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Quantification in Tail-recursive Function Definitions Sandip Ray Department of Computer Science University of Texas at Austin Email: sandip@cs.utexas.edu web: http://www.cs.utexas.edu/users/sandip U NIVERSITY OF T EXAS AT A USTIN D EPARTMENT OF C

  1. Quantification in Tail-recursive Function Definitions Sandip Ray Department of Computer Science University of Texas at Austin Email: sandip@cs.utexas.edu web: http://www.cs.utexas.edu/users/sandip U NIVERSITY OF T EXAS AT A USTIN

  2. D EPARTMENT OF C OMPUTER S CIENCES Prologue “ACL2 is a quantifier-free first order logic of recursive functions.” U NIVERSITY OF T EXAS AT A USTIN 1

  3. D EPARTMENT OF C OMPUTER S CIENCES Prologue “ACL2 is a quantifier-free first order logic of recursive functions.” The Truth: The syntax of ACL2 is quantifier-free, but ACL2 allows us to write quantified predicates via Skolemization. U NIVERSITY OF T EXAS AT A USTIN 2

  4. D EPARTMENT OF C OMPUTER S CIENCES Prologue “ACL2 is a quantifier-free first order logic of recursive functions.” The Truth: The syntax of ACL2 is quantifier-free, but ACL2 allows us to write quantified predicates via Skolemization. (defun-sk exists-foo (x) (exists y (foo x y))) U NIVERSITY OF T EXAS AT A USTIN 3

  5. D EPARTMENT OF C OMPUTER S CIENCES Prologue “ACL2 is a quantifier-free first order logic of recursive functions.” The Truth: The syntax of ACL2 is quantifier-free, but ACL2 allows us to write quantified predicates via Skolemization. (defun-sk exists-foo (x) (exists y (foo x y))) (= (exists-foo x) (foo x (foo-witness x))) (implies (foo x y) (exists-foo x)) U NIVERSITY OF T EXAS AT A USTIN 4

  6. D EPARTMENT OF C OMPUTER S CIENCES A Preliminary Illustration Consider defining a predicate true with the following axiom: (= (true x) (if (done x) t (forall x (true (st x))))) The equation is recursive, but in addition has quantification in the body. U NIVERSITY OF T EXAS AT A USTIN 5

  7. D EPARTMENT OF C OMPUTER S CIENCES A Preliminary Illustration Consider defining a predicate true with the following axiom: (= (true x) (if (done x) t (forall x (true (st x))))) The equation is recursive, but in addition has quantification in the body. ACL2 does not allow us to introduce definitional equations with both recursion and quantification. U NIVERSITY OF T EXAS AT A USTIN 6

  8. D EPARTMENT OF C OMPUTER S CIENCES A Preliminary Illustration But if the axiom is introduced would the resulting theory be inconsistent? No. U NIVERSITY OF T EXAS AT A USTIN 7

  9. D EPARTMENT OF C OMPUTER S CIENCES A Preliminary Illustration But if the axiom is introduced would the resulting theory be inconsistent? No. (encapsulate (((true *) => *)) (local (defun true (x) t)) (defthm true-satisfies-its-equation (= (true x) (if (done x) t (forall x (true (st x))))))) U NIVERSITY OF T EXAS AT A USTIN 8

  10. D EPARTMENT OF C OMPUTER S CIENCES A Preliminary Illustration But if the axiom is introduced would the resulting theory be inconsistent? No. (encapsulate (((true *) => *)) (local (defun true (x) t)) (defthm true-satisfies-its-equation (= (true x) (if (done x) t (forall x (true (st x))))))) ACL2 users have from time to time wanted some form of recursion and quantification together. U NIVERSITY OF T EXAS AT A USTIN 9

  11. D EPARTMENT OF C OMPUTER S CIENCES This Talk We show how to introduce in ACL2 a class of definitional axioms, called extended tail-recursive axioms , that contain both recursion and quantification. U NIVERSITY OF T EXAS AT A USTIN 10

  12. D EPARTMENT OF C OMPUTER S CIENCES This Talk We show how to introduce in ACL2 a class of definitional axioms, called extended tail-recursive axioms , that contain both recursion and quantification. � There is exactly one recursive branch. The defining equation of a predicate Q-iv is extended tail-recursive if � The outermost function call in the recursive branch is Q-iv , possibly enclosed by a sequence of quantifiers. U NIVERSITY OF T EXAS AT A USTIN 11

  13. D EPARTMENT OF C OMPUTER S CIENCES Admissibility of Extended Tail-recursive Definitions Why are extended tail-recursive definitions admissible? U NIVERSITY OF T EXAS AT A USTIN 12

  14. D EPARTMENT OF C OMPUTER S CIENCES Admissibility of Extended Tail-recursive Definitions Why are extended tail-recursive definitions admissible? (= (F-iv1 x) (if (done x) (base x) (forall i (F-iv1 (st1 x i))))) U NIVERSITY OF T EXAS AT A USTIN 13

  15. D EPARTMENT OF C OMPUTER S CIENCES Admissibility of Extended Tail-recursive Definitions Why are extended tail-recursive definitions admissible? (= (F-iv1 x) (if (done x) (base x) (forall i (F-iv1 (st1 x i))))) We view st1 as a transformation function that transforms an object x given a choice i . F-iv1 postulates an invariant over this transformation. U NIVERSITY OF T EXAS AT A USTIN 14

  16. D EPARTMENT OF C OMPUTER S CIENCES Admissibility of Extended Tail-recursive Definitions Why are extended tail-recursive definitions admissible? (= (F-iv1 x) (if (done x) (base x) (forall i (F-iv1 (st1 x i))))) We view st1 as a transformation function that transforms an object x given a choice i . F-iv1 postulates an invariant over this transformation. If (done x) holds the invariant is equal to (base x) . Otherwise the invariant holds for x if and only if it holds for each successor. U NIVERSITY OF T EXAS AT A USTIN 15

  17. D EPARTMENT OF C OMPUTER S CIENCES Admissibility of Extended Tail-recursive Definitions We can introduce the equation by defining a witnessing invariant that posits the same thing a little differently. (defun sn1 (x ch) (if (endp ch) x (sn1 (st1 x (car ch)) (cdr ch)))) (defun n-done (x ch) (if (endp ch) (not (done ch)) (and (not (done x)) (n-done (st1 x (car ch)) (cdr ch))))) (defun done-ch1 (x ch) (and (done (sn1 x ch)) (implies (consp ch) (n-done x (dellast ch))))) (defun-sk F-iv1 (x) (forall ch (implies (done-ch1 x ch) (base (sn1 x ch))))) U NIVERSITY OF T EXAS AT A USTIN 16

  18. D EPARTMENT OF C OMPUTER S CIENCES Admissibility of Extended Tail-recursive Definitions Consider a variant of the above equation. (= (E-iv1 x) (if (done x) (base x) (exists i (E-iv1 (st1 x i))))) U NIVERSITY OF T EXAS AT A USTIN 17

  19. D EPARTMENT OF C OMPUTER S CIENCES Admissibility of Extended Tail-recursive Definitions Consider a variant of the above equation. (= (E-iv1 x) (if (done x) (base x) (exists i (E-iv1 (st1 x i))))) We can introduce the equation the same way as above. ... (defun-sk E-iv1 (x) (exists ch (and (done-ch1 x ch) (sn1 x ch)))) U NIVERSITY OF T EXAS AT A USTIN 18

  20. D EPARTMENT OF C OMPUTER S CIENCES Summing Up the Witnesses (= (F-iv1 x) (if (done x) (base x) (forall i (F-iv1 (st1 x i))))) The witnessing predicate: “For each sequence ch of choices, such the first descendant of x that satisfies done also satisfies base .” Can be expressed in ACL2. U NIVERSITY OF T EXAS AT A USTIN 19

  21. D EPARTMENT OF C OMPUTER S CIENCES Summing Up the Witnesses (= (E-iv1 x) (if (done x) (base x) (exists i (F-iv1 (st x i))))) The witnessing predicate: “There exists a sequence ch of choices, such that the first descendant of x that satisfies done also satisfies base .” Can be expressed in ACL2. U NIVERSITY OF T EXAS AT A USTIN 20

  22. D EPARTMENT OF C OMPUTER S CIENCES Summing Up the Witnesses (= (EF-iv2 x) (if (done x) (base x) (exists i (forall j (F-iv1 (st2 x i j)))))) The witnessing predicate: “ There exists a sequence i-ch of i choices, such that for each sequence j-ch of j choices, the first descendant of x that satisfies done also satisfies base .” Can be expressed in ACL2. U NIVERSITY OF T EXAS AT A USTIN 21

  23. D EPARTMENT OF C OMPUTER S CIENCES Summing Up the Witnesses (= (iv0 x) (if (done x) (base x) (iv0 (st0 x i)))))) The witnessing predicate: “The first descendant of x that satisfies done also satisfies base .” This is essentially the witnessed designed by Manolios and Moore (2000) , to show that tail-recursive equations can always be introduced in ACL2. U NIVERSITY OF T EXAS AT A USTIN 22

  24. D EPARTMENT OF C OMPUTER S CIENCES Logical Impediments We cannot allow arbitrary recursion and quantification. Doing so will violate conservativity. Acknowledgement: This proof is due to an example provided by Matt Kaufmann. (Thanks, Matt!) 1. A truth predicate of Peano arithmetic is not conservative over Peano Arithmetic. 2. If we have both recursion and quantification then we can define a predicate true-formula in ACL2. 3. We can then prove by induction that true-formula holds for all formulas that are provable. 4. Details are in the paper. U NIVERSITY OF T EXAS AT A USTIN 23

  25. D EPARTMENT OF C OMPUTER S CIENCES Upshot of Logical Impediments It is possible to define true-formula if we allow two recursive branches and quantification. Therefore in general a recursive definition containing quantification and more than one recursive branch is not conservative. U NIVERSITY OF T EXAS AT A USTIN 24

  26. D EPARTMENT OF C OMPUTER S CIENCES A Potential Application Moore (2003) showed how to use inductive assertions on operationally modeled sequential programs. U NIVERSITY OF T EXAS AT A USTIN 25

  27. D EPARTMENT OF C OMPUTER S CIENCES A Potential Application Moore (2003) showed how to use inductive assertions on operationally modeled sequential programs. (= (inv s) (if (cutpoint s) (assertion s) (inv (step s)))) Attempting to prove (implies (inv s) (inv (step s))) causes symbolic simulation of the operational semantics from each cutpoint. U NIVERSITY OF T EXAS AT A USTIN 26

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