Forgetting is an important concept in knowledge representation and automated reasoning with widespread applications across a number of disciplines. A standard forgetting operator, characterized in [26] in terms of model-theoretic semantics and primarily focusing on the propositional case, opened up a new research subarea. In this paper, a new operator called weak forgetting, dual to standard forgetting, is introduced and both together are shown to offer a new more uniform perspective on forgetting operators in general. Both the weak and standard forgetting operators are characterized in terms of entailment and inference, rather than a model theoretic semantics. This naturally leads to a useful algorithmic perspective based on quantifier elimination and the use of Ackermann's Lemma and its fixpoint generalization. The strong formal relationship between standard forgetting and strongest necessary conditions and weak forgetting and weakest sufficient conditions is also characterized quite naturally through the entailment-based, inferential perspective used. The framework used to characterize the dual forgetting operators is also generalized to the first-order case and includes useful algorithms for computing first-order forgetting operators in special cases. Practical examples are also included to show the importance of both weak and standard forgetting in modeling and representation.

Inheritance has intensively been studied in both object-oriented programming (Oop) and knowledge representation and reasoning (KRR). On the other hand, the approaches to multiple inheritance and related method resolution, developed in both domains, remain separated. The primary goal of this paper is to demonstrate how these approaches may be integrated using inheritance expressions. In particular, we examine inheritance as a belief bases management machinery designed to operate in dynamically changing environments where objects are embedded and act. We focus on objects that are belief bases containers, potentially participating in complex distributed reasoning scenarios. We show that inheritance expressions, inspired both by Oop and KRR, provide a simple yet flexible and powerful means for expressing inheritance and related belief/knowledge fusion.

Reasoning about uncertainty is one of the main cornerstones of Knowledge Representation. More recently, combining logic with probability has been of major interest. Rough set methods have been proposed for modeling incompleteness and imprecision based on indiscernibility and its generalizations and there is a large body of work in this direction. More recently, the classical theory has been generalized to include probabilistic rough set methods of which there are also a great variety of proposals. Pragmatic, easily accessible, and easy to use tools for specification and reasoning with this wide variety of methods is lacking. It is the purpose of this paper to fill in that gap where the focus will be on probabilistic rough set methods. A landscape of (probabilistic) rough set reasoning methods and the variety of choices involved in specifying them is surveyed first. While doing this, an abstract generalization of all the considered approaches is derived which subsumes each of the methods. One then shows how, via this generalization, one can specify and reason about any of these methods using ProbLog, a popular and widely used probabilistic logic programming language based on Prolog. The paper also considers new techniques in this context such as the use of probabilistic target sets when defining rough sets and the use of partially specified base relations that are also probabilistic. Additionally, probabilistic approaches using tolerance spaces are proposed. The paper includes a rich set of examples and provides a framework based on a library of generic ProbLog relations that make specification of any of these methods, straightforward, efficient and compact. Complete, ready to run ProbLog code is included in the Appendix for all examples considered.

As a side-effect of the Covid-19 pandemic, significant decreases in medical procedures for noncommunicable diseases have been observed. This calls for a decision support assisting in the analysis of opportunities to relocate procedures among hospitals in an efficient or, preferably, optimal manner. In the current paper we formulate corresponding decision problems and develop linear (mixed integer) programming models for them. Since solving mixed integer programming problems is NP-complete, we verify experimentally their usefulness using real-world data about urological procedures. We show that even for large models, with millions of variables, the problems' instances are solved in perfectly acceptable time.

Reasoning about uncertainty is one of the main cornerstones of Knowledge Representation. Formal representations of uncertainty are numerous and highly varied due to different types of uncertainty intended to be modeled such as vagueness, imprecision and incompleteness. There is a rich body of theoretical results that has been generated for many of these approaches. It is often the case though, that pragmatic tools for reasoning with uncertainty lag behind this rich body of theoretical results. Rough set theory is one such approach for modeling incompleteness and imprecision based on indiscernibility and its generalizations. In this paper, we provide a pragmatic tool for constructively reasoning with generalized rough set approximations that is based on the use of Answer Set Programming (Asp). We provide an interpretation of answer sets as (generalized) approximations of crisp sets (when possible) and show how to use Asp solvers as a tool for reasoning about (generalized) rough set approximations situated in realistic knowledge bases. The paper includes generic Asp templates for doing this and also provides a case study showing how these techniques can be used to generate reducts for incomplete information systems. Complete, ready to run clingo Asp code is provided in the Appendix, for all programs considered. These can be executed for validation purposes in the clingo Asp solver.

Organizational structures control goal-directed activities of companies or groups in an efficient manner. These activities are typically described using rules, roles, and responsibilities. In this paper we focus on doxastic reasoning of groups and individuals in organizations situated in dynamic environments. We model organizational structures using partial orders what is general enough to serve our purposes. In particular, we show how belief bases associated with organizational units/entities are to be structured to reflect the induced hierarchies. The main logical tool ensuring efficiency is belief shadowing, a relatively recent approach to transient belief change. As the main contribution, we develop Drocs, a computationally friendly framework for organization-centered doxastic reasoning. An illustrative case study is also provided.

Object-oriented (O2) and declarative rule-based (RB) paradigms are known and intensively used for decades. There have been many approaches to combine O2 and RB concepts in a uniform framework. However, they either focus on embedding rules into O2 languages, or provide a declarative semantics and proof theory for the O2 machinery. In the current paper we focus on understanding objects as basic building blocks for deductive queries. To achieve the goal of object-rule fusion we use carefully chosen elementary constructs, encompassed in a simple and uniform framework easily extendable and adaptable to many existing RB languages.

We apply a non-classical four-valued logic in the process of reasoning regarding strategies for cops in a modified game of “Cops and Robber” played on a graph. We extend the game by introducing uncertainty in a form of random failures of detecting devices. This is realized by allowing that a robber can be detected in a node only with the given probability PA. Additionally, with the probability PF, cops can be given a false-positive, i.e., they are informed that the robber is located at some node, whereas it is located somewhere else. Consequently, non-zero PFintroduces a measurement noise into the system. All the cops have access to information provided by the detectors and can communicate with each other, so they can coordinate the search. By adjusting the number of detectors, PA, and PFwe can achieve a smooth transition between the two well-known variants of the game: “with fully visible robber” and “with invisible robber”. We compare a simple probabilistic strategy for cops with the non-parametric strategy based on reasoning with a four-valued paraconsistent logic. It is shown that this novel approach leads to a good performance, as measured by the required mean catch-time. We conclude that this type of reasoning can be applied in real-world applications where there is no knowledge about the underlying source of errors which is particularly useful in robotics.

This paper discusses an implementation of four speech acts: assert, concede, request and challenge in a paraconsistent framework. A natural four-valued model of interaction yields multiple new cognitive situations. They are analyzed in the context of communicative relations, which partially replace the concept of trust. These assumptions naturally lead to six types of situations, which often require performing conflict resolution and belief revision.

Agents' beliefs can be incomplete and partially inconsistent. The process of agents' belief formation in such contexts has to be supported by suitable tools allowing one to express a variety of inconsistency resolving and nonmonotonic reasoning techniques.

We study Horn fragments of serial multimodal logics which are characterized by regular grammars with converse. Such logics are useful for reasoning about epistemic states of multiagent systems as well as similarity-based approximate reasoning. We provide the first algorithm with PTIME data complexity for checking satisfiability of a Horn knowledge base in a serial regular grammar logic with converse.

This paper focuses on approximate reasoning based on the use of approximation spaces. Approximation spaces and the approximated relations induced by them are a generalization of the rough set-based approximations of Pawlak. Approximation spaces are used to define neighborhoods around individuals and rough inclusion functions. These in turn are used to define approximate sets and relations. In any of the approaches, one would like to embed such relations in an appropriate logical theory which can be used as a reasoning engine for specific applications with specific constraints. We propose a framework which permits a formal study of the relationship between properties of approximations and properties of approximation spaces. Using ideas from correspondence theory, we develop an analogous framework for approximation spaces. We also show that this framework can be strongly supported by automated techniques for quantifier elimination.

In natural language we often use graded concepts, reflecting different intensity degrees of certain features. Whenever such concepts appear in a given real-life context, they need to be appropriately expressed in its models. In this paper, we provide

We reformulate Pratt’s tableau decision procedure of checking satisfiability of a set of formulas in PDL. Our formulation is simpler and its implementation is more direct. Extending the method we give the first ExpTime (optimal) tableau decision procedure not based on transformation for checking consistency of an ABox w.r.t. a TBox in PDL (here, PDL is treated as a description logic). We also prove a new result that the data complexity of the instance checking problem in PDL is coNP-complete.

We present a language for defining paraconsistent rough sets and reasoning about them. Our framework relates and brings together two major fields: rough sets [23] and paraconsistent logic programming [9]. To model inconsistent and incomplete information we use a four-valued logic. The language discussed in this paper is based on ideas of our previous work [21,32,22] developing a four-valued framework for rough sets. In this approach membership function, set containment and set operations are four-valued, where logical values are t (true), f (false), i (inconsistent) and u (unknown). We investigate properties of paraconsistent rough sets as well as develop a paraconsistent rule language, providing basic computational machinery for our approach.

We introduce the notion of a meta-query on relational databases and a technique which can be used to represent and solve a number of interesting problems from the area of knowledge representation using logic. The technique is based on the use of quantifier elimination and may also be used to query relational databases using a declarative query language called SHQL (Semi-Horn Query Language), introduced in [6]. SHQL is a fragment of classical first-order predicate logic and allows us to define a query without supplying its explicit definition. All SHQL queries to the database can be processed in polynomial time (both on the size of the input query and the size of the database). We demonstrate the use of the technique in problem solving by structuring logical puzzles from the Knights and Knaves domain as SHQL meta-queries on relational databases. We also provide additional examples demonstrating the flexibility of the technique. We conclude with a description of a newly developed software tool, The Logic Engineer, which aids in the description of algorithms using transformation and reduction techniques such as those applied in the meta-querying approach.

In this paper, we consider the problem of expressing and computing queries on relational deductive databases in a purely declarative query language, called SHQL (Semi-Horn Query Language). Assuming the relational databases in question are ordered, we show that all SHQL queries are computable in PTIME (polynomial time) and the whole class of PTIME queries is expressible in SHQL. Although similar results have been proven for fixpoint languages and extensions to datalog, the claim is that SHQL has the advantage of being purely declarative, where the negation operator is interpreted as classical negation, mixed quantifiers may be used and a query is simply a restricted first-order theory not limited by the rule-based syntactic restrictions associated with logic programs in general. We describe the PTIME algorithm used to compute queries in SHQL which is based in part on quantifier elimination techniques and also consider extending the method to incomplete relational databases using intuitions related to circumscription techniques.

We first define general domain circumscription (GDC) and provide it with a semantics. GDC subsumes existing domain circumscription proposals in that it allows varying of arbitrary predicates, functions, or constants, to maximize the minimization of the domain of a theory. We then show that for the class of semi-universal theories without function symbols, that the domain circumscription of such theories can be constructively reduced to logically equivalent first-order theories by using an extension of the DLS algorithm, previously proposed by the authors for reducing second-order formulas. We also show that for a certain class of domain circumscribed theories, that any arbitrary second-order circumscription policy applied to these theories is guaranteed to be reducible to a logically equivalent first-order theory. In the case of semi-universal theories with functions and arbitrary theories which are not separated, we provide additional results, which although not guaranteed to provide reductions in all cases, do provide reductions in some cases. These results are based on the use of fixpoint reductions.

Circumscription has been perceived as an elegant mathematical technique for modeling nonmonotonic and commonsense reasoning, but difficult to apply in practice due to the use of second-order formulas. One proposal for dealing with the computational p

In the current paper we present a powerful technique of obtaining natural deduction proof systems for first-order fixpoint logics. The term fixpoint logics refers collectively to a class of logics consisting of modal logics with modalities definable at meta-level by fixpoint equations on formulas. The class was found very interesting as it contains most logics of programs with e.g. dynamic logic, temporal logic and the μ-calculus among them. In this paper we present a technique that allows us to derive automatically natural deduction systems for modal logics from fixpoint equations defining the modalities.

The current paper is devoted to automated techniques in the correspondence theory. The theory we deal with concerns the problem of finding classical first-order axioms corresponding to propositional modal schemas. Given a modal schema and a semantics-based method of translating propositional modal formulae into classical first-order ones, we try to derive automatically classical first-order formulae characterizing precisely the class of frames validating the schema. The technique we consider can, in many cases, be easily applied even without computer support. Although we mainly concentrate on Kripke semantics, the technique we apply is much more general, as it is based on elimination of second-order quantifiers from formulae. We show many examples of application of the method. These can also serve as new, automated proofs of considered correspondences.