Finding Minimal Cost Herbrand Models with Branch-Cut-and-Price

Given (1) a set of clauses $T$ in some first-order language $\cal L$ and (2) a cost function $c : B_{{\cal L}} \rightarrow \mathbb{R}_{+}$, mapping each ground atom in the Herbrand base $B_{{\cal L}}$ to a non-negative real, then the problem of finding a minimal cost Herbrand model is to either find a Herbrand model $\cal I$ of $T$ which is guaranteed to minimise the sum of the costs of true ground atoms, or establish that there is no Herbrand model for $T$. A branch-cut-and-price integer programming (IP) approach to solving this problem is presented. Since the number of ground instantiations of clauses and the size of the Herbrand base are both infinite in general, we add the corresponding IP constraints and IP variables `on the fly' via `cutting' and `pricing' respectively. In the special case of a finite Herbrand base we show that adding all IP variables and constraints from the outset can be advantageous, showing that a challenging Markov logic network MAP problem can be solved in this way if encoded appropriately

Validity, dialetheism and self-reference

It has been argued recently (Beall in Spandrels of truth, Oxford University Press, Oxford, 2009; Beall and Murzi J Philos 110:143–165, 2013) that dialetheist theories are unable to express the concept of naive validity. In this paper, we will show that (Formula presented.) can be non-trivially expanded with a naive validity predicate. The resulting theory, (Formula presented.) reaches this goal by adopting a weak self-referential procedure. We show that (Formula presented.) is sound and complete with respect to the three-sided sequent calculus (Formula presented.). Moreover, (Formula presented.) can be safely expanded with a transparent truth predicate. We will also present an alternative theory (Formula presented.), which includes a non-deterministic validity predicate.Fil: Pailos, Federico Matias. Instituto de Investigaciones Filosóficas - Sadaf; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Towards an efficient prover for the C1 paraconsistent logic

The KE inference system is a tableau method developed by Marco Mondadori which was presented as an improvement, in the computational efficiency sense, over Analytic Tableaux. In the literature, there is no description of a theorem prover based on the KE method for the C1 paraconsistent logic. Paraconsistent logics have several applications, such as in robot control and medicine. These applications could benefit from the existence of such a prover. We present a sound and complete KE system for C1, an informal specification of a strategy for the C1 prover as well as problem families that can be used to evaluate provers for C1. The C1 KE system and the strategy described in this paper will be used to implement a KE based prover for C1, which will be useful for those who study and apply paraconsistent logics.Comment: 16 page

nested PLS

In this note we will introduce a class of search problems, called nested Polynomial Local Search (nPLS) problems, and show that definable NP search problems, i.e., $\Sigma^b_1$-definable functions in $T^2_2$ are characterized in terms of the nested PLS

Practical Reasoning for Very Expressive Description Logics

Description Logics (DLs) are a family of knowledge representation formalisms mainly characterised by constructors to build complex concepts and roles from atomic ones. Expressive role constructors are important in many applications, but can be computationally problematical. We present an algorithm that decides satisfiability of the DL ALC extended with transitive and inverse roles and functional restrictions with respect to general concept inclusion axioms and role hierarchies; early experiments indicate that this algorithm is well-suited for implementation. Additionally, we show that ALC extended with just transitive and inverse roles is still in PSPACE. We investigate the limits of decidability for this family of DLs, showing that relaxing the constraints placed on the kinds of roles used in number restrictions leads to the undecidability of all inference problems. Finally, we describe a number of optimisation techniques that are crucial in obtaining implementations of the decision procedures, which, despite the worst-case complexity of the problem, exhibit good performance with real-life problems