Inconsistency-tolerant temporal reasoning with hierarchical information☆
Introduction
In this paper, we propose a formal method for modeling and verifying inconsistency-tolerant temporal reasoning with hierarchical information. Many logic-based studies have examined handling inconsistency-tolerant reasoning [3], temporal reasoning [6], and reasoning with hierarchical or ontological information [2]. However, to the best of our knowledge, no study has examined integrating these reasoning mechanisms uniformly, i.e., there is no study about inconsistency-tolerant temporal reasoning with hierarchical information. Such a study is required to extend and integrate existing application areas, such as medical diagnosis.
Integration of these useful reasoning mechanisms requires the combination and extension of useful non-classical logics. Combining and extending useful non-classical logics is a very important issue in mathematical logic [4]. Thus, the aim of this paper is to provide a solution for this issue by combining and extending the following useful logics: temporal logic, paraconsistent logic, and new modal logic (with a new modal operator). By combining and extending the above logics, we can integrate existing application areas using these logics.
In this paper, a new temporal logic called sequential paraconsistent computation tree logic (SPCTL), which is an extension of the well-known computation tree logic (CTL) [5], is introduced as a Kripke semantics with a paraconsistent negation connective and sequence modal operators. New illustrative examples of modeling and verification are presented using SPCTL. The validity, satisfiability, and model-checking problems of SPCTL are shown to be EXPTIME-complete, deterministic EXPTIME-complete, and deterministic PTIME-complete, respectively. These complexity results are proved using theorems for embedding SPCTL into a paraconsistent CTL (PCTL) and CTL. These embedding and complexity results for SPCTL allow us to use existing CTL-based algorithms to test satisfiability. Thus, it is shown that SPCTL can be used as an executable logic to model and verify inconsistency-tolerant temporal reasoning with hierarchical information.
Model checking [6] is a formal logic-based method for verifying concurrent systems. Specifications about the underlying system are expressed as temporal logic formulas, and efficient algorithms are used to traverse a model defined by the system and determine whether the specification holds or not. CTL [5] is one of the most useful temporal logics for model checking and one of the most important branching-time temporal logics that uses computation trees to specify and verify concurrent systems.
Some CTL-based model checking frameworks [6] for verifying systems with hierarchical structures are more efficient and suitable than other types of frameworks, such as those based on linear-time temporal logic (LTL) [17] and full computation tree logic (CTL∗) [9], [8]. However, CTL is not suitable for modeling and verifying “inconsistent” systems. Handling inconsistencies in systems requires a paraconsistent logic [16] as a base logic for CTL.
An important feature of CTL is that the existence of paths in computation trees can be specified and verified. A computation tree represents a nondeterministic computation or unwinding of a Kripke structure. A Kripke structure is a directed graph; thus, it can naturally express “simple” hierarchical structures. However, it is unsuitable for representing the “highly complex” and “informative” hierarchical structures of ontologies. This is because “normal” trees are not sufficiently expressive for representing such complex structures. Handling highly complex and informative hierarchies (hereafter hierarchical information) requires modal operators called the sequence modal operators [13].
We use Nelson’s four-valued paraconsistent logic N4 [1], [15] as the base logic for CTL. N4 and its variants have been studied extensively (e.g., [11], [18], [19]) because they have paraconsistency [16]. A satisfaction relation is said to be paraconsistent with respect to a negation connective if the following condition holds: , not-, where s is the state of Kripke structure M. In contrast to N4, classical logic has no paraconsistency because the formula of the form is valid in classical logic.
Paraconsistent logics are more appropriate for inconsistency-tolerant reasoning than other non-classical logics. In addition, paraconsistent logics are useful for modeling and representing medical diagnosis as an example of inconsistency-tolerant reasoning [7], [14].
Here, the usefulness of paraconsistency is explained. For example, it is undesirable that be satisfied for any symptom s and disease d, where means “a person x does not have a symptom s” and means “a person x suffers from a disease d.” Assume a large medical knowledge-base MKB of symptoms and diseases. It can be assumed that MKB is inconsistent in the sense that there is a symptom predicate such that . This assumption is very realistic because symptom is a vague concept that is difficult to determine by any diagnosis; it may be determined true or false by different doctors with different perspectives. Then, the knowledge-base MKB does not derive arbitrary disease , which means “a person x suffers from disease d”; thus, paraconsistent logics ensure the fact that, for some formulas and , the formula is not valid. Therefore, the paraconsistent logic-based MKB is inconsistency-tolerant. In classical logic, the formula is valid for any disease d; thus, the non-paraconsistent formulation based on classical logic is considered inappropriate for application to a medical knowledge base.
Combining N4 and CTL has been studied previously [12]. The resulting combined logic was called PCTL [12], and it was used to verify a medical check up system. SPCTL is obtained from PCTL by adding sequence modal operators.
In this paper, we use a sequence modal operator [13], which represents a sequence b of symbols, to describe the ordered labels in a hierarchy. The reason for using the notion of “sequences” in this modal operator is explained below. The notion of “sequences” is fundamental to practical reasoning in computer science because it can represent concepts such as “data sequences,” “action sequences,” and “time sequences” appropriately. Therefore, the notion of sequences is useful for representing the notions of “information,” “trees,” “orders,” “preferences,” and “ontologies.” Thus, ”hierarchical information” can be represented by sequences. This is plausible because a sequence structure gives a monoid with informational interpretation [19]:
- 1.
M is a set of pieces of (ordered or prioritized) information (i.e., a set of sequences),
- 2.
is a binary operator (on M) that combines two pieces of information (i.e., it is a concatenation operator on sequences),
- 3.
is an empty piece of information (i.e., the empty sequence).
A formula of the form intuitively means that “ is true based on a sequence of (ordered or prioritized) information pieces.” Furthermore, a formula of the form , which coincides with , intuitively means that “ is true without any information (i.e., it is an eternal truth in the sense of classical logic).”
Combining CTL∗ [9], [8] with some sequence modal operators has been studied [13]. The resulting combined logic was called CTLS∗ in [13], and it was used to specify and verify biological ontologies. SPCTL is obtained from a subsystem of CTLS∗ by adding the paraconsistent negation connective .
The remainder of this paper is organized as follows:
In Section 2, SPCTL is introduced as a Kripke semantics by extending CTL with a paraconsistent negation connective and some sequence modal operators.
In Section 3, complexity results for SPCTL are obtained using theorems for embedding SPCTL into PCTL and CTL. First, PCTL and CTL are introduced, and then the embedding theorems are proved. Some translation examples are also addressed.
In Section 4, some modeling and verification examples of SPCTL are presented for reasoning about inconsistency, hierarchical information, and temporal properties.
In Section 5, the paper is concluded with some remarks.
Section snippets
Logics
Formulas of SPCTL are constructed from countably many atomic formulas, (implication), (conjunction), (disjunction), (classical negation), (paraconsistent negation), (next), (globally), (eventually), (until), (release), (all computation paths), (some computation path) and (sequence modal operator) where b is constructed from countably many atomic sequences, the empty sequence and (concatenation). It is remarked that SPCTL has two kinds of negation connectives:
Complexity
In this section, the complexities of the validity, satisfiability and model checking problems for SPCTL and PCTL are shown, and some translation examples are shown for SPCTL.
Modeling examples
Paraconsistent logics are useful for modeling medical reasoning [7], [12], [14]. Here, we provide some new examples. SPCTL can be used to express the negation of vague concepts, such as healthy, happy, disease, and symptom. For example, if we cannot determine if someone is healthy, the vague concept healthy can be represented by asserting the inconsistent formula with a paraconsistent negation connective :This is well formalized becauseis not valid in
Conclusions and remarks
This paper has presented a formal method for inconsistency-tolerant temporal reasoning with hierarchical information. A new logic, SPCTL, was introduced by extending the well-known useful temporal logic CTL. The embedding of SPCTL theorems into PCTL and CTL were proved. By using these theorems, the validity, satisfiability, and model-checking problems of SPCTL were shown to be EXPTIME-complete, deterministic EXPTIME-complete, and deterministic PTIME-complete, respectively. The embedding and
Acknowledgments
This work was supported by Okawa Foundation for Information and Telecommunications, and by JSPS Grant-in-Aid for Scientific Research (C) 26330263.
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