Splitting an operator: Algebraic modularity results for logics with fixpoint semantics
Joost Vennekens, David Gilis, Marc Denecker
备注：
Revised to correct a substantial error in Section 4.2.2 (certain results which only hold for_consistent_ possible world sets were stated to hold in general)
发表信息：
ACM Transactions on Computational Logic, Volume 7, Number 4, 2006
It is well known that, under certain conditions, it is possible to split logic programs under stable model semantics, i.e. to divide such a program into a number of different "levels", such that the models of the entire program can be constructed by incrementally constructing models for each level. Similar results exist for other non-monotonic formalisms, such as auto-epistemic logic and default logic. In this work, we present a general, algebraicsplitting theory for logics with a fixpoint semantics. Together with the framework of approximation theory, a general fixpoint theory for arbitrary operators, this gives us a uniform and powerful way of deriving splitting results for each logic with a fixpoint semantics. We demonstrate the usefulness of these results, by generalizing existing results for logic programming, auto-epistemic logic and default logic.
The paper presents a constructive fixpoint semantics for autoepistemic logic (AEL). This fixpoint characterizes a unique but possibly three-valued belief set of an autoepistemic theory. It may be three-valued in the sense that for a subclass of formulas F, the fixpoint may not specify whether F is believed or not. The paper presents a constructive 3-valued semantics for autoepistemic logic (AEL). We introduce a derivation operator and define the semantics as its least fixpoint. The semantics is 3-valued in the sense that, for some formulas, the least fixpoint does not specify whether they are …
Open Universiteit（开放大学）；Tel-Aviv Academic College（特拉维夫学术学院）；Vrije Universiteit Brussel（布鲁塞尔自由大学）
Approximation fixpoint theory (AFT) is an abstract and general algebraic framework for studying the semantics of nonmonotonic logics. It provides a unifying study of the semantics of different formalisms for nonmonotonic reasoning, such as logic programming, default logic and autoepistemic logic. In this paper, we extend AFT to dealing with non-deterministic constructs that allow to handle indefinite information, represented e.g. by disjunctive formulas. This is done by generalizing the main constructions and corresponding results of AFT to non-deterministic operators, whose ranges are sets of…
We analyze the problem of defining well-founded semantics for ordered logic programs within a general framework based on alternating fixpoint theory. We start by showing that generalizations of existing answer set approaches to preference are too weak in the setting of well-founded semantics. We then specify some informal yet intuitive criteria and propose a semantical framework for preference handling that is more suitable for defining well-founded semantics for ordered logic programs. The suitability of the new approach is convinced by the fact that many attractive properties are satisfied b…
The logic of hybrid MKNF (minimal knowledge and negation as failure) is a powerful knowledge representation language that elegantly pairs ASP (answer set programming) with ontologies. Disjunctive rules are a desirable extension to normal rule-based reasoning and typically semantic frameworks designed for normal knowledge bases need substantial restructuring to support disjunctive rules. Alternatively, one may lift characterizations of normal rules to support disjunctive rules by inducing a collection of normal knowledge bases, each with the same body and a single atom in its head. In this work…
The fixpoint completion fix(P) of a normal logic program P is a program transformation such that the stable models of P are exactly the models of the Clark completion of fix(P). This is well-known and was studied by Dung and Kanchanasut (1989). The correspondence, however, goes much further: The Gelfond-Lifschitz operator of P coincides with the immediate consequence operator of fix(P), as shown by Wendt (2002), and even carries over to standard operators used for characterizing the well-founded and the Kripke-Kleene semantics. We will apply this knowledge to the study of the stable semantics,…
Approximation Fixpoint Theory with Refined Approximation Spaces
Department Of Computer Science, KU Leuven, Belgium（比利时库勒万大学计算机科学系）；Department Of Computer Science, Vrije Universiteit Brussel, Belgium（比利时自由大学布鲁塞尔分校计算机科学系）
The well-founded semantics is one of the most widely studied and used semantics of logic programs with negation. In the case of finite propositional programs, it can be computed in polynomial time, more specifically, in O(|At(P)|size(P)) steps, where size(P) denotes the total number of occurrences of atoms in a logic program P. This bound is achieved by an algorithm introduced by Van Gelder and known as the alternating-fixpoint algorithm. Improving on the alternating-fixpoint algorithm turned out to be difficult. In this paper we study extensions and modifications of the alternating-fixpoint a…
Shanghai University（上海大学）；University of Alberta（阿尔伯塔大学）
Approximation fixpoint theory (AFT) provides an algebraic framework for the study of fixpoints of operators on bilattices and has found its applications in characterizing semantics for various classes of logic programs and nonmonotonic languages. In this paper, we show one more application of this kind: the alternating fixpoint operator by Knorr et al. for the study of the well-founded semantics for hybrid MKNF knowledge bases is in fact an approximator of AFT in disguise, which, thanks to the power of abstraction of AFT, characterizes not only the well-founded semantics but also two-valued as…
We study fixpoints of operators on lattices. To this end we introduce the notion of an approximation of an operator. We order approximations by means of a precision ordering. We show that each lattice operator O has a unique most precise or ultimate approximation. We demonstrate that fixpoints of this ultimate approximation provide useful insights into fixpoints of the operator O. We apply our theory to logic programming and introduce the ultimate Kripke-Kleene, well-founded and stable semantics. We show that the ultimate Kripke-Kleene and well-founded semantics are more precise then their sta…
In this paper, we present two alternative approaches to defining answer sets for logic programs with arbitrary types of abstract constraint atoms (c-atoms). These approaches generalize the fixpoint-based and the level mapping based answer set semantics of normal logic programs to the case of logic programs with arbitrary types of c-atoms. The results are four different answer set definitions which are equivalent when applied to normal logic programs. The standard fixpoint-based semantics of logic programs is generalized in two directions, called answer set by reduct and answer set by complemen…
This technical note describes a monotone and continuous fixpoint operator to compute the answer sets of programs with aggregates. The fixpoint operator relies on the notion of aggregate solution. Under certain conditions, this operator behaves identically to the three-valued immediate consequence operator $Φ^{aggr}_P$ for aggregate programs, independently proposed Pelov et al. This operator allows us to closely tie the computational complexity of the answer set checking and answer sets existence problems to the cost of checking a solution of the aggregates in the program. Finally, we relate th…
Well-known principles of induction include monotone induction and different sorts of non-monotone induction such as inflationary induction, induction over well-founded sets and iterated induction. In this work, we define a logic formalizing induction over well-founded sets and monotone and iterated induction. Just as the principle of positive induction has been formalized in FO(LFP), and the principle of inflationary induction has been formalized in FO(IFP), this paper formalizes the principle of iterated induction in a new logic for Non-Monotone Inductive Definitions (ID-logic). The semantics…
CENTRIA - Centre for Artificial Intelligence（CENTRIA人工智能中心）；Universidade Nova de Lisboa（里斯本新大学）
Even though modularity has been studied extensively in conventional logic programming, there are few approaches on how to incorporate modularity into Answer Set Programming, a prominent rule-based declarative programming paradigm. A major approach is Oikarinnen and Janhunen's Gaifman-Shapiro-style architecture of program modules, which provides the composition of program modules. Their module theorem properly strengthens Lifschitz and Turner's splitting set theorem for normal logic programs. However, this approach is limited by module conditions that are imposed in order to ensure the compatib…
Non deterministic applications arise in many domains, including, stochastic optimization, multi-objectives optimization, stochastic planning, contingent stochastic planning, reinforcement learning, reinforcement learning in partially observable Markov decision processes, and conditional planning. We present a logic programming framework called non deterministic logic programs, along with a declarative semantics and fixpoint semantics, to allow representing and reasoning about inherently non deterministic real-world applications. The language of non deterministic logic programs framework is ext…
Helsinki University of Technology（赫尔辛基理工大学）；Technische Universität Wien（维也纳工业大学）
Practically all programming languages allow the programmer to split a program into several modules which brings along several advantages in software development. In this paper, we are interested in the area of answer-set programming where fully declarative and nonmonotonic languages are applied. In this context, obtaining a modular structure for programs is by no means straightforward since the output of an entire program cannot in general be composed from the output of its components. To better understand the effects of disjunctive information on modularity we restrict the scope of analysis t…
This paper studies the stable model semantics of logic programs with (abstract) constraint atoms and their properties. We introduce a succinct abstract representation of these constraint atoms in which a constraint atom is represented compactly. We show two applications. First, under this representation of constraint atoms, we generalize the Gelfond-Lifschitz transformation and apply it to define stable models (also called answer sets) for logic programs with arbitrary constraint atoms. The resulting semantics turns out to coincide with the one defined by Son et al., which is based on a fixpoi…
University of Lübeck（吕贝克大学）；University of Bamberg（班贝格大学）
In applications that use knowledge representation (KR) techniques, in particular those that combine data-driven and logic methods, the domain of objects is not an abstract unstructured domain, but it exhibits a dedicated, deep structure of geometric objects. One example is the class of convex sets used to model natural concepts in conceptual spaces, which also links via convex optimization techniques to machine learning. In this paper we study logics for such geometric structures. Using the machinery of lattice theory, we describe an extension of minimal orthologic with a partial modularity ru…
The Stable Model Semantics for Higher-Order Logic Programming
Vrije Universiteit Brussel（布鲁塞尔自由大学）；Katholieke Universiteit Leuven（鲁汶大学）；Harokopio University of Athens（雅典哈罗科皮奥大学）；National and Kapodistrian University of Athens（雅典国立与卡波迪斯特里亚大学）
Most modern formalisms used in Databases and Artificial Intelligence for describing an application domain are based on the notions of class (or concept) and relationship among classes. One interesting feature of such formalisms is the possibility of defining a class, i.e., providing a set of properties that precisely characterize the instances of the class. Many recent articles point out that there are several ways of assigning a meaning to a class definition containing some sort of recursion. In this paper, we argue that, instead of choosing a single style of semantics, we achieve better resu…
In this paper we study the uses and the semantics of non-monotonic negation in probabilistic deductive data bases. Based on the stable semantics for classical logic programming, we introduce the notion of stable formula, functions. We show that stable formula, functions are minimal fixpoints of operators associated with probabilistic deductive databases with negation. Furthermore, since a. probabilistic deductive database may not necessarily have a stable formula function, we provide a stable class semantics for such databases. Finally, we demonstrate that the proposed semantics can handle def…
Arizona State University（亚利桑那州立大学）；Universidad Nacional del Sur（国立南方大学）
While deep neural networks have led to major advances in image recognition, language translation, data mining, and game playing, there are well-known limits to the paradigm such as lack of explainability, difficulty of incorporating prior knowledge, and modularity. Neuro symbolic hybrid systems have recently emerged as a straightforward way to extend deep neural networks by incorporating ideas from symbolic reasoning such as computational logic. In this paper, we propose a list desirable criteria for neuro symbolic systems and examine how some of the existing approaches address these criteria.…
In this paper, we examine the concept of modularity, an often cited advantage of the ruled-based representation methodology. We argue that the notion of modularity consists of two distinct concepts which we call syntactic modularity and semantic modularity. We argue that when reasoning under certainty, it is reasonable to regard the rule-based approach as both syntactically and semantically modular. However, we argue that in the case of plausible reasoning, rules are syntactically modular but are rarely semantically modular. To illustrate this point, we examine a particular approach for managi…
We revisit the issue of connections between two leading formalisms in nonmonotonic reasoning: autoepistemic logic and default logic. For each logic we develop a comprehensive semantic framework based on the notion of a belief pair. The set of all belief pairs together with the so called knowledge ordering forms a complete lattice. For each logic, we introduce several semantics by means of fixpoints of operators on the lattice of belief pairs. Our results elucidate an underlying isomorphism of the respective semantic constructions. In particular, we show that the interpretation of defaults as m…
