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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;

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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 approachesto the representation of temporal information within a logical framework;
• Narrowly:themodal-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 Semantics
• Language:¬,∧,G,H

Fϕ =df ¬G¬ϕ

Pϕ =df ¬H¬ϕ

Atemporal 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 ||- ¬ϕiff not M, t╟ϕ,
• M, t ||- ϕ∧ψ iff M, t╟ ϕ and M, t╟ ψ,
• M, t ||- Gϕiff for all s∈T, if t<s then M, s ||- ϕ,
• M, t ||- Hϕ iff for all s∈T, if s<t thenM, s ||- ϕ.
• The definitions of validities are as usual.
2. The Minimal Logic Kt
• Axioms：
• (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.

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 Time
• Linearity: ∀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 Time
• Why 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)).

x

s

t

r

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.

x

s

t

r

2. Definitions of F
• Why consider other definitions?
• The Linear future :
• M, t ||- Fϕiff there exists s∈T, such that t<s and M, s ||- ϕ;
• then
• Fp∨F¬p is valid; Fnp∧Fn¬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∧Fn¬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∧Fn¬p is invalid; {¬Pp, ¬p,¬Fp,PFp}is not satisfiable; Fp∨F¬p is valid.
Analysis
• The 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 BTL
• BTL=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 PBT
• Language:
• 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 OBT
• The 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 ||- ¬ϕiff 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 thenM, s,b||- ϕ.
• M, t, b ||- □ϕ iff for all branches b’⊆T，if t∈b’ thenM, 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. Biff for any branches b’ ∈B, if t∈b’ thenM, 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(Fα∧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, soSOBTLis incomplete for the class of all Ockhamist frames.

The logic of OBT:

• OBTL=SOBTL+LC
• Theorem? 4.5.3
• OBTL is sound and complete for the class of all Ockhamist frames.
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.
VI. References
• [01] J. Burgess, The Unreal Future, Theoria ,44, 157-179,1978.
• [02] J.Burgess, Decidability for Branching Time. StudiaLogica, 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.

[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.

[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.

[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. StudiaLogica, 49, 183–195, 1990.