Towards Locative Inconsistency-tolerant Hierarchical Probabilistic CTL

Model Checking: Survey and Future Work

Norihiro Kamide

and Juan Pedro Altamirano Bernal

Teikyo University, Faculty of Science and Engineering, Department of Information and Electronic Engineering,

Toyosatodai 1-1, Utsunomiya-shi, Tochigi 320-8551, Japan

Panamerican University, Faculty of Engineering, Department of Electronic Engineering and Digital Systems,

Josemaria Escriva de Balaguer No. 101, CP. 20190 Aguascalientes, Aguascalientes, Mexico

Keywords:

Probabilistic Temporal Logic, Inconsistency-tolerant Temporal Logic, Hierarchical Temporal Logic, Locative

Temporal Logic, Probabilistic Model Checking, Inconsistency-tolerant Model Checking, Hierarchical Model

Checking.

Abstract:

A locative inconsistency-tolerant hierarchical probabilistic computation tree logic (LIHpCTL) is introduced

in this paper to establish the logical foundation of a new model checking paradigm. This logic is an extension

of several previously proposed extensions of the standard temporal logic known as CTL, which is widely

used for model checking. The extended model checking paradigm proposed is intended to appropriately

verify locative (spatial), inconsistent, hierarchical, probabilistic (randomized), and time-dependent concurrent

systems. Additionally, a survey of various studies on probabilistic, inconsistency-tolerant, and hierarchical

temporal logics and their applications in model checking is conducted.

1 INTRODUCTION

In this paper, we introduce a locative inconsistency-

tolerant hierarchical probabilistic computation tree

logic (LIHpCTL), which is designed to form the log-

ical foundation of a new model checking paradigm.

This paradigm is intended to appropriately verify

locative (spatial), inconsistent, hierarchical, proba-

bilistic (randomized), and time-dependent concurrent

systems. LIHpCTL is an extension of several pre-

viously proposed locative, inconsistency-tolerant, hi-

erarchical, and probabilistic extensions of the stan-

dard temporal logic known as CTL (Clarke and Emer-

son, 1981), which is widely used for model checking

(Clarke and Emerson, 1981; Clarke et al., 1999; Holz-

mann, 2006; Clarke et al., 2018). Model checking is a

well-known formal and automated technique for ver-

ifying concurrent systems. Although the decidability

of model checking based on LIHpCTL has not been

determined yet, we present an illustrative example of

the novel LIHpCTL-based model checking paradigm.

In what follows, we present a brief explanation

of several standard and extended temporal logics and

their applications to model checking. The follow-

ing are well-known standard temporal logics typi-

cally used in model checking: computation tree logic

(CTL) (Clarke and Emerson, 1981), linear-time tem-

poral logic (LTL) (Pnueli, 1977), and full computa-

tion tree logic (CTL

∗

) (Emerson and Sistla, 1984;

Emerson and Halpern, 1986). The logic CTL (Clarke

and Emerson, 1981) is one of the most useful tem-

poral logics for model checking; it is based on the

branching-time paradigm, which uses computation

trees to represent the passage of time. The logic

LTL (Pnueli, 1977) is another form of temporal logic

widely used for model checking; it is based on the

linear-time paradigm, which uses linear order to rep-

resent the passage of time. The logic CTL

∗

(Emer-

son and Sistla, 1984; Emerson and Halpern, 1986) is

more expressive than LTL and CTL; it is based on the

branching-time paradigm with path quantiﬁers to rep-

resent the passage of time.

To extend the above-mentioned temporal log-

ics, other useful non-classical logics must be com-

bined and integrated in a natural way. This is also

an important issue in mathematical logic (Carnielli

et al., 2008). The following logics can be utilized

for this purpose: probabilistic (probability) logics,

inconsistency-tolerant (paraconsistent) logics, hier-

archical (sequential) logics, and spatial (locative)

logics. By combining and integrating these log-

ics, we can extend and reﬁne the existing standard

model-checking framework. Model checking has

been extended to probabilistic model checking (Aziz

Kamide, N. and Bernal, J.

Towards Locative Inconsistency-tolerant Hierarchical Probabilistic CTL Model Checking: Survey and Future Work.

DOI: 10.5220/0007683808690878

In Proceedings of the 11th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2019), pages 869-878

ISBN: 978-989-758-350-6

869

et al., 1995; Bianco and de Alfaro, 1995; Baier and

Kwiatkowska, 1998; Kwiatkowska et al., 2011; Baier

et al., 2018), inconsistency-tolerant model checking

(Easterbrook and Chechik, 2001; Chen and Wu, 2006;

Kaneiwa and Kamide, 2011b; Kamide and Endo,

2018), and hierarchical model checking (Kamide

and Kaneiwa, 2009; Kaneiwa and Kamide, 2011a;

Kamide, 2015; Kamide and Yano, 2017; Kamide,

2018).

Probabilistic temporal, inconsistency-tolerant

(paraconsistent) temporal, and hierarchical (se-

quential) temporal logics and their applications to

probabilistic, inconsistency-tolerant, and hierarchical

model checking, respectively, have been studied

by many researchers. In the following section, we

present a survey of these previously proposed logics

and their applications in model checking. However,

this survey is not comprehensive; information on

probabilistic temporal logics and their model check-

ing applications in greater detail can be found in

(Hanson, 1994; Hansson and Jonsson, 1994; Aziz

et al., 1995; Bianco and de Alfaro, 1995; Baier

and Kwiatkowska, 1998; Ognjanovic et al., 2011;

Kwiatkowska et al., 2011; Ognjanovic et al., 2012;

Kamide and Koizumi, 2016; Baier et al., 2018).

For inconsistency-tolerant temporal logics and their

model checking applications, information can be

found in greater detail in (Easterbrook and Chechik,

2001; Chen and Wu, 2006; Kamide, 2006a; Kamide

and Wansing, 2011; Kamide and Kaneiwa, 2010;

Kaneiwa and Kamide, 2011b; Kamide, 2015; Kamide

and Koizumi, 2016; Kamide and Endo, 2018). For

hierarchical temporal logics and their model checking

applications, information can be found in greater

detail in (Kamide and Kaneiwa, 2009; Kaneiwa

and Kamide, 2010; Kaneiwa and Kamide, 2011a;

Kamide, 2015; Kamide and Yano, 2017; Kamide,

2018).

The contents of this paper are then summarized

as follows. In Section 2, a survey of previously pro-

posed typical extended temporal logics and their ap-

plications to model checking is performed. In Section

3, the logic LIHpCTL is developed by extending sev-

eral existing typical extensions of CTL. In Section 4,

an illustrative example of model checking based on

LIHpCTL is presented. Section 5 presents the con-

clusion of this study with some remarks.

2 EXISTING LOGICS

2.1 Probabilistic Temporal Logics

In comparison with the standard non-probabilistic

temporal logics CTL

∗

, CTL, and LTL, probabilistic

temporal logics can be effectively used in random-

ized and stochastic situations. Thus, many studies re-

garding probabilistic temporal logics and their appli-

cations, including probabilistic model checking, have

been performed as discussed below. More informa-

tion can be found in (Hanson, 1994; Hansson and Jon-

sson, 1994; Aziz et al., 1995; Bianco and de Alfaro,

1995; Baier and Kwiatkowska, 1998; Ognjanovic

et al., 2011; Kwiatkowska et al., 2011; Ognjanovic

et al., 2012; Kamide and Koizumi, 2016; Baier et al.,

2018).

Probabilistic full computation tree logic (pCTL

∗

)

and its subsystem, probabilistic computation tree

logic (pCTL), have been investigated by Aziz et al.

(Aziz et al., 1995) and Bianco and de Alfaro (Bianco

and de Alfaro, 1995). The logics pCTL

∗

and pCTL

are obtained from CTL

∗

and CTL, respectively, by

adding the probabilistic or probability operator, P

≥x

Formulas with the form P

≥x

α can be interpreted as

follows: the probability that α holds in the future evo-

lution of the system is at least x. In (Bianco and

de Alfaro, 1995), pCTL

∗

and pCTL are introduced

to verify the reliability and performance of systems

modeled by discrete Markov chains. These logics can

appropriately express the quantitative bounds on the

probability of system evolutions. The complexities of

model checking algorithms with respect to the logics

are clariﬁed in (Bianco and de Alfaro, 1995). In (Aziz

et al., 1995), model-checking algorithms for various

extensions of the previous settings of the logics are

proposed to verify the probabilistic non-deterministic

concurrent systems in which probabilistic behavior

coexists with non-determinism. Further, these algo-

rithms are shown to exhibit polynomial-time com-

plexity depending on the sizes of the systems. The

main difference between the pCTL

∗

settings by Aziz

et al. (Aziz et al., 1995) and those by Bianco and de

Alfaro (Bianco and de Alfaro, 1995) is the probability

measure settings in the probabilistic Kripke structure

of pCTL

∗

In (Hansson and Jonsson, 1994), PCTL, a proba-

bilistic and real-time extension of CTL, is investigated

based on an interpretation of discrete time Markov

chains. In contrast to the probabilistic frameworks

of pCTL and pCTL

∗

, the notion of probability in

PCTL is assigned to all of its temporal operators. For

example, a PCTL formula with the form G

≤t

≥p

α im-

plies that α holds continuously for t time units with

ICAART 2019 - 11th International Conference on Agents and Artiﬁcial Intelligence

870

a probability of at least p. In (Hanson, 1994), a

timed probabilistic concurrent computation tree logic

(TPCTL) is introduced and investigated. In (Baier and

Kwiatkowska, 1998), probabilistic branching time

logics (PBTL and PBTL

∗

) are introduced based on the

probabilistic temporal logics TPCTL, PCTL, pCTL,

and pCTL

∗

, and model-checking algorithms based on

PBTL and PBTL

∗

are proposed to automatically ver-

ify randomized distributed systems. In (Ognjanovic

et al., 2011), a propositional discrete probabilistic

branching temporal logic (pBTL) is developed by ex-

tending CTL

∗

. There are two types of novel probabil-

ity operators in pBTL: P

≥r

and P

≥r

, where P

≥r

α im-

plies that the probability that α holds on a randomly

chosen branch is at least γ, and P

≥r

α means that the

probability that α holds on a particular branch is at

least γ. In (Ognjanovic et al., 2012), a propositional

probabilistic logic with discrete linear time is devel-

oped to handle the clinical reasoning with respect to

evidence. More recent developments in probabilistic

model checking based on probabilistic temporal log-

ics are reported in (Baier et al., 2018).

2.2 Inconsistency-tolerant Temporal

Logics

In comparison with the standard non-paraconsistent

temporal logics CTL

∗

, CTL, and LTL, inconsistency-

tolerant (paraconsistent) logics can be appropri-

ately used in inconsistency-tolerant situations (Priest,

2002; da Costa et al., 1995; Wansing, 1993). Typ-

ical examples of non-temporal paraconsistent logics

include Belnap and Dunn’s useful four-valued logic

(Belnap, 1977b; Belnap, 1977a; Dunn, 1976) and

Nelson’s paraconsistent four-valued logic (Almukdad

and Nelson, 1984; Nelson, 1949). Combining these

logics with CTL

∗

, CTL, and LTL has led to the in-

troduction of various inconsistency-tolerant tempo-

ral logics, and inconsistency-tolerant versions (exten-

sions) of CTL, CTL

∗

, and LTL have been developed

by many researchers.

The multi-valued computation tree logic (χCTL)

was introduced by Easterbrook and Chechik (East-

erbrook and Chechik, 2001) as the base logic for

multi-valued model checking, the ﬁrst framework for

inconsistency-tolerant model checking. The quasi-

classical temporal logic (QCTL) was introduced by

Chen and Wu (Chen and Wu, 2006) to verify in-

consistent concurrent systems using inconsistency-

tolerant model checking. The paraconsistent full

computation tree logic (4CTL

∗

) was proposed by

Kamide (Kamide, 2006a) to obtain a logical founda-

tion for inconsistency-tolerant model checking. The

paraconsistent linear-time temporal logic (PLTL)

was introduced by Kamide and Wansing (Kamide

and Wansing, 2011) to obtain a cut-free and com-

plete Gentzen-type sequent calculus. The alternative

paraconsistent computation tree logic (PCTL) was

introduced by Kamide and Kaneiwa (Kamide and

Kaneiwa, 2010; Kaneiwa and Kamide, 2011b) and

offered an alternative inconsistency-tolerant model

checking framework. Kamide (Kamide, 2015) intro-

duced the sequence-indexed paraconsistent computa-

tion tree logic (SPCTL), which can be obtained from

the CTL by adding the paraconsistent negation con-

nective, ∼, and a sequence modal operator, [b]. This

logic was used to verify clinical reasoning through

inconsistency-tolerant hierarchical model checking.

Kamide and Endo (Kamide and Endo, 2018) re-

cently developed pCTL and pLTL, which are alter-

native reﬁned versions of the paraconsistent tempo-

ral logics PCTL and PLTL, respectively, to obtain

various simple and efﬁcient translation methods for

inconsistency-tolerant model checking. These alter-

native reﬁned logics adopt a simple single-satisfaction

relation. The alternative version of PCTL is pCTL,

which has the same name as the probabilistic com-

putation tree logic. Whereas PLTL (Kamide and

Wansing, 2011), PCTL (Kamide and Kaneiwa, 2010;

Kaneiwa and Kamide, 2011b), 4CTL

∗

(Kamide,

2006a), and SPCTL (Kamide, 2015) have two types

of dual-satisfaction relations, |=

(veriﬁcation or jus-

tiﬁcation) and |=

−

(refutation or falsiﬁcation), pCTL

and pLTL (Kamide and Endo, 2018) have a simple

single-satisfaction relation, |=

∗

, which is highly com-

patible with the standard single-satisfaction relations

of LTL, CTL, and CTL

∗

. The single-satisfaction re-

lation can provide simple proofs for embedding theo-

rems, and the connective ∼ can be formalized simply

and handled uniformly.

To deal with open, large, and complex concurrent

systems, such as cloud-based systems and web appli-

cation systems, it is necessary to have extended log-

ics with model checking frameworks that can also si-

multaneously handle both inconsistency-tolerant and

probabilistic reasoning. To this end, in (Kamide

and Koizumi, 2015; Kamide and Koizumi, 2016), a

partial solution was obtained (i.e., an inconsistency-

tolerant extension of the probabilistic computation

tree logic, pCTL, was developed). The paraconsis-

tent probabilistic computation tree logic (PpCTL),

which can be obtained from the probabilistic compu-

tation tree logic, pCTL, by adding a paraconsistent

negation connective ∼, was constructed in (Kamide

and Koizumi, 2015; Kamide and Koizumi, 2016)

based on a probability-measure-independent transla-

tion of PpCTL into pCTL. The theorem for embed-

ding PpCTL into pCTL was proven using this trans-

Towards Locative Inconsistency-tolerant Hierarchical Probabilistic CTL Model Checking: Survey and Future Work

871

lation and entails the relative decidability of PpCTL

with respect to pCTL (i.e., the decidability of pCTL

implies the decidability of PpCTL). This result in-

dicates that we can reuse the existing pCTL-based

model-checking algorithms by Aziz et al. (Aziz et al.,

1995) and Bianco and de Alfaro (Bianco and de Al-

faro, 1995) for the PpCTL-based model-checking al-

gorithms.

2.3 Hierarchical Temporal Logics

In what follows, we present a survey of various hier-

archical (sequential) temporal logics and their appli-

cations. Various extended temporal logics employing

the sequence (hierarchical) modal operator [b], where

b represents a sequence of symbols, have been inves-

tigated to handle hierarchical information (Kamide

and Kaneiwa, 2009; Kaneiwa and Kamide, 2010;

Kaneiwa and Kamide, 2011a; Kamide, 2015; Kamide

and Yano, 2017; Kamide, 2018).

The sequence modal operator [b] can be used

to represent the concepts of hierarchical information

in the following manner: a sequence structure pro-

duces a monoid hM, ;,

0i with the following informa-

tional interpretation (Wansing, 1993): (1) M is a set

of pieces of ordered information (i.e., a set of se-

quences); (2) ‘;’ is a binary operator (on M) that com-

bines two pieces of information (i.e., it is a concate-

nation operator on sequences); (3)

0 is an empty piece

of information (i.e., an empty sequence). Then, for-

mulas with the form [b

; b

;···; b

]α imply that α

is true with the sequence b

; b

;···; b

of ordered

pieces of information. In addition, formulas with the

form [

0]α, which coincide with α, imply that α is true

without any information (i.e., it is an eternal truth in

the sense of classical logic).

Kamide and Kaneiwa (Kamide and Kaneiwa,

2009; Kaneiwa and Kamide, 2011a) introduced

CTLS

∗

, which is an extension of CTL

∗

, by adding

[b] to CTL

∗

. Similarly, the sequence-indexed linear-

time temporal logic (SLTL), which is an extension of

LTL, was introduced in (Kaneiwa and Kamide, 2010)

by adding [b] to LTL. A proof system for SLTL was

developed to verify certain speciﬁcations of secure

authentication systems. The sequence-indexed para-

consistent computation-tree logic (SPCTL), which is

an extension of CTL, was introduced by Kamide in

(Kamide, 2015) by adding [b] and ∼ to CTL. As

explained above, SPCTL was used to verify clini-

cal reasoning through inconsistency-tolerant hierar-

chical model checking. In (Kamide and Yano, 2017;

Kamide, 2018), thesequential linear-time temporal

logic (sLTL) and sequential computation-tree logic

(sCTL) were developed by extending LTL and CTL,

respectively, to obtain the logical foundation of hier-

archical model checking.

The hierarchical (sequential) temporal logics

sCTL and sLTL proposed by Kamide and Yano in

(Kamide and Yano, 2017; Kamide, 2018) adopted a

simple single-satisfaction relation to obtain various

simple and efﬁcient translation methods for hierarchi-

cal model checking. Although the previously pro-

posed logics, CTLS

∗

(Kamide and Kaneiwa, 2009;

Kaneiwa and Kamide, 2011a), SLTL (Kaneiwa and

Kamide, 2010), and SPCTL (Kamide, 2015), have

complex multiple sequence-indexed satisfaction rela-

tions, |=

, sLTL and sCTL (Kamide and Yano, 2017;

Kamide, 2018) have a simple single-satisfaction rela-

tion, |=

, which is highly compatible with the stan-

dard single-satisfaction relations of LTL, CTL, and

CTL

∗

. Using this simple satisfaction relation, embed-

ding theorems can be easily proven, and the operator

[b] can be formalized simply and handled uniformly.

2.4 Locative Temporal Logics

Future studies should aim to extend the previously

proposed standard and extended temporal logics by

adding a location (locative) operator, [l

]. By adding

this operator to previously proposed logics, locative

(spatial) properties within the model checking frame-

work can be appropriately handled. The location

operator was originally introduced by Kobayashi et

al. in (N. Kobayashi and Yonezawa, 1999) using

a structural congruence relation in the formalization

of a distributed concurrent linear logic programming

language. The location operator was also reformu-

lated by Kamide in (Kamide, 2005; Kamide, 2006b)

and the reformulated operator was a variant of the

original setting by Kobayashi et al. (N. Kobayashi

and Yonezawa, 1999). In fact, the framework of the

original operator was improved in (Kamide, 2005;

Kamide, 2006b) as a purely logical formulation with-

out any structural congruence relations. Formulas

with the form [l

]α can be interpreted as follows:

proposition α holds at location l

Various locative (spatial) temporal logics employ-

ing the location operator [l

] have been investigated to

handle distributed concurrent systems. For example,

the locative paraconsistent probabilistic computa-

tion tree logic (LPpCTL) was introduced by Kamide

and Koizumi in (Kamide and Koizumi, 2016) by

adding [l

] to the inconsistency-tolerant (paraconsis-

tent) probabilistic computation tree logic, PpCTL. By

integrating the ﬁndings of the above-mentioned stud-

ies with the ﬁndings in (Kamide and Koizumi, 2016),

we intend to establish a logical foundation for the pro-

posed framework of locative inconsistency-tolerant

ICAART 2019 - 11th International Conference on Agents and Artiﬁcial Intelligence

872

hierarchical probabilistic model checking based on

LIHpCTL employing [l

Assuming a space domain Loc and operator [l]

with l ∈ Loc, the satisfaction relation (s, l) |=

α of

LPpCTL can be interpreted as follows: proposition

α holds at time state s and location l. Then, various

properties and situations with space and time can be

expressed using LPpCTL formulas. The following

liveness property is an example: “If we input the

the mobile computers Comp1 and Comp2, then we

will eventually be able to log in to Comp3.” This can

be formally expressed as AG([comp1]password ∨

[comp2]password→EF[comp3]login), where the

space domain, Loc, is {comp1, comp2, comp3}.

3 PROPOSED LOGIC

Formulas of locative inconsistency-tolerant hierar-

chical probabilistic computation tree logic (LIH-

pCTL) are constructed from countably many propo-

sitional variables, → (implication) ∧ (conjunction),

∨ (disjunction), ¬ (classical negation), ∼ (paracon-

sistent negation), X (next), G (globally), F (eventu-

ally), U (until), R (release), A (all computation paths),

E (some computation path), P

≤x

(less than or equal

probability), P

≥x

(greater than or equal probability),

(less than probability), P

(greater than prob-

ability), [l

] (location operator), and [b] (hierarchical

operator or sequence modal operator) where b is a se-

quence. Sequences are constructed from atomic se-

quences,

0 (empty sequence) and ; (composition).

The set of sequences (including the empty se-

quence

0) is denoted as SE. Lower-case letters b, c, ...

are used to denote sequences. An expression [

0]α

means α, and expressions [

0 ; b]α and [b ;

0]α mean

[b]α. The symbol Φ is used to denote a non-empty set

of propositional variables, the symbol Φ

∼

is used to

denote the set {∼p | p ∈ Φ}, and the symbol Φ

∼[d]

used to denote the set {[d]γ | γ ∈ Φ ∪ Φ

∼

Deﬁnition 3.1. Let Loc be a ﬁnite non-empty set of lo-

cations, and assume l ∈ Loc, and x ∈ [0, 1]. Formulas

α and sequences b of LIHpCTL are deﬁned by the fol-

lowing grammar, assuming p and e represent proposi-

tional variables and atomic sequences, respectively:

α ::= p | α ∧ α | α ∨ α | α→α | ¬α | ∼α |

AXα | EXα | AGα | EGα | AFα | EFα |

A(αUα) | E(αUα) | A(αRα) | E(αRα) |

≤x

α | P

≥x

α | P

α | [l]α | [b]α.

b ::= e |

0 | b ; b.

An expression [d] is used to represent

][d

]···[d

] with i ∈ ω, d

∈ SE and d

≡

The expression [d] can be the empty sequence and is

not uniquely determined.

Deﬁnition 3.2. A structure (Loc, S, S

, R, µ

, L

) is

a locative inconsistency-tolerant hierarchical proba-

bilistic model iff

1. Loc is a ﬁnite non-empty set of locations,

2. S is the set of states,

3. S

is a set of initial states and S

⊆ S,

4. R is a binary relation on S which satisﬁes the con-

dition: ∀s ∈ S ∃s

∈ S [(s, s

) ∈ R],

5. µ

is a certain probability measure concerning s ∈

S: a set of paths beginning at s is mapped into a

real number in [0, 1],

6. L

is a mapping from S to the power set of

d∈SE

∼[d]

A path in a hierarchical probabilistic model is an

inﬁnite sequence of states, π = s

, s

, ... such that

∀i ≥ 0 [(s

, s

i+1

) ∈ R].

Deﬁnition 3.3. (LIHpCTL). A locative

inconsistency-tolerant hierarchical probabilistic

satisfaction relation (M, hs, li) |=

α for any formula

α, where M is a locative inconsistency-tolerant hier-

archical probabilistic model (Loc, S, S

, R, µ

, L

), s

represents a state in S, and l represents a location in

Loc, is deﬁned by:

1. for any γ ∈ Φ

∼[d]

(d ∈ SE), (M, hs, li) |=

γ iff γ ∈

(s),

2. for any p ∈ Φ, (M, hs, li) |=

[d]∼p iff (M, hs, li) |=

∼[d]p,

3. (M, hs, li) |=

[d][b]α iff (M, hs, li) |=

[d ; b]α,

4. (M, hs, li) |=

[d](α ∧ β) iff (M, hs, li) |=

[d]α and

(M, hs, li) |=

[d]β,

5. (M, hs, li) |=

[d](α ∨ β) iff (M, hs, li) |=

[d]α or

(M, hs, li) |=

[d]β,

6. (M, hs, li) |=

[d](α→β) iff (M, hs, li) |=

[d]α implies

(M, hs, li) |=

[d]β,

7. (M, hs, li) |=

[d]¬α iff (M, hs, li) 6|=

[d]α,

8. for any k ∈ Loc, (M, hs, li) |=

[d][k]α iff (M, hs, ki) |=

[d]α,

9. for any x ∈ [0, 1], (M, hs, li) |=

[d]P

≤x

α iff µ

({w ∈

Ω

| (M, hs, li) |=

[d]α}) ≤ x,

10. for any x ∈ [0, 1], (M, hs, li) |=

[d]P

≥x

α iff µ

({w ∈

Ω

| (M, hs, li) |=

[d]α}) ≥ x,

11. for any x ∈ [0, 1], (M, hs, li) |=

[d]P

α iff µ

({w ∈

Ω

| (M, hs, li) |=

[d]α}) < x,

12. for any x ∈ [0, 1], (M, hs, li) |=

[d]P

α iff µ

({w ∈

Ω

| (M, hs, li) |=

[d]α}) > x,

13. (M, hs, li) |=

[d]AXα iff ∀s

∈ S [(s, s

) ∈ R implies

(M, hs

, li) |=

[d]α],

Towards Locative Inconsistency-tolerant Hierarchical Probabilistic CTL Model Checking: Survey and Future Work

873

14. (M, hs, li) |=

[d]EXα iff ∃s

∈ S [(s, s

) ∈ R and

(M, hs

, li) |=

[d]α],

15. (M, hs, li) |=

[d]AGα iff for all paths π ≡ s

, s

, ...,

where s ≡ s

, and all states s

along π, we have

(M, hs

, li) |=

[d]α,

16. (M, hs, li) |=

[d]EGα iff there is a path π ≡

, s

, ..., where s ≡ s

, and for all states s

along

π, we have (M, hs

, li) |=

[d]α,

17. (M, hs, li) |=

[d]AFα iff for all paths π ≡ s

, s

, ...,

where s ≡ s

, there is a state s

along π such that

(M, hs

, li) |=

[d]α,

18. (M, hs, li) |=

[d]EFα iff there is a path π ≡ s

, s

, ...,

where s ≡ s

, and for some state s

along π, we have

(M, hs

, li) |=

[d]α,

19. (M, hs, li) |=

[d]A(αUβ) iff for all paths π ≡

, s

, ..., where s ≡ s

, there is a state s

along

π such that (M, hs

, li) |=

[d]β and ∀0 ≤ k < j

(M, hs

, li) |=

[d]α,

20. (M, hs, li) |=

[d]E(αUβ) iff there is a path π ≡

, s

, ..., where s ≡ s

, and for some state s

along π, we have (M, hs

, li) |=

[d]β and ∀0 ≤ k < j

(M, hs

, li) |=

[d]α,

21. (M, hs, li) |=

[d]A(αRβ) iff for all paths π ≡

, s

, ..., where s ≡ s

, and all states s

along π, we

have (M, hs

, li) |=

[d]β or ∃0 ≤ k < j (M, hs

, li) |=

[d]α,

22. (M, hs, li) |=

[d]E(αRβ) iff there is a path π ≡

, s

, ..., where s ≡ s

, and for all states s

along π, we have (M, hs

, li) |=

[d]β or ∃0 ≤ k < j

(M, hs

, li) |=

[d]α,

23. (M, hs, li) |=

[d]∼∼α iff (M, hs, li) |=

[d]α,

24. (M, hs, li) |=

[d]∼[b]α iff (M, hs, li) |=

[d ; b]∼α,

25. (M, hs, li) |=

[d]∼(α ∧ β) iff (M, hs, li) |=

[d]∼α or

(M, hs, li) |=

[d]∼β,

26. (M, hs, li) |=

[d]∼(α ∨ β) iff (M, hs, li) |=

[d]∼α and

(M, hs, li) |=

[d]∼β,

27. (M, hs, li) |=

[d]∼(α→β) iff (M, hs, li) 6|=

[d]∼α and

(M, hs, li) |=

[d]∼β,

28. (M, hs, li) |=

[d]∼¬α iff (M, hs, li) 6|=

[d]∼α,

29. for any k ∈ Loc, (M, hs, li) |=

[d]∼[k]α

iff (M, hs, ki) |=

[d]∼α,

30. for any x ∈ [0, 1], (M, hs, li) |=

[d]∼P

≤x

α iff µ

({w ∈

Ω

| (M, hs, li) |=

[d]∼α}) > x,

31. for any x ∈ [0, 1], (M, hs, li) |=

[d]∼P

≥x

α iff µ

({w ∈

Ω

| (M, hs, li) |=

[d]∼α}) < x,

32. for any x ∈ [0, 1], (M, hs, li) |=

[d]∼P

α iff µ

({w ∈

Ω

| (M, hs, li) |=

[d]∼α}) ≥ x,

33. for any x ∈ [0, 1], (M, hs, li) |=

[d]∼P

α iff µ

({w ∈

Ω

| (M, hs, li) |=

[d]∼α}) ≤ x,

34. (M, hs, li) |=

[d]∼AXα iff ∃s

∈ S [(s, s

) ∈ R and

(M, hs

, li) |=

[d]∼α],

35. (M, hs, li) |=

[d]∼EXα iff ∀s

∈ S [(s, s

) ∈ R implies

(M, hs

, li) |=

[d]∼α],

36. (M, hs, li) |=

[d]∼AGα iff there is a path π ≡

, s

, ..., where s ≡ s

, and for some state s

along

π, we have (M, hs

, li) |=

[d]∼α,

37. (M, hs, li) |=

[d]∼EGα iff for all paths π ≡

, s

, ..., where s ≡ s

, there is a state s

along π

such that (M, hs

, li) |=

[d]∼α,

38. (M, hs, li) |=

[d]∼AFα iff there is a path π ≡

, s

, ..., where s ≡ s

, and for all states s

along

π, we have (M, hs

, li) |=

[d]∼α,

39. (M, hs, li) |=

[d]∼EFα iff for all paths π ≡ s

, s

, ...,

where s ≡ s

, and all states s

along π, we have

(M, hs

, li) |=

[d]∼α,

40. (M, hs, li) |=

[d]∼A(αUβ) iff there is a path π ≡

, s

, ..., where s ≡ s

, and for all states s

along

π, we have (M, hs = j, li) |=

[d]∼β or ∃0 ≤ k < j

(M, hs

, li) |=

[d]∼α,

41. (M, hs, li) |=

[d]∼E(αUβ) iff for all paths π ≡

, s

, ..., where s ≡ s

, and all states s

along π, we

have (M, hs

, li) |=

[d]∼β or ∃0 ≤ k < j (M, hs

, li) |=

[d]∼α,

42. (M, hs, li) |=

[d]∼A(αRβ) iff there is a path π ≡

, s

, ..., where s ≡ s

, and for some state s

along

π, we have (M, hs

, li) |=

[d]∼β and ∀0 ≤ k < j

(M, hs

, li) |=

[d]∼α,

43. (M, hs, li) |=

[d]∼E(αRβ) iff for all paths π ≡

, s

, ..., where s ≡ s

, there is a state s

along

π such that (M, hs

, li) |=

[d]∼β and ∀0 ≤ k < j

(M, hs

, li) |=

[d]∼α,

44. (M, hs, li) |=

∼[d]∼α iff (M, hs, li) |=

[d]α,

45. (M, hs, li) |=

∼[d][b]α iff (M, hs, li) |=

∼[d ; b]α,

46. (M, hs, li) |=

∼[d](α ∧ β) iff (M, hs, li) |=

∼[d]α or

(M, hs, li) |=

∼[d]β,

47. (M, hs, li) |=

∼[d](α ∨ β) iff (M, hs, li) |=

∼[d]α and

(M, hs, li) |=

∼[d]β,

48. (M, hs, li) |=

∼[d](α→β) iff (M, hs, li) 6|=

∼[d]α and

(M, hs, li) |=

∼[d]β,

49. (M, hs, li) |=

∼[d]¬α iff (M, hs, li) 6|=

∼[d]α,

50. for any k ∈ Loc, (M, hs, li) |=

∼[d][k]α

iff (M, hs, ki) |=

∼[d]α,

51. for any x ∈ [0, 1], (M, hs, li) |=

∼[d]P

≤x

α iff µ

({w ∈

Ω

| (M, hs, li) |=

∼[d]α}) > x,

52. for any x ∈ [0, 1], (M, hs, li) |=

∼[d]P

≥x

α iff µ

({w ∈

Ω

| (M, hs, li) |=

∼[d]α}) < x,

53. for any x ∈ [0, 1], (M, hs, li) |=

∼[d]P

α iff µ

({w ∈

Ω

| (M, hs, li) |=

∼[d]α}) ≥ x,

54. for any x ∈ [0, 1], (M, hs, li) |=

∼[d]P

α iff µ

({w ∈

Ω

| (M, hs, li) |=

∼

[d]α}) ≤ x,

55. (M, hs, li) |=

∼[d]AXα iff ∃s

∈ S [(s, s

) ∈ R and

(M, hs

, li) |=

∼[d]α],

ICAART 2019 - 11th International Conference on Agents and Artiﬁcial Intelligence

874

56. (M, hs, li) |=

∼[d]EXα iff ∀s

∈ S [(s, s

) ∈ R implies

(M, hs

, li) |=

∼[d]α],

57. (M, hs, li) |=

∼[d]AGα iff there is a path π ≡

, s

, ..., where s ≡ s

, and for some state s

along

π, we have (M, hs

, li) |=

∼[d]α,

58. (M, hs, li) |=

∼[d]EGα iff for all paths π ≡

, s

, ..., where s ≡ s

, there is a state s

along π

such that (M, hs

, li) |=

∼[d]α,

59. (M, hs, li) |=

∼[d]AFα iff there is a path π ≡

, s

, ..., where s ≡ s

, and for all states s

along

π, we have (M, hs

, li) |=

∼[d]α,

60. (M, hs, li) |=

∼[d]EFα iff for all paths π ≡ s

, s

, ...,

where s ≡ s

, and all states s

along π, we have

(M, hs

, li) |=

∼[d]α,

61. (M, hs, li) |=

∼[d]A(αUβ) iff there is a path π ≡

, s

, ..., where s ≡ s

, and for all states s

along

π, we have (M, hs

, li) |=

∼[d]β or ∃0 ≤ k < j

(M, hs

, li) |=

∼[d]α,

62. (M, hs, li) |=

∼[d]E(αUβ) iff for all paths π ≡

, s

, ..., where s ≡ s

, and all states s

along π, we

have (M, hs

, li) |=

∼[d]β or ∃0 ≤ k < j (M, hs

, li) |=

∼[d]α,

63. (M, hs, li) |=

∼[d]A(αRβ) iff there is a path π ≡

, s

, ..., where s ≡ s

, and for some state s

along

π, we have (M, hs

, li) |=

∼[d]β and ∀0 ≤ k < j

(M, hs

, li) |=

∼

[d]α,

64. (M, hs, li) |=

∼[d]E(αRβ) iff for all paths π ≡

, s

, ..., where s ≡ s

, there is a state s

along

π such that (M, hs

, li) |=

∼[d]β and ∀0 ≤ k < j

(M, hs

, li) |=

∼[d]α.

A formula α is valid in LIHpCTL iff

(M, hs, li) |=

α holds for any locative inconsistency-

tolerant hierarchical probabilistic model

M := (Loc, S, S

, R, µ

, L

), any s ∈ S, any l ∈ Loc,

and any locative inconsistency-tolerant hierarchical

probabilistic satisfaction relation |=

on M.

Remark 3.4.

1. LIHpCTL is regarded as an extension of the

following temporal logics: probabilistic CTL

(called pCTL) studied in (Aziz et al., 1995;

Bianco and de Alfaro, 1995), inconsistency-

tolerant CTL (also called pCTL, but different

from the above-mentioned probabilistic one) pro-

posed in (Kamide and Endo, 2018), and hierar-

chical CTL (called sCTL) proposed in (Kamide

and Yano, 2017; Kamide, 2018).

2. There are some possibilities of deﬁning a prob-

ability measure µ

. For example, two probabil-

ity measures µ

and µ

−

, which were deﬁned on

a Borel σ-algebra B

, were proposed in (Bianco

and de Alfaro, 1995). A probability measure µ

which is concerned with some discrete Markov

processes, was proposed in (Aziz et al., 1995).

3. The model checking problem for the probabilistic

CTL with the probability measures µ

and µ

−

was

shown to be decidable in (Bianco and de Alfaro,

1995). The model checking problem for the prob-

abilistic CTL with the probability measure µ

was

shown to be decidable in (Aziz et al., 1995).

4. The setting of the conditions concerning the

negated implication and negated negation in LIH-

pCTL is based on the axiom schemes ∼(α→β) ↔

¬∼α ∧ ∼β and ∼¬α ↔ ¬∼α. These axiom

schemes were originally introduced by De and

Omori in (De and Omori, 2015) wherein these are

shown to be natural and plausible from the point

of view of many-valued semantics.

5. The setting of the location operator is the same

as that of the locative paraconsistent probabilis-

tic computation tree logic (called LPpCTL) intro-

duced in (Kamide and Koizumi, 2016). The de-

cidability of the model checking problem for LP-

pCTL has not yet been proved.

6. The single-satisfaction relation |=

of LIHpCTL

is highly compatible with the standard single-

satisfaction relations of CTL. By using this satis-

faction relation, we can simply formalize and uni-

formly handle ∼ and [b].

We then have the following conjecture.

Conjecture 3.5 (Decidability).

1. (Relative decidability for LIHpCTL): If the

model-checking, validity, and satisﬁability prob-

lems for the purely probabilistic fragment of LIH-

pCTL (i.e., it is obtained from CTL by adding the

probability operators) with a certain probability

measure are decidable, then the model-checking,

validity, and satisﬁability problems for LIHpCTL

with the same probability measure as that of the

fragment are also decidable.

2. (Decidability for the probability-free fragment

of LIHpCTL): The model-checking, validity,

and satisﬁability problems for the probability-

operator-free fragment of LIHpCTL (i.e., it is ob-

tained from LIHpCTL by deleting the probability

operators) is decidable.

4 ILLUSTRATIVE EXAMPLE

In what follows, we present an illustrative example

of verifying the reasoning process behind diagnosing

multiple sclerosis (MS), a rare disease, using the pro-

posed LIHpCTL-based model checking. The aim is

to verify mission-critical clinical reasoning with dis-

ease ontology using an extended formal method, such

Towards Locative Inconsistency-tolerant Hierarchical Probabilistic CTL Model Checking: Survey and Future Work

875

[Clinically Isolated Syndrome] [Multiple Sclerosis]

[With early relapses ]

[Without early relapses ]

[Primary Progresive MS]

[Relapsing-remitting MS] [Secondary Progresive MS]

80%

20%

80%

85%

15%

Active

¬Worsening

Worsening

Stable

Active Progression

¬Active

Progression

¬Progretion Active

Stable

¬Progression

Active

Progretion

Progression

¬Active

Healthy

~Healthy

Healthy

MRI detected lesions

~Healthy

Healthy

¬ MRI detected lesions

Stable

～Healthy

Figure 1: An ontological reasoning process model for multiple sclerosis.

as that in our proposal. We consider the scenario pre-

sented in Figure 1, in which a person suffers from MS.

The cause of this disease is not clear and there is no

known cure for it. For more information about MS,

see (Wikipedia, 2018) and the references therein.

MS is more common in people who live far-

ther from the equator; in particular, it is more com-

mon in regions with northern European populations.

Thus, the location operator [l

] in LIHpCTL can be

effectively used to describe this fact. Furthermore,

MS is typically diagnosed based on presenting signs

and symptoms in conjunction with supporting medi-

cal imaging based on Magnetic Resonance Imaging

(MRI) results and laboratory testing. It can be dif-

ﬁcult to conﬁrm, especially in early stages because

the signs and symptoms can be similar to those of

other medical conditions. Thus, the inconsistency-

tolerant (paraconsistent) negation connective ∼ in

LIHpCTL can be effectively used to describe such

symptoms and a patient’s health. For example, if it

cannot be determined whether someone is healthy,

then an ambiguous concept, healthy, can be repre-

sented by asserting the inconsistent formula healthy∧

∼healthy. This is well formalized because (healthy∧

∼healthy)→⊥ is not valid in LHIpCTL.

The United States National Multiple Sclerosis So-

ciety and the Multiple Sclerosis International Fed-

eration describe four types of MS as follows: clin-

ically isolated syndrome (CIS), relapsing-remitting

MS (RRMS), primary progressive MS (PPMS), and

secondary progressive MS (SPMS). The hierarchy of

these types can be effectively described by the se-

quence modal operator [b] in LIHpCTL.

RRMS is characterized by unpredictable relapses

followed by periods of months to years of remission

with no new signs of disease activity. Deﬁcits that

occur during attacks may either resolve or leave per-

manent damage, the latter being the case in approxi-

mately 40% of attacks and being more common the

longer a person has the disease. This process de-

scribes the initial course of 80% of individuals with

MS, and it usually begins with CIS. In CIS, a person

has an attack suggestive of demyelination but does

not fulﬁll the criteria for multiple sclerosis. Of the

individuals experiencing CIS, 30–70% later develop

MS. PPMS occurs in approximately 10–20% of in-

dividuals, with no remission after the initial symp-

toms. SPMS occurs in approximately 65% of indi-

viduals with initial RRMS and involves a progressive

neurologic decline between acute attacks without any

deﬁnite periods of remission. These probabilistic phe-

nomena concerning MS can be effectively expressed

by the probabilistic operators in LIHpCTL.

We can verify the following statements using LIH-

pCTL:

1. “Is there a state in which a person is both healthy

and unhealthy with CIS?”

2. “Is there a state in which a person is both healthy

and unhealthy with RRMS or PPMS?”

The ﬁrst statement is true, while the second statement

is not. These statements are expressed as follows:

1. [Disease ; CIS]EF(healthy ∧ ∼healthy),

2. [Disease ; MS ; RRMS]EF(healthy ∧ ∼healthy) ∨

ICAART 2019 - 11th International Conference on Agents and Artiﬁcial Intelligence

876

[Disease ; MS ; PPMS]EF(healthy ∧ ∼healthy).

We can also verify the following statement using

LIHpCTL:

“If a person living in the USA has CIS and

no lesion has been detected by the MRI, then

there is an approximately 80% chance that the

person will be cured in the near future.”

This statement is true and is expressed as follows:

[USA][Disease ; CIS] (AG(∼healthy ∧

¬MRIdetectedLesion) → EF(P

≤0.85

cure ∧

≥0.75

cure)).

5 CONCLUSION

In this paper, we have introduced a new extended tem-

poral logic, LIHpCTL, to establish the logical founda-

tion of a new model checking paradigm. This model

checking paradigm is called locative inconsistency-

tolerant hierarchical probabilistic CTL model check-

ing, and is intended to effectively verify locative, in-

consistent, hierarchical, probabilistic (randomized),

and time-dependent concurrent systems. LIHpCTL

is an extension of previously proposed locative,

inconsistency-tolerant, hierarchical, and probabilistic

extensions of the standard temporal logic known as

CTL. We have also presented a survey of various stud-

ies on probabilistic, inconsistency-tolerant, and hi-

erarchical temporal logics and their applications to

model checking. Although the decidability of the

model checking problem for LIHpCTL has not yet

been determined, we have presented an illustrative

example for verifying the clinical reasoning process

for a disease, MS, using the LIHpCTL-based model

checking paradigm. We have thus shown that LIH-

pCTL and its model checking framework, which is an

extension of existing frameworks, are useful for a va-

riety of existing and novel applications in computer

science and artiﬁcial intelligence.

ACKNOWLEDGEMENTS

This research was supported by the Kayamori Foun-

dation of Informational Science Advancement. This

research has been partially supported by JSPS KAK-

ENHI Grant Numbers JP18K11171, JP16KK0007

and JSPS Core-to-Core Program (A. Advanced Re-

search Networks).

REFERENCES

Almukdad, A. and Nelson, D. (1984). Constructible falsity

and inexact predicates. Journal of Symbolic Logic,

49:231–233.

Aziz, A., Singhal, V., and Balarin, F. (1995). It usually

works: The temporal logic of stochastic systems. In

Proceedings of the 7th Int. Conf. on Computer Aided

Veriﬁcation (CAV 1995), Lecture Notes in Computer

Science 939, pages 155–165.

Baier, C., de Alfaro, L., Forejt, V., and Kwiatkowska, M.

(2018). Model Checking Probabilistic Systems, In:

Handbook of Model Checking, pp. 963-999. Springer.

Baier, C. and Kwiatkowska, M. (1998). Model checking

for a probabilistic branching time logic with fairnes.

Distributed Computing, 11:125–155.

Belnap, N. (1977a). How a computer should think, in: in:

Contemporary Aspects of Philosophy, (G. Ryle ed.),

pp. 30-56. Oriel Press, Stocksﬁeld.

Belnap, N. (1977b). A useful four-valued logic in: J.M.

Dunn and G. Epstein (eds.), Modern Uses of Multiple-

Valued Logic, pp. 5-37. Reidel, Dordrecht.

Bianco, A. and de Alfaro, L. (1995). Model checking of

probabilistic and nondeterministic systems. In Pro-

ceedings of the 15th Conf. on Foundations of Soft-

ware Technology and Theoretical Computer Science

(FSTTCS 1995), Lecture Notes in Computer Science

1026, pages 499–513.

Carnielli, W., Coniglio, M., Gabbay, D., Gouveia, P., and

Sernadas, C. (2008). Analysis and synthesis of log-

ics: How to cut and paste reasoning systems, Applied

Logic Series, Vol. 35. Springer.

Chen, D. and Wu, J. (2006). Reasoning about inconsistent

concurrent systems: A non-classical temporal logic.

In Lecture Notes in Computer Science, volume 3831,

pages 207–217.

Clarke, E. and Emerson, E. (1981). Design and synthesis of

synchronization skeletons using branching time tem-

poral logic. In Lecture Notes in Computer Science,

volume 131, pages 52–71.

Clarke, E., Grumberg, O., and Peled, D. (1999). Model

checking. The MIT Press.

Clarke, E., Henzinger, T., Veith, H., and Bloem, R. (2018).

Handbook of Model Checking. Springer.

da Costa, N., Beziau, J., and Bueno, O. (1995). Aspects of

paraconsistent logic. Bulletin of the IGPL, 3 (4):597–

614.

De, M. and Omori, H. (2015). Classical negation and ex-

pansions of belnap–dunn logic. Studia Logica, 103

(4):825–851.

Dunn, J. (1976). Intuitive semantics for ﬁrst-degree entail-

ment and ‘coupled trees’. Philosophical Studies, 29

(3):149–168.

Easterbrook, S. and Chechik, M. (2001). A framework for

multi-valued reasoning over inconsistent viewpoints.

In Proceedings of the 23rd International Conference

on Software Engineering (ICSE 2001), pages 411–

420.

Towards Locative Inconsistency-tolerant Hierarchical Probabilistic CTL Model Checking: Survey and Future Work

877

Emerson, E. and Halpern, J. (1986). ‘sometimes’ and ‘not

never’ revisited: on branching versus linear time tem-

poral logic. Journal of the ACM, 33 (1):151–178.

Emerson, E. and Sistla, A. (1984). Deciding full branching

time logic. Information and Control, 61:175–201.

Hanson, H. (1994). Time and probability in formal design of

distributed systems, real-time safety critical systems.

Elsevier.

Hansson, H. and Jonsson, B. (1994). A logic for reasoning

about time and reliability. Formal Aspects of Comput-

ing, 6 (5):512–535.

Holzmann, G. (2006). The SPIN model checker: Primer

and reference manual. Addison-Wesley.

Kamide, N. (2005). A spatial modal logic with a loca-

tion interpretation. Mathematical Logic Quarterly, 51

(4):331–341.

Kamide, N. (2006a). Extended full computation-tree logics

for paraconsistent model checking. Logic and Logical

Philosophy, 15 (3):251–276.

Kamide, N. (2006b). Linear and afﬁne logics with temporal,

spatial and epistemic operators. Theoretical Computer

Science, 353 (1-3):165–207.

Kamide, N. (2015). Inconsistency-tolerant temporal rea-

soning with hierarchical information. Information Sci-

ences, 320:140–155.

Kamide, N. (2018). Logical foundations of hierarchical

model checking. Data Technologies and Applications,

52 (4):539–563.

Kamide, N. and Endo, K. (2018). Logics and transla-

tions for inconsistency-tolerant model checking. In

Proceedings of the 10th International Conference on

Agents and Artiﬁcial Intelligence (ICAART 2018),

volume 2, pages 191–200.

Kamide, N. and Kaneiwa, K. (2009). Extended full

computation-tree logic with sequence modal operator:

Representing hierarchical tree structures. Proceedings

of the 22nd Australasian Joint Conference on Artiﬁ-

cial Intelligence (AI’09), Lecture Notes in Artiﬁcial

Intelligence, 5866:485–494.

Kamide, N. and Kaneiwa, K. (2010). Paraconsistent nega-

tion and classical negation in computation tree logic.

In Proceedings of the 2nd International Conference

on Agents and Artiﬁcial Intelligence (ICAART 2010),

volume 1, pages 464–469.

Kamide, N. and Koizumi, D. (2015). Combining paracon-

sistency and probability in ctl. Proceedings of the 7th

International Conference on Agents and Artiﬁcial In-

telligence (ICAART 2015), 2:285–293.

Kamide, N. and Koizumi, D. (2016). Method for combin-

ing paraconsistency and probability in temporal rea-

soning. Journal of Advanced Computational Intelli-

gence and Intelligent Informatics, 20:813–827.

Kamide, N. and Wansing, H. (2011). A paraconsistent

linear-time temporal logic. Fundamenta Informaticae,

106 (1):1–23.

Kamide, N. and Yano, R. (2017). Logics and translations

for hierarchical model checking. Proceedings of the

21st International Conference on Knowledge-Based

and Intelligent Information and Engineering Systems

(KES2017), Procedia Computer Science, 112:31–40.

Kaneiwa, K. and Kamide, N. (2010). Sequence-indexed

linear-time temporal logic: Proof system and applica-

tion. Applied Artiﬁcial Intelligence, 24 (10):896–913.

Kaneiwa, K. and Kamide, N. (2011a). Conceptual modeling

in full computation-tree logic with sequence modal

operator. International Journal of Intelligent Systems,

26 (7):636–651.

Kaneiwa, K. and Kamide, N. (2011b). Paraconsistent com-

putation tree logic. New Generation Computing, 29

(4):391–408.

Kwiatkowska, M., Norman, G., and Parker, D. (2011).

Prism 4.0: Veriﬁcation of probabilistic real-time sys-

tems. Proceedings of the 23rd International Confer-

ence on Computer Aided Veriﬁcation (CAV 11), Lec-

ture Notes in Computer Science, 6806:585–591.

N. Kobayashi, T. S. and Yonezawa, A. (1999). Distributed

concurrent linear logic programming. Theoretical

Computer Science, 227:185–220.

Nelson, D. (1949). Constructible falsity. Journal of Sym-

bolic Logic, 14:16–26.

Ognjanovic, Z., Doder, D., and Markovic, Z. (2011). A

branching time logic with two types of probability op-

erators. Proceedings of the 5th International Confer-

ence on Scalable Uncertainty Management, Lecture

Notes in Artiﬁcial Intelligence, 6929:219–232.

Ognjanovic, Z., Markovic, Z., Raskovic, M., Doder, D., and

Perovic, A. (2012). A propositional probabilistic logic

with discrete linear time for reasoning about evidence.

Annals of Mathematics and Artiﬁcial Intelligence, 65

(2-3):217–243.

Pnueli, A. (1977). The temporal logic of programs. In Pro-

ceedings of the 18th IEEE Symposium on Foundations

of Computer Science, pages 46–57.

Priest, G. (2002). Paraconsistent logic, Handbook of Philo-

sophical Logic (Second Edition), Vol. 6, D. Gabbay

and F. Guenthner (eds.). Kluwer Academic Publish-

ers, Dordrecht, pp. 287-393.

Wansing, H. (1993). The logic of information structures.

In Lecture Notes in Computer Science, volume 681,

pages 1–163.

Wikipedia (2018). Multiple sclerosis. Wikipedia,

https://en.wikipedia.org/wiki/Multiple_sclerosis.

ICAART 2019 - 11th International Conference on Agents and Artiﬁcial Intelligence

878