Modal Logic S5 Satisfiability in Answer Set Programming

Modal logic S5 has attracted significant attention and has led to several practical applications, owing to its simplified approach to dealing with nesting modal operators. Efficient implementations for evaluating satisfiability of S5 formulas commonly rely on Skolemisation to convert them into propositional logic formulas, essentially by introducing copies of propositional atoms for each set of interpretations (possible worlds). This approach is simple, but often results into large formulas that are too difficult to process, and therefore more parsimonious constructions are required. In this work, we propose to use Answer Set Programming for implementing such constructions, and in particular for identifying the propositional atoms that are relevant in every world by means of a reachability relation. The proposed encodings are designed to take advantage of other properties such as entailment relations of subformulas rooted by modal operators. An empirical assessment of the proposed encodings shows that the reachability relation is very effective and leads to comparable performance to a state-of-the-art S5 solver based on SAT, while entailment relations are possibly too expensive to reason about and may result in overhead.

Characterizations and Classifications of Argumentative Entailments

In this paper we provide a detailed analysis of the inference process induced by logical argumentation frameworks. The frameworks may be defined with respect to any propositional language and logic, different arguments that represent deductions in the logic, various support-based attack relations between arguments, and all the complete Dung-style semantics for the frameworks. We show that, ultimately, for characterizing the inference process with respect to a given framework, extension-based semantics may be divided into two types: single-extension and multiple-extension, which induce respective kinds of entailment relations. These entailments are further classified by the way they tolerate new information (nonmonotonicity-related properties) and maintain conflicts among arguments (inconsistency-related properties).

The real preschoolers of Orange County: Early number learning in a diverse group of children

The authors assessed a battery of number skills in a sample of over 500 preschoolers, including bothmonolingual and bilingual/multilingual learners from households at a range of socio-economic levels.Receptive vocabulary was measured in English for all children, and also in Spanish for those who spoke it.The first goal of the study was to describe entailment relations among numeracy skills: Findings indicatedthat transitive and intransitive counting were jointly required for understanding cardinality; that cardinalityand knowledge of written number symbols were both required for using number lines. The study’s secondgoal was to describe relations between symbolic numeracy and language context (i.e., monolingual vs.bilingual contexts), separating these from well-documented socio-economic influences such as householdincome and parental education: Language context had only a modest effect on numeracy, with nodifferences detectable on most tasks. However, a difference did appear on the scaffolded number-line task,where bilingual learners performed slightly better than monolinguals. The third goal of the study was to findout whether symbolic number knowledge for one subset of children (Spanish/English bilingual learnersfrom low-income households) differed when tested in their home language (Spanish) vs. their language ofpreschool instruction (English): Findings indicated that children performed as well or better in English thanin Spanish for all measures, even when their receptive vocabulary scores in Spanish were higher than inEnglish.

Regular Entailment Relations

Inspired by the work of Lorenzen on the theory of preordered groups in the forties and fifties, we define regular entailment relations and show a crucial theorem for this structure. We also describe equivariant systems of ideals à la Lorenzen and show that the remarkable regularisation process he invented yields a regular entailment relation. By providing constructive objects and arguments, we pursue Lorenzen’s aim of “bringing to light the basic, pure concepts in their simple and transparent clarity”.

A New Approach to Plan-Space Explanation: Analyzing Plan-Property Dependencies in Oversubscription Planning

In many usage scenarios of AI Planning technology, users will want not just a plan π but an explanation of the space of possible plans, justifying π. In particular, in oversubscription planning where not all goals can be achieved, users may ask why a conjunction A of goals is not achieved by π. We propose to answer this kind of question with the goal conjunctions B excluded by A, i. e., that could not be achieved if A were to be enforced. We formalize this approach in terms of plan-property dependencies, where plan properties are propositional formulas over the goals achieved by a plan, and dependencies are entailment relations in plan space. We focus on entailment relations of the form ∧g∈A g ⇒ ⌝ ∧g∈B g, and devise analysis techniques globally identifying all such relations, or locally identifying the implications of a single given plan property (user question) ∧g∈A g. We show how, via compilation, one can analyze dependencies between a richer form of plan properties, specifying formulas over action subsets touched by the plan. We run comprehensive experiments on adapted IPC benchmarks, and find that the suggested analyses are reasonably feasible at the global level, and become significantly more effective at the local level.

A cognitive-pragmatic study of non-scalar implicatures

Whether or not non-entailment relations generate scalar implicatures is a cutting-edge issue in linguistic pragmatics. The present study intends to argue that, based on the Cognitive Grammar paradigm, non-scalar implicatures generated by non-entailment relations are manifested as cognitive defaults which are conventionally incorporated into symbolic units in schema-instance complexes. Conventions provide a shortcut for the hearer to infer non-scalar implicatures in an unconscious, effortless and automatic way. We maintain that, contrary to neo-Gricean pragmatics, non-entailment relations cannot generate (scalar) Q-implicatures.

Contrast and Entailment: Abstract logical relations constrain how 2- and 3-year-old children interpret unknown numbers

Do children understand how different numbers are related before they associate them with specific cardinalities? We explored how children rely on two abstract relations – contrast and entailment – to reason about the meanings of ‘unknown’ number words. Previous studies argue that, because children give variable amounts when asked to give an unknown number, all unknown numbers begin with an existential meaning akin to some. In Experiment 1, we tested an alternative hypothesis, that because numbers belong to a scale of contrasting alternatives, children assign them a meaning distinct from some. In the “Don’t Give-a-Number task”, children were shown three kinds of fruit (apples, bananas, strawberries), and asked to not give either some or a number of one kind (e.g. Give everything, but not [some/five] bananas). While children tended to give zero bananas when asked to not give some, they gave positive amounts when asked to not give numbers. This suggests that contrast – plus knowledge of a number’s membership in a count list – enables children to differentiate the meanings of unknown number words from the meaning of some. Experiment 2 tested whether children’s interpretation of unknown numbers is further constrained by understanding numerical entailment relations – that if someone, e.g. has three, they thereby also have two, but if they do not have three, they also do not have four. On critical trials, children saw two characters with different quantities of fish, two apart (e.g. 2 vs. 4), and were asked about the number in-between – who either has or doesn’t have, e.g. three. Children picked the larger quantity for the affirmative, and the smaller for the negative prompts even when all the numbers were unknown, suggesting that they understood that, whatever three means, a larger quantity is more likely to contain that many, and a smaller quantity is more likely not to. We conclude by discussing how contrast and entailment could help children scaffold their exact meanings of unknown number words.

Extractive multi-document summarization based on textual entailment and sentence compression via knapsack problem

By increasing the amount of data in computer networks, searching and finding suitable information will be harder for users. One of the most widespread forms of information on such networks are textual documents. So exploring these documents to get information about their content is difficult and sometimes impossible. Multi-document text summarization systems are an aid to producing a summary with a fixed and predefined length, while covering the maximum content of the input documents. This paper presents a novel method for multi-document extractive summarization based on textual entailment relations and sentence compression via formulating the problem as a knapsack problem. In this approach, sentences of documents are ranked according to the extended Tf-Idf method, then entailment scores of selected sentences are computed. Through these scores, the final score of each sentence is calculated. Finally, by decreasing the lengths of sentences via sentence compression, the problem has been solved by greedy and dynamic Programming approaches to the knapsack problem. Experiments on standard summarization datasets and evaluating the results based on the Rouge system show that the suggested method, according to the best of our knowledge, has increased F-measure of query-based summarization systems by two per cent and F-measure of general summarization systems by five per cent.

ELIMINATING DISJUNCTIONS BY DISJUNCTION ELIMINATION

Completeness and other forms of Zorn’s Lemma are sometimes invoked for semantic proofs of conservation in relatively elementary mathematical contexts in which the corresponding syntactical conservation would suffice. We now show how a fairly general syntactical conservation theorem that covers plenty of the semantic approaches follows from an utmost versatile criterion for conservation given by Scott in 1974.To this end we work with multi-conclusion entailment relations as extending single-conclusion entailment relations. In a nutshell, the additional axioms with disjunctions in positive position can be eliminated by reducing them to the corresponding disjunction elimination rules, which in turn prove admissible in all known mathematical instances. In deduction terms this means to fold up branchings of proof trees by way of properties of the relevant mathematical structures.Applications include the syntactical counterparts of the theorems or lemmas known under the names of Artin–Schreier, Krull–Lindenbaum, and Szpilrajn. Related work has been done before on individual instances, e.g., in locale theory, dynamical algebra, formal topology and proof analysis.