Inconsistency and Sequentiality in LTL

Norihiro Kamide

Teikyo University, Faculty of Science and Engineering, Department of Human Information Systems,

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

Keywords:

Linear-time Temporal Logic, Paraconsistent Logic, Sequent Calculus, Completeness Theorem, Cut-

elimination Theorem.

Abstract:

Inconsistency-tolerant temporal reasoning with sequential (ordered or hierarchical) information is of gaining

increasing importance in the areas of computer science applications such as medical informatics. A logical

system for representing such reasoning is required for obtaining a theoretical basis for such applications. In

this paper, a new logic called a paraconsistent sequential linear-time temporal logic (PSLTL) is introduced

extending the standard linear-time temporal logic (LTL). PSLTL can appropriately represent inconsistency-

tolerant temporal reasoning with sequential information. The cut-elimination, complexity and completeness

theorems for PSLTL are proved as the main results of this paper.

1 INTRODUCTION

Inconsistency-tolerant temporal reasoning with se-

quential (ordered or hierarchical) information is of

growing importance in the areas of computer science

applications such as medical informatics and agent

communication. A logical system for representing

such reasoning is required for obtaining a concrete

theoretical basis for such applications. But, there was

no logical system that can simultaneously represent

inconsistency, sequentiality and temporality. Thus,

the aim of this paper is to introduce a logical system

for appropriately representing inconsistency-tolerant

temporal reasoning with sequential information.

For this aim, a new logic called a paraconsis-

tent sequential linear-time temporal logic (PSLTL)

is introduced in this paper extending the standard

linear-time temporal logic (LTL) (Pnueli, 1977).

Inconsistency-tolerant reasoning in PSLTL is ex-

pressed by a paraconsistent negation connective, and

sequential information in PSLTL is represented by

some sequence modal operators. Temporal reasoning

in PSLTL is, of course, expressed by some tempo-

ral operators used in LTL. As the main results of this

paper, the cut-elimination, complexity and complete-

ness theorems for PSLTL are proved using some the-

orems for semantically and syntactically embedding

PSLTL into its fragments SLTL and LTL.

The proposed logic PSLTL is regarded as an ex-

tension of both LTL and Nelson’sparaconsistent four-

valued logic with strong negation, N4 (Almukdad and

Nelson, 1984; Kamide and Wansing, 2012; Nelson,

1949; Wansing, 1993). On one hand, LTL is known

to be one of the most useful temporal logics for veri-

fying and specifying concurrentsystems and temporal

reasoning. On the other hand, N4 is known to be one

of the most important base logics for inconsistency-

tolerant reasoning. Combining the logics LTL and

N4 was studied in (Kamide and Wansing, 2011), and

such a combined logic is called a paraconsistent LTL

(PLTL). PSLTL is obtained from PLTL by adding

some sequence modal operators.

Combining LTL with some sequence modal op-

erators was studied in (Kamide, 2010; Kaneiwa and

Kamide, 2010; Kamide, 2013a), and such a combined

logic was called a sequence-indexed LTL (SLTL).

PSLTL is regarded as a modiﬁed paraconsistent ex-

tension of SLTL, and hence PSLTL is a modiﬁed ex-

tension of both PLTL (Kamide and Wansing, 2011)

and SLTL (Kaneiwa and Kamide, 2010). In the fol-

lowing, we explain an important property of the para-

consistent negation connective and a plausible inter-

pretation of sequence modal operators.

The paraconsistent negation connective ∼ used

in PSLTL can suitably be expressed inconsistency-

tolerant reasoning. One reason why ∼ is considered

is that it can be added in such a way that the extended

logics satisfy the property of paraconsistency. A se-

mantic consequence relation |= is called paraconsis-

tent with respect to a negation connective ∼ if there

Kamide N..

Inconsistency and Sequentiality in LTL.

DOI: 10.5220/0005180800460054

In Proceedings of the International Conference on Agents and Artiﬁcial Intelligence (ICAART-2015), pages 46-54

ISBN: 978-989-758-074-1

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

are formulas α,β such that not {α,∼α} |= β. In the

case of LTL, this implies that there is a model M and a

position i of a sequence σ = t

,... of time-points

in M with not [(M, i) |= (α∧ ∼α)→β].

It is known that logical systems with paraconsis-

tency can deal with inconsistency-tolerant and uncer-

tainty reasoning more appropriately than systems that

are non-paraconsistent. For example, we do not desire

that (s(x)∧∼s(x))→d(x) is satisﬁed for any symptom

s and disease d where ∼s(x) means “person x does

not have symptom s” and d(x) means “person x suf-

fers from disease d”, because there may be situations

that support the truth of both s(a) and ∼s(a) for some

individual a but do not support the truth of d(a).

If we cannot determine whether someone is

healthy, then the vague concept healthy can

be represented by asserting the inconsistent for-

mula: healthy( john) ∧ ∼healthy( john). This is

well-formalized in PSLTL because the formula:

healthy( john) ∧ ∼healthy( john)→hasCancer( john)

where hasCancer( john) means John has cancer is

not valid in PSLTL (i.e., PSLTL is inconsistency-

tolerant). On the other hand, the formula

healthy( john) ∧ ¬healthy( john)→hasCancer( john)

where ¬ is the classical negation connective is valid

in classical logic (i.e., inconsistency has undesirable

consequences). For more information on paraconsis-

tency, see e.g., (Priest, 2002).

Some sequence modal operators (Kamide and

Kaneiwa, 2009; Kamide, 2010; Kaneiwa and

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

2013a; Kamide, 2013b) used in PSLTL can suitably

be expressed sequential information. A sequence

modal operator [b] represents a sequence b of sym-

bols. The notion of sequences is useful to represent

the notions of “information,” “trees,” and “ontolo-

gies”. Thus, “sequential (ordered or hierarchical) in-

formation” can be represented by sequences. This is

plausible because a sequence structure gives a monoid

hM,;,

0i with informational interpretation (Wansing,

1993): (1) M is a set of pieces of (ordered or prior-

itized) information (i.e., a set of sequences), (2) ; is

a binary operator (on M) that combines two pieces

of information (i.e., a concatenation operator on se-

quences), and (3)

0 is the empty piece of information

(i.e., the empty sequence).

A formula of the form [b

; b

;· · · ; b

]α in PSLTL

intuitively means that “α is true based on a sequence

; b

;· · · ; b

of (ordered or prioritized) informa-

tion pieces.” Further, a formula of the form [

0]α in

PSLTL, which coincides with α, intuitively means

that “α is true without any information (i.e., it is an

eternal truth in the sense of classical logic).” Using a

sequence modal operator, we can express the formula

[ john ; student ; human]F(happy∧ ∼happy) which

means “a human student, John, will be both happy

and unhappy sometime in the future.” In this formula,

the sequence modal operator [ john ; student ; human]

represents the hierarchy John ⊆ student ⊆ human.

The structure of this paper is then presented as fol-

lows. In Section 2, PSLTL is introduced as a seman-

tics by extending (a semantics of) LTL with a para-

consistent negation connective and some sequence

modal operators. Firstly in this section, LTL is pre-

sented as the standard semantics, and next, SLTL is

presented as the semantics with some sequence modal

operators. Finally, PSLTL is obtained from SLTL by

adding a paraconsistent negation connective similar

to that of N4. In Section 3, a Genten-type sequent

calculus PSLT

for PSLTL is introduced extending a

Gentzen-type sequent calculus LT

for LTL. Firstly

in this section, a Gentzen-type sequent calculus LT

which was introduced by Kawai (Kawai, 1987), is

presented, and next, a Gentzen-type sequent calcu-

lus SLT

for SLTL is presented based on (Kamide,

2010; Kaneiwa and Kamide, 2010). Finally, PSLT

is obtained from SLT

by adding some inference

rules concerning the paraconsistent negation connec-

tive. In Section 4, the cut-elimination, complexity and

completeness theorems for PSLTL (and PSLT

) are

proved using two theorems for semantically and syn-

tactically embedding PSLTL (and PSLT

) into SLTL

(SLT

) and LTL (LT

). In Section 5, this paper is

concluded.

2 SEMANTICS

Formulas of LTL are constructed from countably

many propositional variables, → (implication), ∧

(conjunction), ∨ (disjunction), ¬ (negation), X (next),

G (globally) and F (eventually). Lower-case letters

p,q,... are used to denote propositional variables, and

Greek lower-case letters α,β,... are used to denote

formulas. An expression α ↔ β is used to denote

(α→β)∧(β→α). We write A ≡ B to indicate the syn-

tactical identity between A and B. The symbol ω is

used to represent the set of natural numbers. Lower-

case letters i, j and k are used to denote any natural

numbers. The symbol ≥ or ≤ is used to represent a

linear order on ω.

Deﬁnition 2.1. Formulas of LTL are deﬁned by

the following grammar, assuming p represents

propositional variables: α ::= p | α ∧ α | α ∨

α | α→α | ¬α | Xα | Gα | Fα

Deﬁnition 2.2 (LTL). Let S be a non-empty set of

states. A structure M := (σ, I) is a model if

InconsistencyandSequentialityinLTL

1. σ is an inﬁnite sequence s

,... of states in S,

2. I is a mapping from the set Φ of propositional

variables to the power set of S.

A satisfaction relation (M,i) |= α for any formula

α, where M is a model (σ,I) and i (∈ ω) represents

some position within σ, is deﬁned inductively by

1. for any p ∈ Φ, (M, i) |= p iff s

∈ I(p),

2. (M,i) |= α ∧ β iff (M, i) |= α and (M, i) |= β,

3. (M,i) |= α ∨ β iff (M, i) |= α or (M, i) |= β,

4. (M,i) |= α→β iff (M,i) |= α implies (M,i) |= β,

5. (M,i) |= ¬α iff not-[(M, i) |= α],

6. (M,i) |= Xα iff (M, i+ 1) |= α,

7. (M,i) |= Gα iff ∀ j ≥ i[(M, j) |= α],

8. (M,i) |= Fα iff ∃ j ≥ i[(M, j) |= α].

A formula α is valid in LTL if (M, 0) |= α for any

model M := (σ,I).

Formulas of SLTL are obtained from that of LTL

by adding [b] (sequence modal operator) where b is a

sequence. Sequences are constructed from countable

atomic sequences,

0 (empty sequence) and ; (com-

position). Lower-case letters b,c, ... are used for se-

quences. An expression [

0]α means α, and expres-

sions [

0 ; b]α and [b ;

0]α mean [b]α. The set of

sequences (including

0) is denoted as SE. An ex-

pression

[d] is used to represent [d

][d

]· · · [d

] with

i ∈ ω and d

≡

0. Note that

[d] can be the empty se-

quence. Also, an expression

d is used to represent

; d

; ··· ; d

with i ∈ ω.

Deﬁnition 2.3. Formulas and sequences of SLTL

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 | Xα | Gα | Fα | [b]α. b ::= e |

0 | b ; b.

Deﬁnition 2.4 (SLTL). Let S be a non-empty set of

states. A structure M := (σ,{I

}

d∈SE

) is a sequential

model if

1. σ is an inﬁnite sequence s

,... of states in S,

2. I

(

d ∈ SE) are mappings from the set Φ of propo-

sitional variables to the power set of S.

Satisfaction relations (M,i) |=

α (

d ∈ SE) for

any formula α, where M is a sequential model

(σ,{I

}

d∈SE

) and i (∈ ω) represents some position

within σ, is deﬁned inductively by

1. for any p ∈ Φ, (M, i) |=

p iff s

∈ I

(p),

2. (M,i) |=

α∧ β iff (M,i) |=

α and (M,i) |=

β,

3. (M,i) |=

α∨ β iff (M,i) |=

α or (M, i) |=

β,

4. (M,i) |=

α→β iff (M,i) |=

α implies (M,i) |=

β,

5. (M, i) |=

¬α iff not-[(M,i) |=

α],

6. (M, i) |=

Xα iff (M, i+ 1) |=

α,

7. (M, i) |=

Gα iff ∀ j ≥ i[(M, j) |=

α],

8. (M, i) |=

Fα iff ∃ j ≥ i[(M, j) |=

α].

9. for any atomic sequence e, (M,i) |=

[e]α

iff (M, i) |=

; e

α,

10. (M,i) |=

[b ; c]α iff (M, i) |=

[b][c]α.

A formula α is valid in SLTL if (M, 0) |=

α for

any sequential model M := (σ,{I

}

d∈SE

Some remarks on SLTL are given below.

1. SLTL is an extension of LTL since |=

of SLTL

includes |= of LTL.

2. The following clauses hold for SLTL: For any for-

mula α and any sequences c and

(a) (M, i) |=

[c]α iff (M,i) |=

d ; c

α,

(b) (M, i) |=

[

d]α iff (M,i) |=

α.

3. The following formulas are valid in SLTL: for any

formulas α and β and any b,c ∈ SE,

(a) [b](α◦ β) ↔ ([b]α) ◦ ([b]β)

where ◦ ∈ {∧,∨, →},

(b) [b](♯α) ↔ ♯([b]α) where ♯ ∈ {¬,X,G,F},

Formulas of PSLTL are obtained from that of

SLTL by adding ∼ (paraconsistent negation).

Deﬁnition 2.5. Formulas and sequences of PSLTL

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 | Xα | Gα | Fα | [b]α. b ::=

e |

0 | b ; b.

Deﬁnition 2.6 (PSLTL). Let S be a non-empty set of

states. A structure M := (σ,{I

}

d∈SE

,{I

−

}

d∈SE

) is

a paraconsistent sequential model if

1. σ is an inﬁnite sequence s

,... of states in S,

2. I

∗

(∗ ∈ {+,−},

d ∈ SE) are mappings from the

set Φ of propositional variables to the power set

of S.

Satisfaction relations (M,i) |=

∗

α (∗ ∈ {+,−},

d ∈ SE) for any formula α, where M is a paraconsis-

tent sequential model (σ,{I

}

d∈SE

,{I

−

}

d∈SE

) and

i (∈ ω) represents some position within σ, are deﬁned

1. for any p ∈ Φ, (M, i) |=

p iff s

∈ I

(p),

2. (M, i) |=

α∧β iff (M, i) |=

α and (M,i) |=

β,

ICAART2015-InternationalConferenceonAgentsandArtificialIntelligence

3. (M,i) |=

α∨β iff (M,i) |=

α or (M, i) |=

β,

4. (M,i) |=

α→β iff (M,i) |=

α implies

(M,i) |=

β,

5. (M,i) |=

¬α iff not-[(M,i) |=

α],

6. (M,i) |=

∼α iff (M,i) |=

−

α,

7. (M,i) |=

Xα iff (M, i+ 1) |=

α,

8. (M,i) |=

Gα iff ∀ j ≥ i[(M, j) |=

α],

9. (M,i) |=

Fα iff ∃ j ≥ i[(M, j) |=

α],

10. for any p ∈ Φ, (M,i) |=

−

p iff s

∈ I

−

(p),

11. (M, i) |=

−

α∧β iff (M,i) |=

−

α or (M, i) |=

−

β,

12. (M, i) |=

−

α∨β iff (M, i) |=

−

α and (M,i) |=

−

β,

13. (M, i) |=

−

α→β iff (M,i) |=

α and (M,i) |=

−

β,

14. (M, i) |=

−

¬α iff not-[(M,i) |=

−

α],

15. (M, i) |=

−

∼α iff (M,i) |=

α,

16. (M, i) |=

−

Xα iff (M, i+ 1) |=

−

α,

17. (M, i) |=

−

Gα iff ∃ j ≥ i[(M, j) |=

−

α],

18. (M, i) |=

−

Fα iff ∀ j ≥ i[(M, j) |=

−

α],

19. for any atomic sequence e and any ∗ ∈ {+,−},

(M,i) |=

∗

[e]α iff (M, i) |=

∗

d ; e

α,

20. for any ∗ ∈ {+,−},

(M,i) |=

∗

[b ; c]α iff (M, i) |=

∗

[b][c]α.

A formula α is valid in PSLTL iff (M,0) |=

α for any paraconsistent sequential model M :=

(σ,{I

}

d∈SE

,{I

}

d∈SE

Some remarks on PSLTL are given below.

1. The intuitive meanings of |=

and |=

−

are “ver-

iﬁcation (or justiﬁcation) with sequential infor-

mation” and “refutation (or falsiﬁcation) with se-

quential information,” respectively.

2. F and G are duals of each other not only with re-

spect to ¬ but also with respect to ∼. X is a self

dual not only with respect to ¬ but also with re-

spect to ∼. [b] is a self dual not only with respect

to ¬ but also with respect to ∼. ¬ and ∼ are self-

duals with respect to ∼ and ¬, respectively.

3. The falsiﬁcation conditions for ¬ may be felt to

be in need of some justiﬁcation. Suppose that a is

a person who is neither rich nor poor and that, as

a matter of fact, no one is both rich and poor. Let

p stand for the claim that a is poor and r for the

claim that a is rich. Intuitively, a state deﬁnitely

veriﬁes p iff it falsiﬁes r, and vice versa. Suppose

now that ¬p is indeed falsiﬁed at a state i in model

M: (M, i) |=

−

¬p. This should mean that it is

veriﬁed at i that p is poor or neither poor or rich.

But this is the case iff r is not veriﬁed at i, which

means that p is not falsiﬁed at i.

4. PSLTL is paraconsistent with respect to ∼.

The reason is presented as follows. As-

sume a paraconsistent sequential model M :=

(σ,{I

}

∈SE

,{I

}

∈SE

) such that s

∈ I

(p),

∈ I

−

(p) and s

/∈ I

(q) for a pair of distinct

propositional variables p and q. Then, (M, i) |=

(p∧ ∼p)→q does not hold.

5. The following clauses hold for PSLTL: For any

formula α, any sequences c,

d and any ∗ ∈

{+,−},

(a) (M, i) |=

∗

[c]α iff (M,i) |=

∗

d ; c

α,

(b) (M, i) |=

∗

[

d]α iff (M,i) |=

∗

α.

3 SEQUENT CALCULUS

Greek capital letters Γ, ∆,... are used to represent ﬁ-

nite (possibly empty) sets of formulas. An expression

α for any i ∈ ω is deﬁned inductively by X

α ≡ α

and X

n+1

α ≡ X

Xα. An expression of the form

Γ ⇒ ∆ is called a sequent. An expression L ⊢ S is

used to denote the fact that a sequent S is provable in

a sequent calculus L. A rule R of inference is said to

be admissible in a sequent calculus L if the following

condition is satisﬁed: for any instance

·· ·S

S of R, if

L ⊢ S

for all i, then L ⊢ S.

Kawai’s sequent calculus LT

(Kawai, 1987) for

LTL is presented below.

Deﬁnition 3.1 (LT

). The initial sequents of LT

are

of the form: for any propositional variable p,

p ⇒ X

The structural rules of LT

are of the form:

Γ ⇒ ∆, α α,Σ ⇒ Π

Γ,Σ ⇒ ∆,Π

(cut)

Γ ⇒ ∆

α,Γ ⇒ ∆

(we-left)

Γ ⇒ ∆

Γ ⇒ ∆, α

(we-right).

The logical inference rules of LT

are of the form:

Γ ⇒ Σ, X

α X

β,∆ ⇒ Π

(α→β),Γ, ∆ ⇒ Σ, Π

(→left)

α,Γ ⇒ ∆,X

Γ ⇒ ∆,X

(α→β)

(→right)

α,Γ ⇒ ∆

(α∧ β),Γ ⇒ ∆

(∧left1)

β,Γ ⇒ ∆

(α∧ β),Γ ⇒ ∆

(∧left2)

InconsistencyandSequentialityinLTL

Γ ⇒ ∆, X

α Γ ⇒ ∆,X

Γ ⇒ ∆,X

(α∧ β)

(∧right)

α,Γ ⇒ ∆ X

β,Γ ⇒ ∆

(α∨ β), Γ ⇒ ∆

(∨left)

Γ ⇒ ∆, X

(α∨ β)

(∨right1)

Γ ⇒ ∆, X

(α∨ β)

(∨right2)

Γ ⇒ ∆, X

¬α,Γ ⇒ ∆

(¬left)

α,Γ ⇒ ∆

Γ ⇒ ∆,X

¬α

(¬right)

i+k

α,Γ ⇒ ∆

Gα,Γ ⇒ ∆

(Gleft)

{ Γ ⇒ ∆, X

i+ j

α }

j∈ω

Γ ⇒ ∆,X

Gα

(Gright)

{ X

i+ j

α,Γ ⇒ ∆ }

j∈ω

Fα,Γ ⇒ ∆

(Fleft)

Γ ⇒ ∆,X

i+k

Γ ⇒ ∆, X

Fα

(Fright).

Some remarks on LT

are given below.

1. The rules (Gright) and (Fleft) have inﬁnite

premises.

2. The sequents of the form: X

α ⇒ X

α for any for-

mula α are provable in cut-free LT

. This fact can

be proved by induction on the complexity of α.

3. The cut-elimination and completeness theorems

for LT

were proved by Kawai (Kawai, 1987).

Prior to introduce a sequent calculus for SLTL,

we have to introduce some notations. The symbol

K is used to represent the set {X} ∪ {[b] | b ∈ SE},

and the symbol K

∗

is used to represent the set of all

words of ﬁnite length of the alphabet K. For example,

[b]X

[c] is in K

∗

. Remark that K

∗

includes

0, and

hence {†α | † ∈ K

∗

} includes α. An expression ♯ is

used to represent an arbitrary member of K

∗

A sequent calculus SLT

for SLTL is then intro-

duced below.

Deﬁnition 3.2 (SLT

). The initial sequents of SLT

are of the form: for any propositional variable p,

♯p ⇒ ♯p.

The structural rules of SLT

are (cut), (we-left)

and (we-right) in Deﬁnition 3.1.

The logical inference rules of SLT

are of the

form:

Γ ⇒ Σ, ♯α ♯β,∆ ⇒ Π

♯(α→β),Γ, ∆ ⇒ Σ, Π

(→left

)

♯α,Γ ⇒ ∆,♯β

Γ ⇒ ∆, ♯(α→β)

(→right

)

♯α,Γ ⇒ ∆

♯(α∧ β),Γ ⇒ ∆

(∧left1

)

♯β,Γ ⇒ ∆

♯(α∧ β),Γ ⇒ ∆

(∧left2

)

Γ ⇒ ∆, ♯α Γ ⇒ ∆,♯β

Γ ⇒ ∆, ♯(α∧ β)

(∧right

)

♯α,Γ ⇒ ∆ ♯β,Γ ⇒ ∆

♯(α∨ β), Γ ⇒ ∆

(∨left

)

Γ ⇒ ∆, ♯α

Γ ⇒ ∆,♯(α ∨ β)

(∨right1

)

Γ ⇒ ∆, ♯β

Γ ⇒ ∆, ♯(α∨ β)

(∨right2

)

Γ ⇒ ∆, ♯α

♯¬α,Γ ⇒ ∆

(¬left

)

♯α,Γ ⇒ ∆

Γ ⇒ ∆, ♯¬α

(¬right

)

♯X

α,Γ ⇒ ∆

♯Gα,Γ ⇒ ∆

(Gleft

)

{ Γ ⇒ ∆, ♯X

α }

j∈ω

Γ ⇒ ∆, ♯Gα

(Gright

)

{ ♯X

α,Γ ⇒ ∆ }

j∈ω

♯Fα,Γ ⇒ ∆

(Fleft

)

Γ ⇒ ∆,♯X

Γ ⇒ ∆, ♯Fα

(Fright

)

♯[b]Xα,Γ ⇒ ∆

♯X[b]α,Γ ⇒ ∆

(Xleft)

Γ ⇒ ∆, ♯[b]Xα

Γ ⇒ ∆, ♯X[b]α

(Xright).

The sequence inference rules of SLT

are of the

form:

♯[b][c]α,Γ ⇒ ∆

♯[b ; c]α,Γ ⇒ ∆

(;left)

Γ ⇒ ∆, ♯[b][c]α

Γ ⇒ ∆, ♯[b ; c]α

(;right).

Some remarks on SLT

are given below.

1. The sequents of the form ♯α ⇒ ♯α for any formula

α are provable in cut-free SLT

. This fact can be

proved by induction on the complexity of α.

2. The following rules are admissible in cut-free

SLT

♯X[b]α,Γ ⇒ ∆

♯[b]Xα,Γ ⇒ ∆

(Xleft

−1

)

Γ ⇒ ∆, ♯X[b]α

Γ ⇒ ∆, ♯[b]Xα

(Xright

−1

A sequent calculus PSLT

for PSLTL is intro-

duced below.

Deﬁnition 3.3 (PSLT

). PSLT

is obtained from

SLT

by adding the initial sequents of the form: for

any propositional variable p,

♯∼p ⇒ ♯∼p,

and adding the logical and sequence inference rules

of the form:

♯∼α,Γ ⇒ ∆

∼♯α,Γ ⇒ ∆

(∼♯left)

Γ ⇒ ∆, ♯∼α

Γ ⇒ ∆, ∼♯α

(∼♯right)

♯α,Γ ⇒ ∆

♯∼∼α,Γ ⇒ ∆

(∼∼left)

Γ ⇒ ∆,♯α

Γ ⇒ ∆, ♯∼∼α

(∼∼right)

♯α,Γ ⇒ ∆

♯∼(α→β),Γ ⇒ ∆

(∼→left1)

♯∼β,Γ ⇒ ∆

♯∼(α→β),Γ ⇒ ∆

(∼→left2)

Γ ⇒ ∆, ♯α Γ ⇒ ∆,♯∼β

Γ ⇒ ∆, ♯∼(α→β)

(∼→right)

♯∼α,Γ ⇒ ∆ ♯∼β,Γ ⇒ ∆

♯∼(α∧ β),Γ ⇒ ∆

(∼ ∧ left)

Γ ⇒ ∆,♯∼α

Γ ⇒ ∆, ♯∼(α∧ β)

(∼ ∧ right1)

Γ ⇒ ∆,♯∼β

Γ ⇒ ∆, ♯∼(α∧ β)

(∼ ∧ right2)

ICAART2015-InternationalConferenceonAgentsandArtificialIntelligence

♯∼α,Γ ⇒ ∆

♯∼(α∨ β),Γ ⇒ ∆

(∼ ∨ left1)

♯∼β,Γ ⇒ ∆

♯∼(α∨ β),Γ ⇒ ∆

(∼ ∨ left2)

Γ ⇒ ∆, ♯∼α Γ ⇒ ∆,♯∼β

Γ ⇒ ∆,♯∼(α∨ β)

(∼ ∨ right)

Γ ⇒ ∆, ♯∼α

♯∼¬α,Γ ⇒ ∆

(∼¬left)

♯∼α,Γ ⇒ ∆

Γ ⇒ ∆,♯∼¬α

(∼¬right)

{ ♯X

∼α,Γ ⇒ ∆ }

j∈ω

♯∼Gα,Γ ⇒ ∆

(∼Gleft)

Γ ⇒ ∆, ♯X

∼α

Γ ⇒ ∆, ♯∼Gα

(∼Gright)

♯X

∼α,Γ ⇒ ∆

♯∼Fα,Γ ⇒ ∆

(∼Fleft)

{ Γ ⇒ ∆, ♯X

∼α }

j∈ω

Γ ⇒ ∆, ♯∼Fα

(∼Fright)

♯∼[b][c]α,Γ ⇒ ∆

♯∼[b ; c]α,Γ ⇒ ∆

(∼;left)

Γ ⇒ ∆,♯∼[b][c]α

Γ ⇒ ∆, ♯∼[b ; c]α

(∼;right).

Some remarks on PSLT

are given below.

1. The sequents of the form ♯α ⇒ ♯α for any formula

α are provable in cut-free PSLT

. This fact can be

proved by induction on the complexity of α.

2. The following rules are admissible in cut-free

PSLT

♯X[b]α,Γ ⇒ ∆

♯[b]Xα,Γ ⇒ ∆

(Xleft

−1

)

Γ ⇒ ∆, ♯X[b]α

Γ ⇒ ∆, ♯[b]Xα

(Xright

−1

)

∼♯α,Γ ⇒ ∆

♯∼α,Γ ⇒ ∆

(∼♯left

−1

)

Γ ⇒ ∆,∼♯α

Γ ⇒ ∆,♯∼α

(∼♯right

−1

4 MAIN RESULTS

In this section, we introduce a translation function

f from SLTL into LTL, and a translation function g

from PSLTL into SLTL. Using these functions, we

obtain a translation function gf from PSLTL into

LTL. Using these translation functions, we will show

a theorem for semantically and syntactically embed-

ding PSLTL into SLTL and LTL. Using these em-

bedding theorems, we will show the cut-elimination,

complexity and completeness theorems for PSLTL.

Deﬁnition 4.1 (Translation from SLTL into LTL). Let

Φ be a non-empty set of propositional variables and

be the set {p

| p ∈ Φ} (

d ∈ SE) of propositional

variables where p

:= p (i.e., Φ

:= Φ). The language

(the set of formulas) of SLTL is deﬁned using Φ,

[b], ∧,∨, →,¬, X, F and G. The language L of LTL

is obtained from L

by adding Φ

and deleting [b].

A mapping f from L

to L is deﬁned by:

1. for any p ∈ Φ, f(

[d]p) := p

∈ Φ

, esp., f(p) =

p ∈ Φ

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

3. f(♯†α) := †f(♯α) where † ∈ {¬,X, G,F},

4. f(♯[b ; c]α) := f(♯[b][c]α).

An expression f(Γ) denotes the result of replacing

every occurrenceof a formula α in Γ by an occurrence

of f(α).

Proposition 4.2 ((Kamide, 2010; Kaneiwa and

Kamide, 2010)). Let f be the mapping deﬁned in Def-

inition 4.1.

1. (Semantical embedding): For any formula α, α is

valid in SLTL iff f(α) is valid in LTL.

2. (Syntactical embedding): For any sets Γ and ∆ of

formulas in L

(a) SLT

⊢ Γ ⇒ ∆ iff LT

⊢ f(Γ) ⇒ f(∆),

(b) SLT

− (cut) ⊢ Γ ⇒ ∆ iff LT

− (cut) ⊢

f(Γ) ⇒ f(∆).

3. (Cut-elimination): The rule (cut) is admissible in

cut-free SLT

4. (Completeness): For any formula α, SLT

⊢ ⇒ α

iff α is valid in SLTL.

We now introduce a translation of PSLTL into

SLTL, and by using this translation, we show some

theorems for embedding PSLTL into SLTL. A simi-

lar translation has been used by Vorob’ev (Vorob’ev,

1952), Gurevich (Gurevich, 1977), and Rautenberg

(Rautenberg, 1979) to embed Nelson’s three-valued

constructive logic (Almukdad and Nelson, 1984; Nel-

son, 1949) into intuitionistic logic.

Deﬁnition 4.3 (Translation from PSLTL into SLTL).

Let Φ be a non-empty set of propositional variables

and Φ

′

be the set {p

′

| p ∈ Φ} of propositional vari-

ables. The language L

(the set of formulas) of

PSLTL is deﬁned using Φ, ∼, →,∧,∨,¬, X, F, G and

[b]. The language L

of SLTL is obtained from L

by adding Φ

′

and deleting ∼.

A mapping g from L

to L

is deﬁned by

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

′

∈ Φ

′

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

3. g(†α) := †g(α) where † ∈ {¬,X,F, G, [b]},

4. g(∼∼α) := g(α),

5. g(∼†α) := †g(∼α) where † ∈ {¬,X,[b]},

6. g(∼(α ∧ β)) := g(∼α) ∨ g(∼β),

7. g(∼(α ∨ β)) := g(∼α) ∧ g(∼β),

8. g(∼(α→β)) := g(α) ∧ g(∼β),

9. g(∼Fα) := Gg(∼α),

10. g(∼Gα) := Fg(∼α).

InconsistencyandSequentialityinLTL

We have: g(♯α) = ♯g(α) for any formula α and

any ♯ ∈ K

∗

The following is a translation example from

PSLTL into LTL, by using the translation functions

f and g.

Example 4.4. We consider a formula G(∼([b]p ∧

∼[c]q)) where b,c are atomic sequences, and p, q are

propositional variables.

Firstly, we translate this PSLTL-formula into a

SLTL-formula by the translation function g as follows.

g(G(∼([b]p∧ ∼[c]q)))

= Gg(∼([b]p∧ ∼[c]q))

= G(g(∼[b]p) ∨ g(∼∼[c]q))

= G([b]g(∼p) ∨ g([c]q))

= G([b]p

′

∨ [c]g(q))

= G([b]p

′

∨ [c]q)

where p

′

is a propositional variable in SLTL.

Next, we translate this SLTL-formula into a LTL-

formula by the translation function f as follows.

f(G([b]p

′

∨ [c]q))

= Gf([b]p

′

∨ [c]q)

= G( f([b]p

′

) ∨ f([c]q))

= G(p

′b

∨ q

)

where p

′b

are propositional variables in LTL.

Thus, the formula G(∼([b]p ∧ ∼[c]q)) of PSLTL

is translated into the formula G(p

′b

∨ q

) of LTL.

Next, we will show a theorem for semantically

embedding PSLTL into SLT. To show this theorem,

we need two lemmas which are presented below.

Lemma 4.5. Let g be the mapping deﬁned in Deﬁni-

tion 4.3, and S be a non-empty set of states. For any

paraconsistent sequential model M := (σ, {I

}

d∈SE

−

}

∈SE

) of PSLTL, any satisfaction relations |=

∗

(∗ ∈ {+,−},

d ∈ SE) on M, and any state s

in σ, we

can construct a sequential model N := (σ,{I

}

d∈SE

)

of SLTL and satisfaction relations |=

on N such that

for any formula α in L

1. (M,i) |=

α iff (N,i) |=

g(α).

2. (M,i) |=

−

α iff (N,i) |=

g(∼α).

Proof. Let Φ be a non-empty set of propositional

variables and Φ

′

be the set {p

′

| p ∈ Φ} of proposi-

tional variables. Suppose that M is a paraconsistent

sequential model (σ, { I

}

d∈SE

, {I

−

}

d∈SE

) where

and I

−

are mappings from Φ to the power set of

S. Suppose that N is a sequential model (σ,{I

}

∈SE

)

where I

are mappings from Φ ∪ Φ

′

to the power set

of S. Suppose moreover that M and N satisfy the fol-

lowing conditions: for any s

in σ and any p ∈ Φ,

1. s

∈ I

(p) iff s

∈ I

(p),

2. s

∈ I

−

(p) iff s

∈ I

′

The lemma is then proved by (simultaneous) in-

duction on the complexity of α.

• Base step: Case α ≡ p ∈ Φ: For (1), we obtain:

(M,i) |=

p iff s

∈ I

(p) iff s

∈ I

(p) iff (N,i) |=

p iff (N,i) |=

g(p) (by the def. of g). For (2), we

obtain: (M, i) |=

−

p iff s

∈ I

−

(p) iff s

∈ I

′

) iff

(N,i) |=

′

iff (N,i) |=

g(∼p) (by the def. of g).

• Induction step: We show some cases.

Case α ≡ ∼β: For (1), we obtain: (M, i) |=

∼β

iff (M, i) |=

−

β iff (N, i) |=

g(∼β) (by ind. hypo. for

2). For (2), we obtain: (M,i) |=

−

∼β iff (M, i) |=

iff (N,i) |=

g(β) (by ind. hypo. for 1) iff (N, i) |=

g(∼∼β) (by the def. of g).

Case α ≡ Xβ: For (1), we obtain: (M,i) |=

Xβ iff (M,i + 1) |=

β iff (N,i + 1) |=

g(β) (by

induction hypothesis for 1) iff (N,i) |=

Xg(β) iff

(N,i) |=

g(Xβ) (by the deﬁnition of g). For (2),

we obtain: (M,i) |=

−

Xβ iff (M,i + 1) |=

−

β iff

(N,i + 1) |=

g(∼β) (by induction hypothesis for 2)

iff (N,i) |=

Xg(∼β) iff (N,i) |=

g(∼Xβ) (by the

deﬁnition of g).

Case α ≡ [b]β: For (1), we obtain: (M,i) |=

[b]β iff (M,i) |=

d ; b

β iff (N,i) |=

d ; b

g(β) (by in-

duction hypothesis for 1) iff (N,i) |=

[b]g(β) iff

(N,i) |=

g([b]β) (by the deﬁnition of g). For (2),

we obtain: (M, i) |=

−

[b]β iff (M, i) |=

−

d ; b

β iff

(N,i) |=

d ; b

g(∼β) (by induction hypothesis for 2) iff

(N,i) |=

[b]g(∼β) iff (N,i) |=

g(∼[b]β) (by the def-

inition of g). Q.E.D.

Lemma 4.6. Let g be the mapping deﬁned in Deﬁni-

tion 4.3, and S be a non-empty set of states. For any

sequential model N := (σ,{I

}

d∈SE

) of SLTL and any

satisfaction relations |=

(

d ∈ SE) on N, and any state

in σ, we can construct a paraconsistent sequen-

tial model M := (σ, {I

}

d∈SE

, {I

−

}

d∈SE

) of PSLTL

and satisfaction relations |=

∗

(∗ ∈ {+,−},

d ∈ SE)

on M such that

1. (M, i) |=

α iff (N,i) |=

g(α).

2. (M, i) |=

−

α iff (N,i) |=

g(∼α).

Proof. Similar to the proof of Lemma 4.5. Q.E.D.

Theorem 4.7 (Semantical embedding from PSLTL

into SLTL). Let g be the mapping deﬁned in Deﬁni-

tion 4.3. For any formula α, α is valid in PSLTL iff

g(α) is valid in SLTL.

Proof. By Lemmas 4.5 and 4.6. Q.E.D.

ICAART2015-InternationalConferenceonAgentsandArtificialIntelligence

Theorem 4.8 (Semantical embedding from PSLTL

into LTL). Let f and g be the mappings deﬁned in

Deﬁnitions 4.1 and 4.3, respectively. For any formula

α, α is valid in PSLTL iff f g(α) is valid in LTL.

Proof. By Proposition 4.2 (1) and Theorem 4.7.

Q.E.D.

Theorem 4.9 (Complexity). PSLTL is PSPACE-

complete.

Proof. By decidability of LTL, for each α, it is

possible to decide if fg(α) is valid in LTL. Then,

by Theorem 4.8, PSLTL is also decidable. More-

over the mapping fg is a polynomial time transla-

tion, and LTL is know to be PSPACE-complete (Sistla

and Clarke, 1985). Thus, PSLTL is also PSPACE-

complete. Q.E.D.

Theorem 4.10 (Weak syntactical embedding from

PSLT

into SLT

). Let Γ and ∆ be sets of formulas in

, and g be the mapping deﬁned in Deﬁnition 4.3.

Then:

1. If PSLT

⊢ Γ ⇒ ∆, then SLT

⊢ g(Γ) ⇒ g(∆).

2. If SLT

− (cut) ⊢ g(Γ) ⇒ g(∆), then PSLT

−

(cut) ⊢ Γ ⇒ ∆.

Proof. • (1) : By induction on the proofs P of

Γ ⇒ ∆ in PSLT

. We distinguish the cases according

to the last inference of P, and show some cases.

Case (♯∼p ⇒ ♯∼p): The last inference of P is of

the form: ♯∼p ⇒ ♯∼p. In this case, we obtain the

required fact LT

⊢ g(♯∼p) ⇒ g(♯∼p), since g(♯∼p)

coincides with ♯p

′

by the deﬁnition of g.

Case (∼∼left): The last inference of P is of the

form:

♯α,Γ ⇒ ∆

♯∼∼α,Γ ⇒ ∆

(∼∼left).

By induction hypothesis, we have the required fact:

SLT

⊢ g(♯α),g(Γ) ⇒ g(∆) where g(♯α) coincides

with g(♯∼∼α) by the deﬁnition of g.

Case (∼;left): The last inference of P is of the

form:

♯∼[b][c]α,Γ ⇒ ∆

♯∼[b ; c]α,Γ ⇒ ∆

(∼;left).

By induction hypothesis, we have: SLT

⊢

g(♯∼[b][c]α),g(Γ) ⇒ g(∆) where g(♯∼[b][c]α) coin-

cides with ♯∼[b][c]g(α) by the deﬁnition of g. Then,

we obtain:

♯∼[b][c]g(α),g(Γ) ⇒ g(∆)

♯∼[b ; c]g(α),g(Γ) ⇒ g(∆)

(∼;left)

where ♯∼[b ; c]g(α) coincides with g(♯∼[b ; c]α) by

the deﬁnition of g.

• (2) : By induction on the proofs Q of

g(Γ) ⇒ g(∆) in SLT

. We distinguish the cases ac-

cording to the last inference of Q, and show some

cases.

Case (;left): The last inference of Q is (;left).

Subcase (1): The last inference of Q is of the form:

♯[b][c]g(α),g(Γ) ⇒ g(∆)

♯[b ; c]g(α),g(Γ) ⇒ g(∆)

(;left)

where ♯[b][c]g(α) and ♯[b ; c]g(α) respectively coin-

cide with g(♯[b][c]α) and g(♯[b ; c]α) by the deﬁnition

of g. By induction hypothesis, we have: PSLT

⊢

♯[b][c]α,Γ ⇒ ∆, and hence obtain the required fact:

♯[b][c]α,Γ ⇒ ∆

♯[b ; c]α,Γ ⇒ ∆

(;left).

Subcase (2): The last inference of Q is of the form:

♯[b][c]g(∼α),g(Γ) ⇒ g(∆)

♯[b ; c]g(∼α),g(Γ) ⇒ g(∆)

(;left)

where ♯[b][c]g(∼α) and ♯[b ; c]g(∼α) respectively co-

incide with g(♯∼[b][c]α) and g(♯∼[b ; c]α) by the

deﬁnition of g. By induction hypothesis, we have:

PSLT

⊢ ♯∼[b][c]α,Γ ⇒ ∆, and hence obtain the re-

quired fact:

♯∼[b][c]α,Γ ⇒ ∆

♯∼[b ; c]α,Γ ⇒ ∆

(∼;left).

Q.E.D.

Theorem 4.11 (Cut-elimination). The rule (cut) is

admissible in cut-free PSLT

Proof. Suppose PSLT

⊢ Γ ⇒ ∆. Then, we have

SLT

⊢ f(Γ) ⇒ f(∆) by Theorem 4.10 (1), and hence

SLT

− (cut) ⊢ f(Γ) ⇒ f(∆) by Proposition 4.2 (3).

By Theorem 4.10 (2), we obtain PSLT

− (cut) ⊢

Γ ⇒ ∆. Q.E.D.

Theorem 4.12 (Syntactical embedding from PSLT

into SLT

). Let Γ and ∆ be sets of formulas in L

and g be the mapping deﬁned in Deﬁnition 4.3. Then:

1. PSLT

⊢ Γ ⇒ ∆ iff SLT

⊢ g(Γ) ⇒ g(∆).

2. PSLT

− (cut) ⊢ Γ ⇒ ∆ iff SLT

− (cut) ⊢

g(Γ) ⇒ g(∆).

Proof. • (1). (=⇒): By Theorem 4.10 (1). (⇐=):

Suppose SLT

⊢ g(Γ) ⇒ g(∆). We then have SLT

− (cut) ⊢ g(Γ) ⇒ g(∆) by Proposition 4.2 (3). Thus,

we obtain PSLT

− (cut) ⊢ Γ ⇒ ∆ by Theorem 4.10

(2). Therefore we have PSLT

⊢ Γ ⇒ ∆.

• (2). (=⇒): Suppose PSLT

− (cut) ⊢ Γ ⇒ ∆.

Then we have PSLT

⊢ Γ ⇒ ∆. We then obtain SLT

InconsistencyandSequentialityinLTL

⊢ g(Γ) ⇒ g(∆) by Theorem 4.10 (1). Therefore we

obtain SLT

− (cut) ⊢ g(Γ) ⇒ g(∆) by Proposition

4.2 (3). (⇐=): By Theorem 4.10 (2). Q.E.D.

Theorem 4.13 (Syntactical embedding from PSLT

into LT

). Let Γ and ∆ be sets of formulas in L

Let f and g be the mappings deﬁned in Deﬁnitions

4.1 and 4.3, respectively. Then:

1. PSLT

⊢ Γ ⇒ ∆ iff SLT

⊢ fg(Γ) ⇒ fg(∆).

2. PSLT

− (cut) ⊢ Γ ⇒ ∆ iff SLT

− (cut) ⊢

fg(Γ) ⇒ fg(∆).

Proof. By Proposition 4.2 (2) and Theorem 4.12.

Q.E.D.

Theorem 4.14 (Completeness). For any formula α,

PSLT

⊢ ⇒ α iff α is valid in PSLTL.

Proof. PSLT

⊢ ⇒ α iffSLT

⊢ ⇒ g(α) (by The-

orem 4.12) iff g(α) is valid in SLTL (by Proposition

4.2 (4)) iff α is valid in PSLTL (by Theorem 4.7).

Q.E.D.

5 CONCLUSIONS

In this paper, the logic PSLTL (paraconsistent sequen-

tial linear-time temporal logic) was introduced as a

semantics by extending the standard logic (a seman-

tics of) LTL (linear-time temporal logic). PSLTL can

appropriately represent inconsistency-tolerantreason-

ing by the paraconsistent negation connective, and se-

quential (hierarchical) information by some sequence

modal operators. By using the semantical embedding

theorem of PSLTL into LTL, it was shown that PSLTL

is PSPACE-complete. The Gentzen-type sequent cal-

culus PSLT

for PSLTL was introduced, and the cut-

elimination theorem for this calculus was proved us-

ing the syntactical embedding theorem of PSLT

into

its non-paraconsistent fragment SLT

. The complete-

ness theorem for PSLTL (and PSLT

) was proved us-

ing both syntactical and semantical embedding theo-

rems of PSLTL (and PSLT

) into SLTL (and SLT

It was thus shown in this paper that PSLTL and

PSLT

are a good theoretical basis for inconsistency-

tolerant temporal reasoning with sequential informa-

tion.

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Kamide, N. (2010). A proof system for temporal reason-

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