A functional approach for temporal $\times$ modal logics

Mathematical Logic Qtrly

Muñoz‐Velasco

2010

Self Cite

In previous works, we presented a modification of the usual possible world semantics by introducing an independent temporal structure in each world and using accessibility functions to represent the relation among them. Different properties of the accessibility functions (being injective, surjective, increasing, etc.) have been considered and axiomatic systems (called functional) which define these properties have been given. Only a few of these systems have been proved to be complete. The aim of this paper is to make a progress in the study of completeness for functional systems. For this end, we use indexes as names for temporal flows and give new proofs of completeness. Specifically, we focus our attention on the system which defines injectivity, because the system which defines this property without using indexes was proved to be incomplete in previous works. The only system considered which remains incomplete is the one which defines surjectivity, even if we consider a sequence of natural extensions of the previous one.

Section: For Eachmentioning

confidence: 99%

Section: Soundness and Completeness Of S I -Injmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Analyzing completeness of axiomatic functional systems for temporal × modal logics

Burrieza¹,

Mathematical Logic Qtrly

Muñoz‐Velasco

2010

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“…Basically, this functional approach is based on possible worlds semantics, each world with its own (linear) temporal flow. The most significant feature of this approach is the use of functions (called accessibility functions) to connect the temporal flows in a frame, called functional frame [5][6][7]. Technically, functional frames constitute a generalization of Kamp-frames [6] by establishing more complex comparisons among different temporal flows.…”

Section: Introductionmentioning

confidence: 99%

Functional systems in the context of temporal×modal logics with indexed flows

Burrieza

International Journal of Computer Mathematics

Muñoz‐Velasco

2009

We consider logics which define different properties of functions -such as injectivity, surjectivity, monotonicity, etc. -in the context of temporal × modal logic. In this type of logics, the possible worlds semantics is modified by considering each world as a temporal flow and using accessibility functions to represent the connection among them. This approach is adequate to model interactions between processes with clocks that can be either synchronized or not. We study the definability and give indexed axiomatic systems for these properties.

Completeness of a functional system for surjective functions

Burrieza

Ruiz

Mathematical Logic Qtrly

2017

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Combining modalities has proven to have interesting applications and many approaches that combine time with other types of modalities have been developed. One of these approaches uses accessibility functions between flows of time to study the basic properties of the functions, such as being total or partial, injective, surjective, etc. The completeness of certain systems expressing many of these properties, with the exception of surjectivity, has been proven. In this paper we propose a language with nominals to denote the initial and final sets of each function in a model in order to obtain a complete system for reasoning about surjective partial functions.