Axiomatizations of Temporal Logic

10723029

Xu Zhaoqing

I. Content

Introduction

Basic temporal logic

Branching time logic

Conclusions

II. Introduction

Temporal Logic

Broadly: all approaches to the representation of temporal

information within a logical framework;

Narrowly: the modal-style of temporal logic;

III. Basic Temporal Logic

1. Syntax and semantics

2. The Minimal logic Kt

3. The IRR rule

4. The logic of linear time

1. Syntax and SemanticsLanguage:¬,∧ ,G,H

Fϕ =df ¬G¬ϕ

Pϕ =df ¬H¬ϕ

A temporal frame (or flow of time) F=(T, ＜ ) ， where T is non-

empty，＜ is a binary relation which is irreflexive and transitive;

A valuation V: Ф→P(T) ; A model M=(F,V);

Satisfaction:

M, t ||- p iff t V(p), where p∈ ∈Ф,

M, t ||- ¬ϕ iff not M, t╟ϕ,

M, t ||- ϕ ψ∧ iff M, t╟ ϕ and M, t╟ ψ,

M, t ||- Gϕ iff for all s T, if t<s ∈ then M, s ||- ϕ,

M, t ||- Hϕ iff for all s T, if s<t ∈ then M, s ||- ϕ.

The definitions of validities are as usual.

2. The Minimal Logic KtAxioms：

(1) All classical propositional tautologies;

(2) G(p→q)→(Gp→Gq); and mirror-image;

(3) p→GPp; and mirror-image;

(4) Gp→GGp.

Rules： US: ϕ/ϕ[ψ/p]; MP: ϕ,ϕ→ψ/ψ; TG: ϕ/Gϕ; and ϕ/Hϕ.

The deduction is defined as usual.

Theorem 3.2.1

Kt is sound and complete for the class of all temporal

frames.

3. The IRR Rule

(¬p Gp)→∧ ϕ or alternatively, (H¬p∧ ¬p∧ Gp)→ϕ

ϕ ϕ

where p is an atom and does not appear in ϕ.

Lemma 3.3.1

IRR rule is valid on the class of all temporal frames.

Kt’=Kt+IRR

Theorem 3.3.2

Kt’ is sound and complete for the class of all temporal frames.

4. The Logic of Linear TimeLinearity: x y(x<y∀ ∀ ∨ x=y∨ y<x)

Formulas: a. Fp Fq→F(p Fq) F(p q) F(Fp q);∧ ∧ ∨ ∧ ∨ ∧

b. Pp Pq→P(p Pq) P(p q) P(Pp q);∧ ∧ ∨ ∧ ∨ ∧

Or c. PFp→(Pp p Fp); d.FPp→(Fp p Pp);∨ ∨ ∨ ∨

LTL=Kt+a+b(or +c+d).

Theorem 3.4.1

LTL is sound and complete for the class of all linear

temporal frames.

IV. Branching Time Logic

1. Branching time

2. Definitions of the F

3. The basic branching time logic

4. The logic of Peircean branching time

5. The logic of Ockhamist branching time

1. Branching TimeWhy consider branching time?

The argument for determinism:

1. p→ □p (ANP)

2. Fp→ □Fp

3. F¬p→ □F¬p

4. Fp F¬p (EMP)∨

5. Fp F¬p→ □Fp □F¬p∨ ∨

6. □Fp □F¬p∨

Definition 4.1.1

A treelike frame F= (T, ＜ ) is a temporal frame, where ＜ satisfying

the tree property: x y z(y<x∀ ∀ ∀ ∧ z<x→(y<z y=z z<y)).∨ ∨

s t

r

x

Definition 4.1.2

Where (T, ＜ ) is a treelike frame and t T, a ∈ branch (or

history) b is a maximal linearly ordered subset of T.

s t

r

x

2. Definitions of F

Why consider other definitions?

The Linear future :

M, t ||- Fϕ iff there exists s T, such that t<s ∈ and M, s ||- ϕ;

then

Fp F¬p is valid; F∨ np F∧ n¬p is satisfiable; {¬Pp, ¬p,¬Fp,PFp} is satisfiable.

Other choices:

The Peircean future :

M, t||- Fϕ iff for any branch b through t, there exists s b, such that t<s, and ∈

M, s ||- ϕ;

Then

Fp F¬p is invalid; p||-/PFp; ∨

The Ockhamist future:

M, t, b ||- p iff t V(p), where p∈ ∈Ф,

M, t, b ||- Fϕ iff there exists s b, such that t<s ∈ and M, s,b ||- ϕ.

Then

Fnp F∧ n¬p is invalid; {¬Pp, ¬p,¬Fp,PFp}is not satisfiable;

Fp F¬p is valid.∨

Supervaluation:

M, t||- ϕ iff for any branch b through t, we have M, t, b||- ϕ.

Then

Fnp F∧ n¬p is invalid; {¬Pp, ¬p,¬Fp,PFp}is not satisfiable;

Fp F¬p is valid.∨

AnalysisThe Linear future:

“it possibly will be case”, too weak;

The Peircean future:

“it necessarily will be the case ”, too strong;

The Ockhamist future:

“it will be the case in the actual future”, the most promising.

3. The Basic BTLBTL=Kt+b (or d)+IRR

Theorem 4.3.1

BTL is sound and complete for the class of all treelike

frames.

4. The logic of PBTLanguage:

G, H, F□;

The dual of F□ is defined as:

G◇ϕ=df.¬ F□¬ϕ.

Semantics:

Peircean frame is treelike frame.

For satisfaction, we only add:

M, t||- F□ϕ iff for any branches b through t, there exists t b, such ∈

that t<s and M, s ||- ϕ.

PBTL=BTL+the following axioms:

a. G (p→q)→(F□p→F□q)

b. Hp→Pp ; Gp→F□p

c. Gp→G◇ p

d. F□F□p→F□p

e. Hp→ (p→ (G◇ p→G◇ Hp))

f. F□Gp→GF□p

Theorem 4.4.1

PBTL is sound and complete for the class of all

endless Peircean frames.

Definition 4.4.2

A bundle B on a treelike frame is F=(T, ＜ ) is a collection

of branches through T containing at least one branch

through each t T.∈

Definition 4.4.3

We define weak satisfaction with respect to a bundle B

much as ordinary satisfaction was defined above, changing

only the last clause of the definition:

M, t||- F□ ϕ w.r.t. B iff for any branches b B through t ,there ∈

exists s b with t<s, such that ∈ M, s ||- ϕ w.r.t. B.

Definition 4.4.4

ϕ is weakly satisfiable if M, t||- ϕ w.r.t. B for some M, t and

B; ϕ is strongly valid if ¬ϕ is not weakly satisfiable.

5. The logic of OBTThe language:

G,H,□;

The dual of □ is defined as:

◇ϕ=df.¬ □¬ϕ.

F≤ϕ =df ϕ∨ Fϕ ， G≤ϕ =df ϕ∧ Gϕ ， P≤ϕ =df ϕ∨ Pϕ ，H≤ϕ

=df ϕ∧ Hϕ.

Semantics:

Ockhamist frame is a treelike frame.

We define satisfaction inductively:

M, t, b ||- p iff t V(p), where p∈ ∈Ф,

M, t, b ||- ¬ϕ iff not M, t, b╟ϕ,

M, t, b ||- ϕ ψ∧ iff M, t, b ╟ ϕ and M, t, b||-ψ,

M, t, b ||- Gϕ iff for all s T ∈ ， if s b and t<s ∈ then M, s,b ||- ϕ,

M, t, b ||- Hϕ iff for all s T ∈ ， if s<t then M, s,b||- ϕ.

M, t, b ||- □ϕ iff for all branches b’ T ⊆ ， if t b’ ∈ then M, t,b’ ||- ϕ.

Translation (ϕ)o from Peircean formulas to Ockhamist ones:

The only non-trivial clause of this map concerns the future

operators:

(fϕ)o = □Fϕo and (Gϕ)o = □Gϕo

It is straightforward to prove that for all tree models M, all points

t in M and all branches b with t b, we have that:∈

M, t||- ϕ iff M,t, b||- ϕo

Definition 4.5.1

Weakly satisfaction:

M, t, b||- □ϕ w.r.t. B iff for any branches b’ B, if ∈

t b’ ∈ then M, t,b’ ||- ϕ w.r.t. B.

Strong validity is defined similarly.

The Logic of strong Ockhamist validities(SOBTL):Axioms:

A0. All substitution instance of propositional tautologies; L1: G(α→β)→(Gα→Gβ) and mirror image; L2: Gα→GGα; L3: α→GPα and mirror image; L4: (Fα∧Fβ)(F(α∧Fβ) F(∨ α β∧ ) F(F∨ α β∧ )) and mirror image; BK: □(α→β)→(□α→□β); BT: □α→α; BE: ◇α→□◇α; HN: Pα→□P◇α; MB: G →□G ;⊥ ⊥

Rules: MP, GT, GN, IRR, and ANF: p→□p, for each atomic proposition p.

Theorem 4.5.2

SOBTL is sound and complete for the class of all strong

validities.

We’ve known that every strongly valid Ockhamist formula is

valid, but the converse is not right.

Counterexamples:

□G F□p→ GFp (Burgess, 1978);◇ ◇

GH□FP(H¬p ¬p Gp)→FP FP(¬p □Gp) (Nishimura,1979);∧ ∧ ◇ ∧

(p □GH(p→Fp))→GFp (Thomason,1984);∧

□G(p→ Fp)→ G(p→Fp) (Reynolds,2002).◇ ◇

All formulas above are valid but not strongly valid, so SOBTL is

incomplete for the class of all Ockhamist frames.

□G Fp→ GFp is valid but not strongly valid:◇ ◇

p

p

p

p

p

p

The logic of OBT:

OBTL=SOBTL+LC

Theorem? 4.5.3

OBTL is sound and complete for the class of all Ockhamist

frames.

)α◇F→α◇( ∧G→◇)α◇F→◇α◇(∧G□：LC 1ii1-n0i1ii

1-n0i

V. Conclusion

The most promising suggestion was given by

Reynolds, and if the completeness can be proved, the

long standing open problem gets closed eventually.

Open problems:

Ockhamist logic with until and since connectives;

Ockhamist logics over trees in which all histories have

particular properties such as denseness or being the real

numbers.

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4, pp. 679–697 2002.

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Historical Necessity ， to appear.

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