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Axiomatizations of Temporal Logic

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

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  1. Axiomatizations of Temporal Logic 10723029 Xu Zhaoqing

  2. I. Content • Introduction • Basic temporal logic • Branching time logic • Conclusions

  3. II. Introduction • Temporal Logic • Broadly:all approachesto the representation of temporal information within a logical framework; • Narrowly:themodal-style of temporal logic;

  4. III. Basic Temporal Logic • 1. Syntax and semantics • 2. The Minimal logic Kt • 3. The IRR rule • 4. The logic of linear time

  5. 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);

  6. 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.

  7. 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.

  8. Theorem 3.2.1 • Kt is sound and complete for the class of all temporal frames.

  9. 3. The IRRRule

  10. Lemma 3.3.1 • IRR rule is valid on the class of all temporal frames.

  11. Kt’=Kt+IRR • Theorem 3.3.2 • Kt’ is sound and complete for the class of all temporal frames.

  12. 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);

  13. LTL=Kt+a+b(or +c+d). • Theorem 3.4.1 • LTL is sound and complete for the class of all linear temporal frames.

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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.

  19. 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;

  20. 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.

  21. 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.

  22. 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.

  23. 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.

  24. 4. The logic of PBT • Language: • G, H, F□; • The dual of F□ is defined as: G◇ϕ=df.¬ F□¬ϕ.

  25. 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.

  26. 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

  27. Theorem 4.4.1 • PBTL is sound and complete for the class of all endless Peircean frames.

  28. 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.

  29. 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.

  30. 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.

  31. 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.

  32. 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.

  33. 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

  34. 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.

  35. 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.

  36. Theorem 4.5.2 • SOBTL is sound and complete for the class of all strong validities.

  37. 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.

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

  39. The logic of OBT: • OBTL=SOBTL+LC • Theorem? 4.5.3 • OBTL is sound and complete for the class of all Ockhamist frames.

  40. 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.

  41. 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.

  42. 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.

  43. [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.

  44. [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.

  45. [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.

  46. Thank you!

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