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A Proof from 1988 that PDL has Interpolation? (1) EDV-Beratung Manfred Borzechowski, Berlin, Germany (2) University of Groningen, Groningen, The Netherlands Advances in Modal Logic 2020 2020-08-26 15:00 1 Manfred Borzechowski (1) & Malvin

  1. A Proof from 1988 that PDL has Interpolation? (1) EDV-Beratung Manfred Borzechowski, Berlin, Germany (2) University of Groningen, Groningen, The Netherlands Advances in Modal Logic 2020 2020-08-26 15:00 1 Manfred Borzechowski (1) & Malvin Gattinger (2)

  2. Propositional Dynamic Logic (PDL) x: π‘Ÿ β„³, π‘₯ ⊨ ⟨𝐡⟩(βŸ¨π΅βŸ©Β¬π‘Ÿ ∧ ⟨𝐢⟩[𝐢 βˆ— ]π‘Ÿ) β„³, π‘₯ ⊨ [𝐢 βˆ— ]π‘Ÿ β„³, π‘₯ ⊨ [𝐢]π‘Ÿ β„³, π‘₯ ⊨ ⟨𝐡; πΆβŸ©π‘Ÿ B B A A w: π‘Ÿ Syntax v: π‘ž Example Also expressible: β€œif…then…else…” and β€œwhile…do…” ∢∢= 𝑏 π‘ž ∣ ¬𝜚 ∣ 𝜚 ∧ 𝜚 ∣ [𝑏]𝜚 ∢∢= 𝜚 2 𝐡 ∣ 𝑏; 𝑏 ∣ 𝑏 βˆͺ 𝑏 ∣ 𝑏 βˆ— ∣ 𝜚?

  3. Propositional Dynamic Logic (PDL) x: π‘Ÿ β„³, π‘₯ ⊨ ⟨𝐡⟩(βŸ¨π΅βŸ©Β¬π‘Ÿ ∧ ⟨𝐢⟩[𝐢 βˆ— ]π‘Ÿ) β„³, π‘₯ ⊨ [𝐢 βˆ— ]π‘Ÿ β„³, π‘₯ ⊨ [𝐢]π‘Ÿ β„³, π‘₯ ⊨ ⟨𝐡; πΆβŸ©π‘Ÿ B B A A w: π‘Ÿ Syntax v: π‘ž Example Also expressible: β€œif…then…else…” and β€œwhile…do…” ∢∢= 𝑏 π‘ž ∣ ¬𝜚 ∣ 𝜚 ∧ 𝜚 ∣ [𝑏]𝜚 ∢∢= 𝜚 2 𝐡 ∣ 𝑏; 𝑏 ∣ 𝑏 βˆͺ 𝑏 ∣ 𝑏 βˆ— ∣ 𝜚?

  4. Craig Interpolation πœ„ ⟢ (π‘Ÿ ∨ 𝑠) π‘Ÿ (π‘ž ∧ π‘Ÿ) Example (1918 – 2016) William Craig ⟢ πœ” . We then write 𝜚 𝑀(𝜚) ∢= {π‘ž ∣ π‘ž occurs in 𝜚} - 𝑀(πœ„) βŠ† 𝑀(𝜚) ∩ 𝑀(πœ”) πœ„ β†’ πœ” is valid - is valid - 𝜚 β†’ πœ„ A logic has Craig Interpolation iff for any valid Definition 3 𝜚 β†’ πœ” there is an interpolant πœ„ such that:

  5. Logics that (we know) have Craig Interpolation β€’ Propositional Logic β€’ First-Order Logic β€’ Intuitionistic Logic β€’ Basic and Multi-modal logic (MadarΓ‘sz 1995) β€’ 𝜈 -Calculus (D’Agostino and Hollenberg 2000) What about PDL? Language of a formula := all atomic propositions and programs . Example [(𝐡 βˆͺ 𝐢) βˆ— ](π‘ž ∧ π‘Ÿ) [𝐢 βˆ— ]π‘Ÿ ⟢ [(𝐢; 𝐢) βˆ— ](π‘Ÿ ∨ [𝐷]𝑠) Problem: How to find interpolants for βˆ— systematically? 4

  6. Logics that (we know) have Craig Interpolation β€’ Propositional Logic β€’ First-Order Logic β€’ Intuitionistic Logic β€’ Basic and Multi-modal logic (MadarΓ‘sz 1995) β€’ 𝜈 -Calculus (D’Agostino and Hollenberg 2000) What about PDL? Language of a formula := all atomic propositions and programs . Example [(𝐡 βˆͺ 𝐢) βˆ— ](π‘ž ∧ π‘Ÿ) [𝐢 βˆ— ]π‘Ÿ ⟢ [(𝐢; 𝐢) βˆ— ](π‘Ÿ ∨ [𝐷]𝑠) Problem: How to find interpolants for βˆ— systematically? 4

  7. History β€’ Daniel Leivant: Proof theoretic methodology for propositional dynamic logic . Conference paper in LNCS, 1981. β€’ Manfred Borzechowski: Tableau–KalkΓΌl fΓΌr PDL und Interpolation . Diploma thesis, FU Berlin, 1988. β€’ Tomasz Kowalski: PDL has interpolation . Journal of Symbolic Logic, 2002. Revoked in 2004. β€’ Marcus Kracht: Chapter The open question in Tools and Techniques in Modal Logic , 1999. 5

  8. History β€’ Daniel Leivant: Proof theoretic methodology for propositional dynamic logic . Conference paper in LNCS, 1981. β€’ Manfred Borzechowski: Tableau–KalkΓΌl fΓΌr PDL und Interpolation . Diploma thesis, FU Berlin, 1988. β€’ Tomasz Kowalski: PDL has interpolation . Journal of Symbolic Logic, 2002. Revoked in 2004. β€’ Marcus Kracht: Chapter The open question in Tools and Techniques in Modal Logic , 1999. 5

  9. History β€’ Daniel Leivant: Proof theoretic methodology for propositional dynamic logic . Conference paper in LNCS, 1981. β€’ Manfred Borzechowski: Tableau–KalkΓΌl fΓΌr PDL und Interpolation . Diploma thesis, FU Berlin, 1988. β€’ Tomasz Kowalski: PDL has interpolation . Journal of Symbolic Logic, 2002. Revoked in 2004. β€’ Marcus Kracht: Chapter The open question in Tools and Techniques in Modal Logic , 1999. 5

  10. Borzechowski 1988: Why look at it now? β€’ It seems it was never really published. β€’ We now make an English translation available. β€’ Kracht (1999): not β€œpossible to verify the argument” 6

  11. Outline of the proof attempt 1. Define a tableaux system. 2. Show soundness and completeness. 3. Define interpolants for each node β€œbottom-up”. Problems caused by βˆ— : β€’ How to ensure finite tableaux? β€’ How to define interpolants for βˆ— steps? 7

  12. Tableaux Rules: Part 1/2 (Β¬?) π‘Œ; 𝑄; [𝑏][𝑏 (π‘œ) ]𝑄 (π‘œ) π‘Œ; [𝑏 βˆ— ]𝑄 ∣ π‘Œ; Β¬[𝑏][𝑏 (π‘œ) ]𝑄 π‘Œ; ¬𝑄 (Β¬π‘œ) π‘Œ; Β¬[𝑏 βˆ— ]𝑄 π‘Œ; [𝑏; 𝑐]𝑄 π‘Œ; [𝑅?]𝑄 π‘Œ; [𝑏 βˆͺ 𝑐]𝑄 π‘Œ; Β¬[𝑏; 𝑐]𝑄 classical rules: π‘Œ; 𝑅; ¬𝑄 π‘Œ; Β¬[𝑅?]𝑄 π‘Œ; Β¬(𝑄 ∧ 𝑅) π‘Œ; ¬¬𝑄 (Β¬) π‘Œ; 𝑄 π‘Œ; 𝑄 ∧ 𝑅 ( ∧ ) π‘Œ; 𝑄; 𝑅 8 ∣ π‘Œ; Β¬[𝑐]𝑄 ∣ π‘Œ; ¬𝑅 local rules: π‘Œ; Β¬[𝑏 βˆͺ 𝑐]𝑄 (¬∧) π‘Œ; ¬𝑄 (Β¬βˆͺ) π‘Œ; Β¬[𝑏]𝑄 (Β¬; ) π‘Œ; Β¬[𝑏][𝑐]𝑄 (βˆͺ) π‘Œ; [𝑏]𝑄; [𝑐]𝑄 (?) 𝑦; ¬𝑅 ∣ π‘Œ; 𝑄 (; ) π‘Œ; [𝑏][𝑐]𝑄

  13. Tableaux Rules: Part 2/2 (𝐡𝑒) where (…) (Β¬?) (Β¬π‘œ) (Β¬; ) (Β¬βˆͺ) PDL rules: marked rules: the critical rule. 9 π‘Œ free π‘Œ; Β¬[𝑏]𝑄 (π‘βˆ’) the loading rule , π‘Œ; Β¬[𝑏 0 ] … [𝑏 π‘œ ]𝑄 (𝑁+) the liberation rule , π‘Œ; Β¬[𝑏 0 ] … [𝑏 π‘œ ]𝑄 𝑄 π‘Œ; Β¬[𝑏]𝑄 𝑆 π‘Œ; Β¬[𝐡]𝑄 𝑆 π‘Œ 𝐡 ; ¬𝑄 𝑆\𝑄 π‘Œ; Β¬[𝑏 βˆͺ 𝑐]𝑄 𝑆 π‘Œ; Β¬[𝑏; 𝑐]𝑄 𝑆 π‘Œ; Β¬[𝑏]𝑄 𝑆 ∣ π‘Œ; Β¬[𝑐]𝑄 𝑆 π‘Œ; Β¬[𝑏][𝑐]𝑄 𝑆 π‘Œ; Β¬[𝑏 βˆ— ]𝑄 𝑆 π‘Œ; Β¬[𝑅?]𝑄 𝑆 π‘Œ; ¬𝑄 𝑆\𝑄 ∣ π‘Œ; Β¬[𝑏][𝑏 (π‘œ) ]𝑄 𝑆 π‘Œ; 𝑅; ¬𝑄 𝑆\𝑄 𝑆\𝑄 indicates that 𝑆 is removed iff 𝑆 = 𝑄 .

  14. Tableaux Rules: Extra Conditions 1. On reaching π‘Œ; Β¬[𝐡]𝑄 or π‘Œ; [𝐡]𝑄 , change (π‘œ) back to βˆ— in 𝑄 . 2. Instead of π‘Œ; [𝑏 (π‘œ) ]𝑄 we always obtain π‘Œ . 3. A rule must be applied to an π‘œ -formula whenever it is possible. 4. No rule may be applied to a Β¬[𝑏 (π‘œ) ] -node. 5. To a node obtained using (𝑁+) we may not apply (π‘βˆ’) . 6. If a normal node 𝑒 has a predecessor 𝑑 with the same formulas and the path 𝑑 … 𝑒 uses (𝐡𝑒) and is loaded if 𝑑 is loaded, then 𝑒 is an end node. 7. Every loaded node that is not an end note by 6 has a successor. 10

  15. Tableaux Rules: Extra Conditions 1. On reaching π‘Œ; Β¬[𝐡]𝑄 or π‘Œ; [𝐡]𝑄 , change (π‘œ) back to βˆ— in 𝑄 . 2. Instead of π‘Œ; [𝑏 (π‘œ) ]𝑄 we always obtain π‘Œ . 3. A rule must be applied to an π‘œ -formula whenever it is possible. 4. No rule may be applied to a Β¬[𝑏 (π‘œ) ] -node. 5. To a node obtained using (𝑁+) we may not apply (π‘βˆ’) . path 𝑑 … 𝑒 uses (𝐡𝑒) and is loaded if 𝑑 is loaded, then 𝑒 is an end node. 7. Every loaded node that is not an end note by 6 has a successor. 10 6. If a normal node 𝑒 has a predecessor 𝑑 with the same formulas and the

  16. A Full Proof Example (Β¬π‘œ) x (Β¬?) π‘ž ; Β¬[(𝐡 βˆͺ π‘ž?) x (𝐡𝑒) Β¬[(𝐡 βˆͺ π‘ž?) (Β¬βˆͺ) Β¬[π‘ž?][(𝐡 βˆͺ π‘ž?) (Β¬βˆͺ) Β¬[(𝐡 βˆͺ π‘ž?) Β¬[𝐡][(𝐡 βˆͺ π‘ž?) 11 (𝑁+) x βˆ— ]π‘Ÿ ; [𝐡 βˆ— ]π‘Ÿ (Β¬π‘œ)1. Β¬π‘Ÿ ; π‘Ÿ ; [𝐡][𝐡 βˆ— ]π‘Ÿ Β¬[(𝐡 βˆͺ π‘ž?) (π‘œ) Β¬[(𝐡 βˆͺ π‘ž?)][(𝐡 βˆͺ π‘ž?) Β¬[(𝐡 βˆͺ π‘ž?) βˆ— ]π‘Ÿ π‘Ÿ ; [𝐡 βˆ— ]π‘Ÿ βˆ— ]π‘Ÿ π‘Ÿ ; π‘Ÿ ; [𝐡][𝐡 βˆ— ]π‘Ÿ (π‘œ) ]π‘Ÿ π‘Ÿ ; π‘Ÿ ; [𝐡][𝐡 βˆ— ]π‘Ÿ βˆ— ]π‘Ÿ π‘Ÿ ; π‘Ÿ ; [𝐡][𝐡 βˆ— ]π‘Ÿ (π‘œ) ]π‘Ÿ π‘Ÿ ; π‘Ÿ ; [𝐡][𝐡 βˆ— ]π‘Ÿ π‘Ÿ βˆ— ]π‘Ÿ π‘Ÿ ; [𝐡 βˆ— ] (π‘œ) ]π‘Ÿ π‘Ÿ ; π‘Ÿ ; [𝐡][𝐡 βˆ— ]π‘Ÿ

  17. A Full Proof Example (Β¬π‘œ) closed x (Β¬?) π‘ž ; Β¬[(𝐡 βˆͺ π‘ž?) x (𝐡𝑒) Β¬[(𝐡 βˆͺ π‘ž?) (Β¬βˆͺ) Β¬[π‘ž?][(𝐡 βˆͺ π‘ž?) (Β¬βˆͺ) Β¬[(𝐡 βˆͺ π‘ž?) Β¬[𝐡][(𝐡 βˆͺ π‘ž?) 11 βˆ— ]π‘Ÿ ; [𝐡 βˆ— ]π‘Ÿ x (Β¬π‘œ)1. Β¬[(𝐡 βˆͺ π‘ž?) Β¬π‘Ÿ ; π‘Ÿ ; [𝐡][𝐡 βˆ— ]π‘Ÿ (π‘œ) Β¬[(𝐡 βˆͺ π‘ž?)][(𝐡 βˆͺ π‘ž?) Β¬[(𝐡 βˆͺ π‘ž?) (𝑁+) βˆ— ]π‘Ÿ π‘Ÿ ; [𝐡 βˆ— ]π‘Ÿ βˆ— ]π‘Ÿ π‘Ÿ ; π‘Ÿ ; [𝐡][𝐡 βˆ— ]π‘Ÿ (π‘œ) ]π‘Ÿ π‘Ÿ ; π‘Ÿ ; [𝐡][𝐡 βˆ— ]π‘Ÿ βˆ— ]π‘Ÿ π‘Ÿ ; π‘Ÿ ; [𝐡][𝐡 βˆ— ]π‘Ÿ (π‘œ) ]π‘Ÿ π‘Ÿ ; π‘Ÿ ; [𝐡][𝐡 βˆ— ]π‘Ÿ π‘Ÿ βˆ— ]π‘Ÿ π‘Ÿ ; [𝐡 βˆ— ] (π‘œ) ]π‘Ÿ π‘Ÿ ; π‘Ÿ ; [𝐡][𝐡 βˆ— ]π‘Ÿ

  18. A Full Proof Example (Β¬π‘œ) closed by condition 6 x (Β¬?) π‘ž ; Β¬[(𝐡 βˆͺ π‘ž?) x (𝐡𝑒) Β¬[(𝐡 βˆͺ π‘ž?) (Β¬βˆͺ) Β¬[π‘ž?][(𝐡 βˆͺ π‘ž?) (Β¬βˆͺ) Β¬[(𝐡 βˆͺ π‘ž?) Β¬[𝐡][(𝐡 βˆͺ π‘ž?) 11 βˆ— ]π‘Ÿ ; [𝐡 βˆ— ]π‘Ÿ x (Β¬π‘œ)1. Β¬[(𝐡 βˆͺ π‘ž?) Β¬π‘Ÿ ; π‘Ÿ ; [𝐡][𝐡 βˆ— ]π‘Ÿ (π‘œ) Β¬[(𝐡 βˆͺ π‘ž?)][(𝐡 βˆͺ π‘ž?) Β¬[(𝐡 βˆͺ π‘ž?) (𝑁+) βˆ— ]π‘Ÿ π‘Ÿ ; [𝐡 βˆ— ]π‘Ÿ βˆ— ]π‘Ÿ π‘Ÿ ; π‘Ÿ ; [𝐡][𝐡 βˆ— ]π‘Ÿ (π‘œ) ]π‘Ÿ π‘Ÿ ; π‘Ÿ ; [𝐡][𝐡 βˆ— ]π‘Ÿ βˆ— ]π‘Ÿ π‘Ÿ ; π‘Ÿ ; [𝐡][𝐡 βˆ— ]π‘Ÿ (π‘œ) ]π‘Ÿ π‘Ÿ ; π‘Ÿ ; [𝐡][𝐡 βˆ— ]π‘Ÿ π‘Ÿ βˆ— ]π‘Ÿ π‘Ÿ ; [𝐡 βˆ— ] (π‘œ) ]π‘Ÿ π‘Ÿ ; π‘Ÿ ; [𝐡][𝐡 βˆ— ]π‘Ÿ

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