An Optimal Decision Procedure for MPNL over the Integers

Interval temporal logics provide a natural framework for qualitative and quantitative temporal reason- ing over interval structures, where the truth of formulae is defined over intervals rather than points. In this paper, we study the complexity of the satisfiability problem for Metric Propositional Neigh- borhood Logic (MPNL). MPNL features two modalities to access intervals "to the left" and "to the right" of the current one, respectively, plus an infinite set of length constraints. MPNL, interpreted over the naturals, has been recently shown to be decidable by a doubly exponential procedure. We improve such a result by proving that MPNL is actually EXPSPACE-complete (even when length constraints are encoded in binary), when interpreted over finite structures, the naturals, and the in- tegers, by developing an EXPSPACE decision procedure for MPNL over the integers, which can be easily tailored to finite linear orders and the naturals (EXPSPACE-hardness was already known).Comment: In Proceedings GandALF 2011, arXiv:1106.081

Decidability of the interval temporal logic ABBar over the natural numbers

In this paper, we focus our attention on the interval temporal logic of the Allen's relations "meets", "begins", and "begun by" (ABBar for short), interpreted over natural numbers. We first introduce the logic and we show that it is expressive enough to model distinctive interval properties,such as accomplishment conditions, to capture basic modalities of point-based temporal logic, such as the until operator, and to encode relevant metric constraints. Then, we prove that the satisfiability problem for ABBar over natural numbers is decidable by providing a small model theorem based on an original contraction method. Finally, we prove the EXPSPACE-completeness of the proble

Maximal decidable fragments of Halpern and Shoham's modal logic of intervals

In this paper, we focus our attention on the fragment of Halpern and Shoham's modal logic of intervals (HS) that features four modal operators corresponding to the relations ``meets'', ``met by'', ``begun by'', and ``begins'' of Allen's interval algebra (AAbarBBbar logic). AAbarBBbar properly extends interesting interval temporal logics recently investigated in the literature, such as the logic BBbar of Allen's ``begun by/begins'' relations and propositional neighborhood logic AAbar, in its many variants (including metric ones). We prove that the satisfiability problem for AAbarBBbar, interpreted over finite linear orders, is decidable, but not primitive recursive (as a matter of fact, AAbarBBbar turns out to be maximal with respect to decidability). Then, we show that it becomes undecidable when AAbarBBbar is interpreted over classes of linear orders that contains at least one linear order with an infinitely ascending sequence, thus including the natural time flows N, Z, Q, and R

Maximal decidable fragments of Halpern and Shoham's modal logic of intervals

In this paper, we focus our attention on the fragment of Halpern and Shoham's modal logic of intervals (HS) that features four modal operators corresponding to the relations ``meets'', ``met by'', ``begun by'', and ``begins'' of Allen's interval algebra (AAbarBBbar logic). AAbarBBbar properly extends interesting interval temporal logics recently investigated in the literature, such as the logic BBbar of Allen's ``begun by/begins'' relations and propositional neighborhood logic AAbar, in its many variants (including metric ones). We prove that the satisfiability problem for AAbarBBbar, interpreted over finite linear orders, is decidable, but not primitive recursive (as a matter of fact, AAbarBBbar turns out to be maximal with respect to decidability). Then, we show that it becomes undecidable when AAbarBBbar is interpreted over classes of linear orders that contains at least one linear order with an infinitely ascending sequence, thus including the natural time flows N, Z, Q, and R

Complete and Terminating Tableau for the Logic of Proper Subinterval Structures over Dense Orderings

We introduce special pseudo-models for the interval logic of proper subintervals over dense linear orderings. We prove finite model property with respect to such pseudo-models, and using that result we develop a decision procedure based on a sound, complete, and terminating tableau for that logic. The case of proper subintervals is essentially more complicated than the case of strict subintervals, for which we developed a similar tableau-based decision procedure in a recent work

Crossing the Undecidability Border with Extensions of Propositional Neighborhood Logic over Natural Numbers

Propositional Neighborhood Logic (PNL) is an interval temporal logic featuring two modalities corresponding to the relations of right and left neighborhood between two intervals on a linear order (in terms of Allen's relations, meets and met by). Recently, it has been shown that PNL interpreted over several classes of linear orders, including natural numbers, is decidable (NEXPTIME-complete) and that some of its natural extensions preserve decidability. Most notably, this is the case with PNL over natural numbers extended with a limited form of metric constraints and with the future fragment of PNL extended with modal operators corresponding to Allen's relations begins, begun by, and before. This paper aims at demonstrating that PNL and its metric version MPNL, interpreted over natural numbers, are indeed very close to the border with undecidability, and even relatively weak extensions of them become undecidable. In particular, we show that (i) the addition of binders on integer variables ranging over interval lengths makes the resulting hybrid extension of MPNL undecidable, and (ii) a very weak first-order extension of the future fragment of PNL, obtained by replacing proposition letters by a restricted subclass of first-order formulae where only one variable is allowed, is undecidable (in contrast with the decidability of similar first-order extensions of point-based temporal logics)

A logic for reasoning about knowledge of unawareness

In the most popular logics combining knowledge and awareness, it is not possible to express statements about knowledge of unawareness such as “Ann knows that Bill is aware of something Ann is not aware of” – without using a stronger statement such as “Ann knows that Bill is aware of p and Ann is not aware of p”, for some particular p. In Halpern and Rêgo (2006, 2009b) (revisited in Halpern and Rêgo (2009a, 2013)) Halpern and Rêgo introduced a logic in which such statements about knowledge of unawareness can be expressed. The logic extends the traditional framework with quantification over formulae, and is thus very expressive. As a consequence, it is not decidable. In this paper we introduce a decidable logic which can be used to reason about certain types of unawareness. Our logic extends the traditional framework with an operator expressing full awareness, i.e., the fact that an agent is aware of everything, and another operator expressing relative awareness, the fact that one agent is aware of everything another agent is aware of. The logic is less expressive than Halpern’s and Rêgo’s logic. It is, however, expressive enough to express all of the motivating examples in Halpern and Rêgo (2006, 2009b). In addition to proving that the logic is decidable and that its satisfiability problem is PSPACE-complete, we present an axiomatisation which we show is sound and complete

A general tableau method for propositional interval temporal logics: Theory and implementation

In this paper, we focus our attention on tableau methods for propositional interval temporal logics. These logics provide a natural framework for representing and reasoning about temporal properties in several areas of computer science. However, while various tableau methods have been developed for linear and branching time point-based temporal logics, not much work has been done on tableau methods for interval-based ones. We develop a general tableau method for Venema’s CDT logic interpreted over partial orders (BCDT+ for short). It combines features of the classical tableau method for first-order logic with those of explicit tableau methods for modal logics with constraint label management, and it can be easily tailored to most propositional interval temporal logics proposed in the literature. We prove its soundness and completeness, and we show how it has been implemented

Two Categories of Refutation Decision Procedures for Classical and Intuitionistic Propositional Logic

An automatic theorem prover is a computer program that proves theorems without the assistance of a human being. Theorem proving is an important basic tool in proving theorems in mathematics, establishing the correctness of computer programs, proving the correctness of communication protocols, and verifying integrated circuit designs. This dissertation introduces two new categories of theorem provers, one for classical propositional logic and another for intuitionistic propositional logic. For each logic a container property and generalized algorithm are introduced. Many methods have been developed over the years to prove theorems in propositional logic. This dissertation describes and presents example proofs for five of these methods: natural deduction, Kripke tableau, analytic tableau, matrix, and resolution. Each of these methods uses refutation to prove a theorem. In refutation, the proposed theorem is assumed to be false. The theorem prover is successful, only if the analysis of this assumption leads to a contradiction. Each of these methods, except resolution, share a common algorithm. To prove this, the container is introduced. A data structure used by a method is a container, if it meets a set of properties. A generalized algorithm that proves theorems is introduced. Since each step in this algorithm uses only operations that are provided by the container. The steps it performs can be translated to any method that can be described using a container. This allows the data structures representing a partial proof in one method, to be transformed into the data structures representing the “same ” proof in another method. This can be very beneficial in a situation where another method would be more efficient in advancing the proof. In addition to being able to switch between methods, an heuristic for one method can be examined to see if it can be applied to the other methods. This development is repeated for intuitionistic logic. Each of these methods, except resolution, is modified to prove theorems in intuitionistic logic. An intuitionistic container is presented. Each one of the intuitionistic methods is proven to have the properties of the intuitionistic container. Lastly, a generalized algorithm using the intuitionistic container is presented. This algorithm proves theorems in intuitionistic logic. Examples showing successful and unsuccessful proof attempts are presented