Mak Making Induction Manif ing Induction Manifest in est in - - PowerPoint PPT Presentation

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Mak Making Induction Manif ing Induction Manifest in est in - - PowerPoint PPT Presentation

May 22, 2023 34 likes •545 views

Mak Making Induction Manif ing Induction Manifest in est in Modular Modular ACL2 CL2 Carl Eastlund Matthias Felleisen cce@ccs.neu.edu matthias@ccs.neu.edu Northeastern University Boston, MA, USA 1 Pr Prog ogram V am Verif erifica

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Mak Making Induction Manif ing Induction Manifest in est in Modular Modular ACL2 CL2

Carl Eastlund Matthias Felleisen cce@ccs.neu.edu matthias@ccs.neu.edu Northeastern University Boston, MA, USA

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Pr Prog

  • gram V

am Verif erifica ication in A tion in ACL2 CL2

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Program (C, VHDL) Model (ACL2)

  • ↙
  • Test Suite

Formal Verification

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(defun setp (s) (no-duplicatesp-equal s)) (defun insert (x s) (add-to-set-eql x s)) (defthm insert-preserves-setp (implies (setp s) (setp (insert x s))))

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Termination Argument (Trivial)? ! (defun setp (s) (no-duplicatesp-equal s)) (defun insert (x s) (add-to-set-eql x s)) ! Rewrite Rule. Validity? ! (defthm insert-preserves-setp (implies (setp s) (setp (insert x s)))) ! Rewrite Rule.

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(defun join (l s) (if (endp l) s (insert (car l) (join (cdr l) s)))) (defthm join-preserves-setp (implies (and (true-listp l) (setp s)) (setp (join l s))))

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Termination Argument? ! (defun join (l s) (if (endp l) s (insert (car l) (join (cdr l) s)))) ! Rewrite Rule + Induction Scheme. Validity by Induction? ! (defthm join-preserves-setp (implies (and (true-listp l) (setp s)) (setp (join l s)))) ! Rewrite Rule.

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(defun setp (s) (no-duplicatesp-equal s)) (defun insert (x s) (add-to-set-eql x s)) (defthm insert-preserves-setp (implies (setp s) (setp (insert x s)))) (defun join (l s) (if (endp l) s (insert (car l) (join (cdr l) s)))) (defthm join-preserves-setp (implies (and (true-listp l) (setp s)) (setp (join l s))))

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(defun setp (s) (no-duplicatesp-equal s)) (defun insert (x s) (add-to-set-eql x s)) (defthm insert-preserves-setp (implies (setp s) (setp (insert x s)))) (defun join (l s) (if (endp l) s (insert (car l) (join (cdr l) s)))) (defthm join-preserves-setp (implies (and (true-listp l) (setp s)) (setp (join l s))))

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(defun join (l s) (if (endp l) s (insert (car l) (join (cdr l) s)))) (defthm join-preserves-setp (implies (and (true-listp l) (setp s)) (setp (join l s))))

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? ? ?

(defun join (l s) (if (endp l) s (insert (car l) (join (cdr l) s)))) (defthm join-preserves-setp (implies (and (true-listp l) (setp s)) (setp (join l s))))

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Tak aking a Pr ing a Prog

  • gram Apar

am Apart

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(interface Insert (sig setp (s)) (sig insert (x s)) (con insert-preserves-setp (implies (setp s) (setp (insert x s))))) (interface Join (extend Insert) (sig join (l s)) (con join-preserves-setp (implies (and (true-listp l) (setp s)) (setp (join l s)))))

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(module JoinMod (import Insert) (defun join (l s) (if (endp l) s (insert (car l) (join (cdr l) s)))) (export Join))

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(module JoinMod (import Insert) ! Names + Rewrite Rules. Termination Argument? ! (defun join (l s) (if (endp l) s (insert (car l) (join (cdr l) s)))) ! Rewrite Rule + Induction Scheme. Validity by Induction? ! (export Join))

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(interface BigStep (sig eval (e)) #|contracts|#) (interface SmallStep (sig step (e)) #|contracts|# (sig step-all (e)) #|contracts|#) (interface Equivalence (extend BigStep SmallStep) (con big-step=small-step (implies (expr-p e) (equal (eval e) (step-all e)))))

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(module SmallStepMod (defun step (e) ...) (defun step-all (e) (cond ((integerp e) e) ((calc-p e) (step-all (step e))))) (export SmallStep))

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(module SmallStepMod (defun step (e) ...) Termination Argument? ! (defun step-all (e) (cond ((integerp e) e) ((calc-p e) (step-all (step e))))) ! Rewrite Rule + Induction Scheme. Validity by Induction? ! (export SmallStep))

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(module EquivalenceMod (import BigStep SmallStep) (export Equivalence))

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(module EquivalenceMod (import BigStep SmallStep) ! Names + Rewrite Rules. Validity by Induction? ! (export Equivalence))

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(module EquivalenceMod (import BigStep SmallStep) ! Names + Rewrite Rules. Termination Argument? ! (defun recursion (e) (cond ((integerp e) nil) ((calc-p e) (recursion (step e))))) ! Rewrite Rule + Induction Scheme. Validity by Induction? ! (export Equivalence))

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(interface BigStep (sig eval (e)) #|contracts|#) (interface SmallStep (sig step (e)) #|contracts|# (sig step-all (e)) #|contracts|# ) (interface Equivalence (extend BigStep SmallStep) (con big-step=small-step (implies (expr-p e) (equal (eval e) (step-all e)))))

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(interface BigStep (sig eval (e)) #|contracts|#) (interface SmallStep (sig step (e)) #|contracts|# (sig step-all (e)) #|contracts|# (fun recursion (e) (cond ((integerp e) nil) ((calc-p e) (recursion (step e)))))) (interface Equivalence (extend BigStep SmallStep) (con big-step=small-step (implies (expr-p e) (equal (eval e) (step-all e)))))

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(interface BigStep (sig eval (e)) #|contracts|#) (interface SmallStep (sig step (e)) #|contracts|# (fun step-all (e) (cond ((integerp e) e) ((calc-p e) (step-all (step e))))) ) (interface Equivalence (extend BigStep SmallStep) (con big-step=small-step (implies (expr-p e) (equal (eval e) (step-all e)))))

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(module SmallStepMod (defun step (e) ...) Validity and Termination Argument? ! (export SmallStep) ! Names, Rewrite Rules, and Induction Scheme.)

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(module EquivalenceMod (import BigStep SmallStep) ! Names, Rewrite Rules, and Induction Scheme. Validity by Induction? ! (export Equivalence))

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(defun D (x) d) (defthm E e) (defun F (y) f) (defthm G g) (defun H (z) h) (defthm I i)

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(defun D (x) d) (defthm E e) (defun F (y) f) (defthm G g) (defun H (z) h) (defthm I i)

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(interface A (defun D (x) d) (defthm E e)) (interface B (extend A) (defun F (y) f) (defthm G g)) (interface C (extend A B) (defun H (z) h) (defthm I i))

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(interface A (fun D (x) d) (defthm E e)) (interface B (extend A) (fun F (y) f) (defthm G g)) (interface C (extend A B) (fun H (z) h) (defthm I i))

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(interface A (fun D (x) d) (con E e)) (interface B (extend A) (fun F (y) f) (con G g)) (interface C (extend A B) (fun H (z) h) (con I i))

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(interface A (fun D (x) d) (con E e)) (interface B (extend A) (fun F (y) f) (con G g)) (interface C (extend A B) (fun H (z) h) (con I i)) (module M (export A)) (module N (import A) (export B)) (module O (import A B) (export C))

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Lemma Lemma Modular Modular ACL2 CL2 Optimiz Optimized ed random/type 0.05s 0.05s 0.05s tick/type 0.01s 142.88s 2.00s tick/in-bounds 0.01s 136.67s 2.28s tick/uncrossed 0.02s 320.84s 2.29s

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Putting a Pr Putting a Prog

  • gram Bac

am Back Tog

  • gether

ether

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(link InsertJoinMod (InsertMod JoinMod)) (invoke InsertJoinMod) (join (list 1 2 3) (list 2 3 4))

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(module InsertJoinMod (defun setp (s) (no-duplicatesp-equal s)) (defun insert (x s) (add-to-set-eql x s)) (export Insert) (import Insert) (defun join (l s) (if (endp l) s (insert (car l) (join (cdr l) s)))) (export Join)) (invoke InsertJoinMod) (join (list 1 2 3) (list 2 3 4))

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(module InsertJoinMod (defun setp (s) (no-duplicatesp-equal s)) (defun insert (x s) (add-to-set-eql x s)) (export Insert) (import Insert) (defun join (l s) (if (endp l) s (insert (car l) (join (cdr l) s)))) (export Join)) (invoke InsertJoinMod) (join (list 1 2 3) (list 2 3 4))

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(module InsertJoinMod (defun setp (s) (no-duplicatesp-equal s)) (defun insert (x s) (add-to-set-eql x s)) (export Insert) (import Insert) (defun join (l s) (if (endp l) s (insert (car l) (join (cdr l) s)))) (export Join)) (invoke InsertJoinMod) (join (list 1 2 3) (list 2 3 4))

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(module InsertJoinMod (defun setp (s) (no-duplicatesp-equal s)) (defun insert (x s) (add-to-set-eql x s)) (export Insert) (import Insert) (defun join (l s) (if (endp l) s (insert (car l) (join (cdr l) s)))) (export Join)) (invoke InsertJoinMod) (join (list 1 2 3) (list 2 3 4))

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(module InsertJoinMod (defun setp (s) (no-duplicatesp-equal s)) (defun insert (x s) (add-to-set-eql x s)) (export Insert) (import Insert) (defun join (l s) (if (endp l) s (insert (car l) (join (cdr l) s)))) (export Join)) (invoke InsertJoinMod) (join (list 1 2 3) (list 2 3 4))

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(module InsertJoinMod (defun setp (s) (no-duplicatesp-equal s)) (defun insert (x s) (add-to-set-eql x s)) (export Insert) (import Insert) (defun join (l s) (if (endp l) s (insert (car l) (join (cdr l) s)))) (export Join)) (invoke InsertJoinMod) (join (list 1 2 3) (list 2 3 4))

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(module InsertJoinMod (defun setp (s) (no-duplicatesp-equal s)) (defun insert (x s) (add-to-set-eql x s)) (export Insert) (defun join (l s) (if (endp l) s (insert (car l) (join (cdr l) s)))) (export Join)) (invoke InsertJoinMod) (join (list 1 2 3) (list 2 3 4))

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(module M (export I)) (module N (import I) (export J))

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(module M (export I)) + (module N (import I) (export J)) = (link MN (M N))

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(module M (export I)) + (module N (import I) (export J)) = (module MN (export I) (export J))

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(module M (export I)) + (module N (import I) (export J)) = (module MN (export I) (export J)) I

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(module M (export I)) + (module N (import I) (export J)) = (module MN (export I) (export J)) I , I ⇒ J

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(module M (export I)) + (module N (import I) (export J)) = (module MN (export I) (export J)) I , I ⇒ J

  • I J

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Pr Prog

  • gram

am Modular Modular ACL2 CL2 Worm 135.40s 134.77s Interpreter 116.37s 115.67s Graph (DFS/NLG) 9.00s 9.03s Graph (DFS/ELG) 13.88s 13.82s Graph (BFS/NLG) 158.11s 158.19s Graph (BFS/ELG) 445.15s 444.28s

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Modular A Modular ACL2: CL2: sound, sound, expr xpressiv essive, and ef and efficient. icient.

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Thank Y hank You

  • u

Modular ACL2: http://www.ccs.neu.edu/~cce/acl2/

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