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Logosphere A Digital Library of Formal Proof Carsten Schrmann Processor Verification INTEL (HOL/HOL light). [John Harrison] $500mio Pentium bug. AMD (ACL2, Nqthm). [Matt Kaufmann] Siemens, Microsoft (ASM). [Yuri

  1. Logosphere A Digital Library of Formal Proof Carsten Schürmann

  2. Processor Verification • INTEL (HOL/HOL light). [John Harrison] • $500mio Pentium bug. • AMD (ACL2, Nqthm). [Matt Kaufmann] • Siemens, Microsoft (ASM). [Yuri Gurevich]

  3. What’s Intel up to? HOL [Harrison’03] Actual requirements Mathematical specification Floating Point arithmetic. Mathematical model Actual system Round the same way if lthough analyzing this condition combinatorially |- (precision fmt = 0) ( m. m IN midpoints fmt abs(x - y) < abs(x - m)) (round fmt Nearest x = round fmt Nearest y)

  4. NASA Space Shuttle Huge PVS libraries developed at NASA. Algebra, Real analysis, Complex numbers, Directed graphs, Graph theory, Integer division, Abstract orders, Lattices, Fixed Points, Power sets, Trigonometry, Series, Taylor’s theorem etc. Sharing ok, but how?

  5. Mathematics Four-Color theorem [Appel, Haken 1976] Kepler’s Conjecture 2D [Thue 1890] 3D [Hales 1989]

  6. MIZAR [Trybulec’72] • Reconstruct mathematical vernacular. • Proof verifier. • Large body of mathematical knowledge. • No explicit proof objects. • Journal of formalized mathematics. • On the Hausdorff distance between compact subsets. [Adam Grabowski] • Chains on a grating in Euclidean space. [Freek Wiedijk]

  7. Logic Diversification Tempor SMV SPIN Vampire Nqthm PVS al logic Proposi First- Chaff tional order Otter ACL2 logic logic Higher- Modal MIZAR order OMEGA logic logic Martin Löf's AGDA NUPRL type theory Automath LCF HOL HOL light Calculus of COQ Constructions

  8. Digital Libraries FDL library. [Constable 2000] • Storage, retrieval of mathematical facts. • Logic dependent. Logosphere. [Schürmann 2002] • Logical framework. • Foundationally uncommitted. • Theory morphisms. • Currently under development.

  9. What shall we store? Semantic meaning of a theorem! Formulas alone insufficient. • Logics vary in proof-theoretic strength. • Example: First-order logic vs. impredicative type theory. • Semantics-preserving transformations. | = L 1 F 1 ⇒ | = L 2 F 2 =

  10. Meaning of theorems ... ... are mathematical entities expressed as • Denotations (Domain theory). • Objects (Category theory). • {0,1} (Model theory). • Strategies (Game theory). • Syntactic Proofs (Proof Theory). Large proofs but small trustworthy checkers.

  11. Rest of this talk joint work with Mark-Oliver Stehr • The logic HOL. • Logical framework LF. • Nuprl type theory. • HOL - Nuprl connection. • Open questions.

  13. HOL • Higher-order logic [Church ‘40] • HOL theorem prover [Gordon ‘85] • Flavor: Isabelle/HOL [Paulson, Gordon ‘92] Terms: e 1 , e 2 ::= x | = | ⊃ | e 1 e 2 | λ x : τ .e Types: ::= o | τ 1 → τ 2 τ

  14. HOL (Typing) Judgments: e : τ eq Rules: imp =: τ → τ → o ⊃ : o → o → o u x : τ 1 . . . e : τ 2 e 1 : τ 2 → τ 1 e 2 : τ 2 app lam u e 1 e 2 : τ 1 λ x : τ 1 .e : τ 1 → τ 2

  15. HOL (Proofs) Judgments: � P Rules: � P . . . � Q � P � P ⊃ Q mp disch � Q � P ⊃ Q refl beta � P = P � ( λ x : τ .P ) Q = [ Q/x ] P

  16. HOL (Booleans) bool = o ˆ true = λ x : bool .x = λ x : bool . x ˆ all P = P = λ x : τ . true ˆ false = all ( λ x : bool .x ) ˆ neg P = P ⊃ false ˆ P and Q = all ( λ R : bool . ( P ⊃ Q ⊃ R ) ⊃ R ) ˆ the P (newly declared) ex P = P ( the P ) ˆ

  18. Twelf • Logical framework LF. [Harper ‘93] • Meta-language for deductive systems. • Judgments-as-types, derivations-as-objects. • Representation methodology. • Higher-order abstract syntax. • Captures variable binding.

  19. Twelf (cont’d) Representing numbers in BS (binary strings). � 79 � = ∗ , 1 , 0 , 0 , 1 , 1 , 1 , 1 Representing judgments in LF. � � P � : type = � � P � Representing derivations in LF. � � H 1 H 2 � P ⊃ Q � P � � Q � : mp mp � P � � Q � � H 1 � � H 2 � � Q =

  20. Twelf’s Strength Adequacy Theorem: Every HOL derivation D of P 1 , . . . , P n � Q can be represented in LF as a canonical object � D � � Q � in context : u 1 : � � P 1 � , . . . u n : � � P n � . HOL Logical Framework LF Terms Types Typing Derivability Canonical Definitions objects

  21. Twelf Encoding of HOL bool = o. tp : type. %name tp (A B). true : tm bool = (\ [x : tm bool] x) === (\ [x: tm bool] x). --> : tp -> tp -> tp. %infix right 10 -->. all| : tm ((A --> bool) --> bool) o : tp. = \ [P:tm (A --> bool)] P === \ [x] true. all = [P] all| @ P . tm : tp -> type. %name tm (H G) (x y P Q R). false : tm bool = all (\ [P] P). =>: tm (o --> o --> o). neg : tm (bool --> bool) = \ [P:tm bool] P ==> false. == : tm (A --> A --> o). /|\ : tm (bool --> bool --> bool) @ : tm (A --> B) -> tm A -> tm B. %infix left 15 @. = \ [P:tm bool] \ [Q:tm bool] \ : (tm A -> tm B) -> tm (A --> B). all (\ [R:tm bool] (P ==> Q ==> R) ==> R). ==> = [H:tm o] [G:tm o] => @ H @ G. %infix right 13 ==>. /\ = [P] [Q] /|\ @ P @ Q. %infix right 12 /\. === = [H:tm A] [G:tm A] == @ H @ G. %infix left 14 ===. \|/ : tm (bool --> bool --> bool) = \ [P:tm bool] \ [Q:tm bool] all (\ [R:tm bool] (P ==> R) ==> (Q ==> R) ==> R). |- : tm o -> type. %prefix 10 |-. %name |- D u. \/ = [P] [Q] \|/ @ P @ Q. %infix right 11 \/. mp : |- H -> |- H ==> G -> |- G. the| : tm ((A --> bool) --> A). disch : (|- H -> |- G) -> |- H ==> G. the = [P] the| @ P . refl : |- H === H. ex| : tm ((A --> bool) --> bool) beta : |- (\ H) @ G === (H G). = \ [P:tm (A --> bool)] P @ (the (\ [x] P @ x)). sub : {G:tm A -> tm o} |- H1 === H2 -> |- G H1 -> |- G H2. ex = [P] ex| @ P . abs : |- \ H === \ G <- ({x} |- H x === G x).

  23. Nuprl • Polymorphic extensional type theory. [Constable ‘86] • Judgments establishes equality among terms. • A type is true iff it is inhabited . • Many applications. • Ensemble (TCP/IP stack). [Kreitz ‘04] • Protocol Verification. [Felty et al ‘98]

  25. Translation • Original idea. [Howe ‘98] • Syntactic argument. [Meseguer, Stehr ‘01] • Implemented in Nuprl, replay of proof scripts. [Naumov ‘01] • Formalized and executable specification. [Schürmann, Stehr ‘05]

  26. Translation (cont’d) • Booleans. = unit + unit boolean = tt inl bullet = ff inr bullet = decide e ( λ z. e 1 ) ( λ z. e 2 ) if e e 1 e 2 • Propositions-as-types. = U 1 BOOLEAN = TRUE unit = FALSE void = Π ALL = =n=> Π

  27. Howe’s Observation • Axiom of the excluded middle. inhI � inh # Π x : BOOLEAN . x + ( x → void ) • Lift Booleans to propositions. ↑ ( e ) = if e TRUE FALSE . • Lower propositions to Booleans. ↓ ( P ) = decide ( inh P ) ( λ x. tt ) ( λ y. ff ) . • All important laws verifiable within Nuprl.

  28. Translations-as-Relations • Relations in Twelf. : tp → nuprlterm → type trans-tp tm A → nuprlterm → type : trans-tm : tm o → nuprlterm → type trans-sentence � P → trans-sentence P T → � M # T → type trans-proof : • Defining declarations omitted. • Executable within Twelf. • We can transform HOL proofs into Nuprl.

  30. Conclusion • There is a true need to share mathematical knowledge in form of proofs. • Proof-theory: syntax instead semantics. • Logical framework technology important. • Proof conversion between HOL and Nuprl. • For other systems (PVS), work in progress.

  31. Open Questions • Design of a query language. • Design of the database. • Shared domains, integers, natural numbers, complex numbers. • Partial transformations. • Connection to OMDOC. [Kohlhase 2001] • Formalization of other logics.

  32. www.logosphere.org Thank you!

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