axiomatizations of temporal logic
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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; - PowerPoint PPT PresentationTRANSCRIPT
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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chronological modal logics? Journal of Philosophical Logic, 8, 469–475, 1979.
[07] A. Prior, Past, Present and Future, Oxford University Press, 1967.
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[09] M. Reynolds, An Axiomatization of Prior’s Ockhmist Logic of
Historical Necessity , to appear.
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Philosophical Logic, Vol II: Extensions of Classical Logic, D. Gabbay and
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