Journal of Logic and Computation - Advance Access

Modal Operators over Constructive Logic

Sherkhonov, E. Yu. — 2008-05-05

The article deals with modal extensions of Nelson's constructive logic with strong negation N4. Different logics are constructed with respect to the notions of semantical and formal dualities introduced in Odintsov and Wansing (2004, First-Order Logic Revisited, 269–286) and to a new notion of negative semantical duality. Soundness and completeness are proved for each logic. It is also shown that all the logics are conservative extensions of N4, and they possess the disjunction property and the constructible falsity property.

Aggregating Judgements by Merging Evidence

Williamson, J. — 2008-05-02

The theory of belief revision and merging has recently been applied to judgement aggregation. In this article I argue that judgements are best aggregated by merging the evidence on which they are based, rather than by directly merging the judgements themselves. This leads to a three-step strategy for judgement aggregation. First, merge the evidence bases of the various agents using some method of belief merging. Second, determine which degrees of belief one should adopt on the basis of this merged evidence base, by applying objective Bayesian theory. Third, determine which judgements are appropriate given these degrees of belief by applying a decision-theoretic account of rational judgement formation.

Aggregating Partially Ordered Preferences

Pini, M. S., Rossi, F., Venable, K. B., Walsh, T. — 2008-04-30

Preferences are not always expressible via complete inear orders: sometimes it is more natural to allow for the presence of incomparable outcomes. This may hold both in the agents' preference ordering and in the social order. In this article, we consider this scenario and study what properties it may have. In particular, we show that, despite the added expressivity and ability to resolve conflicts provided by incomparability, classical impossibility results (such as Arrow's theorem, Muller–Satterthwaite's theorem and Gibbard–Satterthwaite's theorem) still hold. We also prove some possibility results, generalizing Sen's theorem for majority voting. To prove these results, we define new notions of unanimity, monotonicity, dictator, triple-wise value-restriction and strategy-proofness, which are suitable and natural generalizations of the classical ones for complete orders.

Geodesic Revision

Georgatos, K. — 2008-04-27

The purpose of this article is to introduce a class of distance-based iterated revision operators generated by minimizing the geodesic distance on a graph. Such operators correspond bijectively to metrics and have a simple finite presentation. As distance is generated by distinguishability, our framework is appropriate for modelling contexts where distance is generated by threshold, and therefore, when measurement is erroneous.

Complete Axiomatization of Discrete-Measure Almost-Everywhere Quantification

Cruz-Filipe, L., Rasga, J., Sernadas, A., Sernadas, C. — 2008-04-23

Following recent developments in the topic of generalized quantifiers, and also having in mind applications in the areas of security and artificial intelligence, a conservative enrichment of (two-sorted) first-order logic (FOL) with almost-everywhere quantification is proposed. The completeness of the axiomatization against the measure-heoretic semantics is carried out using a variant of the Lindenbaum–Henkin technique. The independence of the axioms is analysed, and the almost-everywhere quantifier is compared with related notions of generalized quantification. A suitable fragment of the logic is translated to FOL and validity is shown to be preserved.

The Expansion Problem in Lambda Calculi with Explicit Substitution

Arbiser, A. — 2008-04-23

In this article, we address the problem of expansion with respect to rules of a calculus with explicit substitution. Mainly, we analyse the – and s–calculi sets of terms having the property of expansion to pure terms, as minimal sets of terms for these calculi. We prove that, contrarily to what happens in the x–calculus in which this set is trivial, for and s they are proper and non-recursive, so a calculus based on a minimal set of terms has a syntax which is not context-free and hence cannot be presented in the usual way.

Pedagogical Second-order Propositional Calculi

Colson, L., Michel, D. — 2008-04-22

The present work introduces the notion of pedagogical natural deduction systems, which are natural deduction systems with the following additional constraint: all hypotheses made in a proof must be motivated by an example. Technically speaking, we replace the rule (Hyp):

$$\frac{F\in \Gamma }{\Gamma \vdash F}\mathrm{(Hyp)}$$

with the rule (PHyp):

$$\frac{F\in \Gamma \hspace{1em}\vdash \sigma \cdot \Gamma }{\Gamma \vdash F}(P-\mathrm{Hyp})$$

with denoting a substitution replacing all variables of with an example. This substitution is called the motivation of . These systems are in essence negationless. In the present article, we study the second-order propositional calculus, since it is the simplest non-trivial natural deduction system in which the negation is definable. Some pedagogical versions of the second-order propositional calculus are proposed. We argue that these pedagogical calculi are negationless and we study their expressive power.

Quantifier Elimination for Quantified Propositional Logics on Kripke Frames of Type {omega}

Baaz, M., Preining, N. — 2008-04-14

The minimal extension of intuitionistic propositional language is characterized, where propositional quantifiers are eliminable w.r.t. Kripke frames of type .

Reinforcement Belief Revision

Jin, Y., Thielscher, M. — 2008-04-10

The capability of revising its beliefs upon new information in a rational and efficient way is crucial for an intelligent agent. The classical work in belief revision focuses on idealized models and is not concerned with computational aspects. In particular, many researchers are interested in the logical properties (e.g. the AGM postulates) that a rational revision operator should possess. For the implementation of belief revision, however, one has to consider that any realistic agent is a finite being and that calculations take time. In this article, we introduce a new operation for revising beliefs which we call reinforcement belief revision. The computational model for this operation allows us to assess it in terms of time and space consumption. Moreover, the operation is proved equivalent to a (semantical) model based on the concept of possible worlds, which facilitates showing that reinforcement belief revision satisfies all desirable rationality postulates.

Synthesizing Monadic Predicates

Meghini, C., Spyratos, N. — 2008-04-05

We study the problem of determining a concise, quantifier-free monadic predicate for a given set of objects in a given interpretation. We address both DNF and CNF predicates, as well as important sub-languages thereof. The problem is formalized as the search of a minimal element in a set of predicates equipped with a binary relation. We show that the problem has always a solution, that finding a minimal solution is always hard, but much harder when neither the given set of objects nor its complement are the extent of a formal concept (in the sense of Formal Concept Analysis).

Extracting Frame-Semantics Knowledge using Lattice Theory

Valverde-Albacete, F. J. — 2008-04-04

In this article, we introduce a new representation based on lattice theory for lexical data from a lexical-database embodying the frame-semantic approach to language description, FrameNet. We present proof of the abundance of Concept Lattices as proposed in Formal Concept Analysis both in the theory of frames and in its present-day incarnation, the FrameNet resource, by constructing several types of these. We further argue for the adequacy of such lattices in representing linguistic data with contributions that range from data-visualization to the fine-tuning of some frame-theoretical concepts. We argue finally that FrameNet is better thought of as being a lexical resource rather than an ontology, but we make the case throughout the article for Concept Lattices being a linguistically adequate, formally effective intermediate representations from which knowledge representation languages may draw knowledge-rich, linguistic facts from FrameNet at their convenience.

Adding Intensional Machinery to Hybrid Logic

Brauner, T. — 2008-03-13

In this article we give an intensional version of first-order hybrid logic (which also can be viewed as a hybridized version of Fitting's First-Order Intensional Logic). We consider two different kinds of models—standard models and generalized models. The standard models are the same as Fitting's models for First-Order Intensional Logic. As the name suggests, the generalized models are more general. We give a natural deduction system which is completete with respect to generalized models. The natural deduction system is not complete with respect to standard models, but we show how to extend it with a further rule such that completeness with respect to standard models is obtained.

Degrees of Belief

Levi, I. — 2008-03-04

This article surveys various accounts of degrees of belief and the relation between degrees of belief and full belief or absolute certainty. Corresponding to each notion of degree of belief is a conception of evidential support. Three different kinds of degrees of belief and the corresponding notions of evidential support are considered: probability, evidential support in the maximizing sense and in the satisficing sense. It is argued that probability cannot be the degree of belief or evidential support in either the maximizing or satisficing sense. Reconstructions of maximizing and satisficing degree of belief are proposed, which show that they are ways of evaluating potential answers to questions that demand the inductive expansion of a state K of full belief. These reconstructions are based on an account of inductive expansion briefly summarized in the text that understands inductive expansion to be the choice of a potential answer that maximizes a weighted average of the risk of error and the value of the information acquired. Thus, maximizing this weighted average is an index of degree of evidential support in the maximizing sense. It is explained how an index of evidential support in the satisficing sense can be constructed that achieves the same result. Finally, it is argued that several so called ‘qualitative’ notions of belief other than full belief are deprived of useful application in deliberation and inquiry because they lack the relevance to inductive expansion that maximizing and satisficing evidential support (or degree of belief) has. The discussion should be of interest to students of measures of uncertainty, inductive or non-monotonic reasoning and decision making.

Reconstructing an Agent's Epistemic State from Observations about its Beliefs and Non-beliefs

Booth, R., Nittka, A. — 2008-03-04

We look at the problem in belief revision of trying to make inferences about what an agent believed—or will believe—at a given moment, based on an observation of how the agent has responded to some sequence of previous belief revision inputs over time. We adopt a ‘reverse engineering’ approach to this problem. Assuming a framework for iterated belief revision which is based on sequences, we construct a model of the agent that ‘best explains’ the observation. Further considerations on this best-explaining model then allow inferences about the agent's epistemic behaviour to be made. We also provide an algorithm which computes this best explanation.

An Axiomatic Characterization of Ensconcement-Based Contraction

FermE, E., Krevneris, M., Reis, M. — 2008-02-15

In this article, we propose an axiomatic characterization for ensconcement-based contraction functions, belief base functions proposed by Williams. We relate this function with other kinds of base contraction functions.

On the Hierarchy of Intuitionistic Bounded Arithmetic

Moniri, M. — 2008-02-06

In this article, we study the two hierarchies of intuitionistic bounded arithmetic introduced by Buss and Harnik. Harnik's hierarchy contains the theory $${\mathit{IS}}_{2}^{1}$$ defined and studied by Cook and Urquhart as the first level. We prove level by level equivalence between the two hierarchies (for the first level, the fact was first proved by Cook and Urquhart using realizability and functional interpretation and later by Buss by an elementary method). Next we investigate the question of whether the hierarchy, denoted $${\mathit{IS}}_{2}^{i}$$, collapses. We show that if $${\mathit{IS}}_{2}^{i}\mathrm{\vdash }{\mathit{IS}}_{2}^{i+1}$$, then $${S}_{2}^{i}(\mathit{PV})\mathrm{\vdash }{\Sigma }_{i}^{b}={\Pi }_{i}^{b}$$ and so the polynomial hierarchy collapses to $${\Sigma }_{i}^{p}={\Pi }_{i}^{p}$$. Our proof for this is independent from earlier works on relating the collapse of the hierarchy of classical bounded arithmetic and the collapse of the polynomial hierarchy. We give an elementary model theoretic proof using only the basic properties of the theories $${\mathit{IS}}_{2}^{i}$$ and we do not use results which belong to Cook and Urquhart and also Harnik that characterize the definable functions of these theories with long witnessing proofs.

A Syntactical Proof of the Canonical Reactivity Form for Past Linear Temporal Logic

Guelev, D. P. — 2008-02-06

We present a new proof of the fact that every formula in linear temporal logic with past is equivalent to a formula of the form $${\displaystyle \underset{i}{\bigwedge }\mathrm{\diamond }\square {\alpha }_{i}\Rightarrow \mathrm{\diamond }\square {\beta }_{i},}$$ where _i and β_i are past formulas, which is known as general canonical reactivity form. The original proof is based on the fact that a finite automaton recognizes an LTL-definable -language iff it is counter-free, which was proved in Lenore Zuck's thesis and relies on the theorem of Krohn-Rhodes about cascade decomposition of finite automata. Unlike that, the proof presented in this paper involves only equivalence transformations of LTL formula and makes use of Gabbay's separation theorem, whose proof is based on equivalence transformations too. This makes it possible to obtain the canonical form without resorting to constructions outside LTL with past operators such as automata.

Three Scenarios for the Revision of Epistemic States

Dubois, D. — 2008-02-01

Multi-modal and Temporal Logics with Universal Formula---Reduction of Admissibility to Validity and Unification

Rybakov, V — 2008-01-08

The article studies multi-modal (in particular temporal, tense, logics) possessing universal formulas. We prove (Theorems 11 and 12) that the admissibility problem for inference rules (inference rules with parameters) is decidable for all decidable multi-modal (in particular, temporal) logics possessing an universal formula and decidable w.r.t. unification (unification with parameters). These theorems use characterizations of admissible rules in terms of valid rules and unifiable formulas. Results are applied to wide range of multi-modal logics, including all linear transitive temporal logics, all temporal logics with bounded zigzag, all multi-modal logics with explicit universal modality. As consequence, we show that the admissibility problem for all such logics is decidable (e.g. for all logics of the series S4_nU, n N).

General Models and Completeness of First-Order Modal {micro}-calculus

Kashima, R., Okamoto, K. — 2008-01-08

There is no recursive axiomatization of first-order modal µ-calculus that is complete with respect to usual Kripke models. Then we introduce `general' models, and we prove that the natural axiom system of first-order modal µ-calculus is complete with respect to general models.

Editorial and call for papers

Wansing, H. — 2008-01-08

A Note on Bisimulation Quantifiers and Fixed Points over Transitive Frames

D'Agostino, G., Lenzi, G. — 2007-12-21

We consider three basic questions regarding the extension of modal logic with a special kind of propositional quantifiers, known as bisimulation quantifiers, over arbitrary classes of frames: bisimulation invariance, uniform interpolation, and expressive power. In particular:

– we discuss the relation between bisimulation invariance of bisimulation quantifiers and the semantical notion of amalgamation of the class of frames;

– we consider a strong form of interpolation, uniform interpolation, and its relation with the closure under bisimulation quantifiers;

– we compare bisimulation quantifiers logic with the better known extension of modal logic with extremal fixed points.

In this article we show that the answers to these questions that are valid for the class of all frames do not generalize to arbitrary classes, but they do generalize if we restrict to classes of (finite) transitive or (finite) transitive and reflexive frames.

Lambek Calculus in Natural Deduction

Zimmermann, E. — 2007-12-21

A formulation of Lambek calculus in natural deduction is given. New rules for Lambek's multiplicative, non-commutative conjunction are proposed, rules for Lambek's two implications are standard. Rules for Lambek's conjunction are variants of general elimination rules: a symmetric elimination rule and its specializations, left elimination rule and right elimination rule. Conversions hold for all these rules, but only the symmetric elimination rule is fully permutable. Due to a natural transformation for left and right elimination rules to the symmetric elimination rule with partial empty sequences of assumptions and vice versa, there hold two normalization theorems, one with a minimal set and one with a maximal set of permutations.

Sum and Product in Dynamic Epistemic Logic

van Ditmarsch, H. P., Ruan, J., Verbrugge, R. — 2007-12-21

The Sum-and-Product riddle was first published in the reference H. Freudenthal (1969, Nieuw Archief voor Wiskunde 3, 152) [6]. We provide an overview on the history of the dissemination of this riddle through the academic and puzzle-math community. This includes some references to precursors of the riddle, that were previously (as far as we know) unknown.

We then model the Sum-and-Product riddle in a modal logic called public announcement logic. This logic contains operators for knowledge, but also operators for the informational consequences of public announcements. The logic is interpreted on multi-agent Kripke models. The information in the riddle can be represented in the traditional way by number pairs, so that Sum knows their sum and Product their product, but also as an interpreted system, so that Sum and Product at least know their local state. We show that the different representations are isomorphic. We also provide characteristic formulas of the initial epistemic state of the riddle. We analyse one of the announcements towards the solution of the riddle as a so-called unsuccessful update: a formula that becomes false because it is announced.

The riddle is then implemented and its solution verified in the epistemic model checker DEMO. This can be done, we think, surprisingly elegantly. The results are compared with other work in epistemic model checking and the complexity is experimentally investigated for several representations and parameter settings.

Speech Acts, Epistemic Planning and Grice's Maxims

Ramsay, A., Field, D. — 2007-12-21

Work on speech acts has generally involved the introduction of sets of different actions such as informing, reminding, bluffing and lying. These actions have different preconditions and effects, and hence can be used to achieve a wide variety of different real-world goals. The problem is that they tend to have indistinguishable surface forms. As such, it is extremely difficult for the hearer to decide which action she thinks has been performed, and it is therefore also extremely difficult for the speaker to be confident about how the hearer will respond. We will show how to achieve complex goals on the basis of a very simple set of linguistic actions. These actions have clearly marked surface forms, and hence can easily be distinguishable by a hearer. In order to do this, we have developed an epistemic planner with a number of interesting features, and with a number of optimisations that relate directly to aspects of the task at hand.

Utilizing Natural Language for One-Shot Task Learning

Jung, H., Allen, J., Galescu, L., Chambers, N., Swift, M., Taysom, W. — 2007-12-20

Learning tasks from a single demonstration presents a significant challenge because the observed sequence is specific to the current situation and is inherently an incomplete representation of the procedure. Observation-based machine-learning techniques are not effective without multiple examples. However, when a demonstration is accompanied by natural language explanation, the language provides a rich source of information about the relationships between the steps in the procedure and the decision-making processes that led to them. In this article, we present a one-shot task learning system built on TRIPS, a dialogue-based collaborative problem solving system, and show how natural language understanding can be used for effective one-shot task learning.

Deverbal Nouns in Knowledge Representation

Gurevich, O., Crouch, R., King, T. H., de Paiva, V. — 2007-12-20

Deverbal nouns pose serious challenges for knowledge-representation systems. We present a method of canonicalizing deverbal noun representations, relying on a rich lexicon of verb subcategorization frames, the WordNet database, a large finite-state network for derivational morphology and a series of heuristics for mapping deverbal arguments onto the arguments of corresponding verbs.¹

Testing the Reasoning for Question Answering Validation

Penas, A., Rodrigo, A., Sama, V., Verdejo, F. — 2007-12-13

Question answering (QA) is a task that deserves more collaboration between natural language processing (NLP) and knowledge representation (KR) communities, not only to introduce reasoning when looking for answers or making use of answer type taxonomies and encyclopaedic knowledge, but also, as discussed here, for answer validation (AV), that is to say, to decide whether the responses of a QA system are correct or not. This was one of the motivations for the first Answer Validation Exercise at CLEF 2006 (AVE 2006). The starting point for the AVE 2006 was the reformulation of the answer validation as a recognizing textual entailment (RTE) problem, under the assumption that a hypothesis can be automatically generated instantiating a hypothesis pattern with a QA system answer. The test collections that we developed in seven different languages at AVE 2006 are specially oriented to the development and evaluation of answer validation systems. We show in this article the methodology followed for developing these collections taking advantage of the human assessments already made in the evaluation of QA systems. We also propose an evaluation framework for AV linked to a QA evaluation track. We quantify and discuss the source of errors introduced by the reformulation of the answer validation problem in terms of textual entailment (around 2%, in the range of inter-annotator disagreement). We also show the evaluation results of the first answer validation exercise at CLEF 2006 where 11 groups have participated with 38 runs in seven different languages. The most extensively used techniques were Machine Learning and overlapping measures, but systems with broader knowledge resources and richer representation formalisms obtained the best results.

Linking Semantic and Knowledge Representations in a Multi-Domain Dialogue System

Dzikovska, M. O., Allen, J. F., Swift, M. D. — 2007-12-13

We describe a two-layer architecture for supporting semantic interpretation and domain reasoning in dialogue systems. Building system that supports both semantic interpretation and domain reasoning in a transparent and well-integrated manner is an unresolved problem because of the diverging requirements of the semantic representations used in contextual interpretation versus the knowledge representations used in domain reasoning. We propose an architecture that provides both portability and efficiency in natural language interpretation by maintaining separate semantic and domain knowledge representations, and integrating them via an ontology mapping procedure. The ontology mapping is used to obtain representations of utterances in a form most suitable for domain reasoners and to automatically specialize the lexicon. The use of a linguistically motivated parser for producing semantic representations for complex natural language sentences facilitates building portable semantic interpretation components as well as connections with domain reasoners. Two evaluations demonstrate the effectiveness of our approach: we show that a small number of mapping rules are sufficient for customizing the generic semantic representation to a new domain, and that our automatic lexicon specialization technique improves parser speed and accuracy.

Experience and History: Processes and their Relation to Events

Galton, A. — 2007-12-05

We develop a theory of processes which takes into account the observation that processes differ markedly from events in their relation to change. Whereas events are fixed items of history which cannot be described as undergoing change, processes are more like ordinary objects in that they can be directly present at one time and can undergo change as time proceeds. This leads to a fundamental ontological distinction between EXP, the dynamic experiential world of objects and processes as they exist at one time, and HIST, the static historical overview populated by events that are generated by the ongoing processes in EXP. Formally, this means that terms describing processes can serve as arguments to time-varying predicates, whereas terms describing events cannot. We illustrate this by presenting part of a suitable formalism and using it to give an account of the progressive aspect.

The Semantic Processing of Continuous Quantities for Discrete Terms in Ontologies

Wang, S., Rydeheard, D., Pan, J. Z. — 2007-12-05

We consider continuous quantities that are used to describe the physical world, such as colour, shape, sound, texture and spatial and temporal arrangements. Natural languages are not adept at describing these quantities, nor are they easily incorporated into ontologies in the form of discrete terms. In this article, we analyse the way that natural languages handle continuous quantities, propose a general semantics based on metric spaces, and describe how to treat semantic values computationally, so that we may automate the processing of texts which describe continuous quantities allowing, for example, query evaluation and the integration of multiple texts. This provides a basis for incorporating these quantities into ontologies and combining their semantics with automated reasoning tools. We run a series of experiments to evaluate the semantics, the general framework, and the computational system we have developed.

Call for Papers: Coalgebra & Logic

2007-11-26

One-and-a-halfth-order Logic

Gabbay, M. J., Mathijssen, A. — 2007-11-22

The practice of first-order logic is replete with meta-level concepts. Most notably there are meta-variables ranging over formulae, variables, and terms, and properties of syntax such as alpha-equivalence, capture-avoiding substitution and assumptions about freshness of variables with respect to meta-variables. We present one-and-a-halfth-order logic, in which these concepts are made explicit. We exhibit both sequent and algebraic specifications of one-and-a-halfth-order logic derivability, show them equivalent, show that the derivations satisfy cut-elimination, and prove correctness of an interpretation of first-order logic within it. We discuss the technicalities in a wider context as a case-study for nominal algebra, as a logic in its own right, as an algebraisation of logic, as an example of how other systems might be treated, and also as a theoretical foundation for future implementation.