-monotonic functions ([Z. Ésik and P. Rondogiannis. Theor. Comput. Sci., 574:18–38, 2015]). This theorem proves the existence of a least fixed point of (potentially) non-monotonic functions over specially structured complete lattices, which I will define as stratified complete lattices. When the theorem is applied to monotonic functions is equivalent to Knaster-Tarski theorem. The theorem has direct applications in the semantics of negation-as-failure in logic programming. In particular, it could be used to provide a more direct proof of the least fixed point result of [P. Rondogiannis and W.W. Wadge. ACM Transactions on Computational Logic, 6(2):441–467, 2005]

An analysis of the equational properties of the well-founded fixed point

Well-founded fixed points have been used in several areas of knowledge representation and reasoning and to give semantics to logic programs involving negation. They are an important ingredient of approximation fixed point theory. We study the logical properties of the (parametric) well-founded fixed point operation. We show that the operation satisfies several, but not all of the equational properties of fixed point operations described by the axioms of iteration theories

Evidence for Fixpoint Logic

For many modal logics, dedicated model checkers offer diagnostics (e.g., counterexamples) that help the user understand the result provided by the solver. Fixpoint logic offers a unifying framework in which such problems can be expressed and solved, but a drawback of this framework is that it lacks comprehensive diagnostics generation. We extend the framework with a notion of evidence, which can be specialized to obtain diagnostics for various model checking problems, behavioural equivalence and refinement checking problems. We demonstrate this by showing how our notion of evidence can be used to obtain diagnostics for the problem of deciding stuttering bisimilarity. Moreover, we show that our notion generalizes the existing notions of counterexample and witness for LTL and ACTL* model checking

Overview of an abstract fixed point theory for non-monotonic functions and its applications to logic programming

The purpose of the present paper is to give an overview of our joint work with Zoltán Ésik, namely the development of an abstract fixed point theory for a class of non-monotonic functions [4] and its use in providing a novel denotational semantics for a very broad extension of classical logic programming [1]. Our purpose is to give a high-level presentation of the main developments of these two works, that avoids as much as possible the underlying technical details, and which can be used as a mild introduction to the area