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Superpositionwith DatatypesandCodatatypes Jasmin Blanchette Vrije - PowerPoint PPT Presentation

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Superpositionwith DatatypesandCodatatypes Jasmin Blanchette Vrije Universiteit Amsterdam MPI-INF Saarbrcken Nicolas Peltier Universit Grenoble Alpes Simon Robillard Chalmers University of Technology Superposition with Datatypes and

  1. Superpositionwith DatatypesandCodatatypes Jasmin Blanchette Vrije Universiteit Amsterdam MPI-INF Saarbrücken Nicolas Peltier Université Grenoble Alpes Simon Robillard Chalmers University of Technology

  2. Superposition with Datatypes and Codatatypes Introduction (co)datatypes everywhere! program verification metatheory of programming languages formalization of mathematics . . . Typical application of ATPs 1 / 28

  3. Superposition with Datatypes and Codatatypes Introduction Partial axiomatization? 2 / 28

  4. Superposition with Datatypes and Codatatypes Introduction Partial axiomatization? ✗ Inconvenient 2 / 28

  5. Superposition with Datatypes and Codatatypes Introduction Partial axiomatization? ✗ Inconvenient ✗ Inefficient 2 / 28

  6. Superposition with Datatypes and Codatatypes Introduction Partial axiomatization? ✗ Inconvenient ✗ Inefficient ✗ Incomplete 2 / 28

  7. Superposition with Datatypes and Codatatypes Introduction Example (co)datatype τ = E : τ | F : τ → τ | G : α × τ → τ 3 / 28

  8. Superposition with Datatypes and Codatatypes Introduction Axioms for freely generated (co)datatypes Distinctness ∀ x , E �≈ F ( x ) ∀ ¯ x , F ( x 1 ) �≈ G ( x 2 , x 3 ) ∀ ¯ x , G ( x 1 , x 2 ) �≈ E 4 / 28

  9. Superposition with Datatypes and Codatatypes Introduction Axioms for freely generated (co)datatypes Distinctness ∀ x , E �≈ F ( x ) ∀ ¯ x , F ( x 1 ) �≈ G ( x 2 , x 3 ) ∀ ¯ x , G ( x 1 , x 2 ) �≈ E Injectivity ∀ ¯ x , F ( x 1 ) ≈ F ( x 2 ) → x 1 ≈ x 2 x , G ( x 1 , x ′ 1 ) ≈ G ( x 2 , x ′ 2 ) → x 1 ≈ x 2 ∧ x ′ 1 ≈ x ′ ∀ ¯ 2 4 / 28

  10. Superposition with Datatypes and Codatatypes Introduction Axioms for freely generated (co)datatypes Distinctness ∀ x , E �≈ F ( x ) ∀ ¯ x , F ( x 1 ) �≈ G ( x 2 , x 3 ) ∀ ¯ x , G ( x 1 , x 2 ) �≈ E Injectivity ∀ ¯ x , F ( x 1 ) ≈ F ( x 2 ) → x 1 ≈ x 2 x , G ( x 1 , x ′ 1 ) ≈ G ( x 2 , x ′ 2 ) → x 1 ≈ x 2 ∧ x ′ 1 ≈ x ′ ∀ ¯ 2 Exhaustivity ∀ x ∃ ¯ y , x ≈ E ∨ x ≈ F ( y 1 ) ∨ x ≈ G ( y 2 , y 3 ) 4 / 28

  11. Superposition with Datatypes and Codatatypes Introduction Acyclicity ∀ x , x �≈ F ( x ) ∀ x y , x �≈ G ( y , x ) 5 / 28

  12. Superposition with Datatypes and Codatatypes Introduction Acyclicity ∀ x , x �≈ F ( x ) ∀ x y , x �≈ G ( y , x ) ∀ x , x �≈ F ( F ( x )) ∀ x y , x �≈ F ( G ( y , x )) ∀ x y , x �≈ G ( y , F ( x )) ∀ x ¯ y , x �≈ G ( y 1 , G ( y 2 , x )) 5 / 28

  13. Superposition with Datatypes and Codatatypes Introduction Acyclicity ∀ x , x �≈ F ( x ) ∀ x y , x �≈ G ( y , x ) ∀ x , x �≈ F ( F ( x )) ∀ x y , x �≈ F ( G ( y , x )) ∀ x y , x �≈ G ( y , F ( x )) ∀ x ¯ y , x �≈ G ( y 1 , G ( y 2 , x )) ∀ x , x �≈ F ( F ( F ( x ))) ∀ x y , x �≈ F ( F ( G ( y , x ))) ∀ x y , x �≈ F ( G ( y , F ( x ))) ∀ x ¯ y , x �≈ F ( G ( y 1 , G ( y 2 , x ))) ∀ x y , x �≈ G ( y , F ( F ( F ( x )))) ∀ x ¯ y , x �≈ G ( y 1 , F ( F ( G ( y 2 , x )))) ∀ x ¯ y , x �≈ G ( y 1 , F ( G ( y 2 , F ( x )))) ∀ x ¯ y , x �≈ G ( y 1 , F ( G ( y 2 , G ( y 3 , x )))) ∀ x , x �≈ F ( F ( F ( F ( x )))) ∀ x y , x �≈ F ( F ( F ( G ( y , x )))) 5 / 28

  14. Superposition with Datatypes and Codatatypes Introduction Acyclicity ∀ x , x �≈ F ( x ) ∀ x y , x �≈ G ( y , x ) ∀ x , x �≈ F ( F ( x )) ∀ x y , x �≈ F ( G ( y , x )) ∀ x y , x �≈ G ( y , F ( x )) ∀ x ¯ y , x �≈ G ( y 1 , G ( y 2 , x )) ∀ x , x �≈ F ( F ( F ( x ))) ∀ x , x �≈ Γ[ x ] ∀ x y , x �≈ F ( F ( G ( y , x ))) ∀ x y , x �≈ F ( G ( y , F ( x ))) ∀ x ¯ y , x �≈ F ( G ( y 1 , G ( y 2 , x ))) ∀ x y , x �≈ G ( y , F ( F ( F ( x )))) ∀ x ¯ y , x �≈ G ( y 1 , F ( F ( G ( y 2 , x )))) ∀ x ¯ y , x �≈ G ( y 1 , F ( G ( y 2 , F ( x )))) ∀ x ¯ y , x �≈ G ( y 1 , F ( G ( y 2 , G ( y 3 , x )))) ∀ x , x �≈ F ( F ( F ( F ( x )))) ∀ x y , x �≈ F ( F ( F ( G ( y , x )))) 5 / 28

  15. Superposition with Datatypes and Codatatypes Introduction Codatatype fixpoints ∃ ! x , x ≈ Γ[ x ] 6 / 28

  16. Superposition with Datatypes and Codatatypes Introduction Codatatype fixpoints ∃ ! x , x ≈ Γ[ x ] Example s ≈ F ( G ( a , F ( s ))) ∧ t ≈ F ( G ( a , F ( t ))) implies s ≈ t 6 / 28

  17. Superposition with Datatypes and Codatatypes Axiomatic approach Solution 1 Conservative extension of the theory 7 / 28

  18. Superposition with Datatypes and Codatatypes Axiomatic approach Acyclicity Extra predicate sub ( s , t ) “ s is a subterm of t ” Recursive definition ∀ x , sub ( x , x ) ∀ xy , sub ( x , y ) → sub ( x , F ( y )) Acyclicity ∀ x , ¬ sub ( F ( x ) , x ) 8 / 28

  19. Superposition with Datatypes and Codatatypes Axiomatic approach Fixpoints Extra sort G ( • , E ) context = term with hole(s) Application function app : context × term → term Example app ( G ( • , E ) , F ( E )) ≈ G ( F ( E ) , E ) 9 / 28

  20. Superposition with Datatypes and Codatatypes Axiomatic approach Existence of fixpoints Extra function cyc : context → term ∀ x , cyc ( x ) ≈ app ( x , cyc ( x )) 10 / 28

  21. Superposition with Datatypes and Codatatypes Axiomatic approach Existence of fixpoints Extra function cyc : context → term ∀ x , cyc ( x ) ≈ app ( x , cyc ( x )) Example with x := G ( • , E ) cyc ( G ( • , E )) ≈ app ( G ( • , E ) , cyc ( G ( • , E ))) ≈ G ( cyc ( G ( • , E )) , E ) cyc (Γ) is the solution of y ≈ Γ[ y ] 10 / 28

  22. Superposition with Datatypes and Codatatypes Axiomatic approach Existence of fixpoints Extra function cyc : context → term ∀ x , cyc ( x ) ≈ app ( x , cyc ( x )) Example with x := G ( • , E ) cyc ( G ( • , E )) ≈ app ( G ( • , E ) , cyc ( G ( • , E ))) ≈ G ( cyc ( G ( • , E )) , E ) cyc (Γ) is the solution of y ≈ Γ[ y ] Uniqueness ∀ xy , y �≈ • ∧ x ≈ app ( y , x ) → x ≈ cyc ( y ) 10 / 28

  23. Superposition with Datatypes and Codatatypes Axiomatic approach Mutually recursive types (co)datatype α = E : α | F : β → α and β = G : α → β Solution Datatypes sub αα sub αβ sub βα sub ββ Codatatypes α β -contexts with holes for α β -terms 11 / 28

  24. Superposition with Datatypes and Codatatypes Axiomatic approach Completeness First-order theory ≈ No uninterpreted functions Complete, but not finitely axiomatizable Conservative extension Extra symbols ✓ Encode cyclicity properties ✗ Shouldn’t be used in conjecture 12 / 28

  25. Superposition with Datatypes and Codatatypes Axiomatic approach Conservative extension of the theory ✓ Complete ✓ Easy to implement But can we improve proof search? 13 / 28

  26. Superposition with Datatypes and Codatatypes Inference rules Solution 2 Dedicated inference rules 14 / 28

  27. Superposition with Datatypes and Codatatypes Inference rules Chains and cycles a ≈ F ( b ) 15 / 28

  28. Superposition with Datatypes and Codatatypes Inference rules Chains and cycles a ≈ F ( b ) b ≈ G ( F ( c ) , d ) 15 / 28

  29. Superposition with Datatypes and Codatatypes Inference rules Chains and cycles a ≈ F ( b ) b ≈ G ( F ( c ) , d ) c ≈ F ( h ( e )) 15 / 28

  30. Superposition with Datatypes and Codatatypes Inference rules Chains and cycles a ≈ F ( b ) b ≈ G ( F ( c ) , d ) c ≈ F ( h ( e )) h ( x ) ≈ G ( a , e ) 15 / 28

  31. Superposition with Datatypes and Codatatypes Inference rules Chains and cycles a ≈ F ( b ) b ≈ G ( F ( c ) , d ) c ≈ F ( h ( e )) h ( x ) ≈ G ( a , e ) a ≈ F ( G ( F ( F ( G ( a , e ))) , d ) under unifier { x ← e } 15 / 28

  32. Superposition with Datatypes and Codatatypes Inference rules The acyclicity rule s 1 ≈ Γ 1 [ s ′ s 2 ≈ Γ 2 [ s ′ s n ≈ Γ n [ s ′ 2 ] ∨ C 1 3 ] ∨ C 2 1 ] ∨ C n . . . ( C 1 ∨ C 2 ∨ · · · ∨ C n ) θ 16 / 28

  33. Superposition with Datatypes and Codatatypes Inference rules The acyclicity rule s 1 ≈ Γ 1 [ s ′ s 2 ≈ Γ 2 [ s ′ s n ≈ Γ n [ s ′ 2 ] ∨ C 1 3 ] ∨ C 2 1 ] ∨ C n . . . ( C 1 ∨ C 2 ∨ · · · ∨ C n ) θ mgu { s 1 ≈ s ′ 1 , . . . , s n ≈ s ′ n } 16 / 28

  34. Superposition with Datatypes and Codatatypes Inference rules The acyclicity rule maximal literals s 1 ≈ Γ 1 [ s ′ s 2 ≈ Γ 2 [ s ′ s n ≈ Γ n [ s ′ 2 ] ∨ C 1 3 ] ∨ C 2 1 ] ∨ C n . . . ( C 1 ∨ C 2 ∨ · · · ∨ C n ) θ mgu { s 1 ≈ s ′ 1 , . . . , s n ≈ s ′ n } 16 / 28

  35. Superposition with Datatypes and Codatatypes Inference rules The acyclicity rule maximal literals s 1 ≈ Γ 1 [ s ′ s 2 ≈ Γ 2 [ s ′ s n ≈ Γ n [ s ′ 2 ] ∨ C 1 3 ] ∨ C 2 1 ] ∨ C n . . . ( C 1 ∨ C 2 ∨ · · · ∨ C n ) θ mgu orientation of equalities { s 1 ≈ s ′ 1 , . . . , s n ≈ s ′ n } s i � � Γ i [ s i + 1 ] 16 / 28

  36. Superposition with Datatypes and Codatatypes Inference rules Trouble with the variables { t ≈ F ( x ) ∨ p ( x ) ??? 17 / 28

  37. Superposition with Datatypes and Codatatypes Inference rules Trouble with the variables unifier = { x ← t } t ≈ F ( x ) ∨ p ( x ) p ( t ) 17 / 28

  38. Superposition with Datatypes and Codatatypes Inference rules Trouble with the variables unifier = { x ← Γ[ t ] } t ≈ F ( x ) ∨ p ( x ) p (Γ[ t ]) 17 / 28

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