A Vernacular for Coherent Logic

We propose a simple, yet expressive proof representation from which proofs for different proof assistants can easily be generated. The representation uses only a few inference rules and is based on a frag- ment of first-order logic called coherent logic. Coherent logic has been recognized by a number of researchers as a suitable logic for many ev- eryday mathematical developments. The proposed proof representation is accompanied by a corresponding XML format and by a suite of XSL transformations for generating formal proofs for Isabelle/Isar and Coq, as well as proofs expressed in a natural language form (formatted in LATEX or in HTML). Also, our automated theorem prover for coherent logic exports proofs in the proposed XML format. All tools are publicly available, along with a set of sample theorems.Comment: CICM 2014 - Conferences on Intelligent Computer Mathematics (2014

Distributed First Order Logic

Distributed First Order Logic (DFOL) has been introduced more than ten years ago with the purpose of formalising distributed knowledge-based systems, where knowledge about heterogeneous domains is scattered into a set of interconnected modules. DFOL formalises the knowledge contained in each module by means of first-order theories, and the interconnections between modules by means of special inference rules called bridge rules. Despite their restricted form in the original DFOL formulation, bridge rules have influenced several works in the areas of heterogeneous knowledge integration, modular knowledge representation, and schema/ontology matching. This, in turn, has fostered extensions and modifications of the original DFOL that have never been systematically described and published. This paper tackles the lack of a comprehensive description of DFOL by providing a systematic account of a completely revised and extended version of the logic, together with a sound and complete axiomatisation of a general form of bridge rules based on Natural Deduction. The resulting DFOL framework is then proposed as a clear formal tool for the representation of and reasoning about distributed knowledge and bridge rules

Using a Logic Programming Framework to Control Database Query Dialogues in Natural Language

We present a natural language question/answering system to interface the University of Évora databases that uses clarification dialogs in order to clarify user questions. It was developed in an integrated logic programming framework, based on constraint logic programming using the GnuProlog(-cx) language [2,11] and the ISCO framework [1]. The use of this LP framework allows the integration of Prolog-like inference mechanisms with classes and inheritance, constraint solving algorithms and provides the connection with relational databases, such as PostgreSQL. This system focus on the questions’ pragmatic analysis, to handle ambiguity, and on an efficient dialogue mechanism, which is able to place relevant questions to clarify the user intentions in a straightforward manner. Proper Nouns resolution and the pp-attachment problem are also handled. This paper briefly presents this innovative system focusing on its ability to correctly determine the user intention through its dialogue capability

Subformula and separation properties in natural deduction via small Kripke models

Various natural deduction formulations of classical, minimal, intuitionist, and intermediate propositional and first-order logics are presented and investigated with respect to satisfaction of the separation and subformula properties. The technique employed is, for the most part, semantic, based on general versions of the Lindenbaum and Lindenbaum–Henkin constructions. Careful attention is paid (i) to which properties of theories result in the presence of which rules of inference, and (ii) to restrictions on the sets of formulas to which the rules may be employed, restrictions determined by the formulas occurring as premises and conclusion of the invalid inference for which a counterexample is to be constructed. We obtain an elegant formulation of classical propositional logic with the subformula property and a singularly inelegant formulation of classical first-order logic with the subformula property, the latter, unfortunately, not a product of the strategy otherwise used throughout the article. Along the way, we arrive at an optimal strengthening of the subformula results for classical first-order logic obtained as consequences of normalization theorems by Dag Prawitz and Gunnar Stalmarck

Modal and Relevance Logics for Qualitative Spatial Reasoning

Qualitative Spatial Reasoning (QSR) is an alternative technique to represent spatial relations without using numbers. Regions and their relationships are used as qualitative terms. Mostly peer qualitative spatial reasonings has two aspect: (a) the first aspect is based on inclusion and it focuses on the ”part-of” relationship. This aspect is mathematically covered by mereology. (b) the second aspect focuses on topological nature, i.e., whether they are in ”contact” without having a common part. Mereotopology is a mathematical theory that covers these two aspects. The theoretical aspect of this thesis is to use classical propositional logic with non-classical relevance logic to obtain a logic capable of reasoning about Boolean algebras i.e., the mereological aspect of QSR. Then, we extended the logic further by adding modal logic operators in order to reason about topological contact i.e., the topological aspect of QSR. Thus, we name this logic Modal Relevance Logic (MRL). We have provided a natural deduction system for this logic by defining inference rules for the operators and constants used in our (MRL) logic and shown that our system is correct. Furthermore, we have used the functional programming language and interactive theorem prover Coq to implement the definitions and natural deduction rules in order to provide an interactive system for reasoning in the logic