Borderline vs. unknown: comparing three-valued representations of imperfect information

In this paper we compare the expressive power of elementary representation formats for vague, incomplete or conflicting information. These include Boolean valuation pairs introduced by Lawry and González-Rodríguez, orthopairs of sets of variables, Boolean possibility and necessity measures, three-valued valuations, supervaluations. We make explicit their connections with strong Kleene logic and with Belnap logic of conflicting information. The formal similarities between 3-valued approaches to vagueness and formalisms that handle incomplete information often lead to a confusion between degrees of truth and degrees of uncertainty. Yet there are important differences that appear at the interpretive level: while truth-functional logics of vagueness are accepted by a part of the scientific community (even if questioned by supervaluationists), the truth-functionality assumption of three-valued calculi for handling incomplete information looks questionable, compared to the non-truth-functional approaches based on Boolean possibility–necessity pairs. This paper aims to clarify the similarities and differences between the two situations. We also study to what extent operations for comparing and merging information items in the form of orthopairs can be expressed by means of operations on valuation pairs, three-valued valuations and underlying possibility distributions.     

A computational glimpse at the Leibniz and Frege hierarchies

In this paper we consider, from a computational point of view, the problem of classifying logics within the Leibniz and Frege hierarchies typical of abstract algebraic logic. The main result states that, for logics presented syntactically, this problem is in general undecidable. More precisely, we show that there is no algorithm that classifies the logic of a finite consistent Hilbert calculus in the Leibniz and in the Frege hierarchies.     

A simple logic for reasoning about incomplete knowledge

The semantics of modal logics for reasoning about belief or knowledge is often described in terms of accessibility relations, which is too expressive to account for mere epistemic states of an agent. This paper proposes a simple logic whose atoms express epistemic attitudes about formulae expressed in another basic propositional language, and that allows for conjunctions, disjunctions and negations of belief or knowledge statements. It allows an agent to reason about what is known about the beliefs held by another agent. This simple epistemic logic borrows its syntax and axioms from the modal logic KD. It uses only a fragment of the S5 language, which makes it a two-tiered propositional logic rather than as an extension thereof. Its semantics is given in terms of epistemic states understood as subsets of mutually exclusive propositional interpretations. Our approach offers a logical grounding to uncertainty theories like possibility theory and belief functions. In fact, we define the most basic logic for possibility theory as shown by a completeness proof that does not rely on accessibility relations.     

Tractability frontiers in probabilistic team semantics and existential second-order logic over the reals

Probabilistic team semantics is a framework for logical analysis of probabilistic dependencies. Our focus is on the axiomatizability, complexity, and expressivity of probabilistic inclusion logic and its extensions. We identify a natural fragment of existential second-order logic with additive real arithmetic that captures exactly the expressivity of probabilistic inclusion logic. We furthermore relate these formalisms to linear programming, and doing so obtain PTIME data complexity for the logics. Moreover, on finite structures, we show that the full existential second-order logic with additive real arithmetic can only express NP properties. Lastly, we present a sound and complete axiomatization for probabilistic inclusion logic at the atomic level.     

Bounded linear-time temporal logic: A proof-theoretic investigation

Propositional and first-order bounded linear-time temporal logics (BLTL and FBLTL, respectively) are introduced by restricting Gentzen type sequent calculi for linear-time temporal logics. The corresponding Robinson type resolution calculi, RC and FRC for BLTL and FBLTL respectively are obtained. To prove the equivalence between FRC and FBLTL, a temporal version of Herbrand theorem is used. The completeness theorems for BLTL and FBLTL are proved for simple semantics with both a bounded time domain and some bounded valuation conditions on temporal operators. The cut-elimination theorems for BLTL and FBLTL are also proved using some theorems for embedding BLTL and FBLTL into propositional (first-order, respectively) classical logic. Although FBLTL is undecidable, its monadic fragment is shown to be decidable.     

Quantifier-free epistemic term-modal logic with assignment operator

In standard epistemic logic, the names and the existence of agents are usually assumed to be common knowledge implicitly. This is unreasonable for various applications in computer science and philosophy. Inspired by term-modal logic and assignment operators in dynamic logic, we introduce a lightweight modal predicate logic where names can be non-rigid, and the existence of agents can be uncertain. The language can handle various de dicto/de re distinctions in a natural way. We characterize the expressive power of our language, obtain complete axiomatisations of the logics over several classes of varying-domain/constant-domain epistemic models, and show their (un)decidability.     

A logic of soft constraints based on partially ordered preferences

Representing and reasoning with an agent’s preferences is important in many applications of constraints formalisms. Such preferences are often only partially ordered. One class of soft constraints formalisms, semiring-based CSPs, allows a partially ordered set of preference degrees, but this set must form a distributive lattice; whilst this is convenient computationally, it considerably restricts the representational power. This paper constructs a logic of soft constraints where it is only assumed that the set of preference degrees is a partially ordered set, with a maximum element 1 and a minimum element 0. When the partially ordered set is a distributive lattice, this reduces to the idempotent semiring-based CSP approach, and the lattice operations can be used to define a sound and complete proof theory. A generalised possibilistic logic, based on partially ordered values of possibility, is also constructed, and shown to be formally very strongly related to the logic of soft constraints. It is shown how the machinery that exists for the distributive lattice case can be used to perform sound and complete deduction, using a compact embedding of the partially ordered set in a distributive lattice.     

Separation logic and logics with team semantics

Separation logic is a successful logical system for formal reasoning about programs that mutate their data structures. Team semantics, on the other side, is the basis of modern logics of dependence and independence. Separation logic and team semantics have been introduced with quite different motivations, and are investigated by research communities with rather different backgrounds and objectives. Nevertheless, there are obvious similarities between these formalisms. Both separation logic and logics with team semantics involve the manipulation of second-order objects, such as heaps and teams, by first-order syntax without reference to second-order variables. Moreover, these semantical objects are closely related; it is for instance obvious that a heap can be seen as a team, and the separating conjunction of separation logic is (essentially) the same as the team-semantical disjunction. Based on such similarities, the possible connections between separation logic and team semantics have been raised as a question at several occasions, and lead to informal discussions between these research communities. The objective of this paper is to make this connection precise, and to study its potential but also its obstacles and limitations.     

Sequent Calculi with Analytic Cut for Logics of Time and Knowledge with Perfect Recall

In this paper, we consider two logics of time and knowledge. These logics involve the discrete time linear temporal logic operators ``next' and ``until'. In addition, they contain an indexed set of unary epistemic modalities ``agent $i$ knows'. In these logics, the temporal and epistemic dimensions may interact. The particular interactions we consider capture perfect recall. We consider perfect recall in synchronously distributed systems and in systems without any assumptions. For these logics, we present sequent calculi with an analytic cut rule. Thus, we get proof systems where proof-search becomes decidable. The soundness and completeness of these calculi are proved.     

A case study in logical deconstruction: Formalizing J.D. Thompson's Organizations in Action in a multi-agent action logic

Logic is a popular word in the social sciences, but it is rarely used as a formal tool. In the past, the logical formalisms were cumbersome and difficult to apply to domains of purposeful action. Recent years, however, have seen the advance of new logics specially designed for representing actions. We present such a logic and apply it to a classical organization theory, J.D. Thompson's Organizations in Action. The working hypothesis is that formal logic draws attention to some finer points in the logical structure of a theory, points that are easily neglected in the discursive reasoning typical for the social sciences. Examining Organizations in Action we find various problems in its logical structure that should, and, as we argue, could be addressed.     

Extending possibilistic logic over Gödel logic

In this paper we present several fuzzy logics trying to capture different notions of necessity (in the sense of possibility theory) for Gödel logic formulas. Based on different characterizations of necessity measures on fuzzy sets, a group of logics with Kripke style semantics is built over a restricted language, namely, a two-level language composed of non-modal and modal formulas, the latter, moreover, not allowing for nested applications of the modal operator N. Completeness and some computational complexity results are shown.     

Relating and extending semantical approaches to possibilistic reasoning

Two main semantical approaches to possibilistic reasoning with classical propositions have been proposed in the literature. Namely, Dubois-Prade's approach known as possibilistic logic, whose semantics is based on a preference ordering in the set of possible worlds, and Ruspini's approach that we redefine and call similarity logic, which relies on the notion of similarity or resemblance between worlds. In this article we put into relation both approaches, and it is shown that the monotonic fragment of possibilistic logic can be semantically embedded into similarity logic. Furthermore, to extend possibilistic reasoning to deal with fuzzy propositions, a semantical reasoning framework, called fuzzy truth-valued logic, is also introduced and proved to capture the semantics of both possibilistic and similarity logics.     

Mapping deontic operators to abductive expectations

Deontic concepts and operators have been widely used in several fields where representation of norms is needed, including legal reasoning and normative multi-agent systems. The EU-funded SOCS project has provided a language to specify the agent interaction in open multi-agent systems. The language is equipped with a declarative semantics based on abductive logic programming, and an operational semantics consisting of a (sound and complete) abductive proof procedure. In the SOCS framework, the specification is used directly as a program for the verification procedure. In this paper, we propose a mapping of the usual deontic operators (obligations, prohibition, permission) to language entities, called expectations, available in the SOCS social framework. Although expectations and deontic operators can be quite different from a philosophical viewpoint, we support our mapping by showing a similarity between the abductive semantics for expectations and the Kripke semantics that can be given to deontic operators. The main purpose of this work is to make the computational machinery from the SOCS social framework available for the specification and verification of systems by means of deontic operators. Marco Alberti received his laurea degree in Electronic Engineering in 2001 and his Ph.D. in Information Engineering in 2005 from the University of Ferrara, Italy. His research interests include constraint logic programming and abductive logic programming, applied in particular to the specification and verification of multi-agent systems. He has been involved as a research assistants in national and European research projects. He currently has a post-doc position in the Department of Engineering at the University of Ferrara. Marco Gavanelli is currently assistant professor in the Department of Engineering at the University of Ferrara, Italy. He graduated in Computer Science Engineering in 1998 at the University of Bologna, Italy. He got his Ph.D. in 2002 at Ferrara University. His research interest include Artificial Intelligence, Constraint Logic Programming, Multi-criteria Optimisation, Abductive Logic Programming, Multi-Agent Systems. He is a member of ALP (the Association for Logic Programming) and AI*IA (the Italian Association for Artificial Intelligence). He has organised workshops, and is author of more than 30 publications between journals and conference proceedings. Evelina Lamma received her degree in Electronic Engineering from University of Bologna, Italy, in 1985 and her Ph.D. degree in Computer Science in 1990. Currently she is Full Professor at the Faculty of Engineering of the University of Ferrara where she teaches Artificial Intelligence and Foundations of Computer Science. Her research activity focuses around: – programming languages (logic languages, modular and object-oriented programming); – artificial intelligence; – knowledge representation; – intelligent agents and multi-agent systems; – machine learning. Her research has covered implementation, application and theoretical aspects. She took part to several national and international research projects. She was responsible of the research group at the Dipartimento di Ingegneria of the University of Ferrara in the UE ITS-2001-32530 Project (named SOCS), in the the context of the UE V Framework Programme - Global Computing Action. Paola Mello received her degree in Electronic Engineering from the University of Bologna, Italy, in 1982, and her Ph.D. degree in Computer Science in 1989. Since 1994 she has been Full Professor. She is enrolled, at present, at the Faculty of Engineering of the University of Bologna (Italy), where she teaches Artificial Intelligence. Her research activity focuses on programming languages, with particular reference to logic languages and their extensions, artificial intelligence, knowledge representation, expert systems with particular emphasis on medical applications, and multi-agent systems. Her research has covered implementation, application and theoretical aspects and is presented in several national and international publications. She took part to several national and international research projects in the context of computational logic. Giovanni Sartor is Marie-Curie professor of Legal informatics and Legal Theory at the European University Institute of Florence and professor of Computer and Law at the University of Bologna (on leave), after obtaining a PhD at the European University Institute (Florence), working at the Court of Justice of the European Union (Luxembourg), being a researcher at the Italian National Council of Research (ITTIG, Florence), and holding the chair in Jurisprudence at Queen’s University of Belfast (where he now is honorary professor). He is co-editor of the Artificial Intelligence and Law Journal and has published widely in legal philosophy, computational logic, legislation technique, and computer law. Paolo Torroni is Assistant Professor in computing at the Faculty of Engineering of the University of Bologna, Italy. He obtained a PhD in Computer Science and Electronic Engineering in 2002, with a dissertation on logic-based agent reasoning and interaction. His research interests mainly focus on computational logic and multi-agent systems research, including logic programming, abductive and hypothetical reasoning, agent interaction, dialogue, negotiation, and argumentation. He is in the steering committee of the CLIMA and DALT international workshops and of the Italian logic programming interest group GULP.     

Knowledge Representation, Reasoning and Integration Using Temporal Logic with Clocks

Representation, reasoning about and integrating knowledge based on multiple time granularities in knowledge-based systems is important, especially when talking about events that take place in the real world. Formal approaches based on temporal logics have been successfully applied in many application domains of knowledge-based systems where the evolution of a system and its environment through time is central. This paper presents a methodology based on temporal logic to deal with knowledge based on multiple time granularities in knowledge-based systems. The temporal logic we consider is especially suitable for modelling events with different rates and/or scales of progress. The methodology includes an approach to the representation of timing systems, a method used for representing facts and rules in a knowledge-based system that involve multiple time granularities using temporal logic, and several deductive reasoning techniques. The work presented in this article has been supported in part by The Australian Research Council and Macquarie University. Note that this paper is an extended and revised version of Orgun, Liu and Nayak [37].     

Sequent calculi for branching time temporal logics of knowledge and beliefwith awareness: Completeness and decidability

In this paper, we consider branching time temporal logic CT L with epistemic modalities for knowledge (belief) and with awareness operators. These logics involve the discrete-time linear temporal logic operators “next” and “until” with the branching temporal logic operator “on all paths”. In addition, the temporal logic of knowledge (belief) contains an indexed set of unary modal operators “agent i knows” (“agent i believes”). In a language of these logics, there are awareness operators. For these logics, we present sequent calculi with a restricted cut rule. Thus, we get proof systems where proof-search becomes decidable. The soundness and completeness for these calculi are proved. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 3, pp. 328–340, July–September, 2007.     

Infinitary S5-Epistemic Logic

It is known that a theory in S5-epistemic logic with several agents may have numerous models. This is because each such model specifies also what an agent knows about infinite intersections of events, while the expressive power of the logic is limited to finite conjunctions of formulas. We show that this asymmetry between syntax and semantics persists also when infinite conjunctions (up to some given cardinality) are permitted in the language. We develop a strengthened S5-axiomatic system for such infinitary logics, and prove a strong completeness theorem for them. Then we show that in every such logic there is always a theory with more than one model.     

An Infinitary Graded Modal Logic (Graded Modalities VI)

We prove a completeness theorem for K, the infinitary extension of the graded version K⁰ of the minimal normal logic K, allowing conjunctions and disjunctions of countable sets of formulas. This goal is achieved using both the usual tools of the normal logics with graded modalities and the machinery of the predicate infinitary logics in a version adapted to modal logic.     

Rudimentary Languages and Second-Order Logic

The aim of this paper is to point out the equivalence between three notions respectively issued from recursion theory, computational complexity and finite model theory. One the one hand, the rudimentary languages are known to be characterized by the linear hierarchy. On the other hand, this complexity class can be proved to correspond to monadic second-order logic with addition. Our viewpoint sheds some new light on the close connection between these domains: We bring together the two extremal notions by providing a direct logical characterization of rudimentary languages and a representation result of second-order logic into these languages. We use natural arithmetical tools, and our proofs contain no ingredient from computational complexity.