First-order Predicate Logic Research Articles

Meaningful web annotations for humans and machines using controlled natural language

Abstract: In this paper we present a novel approach that allows humans to create meaningful web annotations in controlled natural language. The controlled natural language serves as a high-level interface language which enables human annotators to summarize individual web pages of a website and to express domain-specific ontological knowledge about that website in an unambiguous subset of English. The annotation process is backed up by an intelligent text editor which supports the writing process of the controlled natural language with the help of predictive interface techniques. The text editor runs as a Java applet and is connected over the Internet to a controlled natural language processor and to a reasoning service (consisting of a theorem prover and a model builder). The controlled language processor translates the summaries of web pages and the ontological knowledge about a website into first-order predicate logic and the reasoning service combines this information into a set of micro-theories for consistency and informativity checking as well as for question-answering. Specification texts written in controlled natural language are both human-readable and machine-processable and can be easily exported and distributed as web feeds.

On contraction and the modal fragment

AbstractWe observe that removing contraction from a standard sequent calculus for first‐order predicate logic preserves completeness for the modal fragment. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A-Logic With Arrows

We present an extension of first-order predicate logic with a novel predicate ‘att’ meaning intuitively “this term is a variable symbol”. We give simple sequent proof-rules for it, we demonstrate cut-elimination for the resulting logic, and we give a semantics for which the logic is sound and complete. Because we can now make assertions about what would normally be considered an intensional property of a term (being a variable symbol) we can now express inside the logic, properties of its terms and predicates which would normally be external to the logic. We give axiomatisations in a-logic, including of the lambda-calculus, and discuss what relevance this might have to logic programming.

Connectionist model generation: A first-order approach

Knowledge-based artificial neural networks have been applied quite successfully to propositional knowledge representation and reasoning tasks. However, as soon as these tasks are extended to structured objects and structure-sensitive processes as expressed e.g., by means of first-order predicate logic, it is not obvious at all what neural-symbolic systems would look like such that they are truly connectionist, are able to learn, and allow for a declarative reading and logical reasoning at the same time. The core method aims at such an integration. It is a method for connectionist model generation using recurrent networks with feed-forward core. We show in this paper how the core method can be used to learn first-order logic programs in a connectionist fashion, such that the trained network is able to do reasoning over the acquired knowledge. We also report on experimental evaluations which show the feasibility of our approach.

Completeness of Modal Proofs in First-Order Predicate Logic

Completeness of Modal Proofs in First-Order Predicate Logic

The Grammatical Background of Kant's General Logic

In theCritique of Pure Reason, Kant conceives of general logic as a set of universal and necessary rules for the possibility of thought, or as a set of minimal necessary conditions for ascribing rationality to an agent (exemplified by the principle of non-contradiction). Such a conception, of course, contrasts with contemporary notions of formal, mathematical or symbolic logic. Yet, in so far as Kant seeks to identify those conditions that must hold for the possibility of thought in general, such conditions must holda fortiorifor any specific model of thought, including axiomatic treatments of logic and standard natural deduction models of first-order predicate logic. Kant's general logic seeks to isolate those conditions by thinking through – or better, reflecting on – those conditions that themselves make thought possible.

From Logic Specification to y-calculus

A program construction method based on ?-calculus is proposed: The problem to be solved is specified by first-order predicate logic, and a semantic verification program is constructed directly from the specification. We exploit this method in synthesizing the architectural specifications of multiagent systems (MAS) in ?-calculus based on the logic specifications of the MAS. By enabling the transformation from the logic specifications to operational specifications of MAS, this method allows the design of the MAS to be focused on the architectural definition level. It benefits the development of MAS by enabling logic deduction on behaviors of the MAS, and a design methodology in an incremental fashion.

An Institution-Independent Proof of the Robinson Consistency Theorem

We prove an institutional version of A. Robinson’s Consistency Theorem. This result is then appliedto the institution of many-sorted first-order predicate logic and to two of its variations, infinitary and partial, obtaining very general syntactic criteria sufficient for a signature square in order to satisfy the Robinson consistency and Craig interpolation properties.

Completeness of Modal Proofs in First-Order Predicate Logic

Completeness of Modal Proofs in First-Order Predicate Logic

SPECIFICATION AND VERIFICATION OF A PARAMETRIC HANDOVER PROCEDURE USING TLA

In mobile systems, some nodes link dynamically through different channels at different times. The π-calculus is a popular process algebra for specification and verification of systems with dynamic linking and synchronous, rendezvous message passing communication of component processes. Dynamic linking can be elegantly specified in it by sending channels to be used for further communication as messages. However, in order to be able to use the π-calculus, one has to learn its process-algebraic concepts and laws. On the other hand, specification and verification in the Temporal Logic of Actions (TLA) is almost entirely based on ordinary mathematics (i.e., first-order predicate logic, set theory, etc.). Therefore, this paper proposes a TLA approach to the specification and verification of systems with synchronous communication and dynamic linking similar to the π-calculus one. It is presented with help of a TLA specification and verification of a parametric mobile network handover procedure which has previously been specified with the π-calculus. In fact, TLA+ is used as the specification language, thus achieving completely formal specifications.

Now is the time1

The aim of this paper is to consider some logical aspects of the debate between the view that the present is the only ‘real’ time, and the view that the present is not in any way metaphysically privileged. In particular I shall set out a language of first-order predicate tense logic with a now predicate, and a first order (extensional) language with an abstraction operator, in such a way that each language can be shewn to be exactly translatable into the other. I shew that this translation is preserved at the metalinguistic level, so that equivalent truth conditions can be defined in a tensed metalanguage or an indexical metalanguage. I then make some remarks about the connection between proofs of relative consistency and metaphysical truth; and some historical remarks about Arthur Prior's use of formal logic in expressing his presentist views.

Ontological Investigation of a Multiscale Ecosystem Classification Using the “National Hierarchical Framework of Ecological Units” as an Example

This paper presents a formalized ontological framework for the analysis of multiscale classifications of geographic objects. We propose a set of logical principles that guide such geographic classifications. These principles are represented as definitions and axioms of the first-order predicate logic. Then we demonstrate an application of these principles with a practical example of the "National Hierarchical Framework of Ecological Units." The study presents a minimal set of entities and relation need for such formalizations and emphasizes the need to distinguish clearly between classes of ecosystems and ecosystems-individuals physically located on the Earth surface. The ontological framework proposed in the paper has the potential to be used to facilitate interoperability between such geographic classifications.

A Speedup Algorithm for Repetition of Hypothetical Reasoning

This paper presents performance evaluations of a speedup method for hypothetical reasoning to a first-order predicate logic knowledge base in the case when reasoning is executed repeatedly replacing hypothetical knowledge and keeping background knowledge and the goal intact. The proposed method consists of substituting hypotheses for predicate knowledge which has recursive structures, deriving bit patterns of solutions from background knowledge and the goal in advance when all the hypotheses are true, and finding actual solutions from inclusion relation with bit patterns of hypothetical knowledge.

Completeness and Counter-Example Generations of a Basic Protocol Logic: (Extended Abstract)

We give an axiomatic system in first-order predicate logic with equality for proving security protocols correct. Our axioms and inference rules derive the basic inference rules, which are explicitly or implicitly used in the literature of protocol logics, hence we call our axiomatic system Basic Protocol Logic (or BPL, for short). We give a formal semantics for BPL, and show the completeness theorem such that for any given query (which represents a correctness property) the query is provable iff it is true for any model. Moreover, as a corollary of our completeness proof, the decidability of provability in BPL holds for any given query. In our formal semantics we consider a “trace” any kind of sequence of primitive actions, counter-models (which are generated from an unprovable query) cannot be immediately regarded as realizable traces (i.e., attacked processes on the protocol in question). However, with the aid of Comon-Treinen's algorithm for the intruder deduction problem, we can determine whether there exists a realizable trace among formal counter-models, if any, generated by the proof-search method (used in our completeness proof). We also demonstrate that our method is useful for both proof construction and flaw analysis by using a simple example.

Reducing Symmetries to Generate Easier SAT Instances

Finding countermodels is an effective way of disproving false conjectures. In first-order predicate logic, model finding is an undecidable problem. But if a finite model exists, it can be found by exhaustive search. The finite model generation problem in the first-order logic can also be translated to the satisfiability problem in the propositional logic. But a direct translation may not be very efficient. This paper discusses how to take the symmetries into account so as to make the resulting problem easier. A static method for adding constraints is presented, which can be thought of as an approximation of the least number heuristic (LNH). Also described is a dynamic method, which asks a model searcher like SEM to generate a set of partial models, and then gives each partial model to a propositional prover. The two methods are analyzed, and compared with each other.

The semantics of augmented constraint diagrams

Constraint diagrams are a diagrammatic notation which may be used to express logical constraints. They generalize Venn diagrams and Euler circles, and include syntax for quantification and navigation of relations. The notation was designed to complement the Unified Modelling Language in the development of software systems. Since symbols representing quantification in a diagrammatic language can be naturally ordered in multiple ways, some constraint diagrams have more than one intuitive meaning in first-order predicate logic. Any equally expressive notation which is based on Euler diagrams and conveys logical statements using explicit quantification will have to address this problem. We explicitly augment constraint diagrams with reading trees, which provides a partial ordering for the quantifiers (determining their scope as well as their relative ordering). Alternative approaches using spatial arrangements of components, or alphabetical ordering of symbols, for example, can be seen as implicit representations of a reading tree. Whether the reading tree accompanies the diagram explicitly (optimizing expressiveness) or implicitly (simplifying diagram syntax), we show how to construct unambiguous semantics for the augmented constraint diagram.

Extracting Herbrand disjunctions by functional interpretation

Carrying out a suggestion by Kreisel, we adapt Godel’s functional interpretation to ordinary first-order predicate logic(PL) and thus devise an algorithm to extract Herbrand terms from PL-proofs. The extraction is carried out in an extension of PL to higher types. The algorithm consists of two main steps: first we extract a functional realizer, next we compute the β-normal-form of the realizer from which the Herbrand terms can be read off. Even though the extraction is carried out in the extended language, the terms are ordinary PL-terms. In contrast to approaches to Herbrand’s theorem based on cut elimination orɛ-elimination this extraction technique is, except for the normalization step, of low polynomial complexity, fully modular and furthermore allows an analysis of the structure of the Herbrand terms, in the spirit of Kreisel ([13]), already prior to the normalization step. It is expected that the implementation of functional interpretation in Schwichtenberg’s MINLOG system can be adapted to yield an efficient Herbrand-term extraction tool.

The Expressiveness of Spider Diagrams

Spider diagrams are a visual language for expressing logical statements. In this paper we identify a well-known fragment of first-order predicate logic that we call MFOL=, equivalent in expressive power to the spider diagram language. The language MFOL= is monadic and includes equality but has no constants or function symbols. To show this equivalence, in one direction, for each diagram we construct a sentence in MFOL= that expresses the same information. For the more challenging converse we prove that there exists a finite set of models for a sentence S that can be used to classify all the models for S. Using these classifying models we show that there is a diagram expressing the same information as S.

Possibility Semantics for Intuitionistic Logic

The paper investigates interpretations of propositional and first-order logic in which validity is defined in terms of partial indices; sometimes called possibilities but here understood as non-empty subsets of a set W of possible worlds. Truth at a set of worlds is understood to be truth at every world in the set. If all subsets of W are permitted the logic so determined is classical first-order predicate logic. Restricting allowable subsets and then imposing certain closure conditions provides a modelling for intuitionistic predicate logic. The same semantic interpretation rules are used in both logics for all the operators.

Is “being” predicated in only one sense, after all?

In this essay, I argue that for sentences of form “A is B” there is a distinction between identity and “mere” predication to be made, and that Leśniewski’s Ontology puts us in a better position to make this distinction than first-order predicate logic. I also gesture at how Ontology could help us to decide questions of identity. The nub of the matter seems to be a “primordial” sense of the copula that Ontology has at its basis.