Luis Fariñas del Cerro

A simple deduction method for modal logic

We defme formulas as words in a furite alphabet. A modal formula is either a literal (propositional variable or negation of propositional variable) or has the form: (A & B), A or LA (L means necessary), where A and B are modal formulas. MA (M means possible) is defined as usual as L A. A modal system is a set of modal formulas. The axiomatization of the modal propositional system K is obtained by adding to the habitual formalization of the propositional system (IT) the following axiom scIleme and the inference rule:

The modal logic of equilibrium models

Here-and-there models and equilibrium models were investigated as a semantical framework for answer set programming by Pearce, Cabalar, Lifschitz, Ferraris and others. The semantics of equilibrium logic is indirect in that the notion of satisfiability is defined in terms of satisfiability in the logic of here-and-there. We here give a direct semantics of equilibrium logic, stated in terms of a modal language into which the language of equilibrium logic can be embedded.

Towards a Logical Characterization of Sentences of the Kind "Sentence p is about Object c"

There are many dynamic applications where the set of represented objects may change. Some new objects are introduced in the representations, others are ruled out. For instance, an aircraft may be traced by a radar, and at a given time, for some technical reasons, the aircraft is no more in the scope of the radar. Another example may be found in a database which is used for personnel management in a company. When an employee leaves the company we may want to “erase” the overall information about this employee, because this information is no longer of interest. A similar situation may happen in the case of a patient who leaves a hospital. For privacy reasons, we may desire that the patient no longer “exists” in the database. In all these examples, we want to characterise a situation where there is an object that disappears, in the sense that in our representation of the world we have no more pieces of information about this object. Then, an important problem is to find a formal definition of the property that in a set of sentences that represents our knowledge about the world there is no sentence about a given object. Another important related problem, which is not investigated in this paper, is to determine how to change this set of sentences when an object disappears.

Contingency-based equilibrium logic

We investigate an alternative language for equilibrium logic that is based on the concept of positive and negative contingency. Beyond these two concepts our language has the modal operators of necessity and impossibility and the Boolean operators of conjunction and disjunction. Neither negation nor implication are available. Our language is just as expressive as the standard language of equilibrium logic (that is based on conjunction and intuitionistic implication).

Finite non-deterministic semantics for some modal systems

Trying to overcome Dugundji’s (1940) result on uncharacterisability of modal logics by finite logical matrices, Kearns (1981) and Ivlev (1988) proposed, independently, a characterisation of some modal systems by means of four-valued multivalued truth-functions (by restricting the valuations using level valuations, in Kearns’s approach), as an alternative to Kripke semantics. This constitutes an antecedent of the non-deterministic matrices introduced by Avron and Lev (2001). In this paper we propose a reconstruction of Kearns’s and Ivlev’s results (which did not have the dissemination or impact they deserved) in a uniform way, obtaining an extension to another modal systems. The first part of the paper is devoted to four-valued Nmatrices, including Kearns’s and Ivlev’s. Besides proving with full details Kearns’s results for T, S4 and S5, we also obtain a characterisation of the system B by four-valued Nmatrices with level valuations. Concerning Ivlev’s results, two new modal systems are introduced and characterised by Nmatrices. In the second part of this paper, six-valued Nmatrices are introduced which characterise a variant of the eight systems studied in the first part, by replacing axiom with axiom . As a by-product, novel decision procedures for T, S4, S5, D, KDB, KD4 and KD45 are obtained, which open up interesting possibilities in the study of the complexity of modal logics and, in particular, of intuitionistic propositional logic (IPC), taking into account the Gödel-McKinsey-Tarski translation between IPC and S4.

Combining Equilibrium Logic and Dynamic Logic

We extend the language of here-and-there logic by two kinds of atomic programs allowing to minimally update the truth value of a propositional variable here or there, if possible. These atomic programs are combined by the usual dynamic logic program connectives. We investigate the mathematical properties of the resulting extension of equilibrium logic: we prove that the problem of logical consequence in equilibrium models is EXPTIME complete by relating equilibrium logic to dynamic logic of propositional assignments.

Temporal Logic Modeling of Biological Systems

Metabolic networks, formed by a series of metabolic pathways, are made of intracellular and extracellular reactions that determine the biochemical properties of a cell, and by a set of interactions that guide and regulate the activity of these reactions. Cancer, for example, can sometimes appear in a cell as a result of some pathology in a metabolic pathway. Most of these pathways are formed by an intricate and complex network of chain reactions, and can be represented in a human readable form using graphs which describe the cell signaling pathways. In this paper, we define a logic, called Molecular Interaction Logic (MIL), able to represent these graphs and we present a method to automatically translate graphs into MIL formulas. Then we show how MIL formulas can be translated into linear time temporal logic, and then grounded into propositional classical logic. This enables us to solve complex queries on graphs using only propositional classical reasoning tools such as SAT solvers.

Translation of first order formulas into ground formulas via a completion theory

A translation technique is presented which transforms a class of First Order Logic formulas, called Restricted formulas, into ground formulas. For the formulas in this class the range of quantified variables is restricted by Domain formulas.If we have a complete knowledge of the predicates involved in the Domain formulas their extensions can be evaluated with the Relational Algebra and these extensions are used to transform universal (respectively existential) quantifiers into finite conjunctions (respectively disjunctions).It is assumed that the complete knowledge is represented by Completion Axioms and Unique Name Axioms a la Reiter. These axioms involve the equality predicate. However, the translation allows to remove the equality in the ground formulas and for a large class of formulas their consequences are the same as the initial First Order formulas. This result open the door for the design of efficient deduction techniques.

Modal Tableaux for Reasoning About Diagrams

This paper, we propose a modal logic satisfying minimal requirements for reasoning about diagrams via collection of sets and relations between them, following Harel’s proposal. We first give an axiomatics of such a theory and then provide its Kripke semantics. Then we extend previous works of ours in order to obtain a decision procedure based on tableaux for this logic. Beside soundness and completeness of our tableaux, we manage to define a strategy of rule application ensuring termination by extending the usual loop test of modal logic S4 to whole sub-structures of the model being computed.

Argumentation Frameworks with Recursive Attacks and Evidence-Based Supports

The purpose of this work is to study a generalisation of Dung’s abstract argumentation frameworks that allows representing positive interactions (called supports). The notion of support studied here is based in the intuition that every argument must be supported by some chain of supports from some special arguments called prima-facie. The theory developed also allows the representation of both recursive attacks and supports, that is, a class of attacks or supports whose targets are other attacks or supports. We do this by developing a theory of argumentation where the classic role of attacks in defeating arguments is replaced by a subset of them, which is extension dependent and which, intuitively, represents a set of “valid attacks” with respect to the extension. Similarly, only the subset of “valid supports” is allowed to support other elements (arguments, attacks or supports). This theory displays a conservative generalisation of Dung’s semantics (complete, preferred and stable) and also of their principles (conflict-freeness, acceptability and admissibility). When restricted to finite non-recursive frameworks, we are also able to prove a one-to-one correspondence with Evidence-Based Argumentation (EBA). When supports are ignored a one-to-one correspondence with Argumentation Frameworks with Recursive Attacks (AFRA) semantics is also established.