Unsorted Functional Translations

AbstractIn this article we first show how the functional and the optimized functional translation from modal logic to many-sorted first-order logic can be naturally extended to the hybrid language H(@,↓). The translation is correct not only when reasoning over the class of all models, but for any first-order definable class. We then show that sorts can be safely removed (i.e., without affecting the satisfiability status of the formula) for frame classes that can be defined in the basic modal language, and show a counterexample for a frame class defined using nominals

Faithful (meta-)encodings of programmable strategies into term rewriting systems

Rewriting is a formalism widely used in computer science and mathematical logic. When using rewriting as a programming or modeling paradigm, the rewrite rules describe the transformations one wants to operate and rewriting strategies are used to con- trol their application. The operational semantics of these strategies are generally accepted and approaches for analyzing the termination of specific strategies have been studied. We propose in this paper a generic encoding of classic control and traversal strategies used in rewrite based languages such as Maude, Stratego and Tom into a plain term rewriting system. The encoding is proven sound and complete and, as a direct consequence, estab- lished termination methods used for term rewriting systems can be applied to analyze the termination of strategy controlled term rewriting systems. We show that the encoding of strategies into term rewriting systems can be easily adapted to handle many-sorted signa- tures and we use a meta-level representation of terms to reduce the size of the encodings. The corresponding implementation in Tom generates term rewriting systems compatible with the syntax of termination tools such as AProVE and TTT2, tools which turned out to be very effective in (dis)proving the termination of the generated term rewriting systems. The approach can also be seen as a generic strategy compiler which can be integrated into languages providing pattern matching primitives; experiments in Tom show that applying our encoding leads to performances comparable to the native Tom strategies

Efficient Encodings of First-Order Horn Formulas in Equational Logic

We present several translations from first-order Horn formulas to equational logic. The goal of these translations is to allow equational theorem provers to efficiently reason about non-equational problems. Using these translations we were able to solve 37 problems of rating 1.0 (i.e. which had not previously been automatically solved) from the TPTP

Extending the Automated Reasoning Toolbox

Due to the semi-decidable nature of first-order logic, it can be desirable to address a wider range of problems than the standard ones of satisfiability and derivability. We extend the automated reasoning toolbox by introducing three new tools for analysing problems in first-order logic. Infinox aims to show finite unsatisfiability, i.e. the absence of models with finite domains, and is a useful complement to finite model-finding. Infinox can also be used to reason about the relative sizes of model domains in sorted first-order logic. Monotonox uses a novel analysis that can identify sorts with extendable domains, improving on well-known existing translations between sorted and unsorted logic. This enables reasoning tools for unsorted logic to tackle problems in sorted logic. Conversely, finite model finders benefit from sort information which Monotonox can add to unsorted problems. Equalox, the third tool in our toolbox, can improve the per- formance of first-order provers on problems involving transitive relations. The insight is that first-order provers are poor at applying the transitivity axiom effectively, but that the problem can always be transformed to safely remove the transitivity axiom. Finally, we explore the field of computational linguistics as an application of automated reasoning. The tool Morfar uses a constraint solver to analyse the morphology of an input language. The result is a novel automatic method for segmentation and labelling that works well even when there is very little training data available

Constraining Montague Grammar for computational applications

This work develops efficient methods for the implementation of Montague Grammar on a computer. It covers both the syntactic and the semantic aspects of that task. Using a simplified but adequate version of Montague Grammar it is shown how to translate from an English fragment to a purely extensional first-order language which can then be made amenable to standard automatic theorem-proving techniques. Translating a sentence of Montague English into the first-order predicate calculus usually proceeds via an intermediate translation in the typed lambda calculus which is then simplified by lambda-reduction to obtain a first-order equivalent. If sufficient sortal structure underlies the type theory for the reduced translation to always be a first-order one then perhaps it should be directly constructed during the syntactic analysis of the sentence so that the lambda-expressions never come into existence and no further processing is necessary. A method is proposed to achieve this involving the unification of meta-logical expressions which flesh out the type symbols of Montague's type theory with first-order schemas. It is then shown how to implement Montague Semantics without using a theorem prover for type theory. Nothing more than a theorem prover for the first-order predicate calculus is required. The first-order system can be used directly without encoding the whole of type theory. It is only necessary to encode a part of second-order logic and this can be done in an efficient, succinct, and readable manner. Furthermore the pseudo-second-order terms need never appear in any translations provided by the parser. They are vital just when higher-order reasoning must be simulated. The foundation of this approach is its five-sorted theory of Montague Semantics. The objects in this theory are entities, indices, propositions, properties, and quantities. It is a theory which can be expressed in the language of first-order logic by means of axiom schemas and there is a finite second-order axiomatisation which is the basis for the theorem-proving arrangement. It can be viewed as a very constrained set theory

Satisfiability for relation-changing logics

Relation-changing modal logics (RC for short) are extensions of the basic modal logic with dynamic operators that modify the accessibility relation of a model during the evaluation of a formula. These languages are equipped with dynamic modalities that are able e.g. to delete, add and swap edges in the model, both locally and globally. We study the satisfiability problem for some of these logics.We first show that they can be translated into hybrid logic. As a result, we can transfer some results from hybrid logics to RC. We discuss in particular decidability for some fragments. We then show that satisfiability is, in general, undecidable for all the languages introduced, via translations from memory logics.Fil: Areces, Carlos Eduardo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física. Sección Ciencias de la Computación; ArgentinaFil: Fervari, Raul Alberto. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física. Sección Ciencias de la Computación; ArgentinaFil: Hoffmann, Guillaume Emmanuel. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física. Sección Ciencias de la Computación; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Martel, Mauricio. Universitat Bremen; Alemani

Efficient Management of Short-Lived Data

Motivated by the increasing prominence of loosely-coupled systems, such as mobile and sensor networks, which are characterised by intermittent connectivity and volatile data, we study the tagging of data with so-called expiration times. More specifically, when data are inserted into a database, they may be tagged with time values indicating when they expire, i.e., when they are regarded as stale or invalid and thus are no longer considered part of the database. In a number of applications, expiration times are known and can be assigned at insertion time. We present data structures and algorithms for online management of data tagged with expiration times. The algorithms are based on fully functional, persistent treaps, which are a combination of binary search trees with respect to a primary attribute and heaps with respect to a secondary attribute. The primary attribute implements primary keys, and the secondary attribute stores expiration times in a minimum heap, thus keeping a priority queue of tuples to expire. A detailed and comprehensive experimental study demonstrates the well-behavedness and scalability of the approach as well as its efficiency with respect to a number of competitors.Comment: switched to TimeCenter latex styl

Formalization of Universal Algebra in Agda

In this work we present a novel formalization of universal algebra in Agda. We show that heterogeneous signatures can be elegantly modelled in type-theory using sets indexed by arities to represent operations. We prove elementary results of heterogeneous algebras, including the proof that the term algebra is initial and the proofs of the three isomorphism theorems. We further formalize equational theory and prove soundness and completeness. At the end, we define (derived) signature morphisms, from which we get the contravariant functor between algebras; moreover, we also proved that, under some restrictions, the translation of a theory induces a contra-variant functor between models.Fil: Gunther, Emmanuel. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Gadea, Alejandro Emilio. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Pagano, Miguel Maria. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentin