MAT Logic: A Temporal×Modal Logic with Non-deterministic Operators to Deal with Interactive Systems in Communication Technologies

Aguilera, Gabriel; Burrieza, Alfredo; Guzmán, Inman P. de; Muñoz, Emilo

doi:10.1007/11874850_64

Cited by 3 publications

(6 citation statements)

References 17 publications

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“…Another possibility would be to consider double indexes in order to represent the domain and image of each accessibility function. Other future works are related to the extension of this approach to more properties of accessibility functions, and the consideration of non-deterministic operators, in the line of [1,11].…”

Section: Discussionmentioning

confidence: 99%

Analyzing completeness of axiomatic functional systems for temporal × modal logics

Burrieza¹,

Guzmán

Muñoz‐Velasco

2010

Mathematical Logic Qtrly

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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.

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Section: Discussionmentioning

confidence: 99%

Analyzing completeness of axiomatic functional systems for temporal × modal logics

Burrieza¹,

Guzmán

Muñoz‐Velasco

2010

Mathematical Logic Qtrly

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“…If we assume that none of the conditions (i) to (iii) hold, then we have a) there exists a formula ♦ 4 . By modal logic and Proposition 5.4 (1)…”

Section: Proposition 54mentioning

confidence: 94%

“…P r o o f . Assume that 1 1 is a traced frame, where 1 is an unnamed simple frame and 1 is quasi-coherent. Since 1 is unnamed, it is of the form ({w}, I 1 , {(T w , < w )}, ∅) and there is no i ∈ I such that I 1 (i) = w. Assume also that ♦ j i A is the formula in the antecedent of (α) 1 , where i, j ∈ I (we reason in the same manner if the formula is of the form j i A).…”

Section: Definition 532mentioning

confidence: 99%

“…Lemma 5.34 (Exhausting Lemma) Let 1 1 be a traced frame. If (α) 1 is a conditional active for 1 , then there is a traced frame 2 2 , extension of 1 1 , such that (α) 2 is a conditional exhausted for 2 . Moreover,…”

Section: Definition 532mentioning

confidence: 99%

“…(a) n = 0. We extend 1 1 by introducing a new coordinate associated with as the immediate successor of r i in T i . Thus, we have to extend 1 and 1 as follows: W 2 = W 1 and I 2 = I 1 ;…”

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confidence: 99%

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Completeness of a functional system for surjective functions

Burrieza

Ruiz

Guzmán

2017

Mathematical Logic Qtrly

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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.

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