Inconsistency-tolerant Hierarchical Probabilistic Computation Tree

Logic and Its Application to Model Checking

Norihiro Kamide

and Noriko Yamamoto

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

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

Teikyo University, Graduate School of Science and Engineering, Division of Information Science,

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

Keywords:

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

Probabilistic Model Checking, Inconsistency-tolerant Model Checking, Hierarchical Model Checking.

Abstract:

An inconsistency-tolerant hierarchical probabilistic computation tree logic (IHpCTL) is developed to establish

a new extended model checking paradigm referred to as IHpCTL model checking, which is intended to verify

randomized, open, large, and complex concurrent systems. The proposed IHpCTL is constructed based on

several previously established extensions of the standard probabilistic temporal logic known as probabilistic

computation tree logic (pCTL), which is widely used for probabilistic model checking. IHpCTL is shown to

be embeddable into pCTL and is relatively decidable with respect to pCTL. This means that the decidability of

pCTL with certain probability measures implies the decidability of IHpCTL. The results indicate that we can

effectively reuse the previously proposed pCTL model-checking algorithms for IHpCTL model checking.

1 INTRODUCTION

Model Checking: is a computer-assisted method used

to verify concurrent systems that can be modeled by

state-transition systems (Clarke and Emerson, 1981;

Clarke et al., 1999; Holzmann, 2006; Clarke et al.,

2018). The aim of this study is to develop a new tem-

poral logic that can establish a logical foundation of

extended model checking to verify randomized, open,

large, and complex concurrent systems. To develop

this type of temporal logic, combining and integrat-

ing probabilistic, inconsistency-tolerant, and hierar-

chical reasoning mechanisms into a single logic are

required. The reasons for these requirements are as

follows: (1) verifying randomized concurrent systems

(e.g., fault-tolerant communication systems over un-

reliable channels) requires the handling of probabilis-

tic reasoning (Bianco and de Alfaro, 1995), (2) ver-

ifying open and large concurrent systems (e.g., web

and cloud application systems) requires the handling

of inconsistency-tolerant reasoning (Chen and Wu,

2006), and (3) verifying complex concurrent systems

(e.g., web systems with wide tree structures) requires

the handling of hierarchical reasoning (Kaneiwa and

Kamide, 2011a).

To develop this type of logic , we combine and

integrate the following useful non-classical logics:

temporal logics, probabilistic (or probability) logics,

inconsistency-tolerant (or paraconsistent) logics, and

hierarchical (or sequential) logics. By combining and

integrating these non-classical logics, we can extend

and reﬁne the previously established model-checking

frameworks (Clarke and Emerson, 1981; Clarke et al.,

1999; Holzmann, 2006; Clarke et al., 2018), which

are well-known as formal and automated techniques

for verifying concurrent systems. Model checking has

been extended to probabilistic model checking (Aziz

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). Therefore, by developing this combined and

integrated logic, we can combine and integrate these

extended model-checking frameworks.

In this study, we develop a new combined and in-

tegrated computation tree logic called inconsistency-

tolerant hierarchical probabilistic computation tree

logic (IHpCTL). This IHpCTL is developed to es-

tablish a new extended model checking paradigm re-

ferred to as IHpCTL model checking, which is in-

490

Kamide, N. and Yamamoto, N.

Inconsistency-tolerant Hierarchical Probabilistic Computation Tree Logic and Its Application to Model Checking.

DOI: 10.5220/0010181604900499

In Proceedings of the 13th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2021) - Volume 2, pages 490-499

ISBN: 978-989-758-484-8

tended to verify randomized, open, large, and com-

plex concurrent systems, including clinical reasoning

systems. We construct IHpCTL by combining and

integrating several previously established extensions

of the standard probabilistic temporal logic known

as probabilistic computation tree logic (pCTL) (Aziz

et al., 1995; Bianco and de Alfaro, 1995), which is

widely used for probabilistic model checking. As a

main contribution of this study, IHpCTL is shown

to be embeddable into pCTL and is relatively de-

cidable with respect to pCTL. This means that the

decidability of pCTL with certain probability mea-

sures implies the decidability of IHpCTL. These re-

sults indicate that we can effectively reuse the pre-

viously proposed pCTL model-checking algorithms

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

IHpCTL model checking.

We next explain pCTL and its probabilistic model-

checking framework. pCTL is an extension of the

standard temporal logic known as computation tree

logic (CTL) (Clarke and Emerson, 1981) for model

checking. It is obtained from CTL by adding the

probabilistic or probability operator P

≥x

. The formu-

las in the form of P

≥x

α are intended to be read as “the

probability of α holding in the future evolution of the

system is at least x.” pCTL was previously investi-

gated by Aziz et al. (Aziz et al., 1995) and Bianco and

de Alfaro (Bianco and de Alfaro, 1995). In (Bianco

and de Alfaro, 1995), pCTL was introduced to verify

the reliability properties and performances of the sys-

tems modeled by discrete Markov chains. In (Bianco

and de Alfaro, 1995), the complexities of model-

checking algorithms with respect to this logic were

clariﬁed. In (Aziz et al., 1995), model-checking algo-

rithms for various extensions of the previous settings

of pCTL were proposed to verify probabilistic non-

deterministic concurrent systems. These algorithms

were shown to exhibit polynomial-time complexity

depending on the different sizes of the systems. The

main difference between the approaches of Aziz et al.

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

and de Alfaro, 1995) is the settings of the probability

measures in the probabilistic Kripke models of pCTL.

Although, as previously mentioned, pCTL and its

probabilistic model-checking framework are useful,

they are insufﬁcient for handling open, large, and

complex concurrent systems such as very large and

complex cloud-based systems. Verifying these sys-

tems requires the handling of inconsistency-tolerant

reasoning. This is because in open and large concur-

rent systems, inconsistencies are inevitable and ap-

pear often (Chen and Wu, 2006). Verifying these sys-

tems also requires the handling of hierarchical reason-

ing, as complex concurrent systems are constructed

based on certain hierarchies (Kaneiwa and Kamide,

2011a). In addition, verifying clinical reasoning sys-

tems with complex disease ontologies, for example,

requires the handling of both inconsistency-tolerant

and hierarchical reasoning, as these types of systems

consist of both open data related to vague concepts

of symptoms and complex hierarchical structures of

disease ontologies (Kamide and Bernal J.P.A., 2019).

Thus, an extended logic with an extended model-

checking framework is needed that can also simul-

taneously handle inconsistency-tolerant, hierarchical,

and probabilistic reasoning.

For this direction, a few partial solutions were

obtained in some previous studies (Kamide and

Koizumi, 2015; Kamide and Koizumi, 2016; Kamide

and Yano, 2019; Kamide and Bernal J.P.A., 2019). An

inconsistency-tolerant (or paraconsistent) probabilis-

tic computation tree logic (PpCTL), which was ob-

tained from pCTL by adding the paraconsistent nega-

tion connective ∼, was developed in (Kamide and

Koizumi, 2015; Kamide and Koizumi, 2016) based

on a probability-measure-independent translation of

PpCTL to pCTL. A theorem for embedding PpCTL

into pCTL was proved using this translation and en-

tailed the relative decidability of PpCTL with respect

to pCTL. A hierarchical probabilistic computation

tree logic (HpCTL), which was obtained from pCTL

by adding the hierarchical (or sequence) modal oper-

ator [b], was developed in (Kamide and Yano, 2019)

based on a probability-measure-independent trans-

lation of HpCTL to pCTL. The same theorems as

those for PpCTL were obtained for HpCTL. A loca-

tive inconsistency-tolerant hierarchical probabilistic

computation tree logic (LIHpCTL), which is regarded

as an extension of both PpCTL and HpCTL with the

addition of the location operator [l

] introduced in

(N. Kobayashi and Yonezawa, 1999), was considered

in (Kamide and Bernal J.P.A., 2019).

However, the embedding and relative decidabil-

ity theorems for LIHpCTL proposed in (Kamide and

Bernal J.P.A., 2019) have not yet been proved, as

some technical difﬁculties remain. Thus, the objec-

tive of this study is to make progress in this direc-

tion. The current study proves the embedding and

relative decidability theorems for the proposed logic

IHpCTL, which is considered to be a modiﬁed ver-

sion of the location-operator-free subsystem of LIH-

pCTL. To prove these theorems, we need to overcome

some technical difﬁculties in formalizing and deﬁn-

ing a satisfaction relation and proving some key lem-

mas for the embedding theorem. For example, some

previously proposed extended CTLs with the hierar-

chical modal operator [b] (see, for example, (Kamide

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

Inconsistency-tolerant Hierarchical Probabilistic Computation Tree Logic and Its Application to Model Checking

491

Kaneiwa and Kamide, 2010; Kamide, 2015)) were

shown to have complex multiple sequence-indexed

satisfaction relations |=

, where

d represents se-

quences. The proposed IHpCTL has a simple sin-

gle satisfaction relation |=

, which is compatible with

the standard single satisfaction relation of CTL. Using

this simple satisfaction relation, we can naturally for-

malize the interaction between [b] and ∼ in IHpCTL,

and through this natural formulation, we can prove

the required embedding theorem. However, a careful

treatment of the interaction of [b] and ∼ is required to

prove some key lemmas of the embedding theorem.

This rigorous treatment represents a technical contri-

bution of this study.

The remainder of this paper is organized as fol-

lows. In Section 2, we introduce the logic IHpCTL

and introduce some basic propositions for IHpCTL.

In Section 3, we prove the theorems for embedding

IHpCTL into HpCTL and pCTL, and using these em-

bedding theorems, we prove the theorems for relative

decidabilities for IHpCTL with respect to HpCTL and

pCTL. In Section 4, we conclude our study and ad-

dress some illustrative examples.

2 LOGIC

Formulas of inconsistency-tolerant hierarchical prob-

abilistic computation tree logic (IHpCTL) are con-

structed from countably many propositional variables

by → (implication), ∧ (conjunction), ∨ (disjunction),

¬ (classical negation), ∼ (paraconsistent negation),

X (next time), G (globally in the future), F (even-

tually in the future), 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), P

(less than probability), P

(greater than probability), and [b] (hierarchical or se-

quence modal operator) where b is a sequence. Se-

quences are constructed from countably many atomic

sequences and

0 (empty sequence) by ; (composition).

The symbols X, G, F, U, and R are called tempo-

ral operators, the symbols A and E are called path

quantiﬁers, and the symbols P

≤x

, P

≥x

, P

, and P

are called probabilistic or probability operators. We

use lower-case letters p, q,r, ... to denote propositional

variables, Greek small letters α, β,γ, ... to denote for-

mulas, and lower-case letters b,c, d, ... to denote se-

quences. we use an expression α ↔ β to denote the

formula (α→β)∧(β→α) and an expression A ≡ B to

denote the syntactical identity between A and B. An

expression [

0]α means α, and expressions [

0 ; b]α and

[b ;

0]α mean [b]α. We use the symbol SE to de-

note the set of all sequences (including the empty se-

quence

0) and the symbol ω to denote the set of all

natural numbers. Furthermore, we use the symbol Φ

to denote a non-empty set of propositional variables,

the symbol Φ

to denote the set {p

| p ∈ Φ} of new

propositional variable, the symbol Φ

∼

is used to de-

note the set {∼p | p ∈ Φ}, the symbol Φ

[d]

to denote

the set {[d]p | p ∈ Φ}, the symbol Φ

0[d]

to denote the

set {[d]γ | γ ∈ Φ ∪ Φ

}, the symbol Φ

[d]

to denote the

set {γ

| γ ∈ Φ ∪ Φ

[d]

}, the symbol Φ

∼[d]

to denote

the set {[d]γ | γ ∈ Φ ∪ Φ

∼

}, and the symbol Φ

[d]∼

denote the set {∼γ | γ ∈ Φ∪Φ

[d]

}. We assume the fol-

lowing commutativity condition: For any p ∈ Φ and

any d ∈ SE, ([d]p)

= [d](p

) (i.e., it can simply be

denoted as [d]p

). Then, we have the following fact

by this commutativity condition: Φ

0[d]

= Φ

[d]

Deﬁnition 2.1. Let x be in [0, 1]. Formulas α and

sequences b of IHpCTL are deﬁned by the following

grammar, assuming p and e represent propositional

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 | [b]α.

b ::= e |

0 | b ; b.

We use an expression

[d] to denote [d

][d

]· ··[d

] with

i ∈ ω, d

∈ SE and d

≡

0. The expression [d] can be

the empty sequence and is not uniquely determined.

For example, if d ≡ d

; d

where d

, d

and

are atomic sequences, then [d] means [d

][d

; d

][d

], [d

][d

; d

] or [d

; d

]. Note that

[d]

can be [d] (i.e., [d] includes [d]).

Remark 2.2. We make the following remarks.

1. The inconsistency-tolerant negation connective

∼ characterizes inconsistency-tolerant logics

(also referred to as paraconsistent logics) (Priest,

2002; da Costa et al., 1995) that reject the law

(α∧∼α)→β of explosion. In comparison with

other logics, inconsistency-tolerant logics can

be used in inconsistency-tolerant reasoning. For

example, the following scenario is undesirable

in a realistic situation. The formulas of the

form (s(x)∧∼s(x))→d(x) are valid for any

symptom s and disease d, where ∼s(x) means

that “a person x does not have a symptom

s” and d(x) means that “a person x suffers

from a disease d.” The scenario described

as melancholia( john)∧∼melancholia( john)

will naturally emerge from the vague def-

inition of melancholia (i.e., the statement

“John has melancholia” may be judged

as true or false based on the perception

of different doctors or pathologists). In

ICAART 2021 - 13th International Conference on Agents and Artiﬁcial Intelligence

492

this case, the formula (melancholia( john)∧

∼melancholia( john))→cancer( john) is valid in

non-inconsistency-tolerant logics, but invalid in

inconsistency-tolerant logics.

2. The hierarchical modal operator [b] can be used

to represent the concepts of hierarchical informa-

tion in the following manner: a sequence struc-

ture produces a monoid hM, ;,

0i with the follow-

ing informational interpretation (Wansing, 1993):

(1) M is a set of pieces of ordered information

(i.e., a set of sequences); (2) ‘;’ is a binary op-

erator (on M) that combines two pieces of infor-

mation (i.e., it is a concatenation operator on se-

quences); (3)

0 is an empty piece of information

(i.e., an empty sequence). Then, formulas of 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).

We deﬁne the logic IHpCTL as follows.

Deﬁnition 2.3 (IHpCTL). A structure (S,S

,R, µ

)

is an inconsistency-tolerant hierarchical probabilistic

model iff

1. S is the set of states,

2. S

is a set of initial states and S

⊆ S,

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

dition:

∀s ∈ S ∃s

∈ S [(s,s

) ∈ R],

4. 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] (i.e., any sets of paths start-

ing from s are measurable),

5. L

is a mapping from S to the power set of

[

d∈SE

∼[d]

A path in an inconsistency-tolerant hierarchical

probabilistic model is an inﬁnite sequence of states,

π = s

,... such that ∀i ≥ 0 [(s

i+1

) ∈ R]. We

use the symbol Ω

to denote the set of all paths begin-

ning at s.

An inconsistency-tolerant hierarchical probabilis-

tic satisfaction relation (M, s) |=

α for any formula

α, where M is an inconsistency-tolerant hierarchical

probabilistic model (S, S

,R, µ

), and s represents

a state in S, is deﬁned by the following clauses:

1. for any γ ∈ Φ

∼[d]

, (M, s) |=

γ iff γ ∈ L

(s),

2. for any p ∈ Φ, (M, s) |=

[d]∼p iff (M,s) |=

∼[d]p,

3. (M, s) |=

[d][b]α iff (M,s) |=

[d ; b]α,

4. (M, s) |=

[d](α∧β) iff (M, s) |=

[d]α and (M,s) |=

[d]β,

5. (M, s) |=

[d](α∨β) iff (M, s) |=

[d]α or (M,s) |=

[d]β,

6. (M, s) |=

[d](α→β) iff (M, s) |=

[d]α implies

(M, s) |=

[d]β,

7. (M, s) |=

[d]¬α iff (M,s) 6|=

[d]α,

8. for any x ∈ [0,1], (M, s) |=

[d]P

≤x

α iff

({w ∈ Ω

| (M, s) |=

[d]α}) ≤ x,

9. for any x ∈ [0,1], (M, s) |=

[d]P

≥x

α iff

({w ∈ Ω

| (M, s) |=

[d]α}) ≥ x,

10. for any x ∈ [0,1], (M, s) |=

[d]P

α iff

({w ∈ Ω

| (M, s) |=

[d]α}) < x,

11. for any x ∈ [0,1], (M, s) |=

[d]P

α iff

({w ∈ Ω

| (M, s) |=

[d]α}) > x,

12. (M, s) |=

[d]AXα iff ∀s

∈ S [(s, s

) ∈ R implies

(M, s

) |=

[d]α],

13. (M, s) |=

[d]EXα iff ∃s

∈ S [(s,s

) ∈ R and

(M, s

) |=

[d]α],

14. (M, s) |=

[d]AGα iff for all paths π ≡ s

,...,

where s ≡ s

, and all states s

along π, we have

(M, s

) |=

[d]α,

15. (M, s) |=

[d]EGα iff there is a path π ≡ s

,...,

where s ≡ s

, and for all states s

along π, we have

(M, s

) |=

[d]α,

16. (M, s) |=

[d]AFα iff for all paths π ≡ s

,...,

where s ≡ s

, there is a state s

along π such that

(M, s

) |=

[d]α,

17. (M, s) |=

[d]EFα iff there is a path π ≡ s

,...,

where s ≡ s

, and for some state s

along π, we have

(M, s

) |=

[d]α,

18. (M, s) |=

[d]A(αUβ) iff for all paths π ≡ s

,...,

where s ≡ s

, there is a state s

along π such that

(M, s

) |=

[d]β and ∀0 ≤ k < j (M, s

) |=

[d]α,

19. (M, s) |=

[d]E(αUβ) iff there is a path π ≡ s

,...,

where s ≡ s

, and for some state s

along π, we have

(M, s

) |=

[d]β and ∀0 ≤ k < j (M, s

) |=

[d]α,

20. (M, s) |=

[d]A(αRβ) iff for all paths π ≡ s

,...,

where s ≡ s

, and all states s

along π, we have

(M, s

) |=

[d]β or ∃0 ≤ k < j (M, s

) |=

[d]α,

21. (M, s) |=

[d]E(αRβ) iff there is a path π ≡ s

,...,

where s ≡ s

, and for all states s

along π, we have

(M, s

) |=

[d]β or ∃0 ≤ k < j (M, s

) |=

[d]α,

22. (M, s) |=

[d]∼∼α iff (M,s) |=

[d]α,

23. (M, s) |=

[d]∼[b]α iff (M,s) |=

[d ; b]∼α,

24. (M, s) |=

[d]∼(α∧β) iff (M, s) |=

[d]∼α or (M, s) |=

[d]∼β,

25. (M, s) |=

[d]∼(α∨β) iff (M,s) |=

[d]∼α and

(M, s) |=

[d]∼β,

26. (M, s) |=

[d]∼(α→β) iff (M, s) 6|=

[d]∼α and

(M, s) |=

[d]∼β,

27. (M, s) |=

[d]∼¬α iff (M,s) 6|=

[d]∼α,

Inconsistency-tolerant Hierarchical Probabilistic Computation Tree Logic and Its Application to Model Checking

493

28. for any x ∈ [0, 1], (M, s) |=

[d]∼P

≤x

α iff

({w ∈ Ω

| (M, s) |=

[d]∼α}) > x,

29. for any x ∈ [0, 1], (M, s) |=

[d]∼P

≥x

α iff

({w ∈ Ω

| (M, s) |=

[d]∼α}) < x,

30. for any x ∈ [0, 1], (M, s) |=

[d]∼P

α iff

({w ∈ Ω

| (M, s) |=

[d]∼α}) ≥ x,

31. for any x ∈ [0, 1], (M, s) |=

[d]∼P

α iff

({w ∈ Ω

| (M, s) |=

[d]∼α}) ≤ x,

32. (M, s) |=

[d]∼AXα iff ∃s

∈ S [(s,s

) ∈ R and

(M, s

) |=

[d]∼α],

33. (M, s) |=

[d]∼EXα iff ∀s

∈ S [(s, s

) ∈ R implies

(M, s

) |=

[d]∼α],

34. (M, s) |=

[d]∼AGα iff there is a path π ≡ s

,...,

where s ≡ s

, and for some state s

along π, we have

(M, s

) |=

[d]∼α,

35. (M, s) |=

[d]∼EGα iff for all paths π ≡ s

,...,

where s ≡ s

, there is a state s

along π such that

(M, s

) |=

[d]∼α,

36. (M, s) |=

[d]∼AFα iff there is a path π ≡ s

,...,

where s ≡ s

, and for all states s

along π, we have

(M, s

) |=

[d]∼α,

37. (M, s) |=

[d]∼EFα iff for all paths π ≡ s

,...,

where s ≡ s

, and all states s

along π, we have

(M, s

) |=

[d]∼α,

38. (M, s) |=

[d]∼A(αUβ) iff there is a path π ≡

,..., where s ≡ s

, and for all states s

along

π, we have (M, s

) |=

[d]∼β or ∃0 ≤ k < j (M, s

) |=

[d]∼α,

39. (M, s) |=

[d]∼E(αUβ) iff for all paths π ≡ s

,...,

where s ≡ s

, and all states s

along π, we have

(M, s

) |=

[d]∼β or ∃0 ≤ k < j (M, s

) |=

[d]∼α,

40. (M, s) |=

[d]∼A(αRβ) iff there is a path π ≡

,..., where s ≡ s

, and for some state s

along

π, we have (M,s

) |=

[d]∼β and ∀0 ≤ k < j (M,s

) |=

[d]∼α,

41. (M, s) |=

[d]∼E(αRβ) iff for all paths π ≡ s

,...,

where s ≡ s

, there is a state s

along π such that

(M, s

) |=

[d]∼β and ∀0 ≤ k < j (M, s

) |=

[d]∼α,

42. (M, s) |=

∼[d]∼α iff (M,s) |=

[d]α,

43. (M, s) |=

∼[d][b]α iff (M,s) |=

∼[d ; b]α,

44. (M, s) |=

∼[d](α∧β) iff (M, s) |=

∼[d]α or (M, s) |=

∼[d]β,

45. (M, s) |=

∼[d](α∨β) iff (M,s) |=

∼[d]α and

(M, s) |=

∼[d]β,

46. (M, s) |=

∼[d](α→β) iff (M, s) 6|=

∼[d]α and

(M, s) |=

∼[d]β,

47. (M, s) |=

∼[d]¬α iff (M,s) 6|=

∼[d]α,

48. for any x ∈ [0, 1], (M, s) |=

∼[d]P

≤x

α iff

({w ∈ Ω

| (M, s) |=

∼[d]α}) > x,

49. for any x ∈ [0, 1], (M, s) |=

∼[d]P

≥x

α iff

({w ∈ Ω

| (M, s) |=

∼[d]α}) < x,

50. for any x ∈ [0, 1], (M, s) |=

∼[d]P

α iff

({w ∈ Ω

| (M, s) |=

∼[d]α}) ≥ x,

51. for any x ∈ [0, 1], (M, s) |=

∼[d]P

α iff

({w ∈ Ω

| (M, s) |=

∼[d]α}) ≤ x,

52. (M, s) |=

∼[d]AXα iff ∃s

∈ S [(s,s

) ∈ R and

(M, s

) |=

∼[d]α],

53. (M, s) |=

∼[d]EXα iff ∀s

∈ S [(s, s

) ∈ R implies

(M, s

) |=

∼[d]α],

54. (M, s) |=

∼[d]AGα iff there is a path π ≡ s

,...,

where s ≡ s

, and for some state s

along π, we have

(M, s

) |=

∼[d]α,

55. (M, s) |=

∼[d]EGα iff for all paths π ≡ s

,...,

where s ≡ s

, there is a state s

along π such that

(M, s

) |=

∼[d]α,

56. (M, s) |=

∼[d]AFα iff there is a path π ≡ s

,...,

where s ≡ s

, and for all states s

along π, we have

(M, s

) |=

∼[d]α,

57. (M, s) |=

∼[d]EFα iff for all paths π ≡ s

,...,

where s ≡ s

, and all states s

along π, we have

(M, s

) |=

∼[d]α,

58. (M, s) |=

∼[d]A(αUβ) iff there is a path π ≡

,..., where s ≡ s

, and for all states s

along

π, we have (M, s

) |=

∼[d]β or ∃0 ≤ k < j (M, s

) |=

∼[d]α,

59. (M, s) |=

∼[d]E(αUβ) iff for all paths π ≡ s

,...,

where s ≡ s

, and all states s

along π, we have

(M, s

) |=

∼[d]β or ∃0 ≤ k < j (M, s

) |=

∼[d]α,

60. (M, s) |=

∼[d]A(αRβ) iff there is a path π ≡

,..., where s ≡ s

, and for some state s

along

π, we have (M,s

) |=

∼

[d]β and ∀0 ≤ k < j (M, s

) |=

∼[d]α,

61. (M, s) |=

∼[d]E(αRβ) iff for all paths π ≡ s

,...,

where s ≡ s

, there is a state s

along π such that

(M, s

) |=

∼[d]β and ∀0 ≤ k < j (M, s

) |=

∼[d]α.

A formula α is valid in IHpCTL iff (M,s) |=

α holds for any inconsistency-tolerant hierarchical

probabilistic model M := (S,S

,R, µ

), any s ∈ S,

and any inconsistency-tolerant hierarchical proba-

bilistic satisfaction relation |=

on M.

Deﬁnition 2.4. Let M be an inconsistency-

tolerant hierarchical probabilistic model

M := (S, S

,R, µ

) for IHpCTL, and |=

an inconsistency-tolerant hierarchical probabilistic

satisfaction relation on M. Then, the model checking

problem for IHpCTL is deﬁned as follows. For any

formula α, ﬁnd the set {s ∈ S | M, s |=

α}.

Remark 2.5. We make the following remarks.

1. The logic IHpCTL is an extension of the following

temporal logics: Probabilistic computation tree

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

and de Alfaro, 1995), inconsistency-tolerant com-

putation tree logic (also referred to as pCTL,

ICAART 2021 - 13th International Conference on Agents and Artiﬁcial Intelligence

494

but different from the aforementioned probabilis-

tic one) proposed in (Kamide and Endo, 2018),

and hierarchical computation tree logic (sCTL)

proposed in (Kamide and Yano, 2017; Kamide,

2018).

2. The deﬁnition of µ

is not precisely and explic-

itly given in this study, because (1) the proposed

translation from IHpCTL to HpCTL is indepen-

dent of the setting of µ

and (2) there are some

different ways of deﬁning µ

3. There are some ways of deﬁning a probability

measure µ

. For example, two probability mea-

sures µ

and µ

−

, which were deﬁned on a Borel σ-

algebra B

(⊆ 2

Ω

), were proposed in (Bianco and

de Alfaro, 1995) for pCTL. A probability mea-

sure µ

, which is concerned with certain discrete

Markov processes, was proposed in (Aziz et al.,

1995) for pCTL.

4. The setting of the conditions concerning the

negated implication and negated negation in IH-

pCTL adopts 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).

5. The single-satisfaction relation |=

of IHpCTL is

compatible with the standard single-satisfaction

relation of CTL. By using this satisfaction rela-

tion, we can simply and uniformly handle both ∼

and [b].

Proposition 2.6. Let M be an inconsistency-tolerant

hierarchical probabilistic model (S,S

,R, µ

), and

let s be a state in S, Then, the following clauses hold

for IHpCTL: For any formula α and any b, c, d ∈ SE,

1. (M, s) |=

[b][c]α iff (M, s) |=

[b ; c]α,

2. (M, s) |=

[d]α iff (M, s) |=

[d]α,

3. (M, s) |=

[d]∼α iff (M, s) |=

∼[d]α.

Proof. By induction on α.

3 EMBEDDABILITY AND

RELATIVE DECIDABILITY

In order to prove the relative decidability theorem for

IHpCTL with respect to pCTL, we need a theorem

for embedding IHpCTL into the ∼-free part HpCTL

of IHpCTL. By combining this theorem for embed-

ding IHpCTL into HpCTL and the previously proved

theorem in (Kamide and Yano, 2019) for embedding

HpCTL into pCTL, we can obtain the relative decid-

ability of IHpCTL with respect to pCTL. Thus, we

introduce HpCTL below. The language and formu-

las of HpCTL are respectively obtained from those of

IHpCTL by deleting ∼.

Deﬁnition 3.1 (HpCTL). A structure (S, S

,R, µ

,L)

is a hierarchical probabilistic model iff S, S

, R, and

are the same as those in Deﬁnition 2.3, and

L is a mapping from S to the power set of

[

d∈SE

[d]

A path in a hierarchical probabilistic model is deﬁned

in a similar way as in Deﬁnition 2.3.

A hierarchical probabilistic satisfaction relation

(M, s) |= α for any formula α, where M is a hierar-

chical probabilistic model (S,S

,R, µ

,L) and s repre-

sents a state in S, is deﬁned inductively by the same

clauses 3 – 21 in Deﬁnition 2.3 (but |=

in Deﬁni-

tion 2.3 should be replaced with |=) and the following

clause:

For any p ∈ Φ, (M, s) |= [d]p iff [d]p ∈ L(s).

A formula α is valid in HpCTL iff (M,s) |= α

holds for any hierarchical probabilistic model M :=

(S, S

,R, µ

,L), any s ∈ S, and any hierarchical prob-

abilistic satisfaction relation |= on M.

We deﬁne the logic pCTL, which was originally

studied in (Aziz et al., 1995; Bianco and de Alfaro,

1995).

Deﬁnition 3.2 (pCTL). The logic pCTL is deﬁned as

the [b]-free part of HpCTL (i.e., it is obtained from

HpCTL by replacing the sequences d and b with the

empty sequence

0).

We deﬁne a translation from IHpCTL to HpCTL.

Deﬁnition 3.3. The language L

(the set of formulas)

of IHpCTL is deﬁned using Φ, ∧, ∨, →,¬, ∼, X, G,

F, U, R, A, E, P

≤x

≥x

, and [b]. The lan-

guage L of HpCTL is obtained from L

by adding Φ

and deleting ∼. A mapping f from L

to L is deﬁned

inductively by:

1. for any p ∈ Φ, f (p) := p and f (∼p) := p

∈ Φ

2. f (α ] β) := f (α) ] f (β) where ] ∈ {∧,∨, →},

3. f (]α) := ] f (α) where ] ∈

{¬,AX, EX, AG, EG,AF,EF, P

≤x

≥x

,[b]},

4. f (A(α ] β)) := A( f (α) ] f (β)) where ] ∈ {U, R},

5. f (E(α ] β)) := E( f (α) ] f (β)) where ] ∈ {U,R},

6. f (∼∼α) := f (α),

7. f (∼(α∧β)) := f (∼α)∨ f (∼β),

8. f (∼(α∨β)) := f (∼α)∧ f (∼β),

9. f (∼(α→β)) := ¬ f (∼α)∧ f (∼β),

10. f (∼]α) := ] f (∼α) where ] ∈ {¬, [b]},

11. f (∼AXα) := EX f (∼α),

12. f (∼EXα) := AX f (∼α),

Inconsistency-tolerant Hierarchical Probabilistic Computation Tree Logic and Its Application to Model Checking

495

13. f (∼AGα) := EF f (∼α),

14. f (∼EGα) := AF f (∼α),

15. f (∼AFα) := EG f (∼α),

16. f (∼EFα) := AG f (∼α),

17. f (∼A(αUβ)) := E( f (∼α)R f (∼β)),

18. f (∼E(αUβ)) := A( f (∼α)R f (∼β)),

19. f (∼A(αRβ)) := E( f (∼α)U f (∼β)),

20. f (∼E(αRβ)) := A( f (∼α)U f (∼β)),

21. f (∼P

≤x

α) := P

f (∼α),

22. f (∼P

≥x

α) := P

f (∼α),

23. f (∼P

α) := P

≥x

f (∼α),

24. f (∼P

α) := P

≤x

f (∼α).

Lemma 3.4. Let f be the mapping deﬁned in Def-

inition 3.3. For any inconsistency-tolerant hierar-

chical probabilistic model M := (S,S

,R, µ

) of

IHpCTL, and any inconsistency-tolerant hierarchi-

cal probabilistic satisfaction relation |=

on M, we

can construct a hierarchical probabilistic model N :=

(S, S

,R, µ

,L) of HpCTL and a hierarchical proba-

bilistic satisfaction relation |= on N such that for any

formula α in L

, any sequence d in L

, and any state

s in S, (M, s) |=

[d]α iff (N, s) |= [d] f (α).

Proof. Suppose that M is an inconsistency-tolerant

hierarchical probabilistic model (S, S

,R, µ

) s.t.

is a mapping from S to the power set of

[

d∈SE

∼[d]

We then deﬁne a hierarchical probabilistic model

N := (S,S

,R, µ

,L) such that

1. L is a mapping from S to the power set of

[

d∈SE

0[d]

2. for any s ∈ S, any p ∈ Φ, and any c ∈ SE,

(a) [c]p ∈ L

(s) iff [c]p ∈ L(s),

(b) [c]∼p ∈ L

(s) iff [c]p

∈ L(s).

This lemma is then proved by induction on α.

• Base step:

1. Case α ≡ p ∈ Φ: We obtain: (M, s) |=

[d]p iff

[d]p ∈ L

(s) iff [d]p ∈ L(s) iff (N, s) |= [d]p iff

(N,s) |= [d] f (p) (by the deﬁnition of f ).

2. Case α ≡ ∼p ∈ Φ: We obtain: (M, s) |=

[d]∼p

iff [d]∼p ∈ L

(s) iff [d]p

∈ L(s) iff (N, s) |= [d]p

iff (N,s) |= [d] f (∼p) (by the deﬁnition of f ).

• Induction step: We show some cases.

1. Case α ≡ [b]β (b ∈SE): We obtain: (M,s) |=

[d][b]β iff (M, s) |=

[d ; b]β iff (N,s) |=

[d ; b] f (β) (by induction hypothesis) iff (N,s) |=

[d][b] f (β) iff (N, s) |= [d] f ([b]β) (by the deﬁni-

tion of f ).

2. Case α ≡ P

≤x

β: We obtain: (M, s) |=

[d]P

≤x

β iff

({w ∈ Ω

| (M, w) |=

[d]β}) ≤ x iff µ

({w ∈

Ω

| (N, w) |= [d] f (β)}) ≤ x (by induction hy-

pothesis) iff (N, s) |= [d]P

≤x

f (β) iff (N, s) |=

[d] f (P

≤x

β) (by the deﬁnition of f ).

3. Case α ≡ ∼[b]β (b ∈SE): We obtain: (M, s) |=

[d]∼[b]β iff (M, s) |=

∼[d][b]β (by Proposition

2.6 (3)) iff (M, s) |=

∼[d ; b]β iff (M, s) |=

[d ; b]∼β (by Proposition 2.6 (3)) iff (N, s) |=

[d ; b] f (∼β) (by induction hypothesis) iff

(N,s) |= [d][b] f (∼β) iff (N, s) |= [d] f (∼[b]β) (by

the deﬁnition of f ).

4. Case α ≡ ∼∼β: We obtain: (M, s) |=

[d]∼∼β iff

(M, s) |=

[d]β iff (N, s) |= [d] f (β) (by induction

hypothesis) (N, s) |= [d] f (∼∼β) (by the deﬁnition

of f ).

5. Case α ≡ ∼¬β: We obtain: (M,s) |=

[d]∼¬β

iff (M,s) 6|=

[d]∼β iff (N, s) 6|= [d] f (∼β) (by in-

duction hypothesis) iff (N, s) |= [d]¬ f (∼β) iff

(N,s) |= [d] f (∼¬β) (by the deﬁnition of f ).

6. Case α ≡ ∼(β→γ): We obtain: (M,s) |=

[d]∼(β→γ) iff (M, s) 6|=

[d]∼β and (M, s) |=

[d]∼γ iff (N, s) 6|= [d] f (∼β) and (N,s) |=

[d] f (∼γ) (by induction hypothesis) iff (N, s) |=

[d](¬ f (∼β)∧ f (∼γ)) iff (N,s) |= [d] f (∼(β→γ))

(by the deﬁnition of f ).

7. Case α ≡ ∼A(βUγ): We obtain:

(M, s) |=

[d]∼A(βUγ)

iff there is a path π ≡ s

,..., where s ≡ s

and for all states s

along π, we have (M, s

) |=

[d]∼γ or ∃0 ≤ k < j (M,s

) |=

[d]∼β

iff there is a path π ≡ s

,..., where s ≡ s

and for all states s

along π, we have (N,s

) |=

[d] f (∼γ) or ∃0 ≤ k < j (N, s

) |= [d] f (∼β) (by

induction hypothesis)

iff (N, s) |= [d](E( f (∼β)R f (∼γ)))

iff (N, s) |= [d] f (∼A(βUγ)) (by the deﬁnition of

f ).

8. Case α ≡ ∼P

≤x

β: We obtain: (M,s) |=

[d]∼P

≤x

iff µ

({w ∈ Ω

| (M, w) |=

[d]∼β}) > x iff

({w ∈ Ω

| (N, w) |= [d] f (∼β)}) > x (by in-

duction hypothesis) iff (N,s) |= [d]P

f (∼β) iff

(N,s) |= [d] f (∼P

≤x

β) (by the deﬁnition of f ).

Lemma 3.5. Let f be the mapping deﬁned in Deﬁ-

nition 3.3. For any hierarchical probabilistic model

ICAART 2021 - 13th International Conference on Agents and Artiﬁcial Intelligence

496

N := (S, S

,R, µ

,L) of HpCTL, and any hierarchi-

cal probabilistic satisfaction relation |= on N, we

can construct an inconsistency-tolerant hierarchical

probabilistic model M := (S,S

,R, µ

) of IHpCTL

and an inconsistency-tolerant hierarchical proba-

bilistic satisfaction relation |=

on M such that for

any formula α in L

, any d in L

, and any state s in S,

(N,s) |= [d] f (α) iff (M, s) |=

[d]α.

Proof. Similar to the proof of Lemma 3.4.

We then obtain the following theorem.

Theorem 3.6 (Embedding from IHpCTL into

HpCTL). IHpCTL is embeddable into HpCTL.

Namely, we have the following. Let f be the map-

ping deﬁned in Deﬁnition 3.3. For any formula α, α

is valid in IHpCTL iff f (α) is valid in HpCTL.

Proof. By Lemmas 3.4 and 3.5 (by taking d as

0).

We can also obtain the following theorem.

Theorem 3.7 (Embedding from IHpCTL into pCTL).

IHpCTL is embeddable into pCTL.

Proof. By combining Theorem 3.6 and the theorem

proved in (Kamide and Yano, 2019) for embedding

HpCTL into pCTL.

We then obtain the following theorem.

Theorem 3.8 (Relative decidability for IHpCTL with

respect to HpCTL). If the model-checking, validity,

and satisﬁability problems for HpCTL with a certain

probability measure are decidable, then the model-

checking, validity, and satisﬁability problems for IH-

pCTL with the same probability measure as that of

HpCTL are also decidable.

Proof. Suppose that the probability measure µ

in the

underlying inconsistency-tolerant hierarchical proba-

bilistic model (S,S

,R, µ

, L

) of IHpCTL is the same

as the underlying hierarchical probabilistic model

(S, S

,R, µ

, L) of HpCTL. Suppose also that HpCTL

with µ

is decidable. Then, by the mapping f , a

formula α of IHpCTL can be transformed into the

corresponding formula f (α) of HpCTL. By Lem-

mas 3.4 and 3.5 and Theorem 3.6, the model check-

ing, validity and satisﬁability problems for IHpCTL

can be transformed into those of HpCTL. Since the

model checking, validity and satisﬁability problems

for HpCTL with µ

are decidable by the assump-

tion, the problems for IHpCTL with µ

are also de-

cidable.

We can also obtain the following theorem.

Theorem 3.9 (Relative decidability for IHpCTL with

respect to pCTL). If the model-checking, validity,

and satisﬁability problems for pCTL with a certain

probability measure are decidable, then the model-

checking, validity, and satisﬁability problems for IH-

pCTL with the same probability measure as that of

pCTL are also decidable.

Proof. By combining Theorem 3.8 and the theorem

proved in (Kamide and Yano, 2019) for the relative

decidability of HpCTL with respect to pCTL.

Remark 3.10. The model checking problem for the

logic pCTL with the probability measures µ

and µ

−

introduced by Bianco and de Alfaro was shown to be

decidable in (Bianco and de Alfaro, 1995). The model

checking problem for the logic pCTL with the proba-

bility measure µ

introduced by Aziz et al. was shown

to be decidable in (Aziz et al., 1995). Thus, an ex-

tended IHpCTL with the above-mentioned probabil-

ity measures by Bianco and de Alfaro or by Aziz et al.

is also decidable by Theorem 3.9. If we consider a

sublogic without any probability measures, the deci-

sion problems for such a logic are decidable.

4 CONCLUSION AND REMARKS

In this study, the inconsistency-tolerant hierarchical

probabilistic computation tree logic IHpCTL was in-

troduced to establish a new extended model check-

ing paradigm referred to as IHpCTL model checking,

which can verify randomized, open, large, and com-

plex concurrent systems. The proposed logic IHpCTL

was shown to be embeddable into HpCTL and pCTL

and relatively decidable with respect to HpCTL and

pCTL. This means that the decidabilities of HpCTL

and pCTL with certain probability measures imply

the decidability of IHpCTL. This study thus showed

that we can effectively reuse the previously proposed

pCTL model-checking algorithms for IHpCTL model

checking.

As an example of IHpCTL model checking, we

next consider a lung cancer model presented as Fig-

ure 1. Lung cancer, also known as lung carcinoma,

is a disease or malignant lung tumor characterized

by uncontrolled cell growth in tissues of the lung

(Wikipedia, 2020). This growth can spread beyond

the lungs by the process of metastasis into nearby tis-

sue or other parts of the body. In this example, we can

express the following formula:

[Cancer ; LungCancer]AG(stage4∧

hasMalignantTumor∧pain f ul∧∼healthy

→ EF(P

≤0.89

death∧P

≥0.87

death)

Inconsistency-tolerant Hierarchical Probabilistic Computation Tree Logic and Its Application to Model Checking

497

Figure 1: Lung cancer model.

which implies the following:

“If a person with lung cancer Stage 4 has

a malignant tumor (i.e., lung cancer), is not

healthy, and has pain, then there is a probabil-

ity of approximately 88 percent exists that this

person will die in the near future.”

This statement is true and we can verify it using the

translation from IHpCTL to pCTL. In other words,

the IHpCTL formula can be veriﬁed using the cor-

responding pCTL formula by the translation. Us-

ing the translation, we can obtain the corresponding

pCTL formula as follows: AG(q

∧ q

→

EF(P

≤0.89

∧P

≥0.87

) where q

, and q

are distinct propositional variables.

Finally, we make the following remarks on re-

lated works. Probabilistic temporal logics, includ-

ing pCTL, inconsistency-tolerant (or paraconsistent)

temporal logics, hierarchical (or sequential) tem-

poral logics, and their applications to probabilistic,

inconsistency-tolerant, and hierarchical model check-

ing, have been investigated by many researchers. For

more information on probabilistic temporal logics

and their model-checking applications, see (Hanson,

1994; Hansson and Jonsson, 1994; Aziz et al., 1995;

Bianco and de Alfaro, 1995; Baier and Kwiatkowska,

1998; Kwiatkowska et al., 2011; Kamide and

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

J.P.A., 2019; Kamide and Yano, 2019). For more

information on inconsistency-tolerant temporal log-

ics and their model-checking applications, see (East-

erbrook and Chechik, 2001; Chen and Wu, 2006;

Kamide, 2006; Kamide and Wansing, 2011; Kamide

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

Kamide, 2015; Kamide and Koizumi, 2016; Kamide

and Endo, 2018). For more information on hierarchi-

cal temporal logics and their model-checking appli-

cations, see (Kamide and Kaneiwa, 2009; Kaneiwa

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

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

2018; Kamide and Yano, 2019). Finally, for a sur-

vey of a few closely related studies on probabilistic,

inconsistency-tolerant, and hierarchical temporal log-

ics and their applications, see (Kamide and Bernal

J.P.A., 2019).

ACKNOWLEDGEMENTS

We would like to thank the anonymous referees for

their valuable comments. This research was sup-

ported by Grant-in-Aid for the Japan Research Insti-

tute of Industrial Science and JSPS KAKENHI Grant

Numbers JP18K11171 and JP16KK0007.

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