axiomatizations of temporal logic
DESCRIPTION
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
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10723029
Xu Zhaoqing
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I. Content
Introduction
Basic temporal logic
Branching time logic
Conclusions
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II. Introduction
Temporal Logic
Broadly: all approaches to the representation of temporal
information within a logical framework;
Narrowly: the modal-style of temporal logic;
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III. Basic Temporal Logic
1. Syntax and semantics
2. The Minimal logic Kt
3. The IRR rule
4. The logic of linear time
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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);
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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.
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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.
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Theorem 3.2.1
Kt is sound and complete for the class of all temporal
frames.
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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.
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Lemma 3.3.1
IRR rule is valid on the class of all temporal frames.
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Kt’=Kt+IRR
Theorem 3.3.2
Kt’ is sound and complete for the class of all temporal frames.
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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);∨ ∨ ∨ ∨
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LTL=Kt+a+b(or +c+d).
Theorem 3.4.1
LTL is sound and complete for the class of all linear
temporal frames.
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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
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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∨
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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
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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
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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.
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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; ∨
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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.∨
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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.∨
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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.
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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.
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4. The logic of PBTLanguage:
G, H, F□;
The dual of F□ is defined as:
G◇ϕ=df.¬ F□¬ϕ.
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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.
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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
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Theorem 4.4.1
PBTL is sound and complete for the class of all
endless Peircean frames.
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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.∈
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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.
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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.
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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.
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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.
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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
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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.
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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.
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Theorem 4.5.2
SOBTL is sound and complete for the class of all strong
validities.
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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.
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□G Fp→ GFp is valid but not strongly valid:◇ ◇
p
p
p
p
p
p
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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
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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.
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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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VI. References [01] J. Burgess, The Unreal Future, Theoria ,44, 157-179,1978.
[02] J.Burgess, Decidability for Branching Time. Studia Logica, 39, 203–218,
1980.
[03] D.Gabbay, I. Hodkinson, and M. Reynolds, Temporal Logic: Mathematical
Foundations and Computational Aspects, Volume 1. Oxford University Press, 1994.
[04] R. Goldblatt. Logics of Time and Computation. CSLI Lecture Notes. Center for
the Study of Language and Information, Stanford University, second edition, 1987.
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[05] Y. Gurevich and S. Shelah. The decision problem for branching time logic. In
The Journal of Symbolic Logic, 50, 668-681,1985.
[06] H. Nishimura. Is the semantics of branching structures adequate for
chronological modal logics? Journal of Philosophical Logic, 8, 469–475, 1979.
[07] A. Prior, Past, Present and Future, Oxford University Press, 1967.
[08] M. Reynolds. Axioms for branching time. Logic and Computation, Vol. 12 No.
4, pp. 679–697 2002.
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[09] M. Reynolds, An Axiomatization of Prior’s Ockhmist Logic of
Historical Necessity , to appear.
[10] R. Thomason, Indeterminist Time and Truth-value Gaps. Theoria, 36,
264–281, 1970.
[11] R. Thomason. Combinations of tense and modality. In Handbook of
Philosophical Logic, Vol II: Extensions of Classical Logic, D. Gabbay and
F. Guenthner, eds, pp. 135–165. Reidel, Dordrecht, 1984.
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[12] Y. Venema, Temporal Logic, in The Blackwell Guide to Philosophical
Logic, Blackwell publishers, 2001.
[13] A. Zanardo. A finite axiomatization of the set of strongly valid
Ockamist formulas. Journal of Philosophical Logic, 14, 447–468, 1985.
[14] A. Zanardo. Axiomatization of ‘Peircean’ branching-time logic.
Studia Logica, 49, 183–195, 1990.
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