113,363 research outputs found
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
A Characterization Result for Non-Distributive Logics
Recent published work has addressed the Shalqvist correspondence problem for
non-distributive logics. The natural question that arises is to identify the
fragment of first-order logic that corresponds to logics without distribution,
lifting van Benthem's characterization result for modal logic to this new
setting. Carrying out this project is the contribution of the present article.
The article is intended as a demonstration and application of a project of
reduction of non-distributive logics to (sorted) residuated modal logics. The
reduction is an application of recent representation results by this author for
normal lattice expansions and a generalization of a canonical and fully
abstract translation of the language of substructural logics into the language
of their companion sorted, residuated modal logics. The reduction of
non-distributive logics to sorted modal logics makes the proof of a van Benthem
characterization of non-distributive logics nearly effortless, by adapting and
reusing existing results, demonstrating the usefulness and suitability of this
approach in studying logics that may lack distribution
Spatial Interpolants
We propose Splinter, a new technique for proving properties of
heap-manipulating programs that marries (1) a new separation logic-based
analysis for heap reasoning with (2) an interpolation-based technique for
refining heap-shape invariants with data invariants. Splinter is property
directed, precise, and produces counterexample traces when a property does not
hold. Using the novel notion of spatial interpolants modulo theories, Splinter
can infer complex invariants over general recursive predicates, e.g., of the
form all elements in a linked list are even or a binary tree is sorted.
Furthermore, we treat interpolation as a black box, which gives us the freedom
to encode data manipulation in any suitable theory for a given program (e.g.,
bit vectors, arrays, or linear arithmetic), so that our technique immediately
benefits from any future advances in SMT solving and interpolation.Comment: Short version published in ESOP 201
Hierarchic Superposition Revisited
Many applications of automated deduction require reasoning in first-order
logic modulo background theories, in particular some form of integer
arithmetic. A major unsolved research challenge is to design theorem provers
that are "reasonably complete" even in the presence of free function symbols
ranging into a background theory sort. The hierarchic superposition calculus of
Bachmair, Ganzinger, and Waldmann already supports such symbols, but, as we
demonstrate, not optimally. This paper aims to rectify the situation by
introducing a novel form of clause abstraction, a core component in the
hierarchic superposition calculus for transforming clauses into a form needed
for internal operation. We argue for the benefits of the resulting calculus and
provide two new completeness results: one for the fragment where all
background-sorted terms are ground and another one for a special case of linear
(integer or rational) arithmetic as a background theory
Hierarchic Superposition Revisited
Many applications of automated deduction require reasoning in first-order logic modulo background theories, in particular some form of integer arithmetic. A major unsolved research challenge is to design theorem provers that are "reasonably complete" even in the presence of free function symbols ranging into a background theory sort. The hierarchic superposition calculus of Bachmair, Ganzinger, and Waldmann already supports such symbols, but, as we demonstrate, not optimally. This paper aims to rectify the situation by introducing a novel form of clause abstraction, a core component in the hierarchic superposition calculus for transforming clauses into a form needed for internal operation. We argue for the benefits of the resulting calculus and provide two new completeness results: one for the fragment where all background-sorted terms are ground and another one for a special case of linear (integer or rational) arithmetic as a background theory
A short overview of Hidden Logic
In this paper we review a hidden (sorted) generalization of k-deductive systems - hidden k-logics. They encompass deductive systems as
well as hidden equational logics and inequational logics. The special case of
hidden equational logics has been used to specify and to verify properties in
program development of behavioral systems within the dichotomy visible vs.
hidden data. We recall one of the main applications of this work - the study
of behavioral equivalence. Related results are obtained through combinatorial
properties of the Leibniz congruence relation.
In addition we obtain a few new developments concerning hidden equational
logic, namely we present a new characterization of the behavioral consequences of a theory
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