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Input, Output, and Automation in x86 Proved

Input, Output, and Automation in x86 Proved. Jason Gross Hosted by Andrew Kennedy Summer 2014. Verified I/O Programs. “. When doing formal verification, come up with the simplest non-trivial example you can. Then start with something simpler. —Adam Chlipala (paraphrased). ”.

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Input, Output, and Automation in x86 Proved

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  1. Input, Output, and Automation in x86 Proved Jason Gross Hosted by Andrew Kennedy Summer 2014

  2. Verified I/O Programs “ When doing formal verification, come up with the simplest non-trivial example you can. Then start with something simpler. —Adam Chlipala (paraphrased) ” Precondition Postcondition {true}{false} while true do skip done [] I/O behavior

  3. Verified I/O Programs: Trivial Loop {true}{false} while true do skip done [] Examplesafe_loop_whileeax: ⊢ basic (EAX ≅ eax⋆OSZCP?) (while (TESTEAX, EAX) CC_Ofalseprog_skip) [] false. Proof. basic apply (while_rule_ro(I := bb = false ∧ EAX? ⋆ SF? ⋆ ZF? ⋆ CF? ⋆ PF?)) //=; rewrite /stateIsAny; specintros; basic apply . Qed.

  4. Verified I/O Programs: Trivial Loop {true}{false} while true do skip done [] I/O Spec Postcondition Precondition Code Examplesafe_loop_whileeax: ⊢ basic (EAX ≅ eax⋆OSZCP?) (while (TESTEAX, EAX) CC_Ofalseprog_skip) [] false. Proof. basic apply (while_rule_ro(I := bb = false ∧ EAX? ⋆ SF? ⋆ ZF? ⋆ CF? ⋆ PF?)) //=; rewrite /stateIsAny; specintros; basic apply . Qed. Loop invariant

  5. Verified I/O Programs: Eternal Output {true}{false} while true doout 1 done 1* Kleene star Exampleloop_forever_onechannel: ⊢ basic ( AL?) ( MOV AL, 1;; LOCAL LOOP; LOOP:;; OUT channel, AL;; JMP LOOP) 1* false. Proof. (elided) Qed.

  6. Verified I/O Programs: Eternal Output {true}{false} while true doout 1 done 1* Exampleloop_forever_onechannel: ⊢ loopy_basic ( AL?) ( MOV AL,(#1: DWORD);; LOCAL LOOP; LOOP:;; OUT channel, AL;; JMP LOOP) (starOP(outOP(zeroExtend n8channel) (#1 : BYTE))) lfalse. Proof. (still elided) Qed.

  7. Verified I/O Programs: Echo Once v, {true} {true} v ← input; out v In v; Out v Examplesafe_echo_oncein_channelout_channel: ⊢ ∀ v, basic ( AL?) ( IN in_channel, AL;; OUT out_channel, AL) [Inv; Outv] (AL v). Proof. rewrite/stateIsAny;specintrosal v. basic apply . basic apply . Qed.

  8. Digression: Hoare Rule for While Standard rule from Wikipedia What’s ? Desired output annotations The I/O doesn’t get to talk about state!

  9. Digression: Hoare Rule for While The test of had value lets us “negate” , (which is code)

  10. Digression: Hoare Rule for While Example We can drop B, which is technical info about flags

  11. Digression: Hoare Rule for While Example

  12. Digression: Hoare Rule for While Example

  13. Digression: Hoare Rule for While Example with output Worry about side conditions later

  14. Digression: Hoare Rule for While Example with output Uh-oh! Worry about side conditions later

  15. Digression: Hoare Rule for While Eliding for space “transition function”

  16. Digression: Hoare Rule for While Can’t talk about state here

  17. Digression: Hoare Rule for While Logical part of

  18. Digression: Hoare Rule for While Example with output On-the-fly demo at the end, time and interest permitting

  19. Verified I/O Programs: Echo : stream, {true} {true} while true do (v ← input; out v) done map ( In ; Out ) Examplesafe_echoeaxin_channelout_channel: ⊢ ∀ vs, basic ( AL? ⋆ EAXeax⋆ OSZCP?) ( while (TESTEAX, EAX) CC_Ofalse ( IN in_channel, AL;; OUT out_channel, AL ) (stream_Opred_map ( [Inv; Outv]) ) lfalse. Proof. (elided; 8 lines of filling in arguments to the while rule, 9 lines of automation about specs) Qed.

  20. Verified I/O Programs: Accumulator : list (non-zero BYTE), {acc=x} {acc = fold accumulate x vs} while ((v ← input) 0) do (acc = accumulate acc v) done map ( In ) ExampleaddB_until_zero_prog_safech o s z c p S al : S(initial (x : BYTE) (xs: seq BYTE) (pf1 : only_last(t : BYTE⇒ t == #0) x xs), (loopy_basic(AH≅ initial ⋆ ALal ⋆ OSZCPo s z c p) ( IN ch, AL;; while (CMP AL,#0) CC_Zfalse (ADDAH, AL;; INch, AL)) (foldrcatOPempOP(map (inOP(zeroExtend8 ch)) (x::xs))) ((AH ≅ (foldladdBinitial (drop_lastx xs))) ⋆ AL ≅ #0 ⋆ OF? ⋆ SF? ⋆ ZF ≅ true ⋆ CF? ⋆ PF?))). Proof. specintros⇒ ∗. basic apply (@accumulate_until_zero_prog_safe_ (𝜆 x ⇒ AH≅ x)) ⇒ ∗; first assumption. basic apply ∗. Qed.

  21. Verified I/O Programs: Next Steps • readline (via accumulator template) • prime number printer • text adventure? • use memory-mapped I/O rather than IN and OUT

  22. Automation

  23. Instruction Automation: Ideal • Define the instruction in the model • State the high-level (e.g., Hoare) rule • Push-button verification Maybe even omit 2, if the inference is good enough. Image from https://flic.kr/p/3KYpfJ, “State of Mind”, tshein

  24. Instruction Automation: Reality • Define the instruction in the model DefinitionevalBinOp{n} op : BITSn→ BITSn→ ST (BITSn) := matchopwith | OP_XOR ⇒ evalLogicalOpxorB … end. Definition evalLogicalOp{n} (f : BITSn → BITSn → BITSn) arg1 arg2 := letresult := f arg1 arg2in do!updateFlagInProcStateCFfalse; do!updateFlagInProcStateOF false; do!updateZPSresult; retnresult.

  25. Instruction Automation: Reality • State the high-level (e.g., Hoare) rule LemmaXOR_RR_rules (r1 r2:VRegs) v1 (v2:VWORDs): basic (VRegIsr1 v1⋆ VRegIsr2 v2⋆ OSZCP?) (XORr1, r2) [] (VRegIsr1 (xorBv1 v2) ⋆ VRegIsr2 v2 ⋆ OSZCP false (msb(xorBv1 v2)) (xorBv1 v2 == #0) false (lsb(xorBv1 v2))).

  26. Instruction Automation: Reality • Push-button verification LemmaXOR_RR_rules (r1 r2:VRegs) v1 (v2:VWORDs): basic (VRegIsr1 v1⋆ VRegIsr2 v2⋆ OSZCP?) (XORr1, r2) [] (VRegIsr1 (xorBv1 v2) ⋆ VRegIsr2 v2 ⋆ OSZCP false (msb(xorBv1 v2)) (xorBv1 v2 == #0) false (lsb(xorBv1 v2))). Proof. destruct s; do_instrrule_triple. Qed.

  27. Instruction Automation: Old Reality • Push-button verification

  28. Instruction Automation: Old Reality

  29. Instruction Automation: Mechanics • Lookup • Application • Heuristics LemmaXOR_RR_rules (r1 r2:VRegs) v1 (v2:VWORDs): basic (VRegIsr1 v1⋆ VRegIsr2 v2⋆ OSZCP?) (XORr1, r2) [] (VRegIsr1 (xorBv1 v2) ⋆ VRegIsr2 v2 ⋆ OSZCP false (msb(xorBv1 v2)) (xorBv1 v2 == #0) false (lsb(xorBv1 v2))). Proof. destruct s; do_instrrule_triple. Qed.

  30. Instruction Automation: Mechanics Automated timing scripts helped ensure that the automation didn’t slow things down unreasonably. Example diff After | File Name | Before || Change ------------------------------------------------------------ 17m33.61s | Total | 18m51.54s || -1m17.92s ------------------------------------------------------------ 0m35.85s | examples/mulc | 0m42.18s || -0m06.32s 0m33.42s | examples/specexamples | 0m27.19s || +0m06.23s 0m24.29s | x86/lifeimp | 0m30.41s || -0m06.12s 0m17.60s | x86/inlinealloc | 0m21.79s || -0m04.18s 0m27.75s | x86/imp | 0m31.41s || -0m03.66s 0m17.56s | x86/instrrules/mov | 0m20.79s || -0m03.23s 0m15.76s | x86/call | 0m19.29s || -0m03.52s 0m51.82s | x86/instrrules/addsub | 0m54.54s || -0m02.71s 0m33.68s | x86/instrrules/cmp | 0m35.90s || -0m02.21s 0m27.69s | x86/cfuncspec | 0m29.79s || -0m02.09s 0m25.84s | x86/screenproof | 0m28.47s || -0m02.62s 0m19.90s | x86/instrcodec | 0m22.39s || -0m02.49s 0m26.98s | triple/set | 0m28.91s || -0m01.92s

  31. Program Automation: Ideal • Write a program • State the spec • Sprinkle annotations • Push-button verification Maybe even omit 3, if the inference is good enough. Image from https://flic.kr/p/3KYpfJ, “State of Mind”, tshein

  32. Program Automation: Reality specapply∗ takes care of essentially all of the unstructured code reasoning Examplesafe_echoeaxin_cout_c: ⊢ ∀ vs, basic ( AL? ⋆ EAXeax⋆ OSZCP?) ( while (TESTEAX, EAX) CC_Ofalse ( IN in_c, AL;; OUT out_c, AL ) (eq_opred_stream(stream_to_in_outin_cout_c)) lfalse. Proof. eapply@while_rule_ind with (I_logic := 𝜆 _ b⇒ false == b) (Otest := 𝜆 _ ⇒ empOP) (Obody := 𝜆 s ⇒ echo_once_OP_specin_cout_c(hds)) (I_state := 𝜆 s_ ⇒ EAXeax⋆ AL? ⋆ SF? ⋆ ZF? ⋆ CF? ⋆ PF?) (transition_body := @tl _) (O_after_test := 𝜆 s ⇒ default_PointedOPred (catOP(echo_once_OP_specin_cout_c(hds)) (eq_opred_stream(stream_to_in_outin_cout_c(tls))))); do ![ progress rewrite → ?empOPL, → ?eq_opred_stream__echo_once | specintros⇒ ∗ | done | byssimpl | basic apply ∗ | progress simplOPred_pred | progress move ⇒ ∗ | progress rewrite /stateIsAny | reflexivity ]. Qed. basic apply ∗ takes care of all of the code reasoning here

  33. Program Automation: Mechanics • Lookup (via typeclasses) • Application (via helper lemmas) • Heuristics (for side conditions)

  34. Open Issues

  35. Open Issues: Pointy Predicates DefinitionbasicP (c : T) (O : OPred) Q : spec := (i j : DWORD) (O': OPred), (obsO' @ (EIPjQ) obs(O + O') @ (EIPiP)) <@ (i -- jc). Definitionloopy_basicP (c : T) (O : OPred) Q : spec := (i j : DWORD) (O' : PointedOPred), (obsO' @ (EIP jQ) obs(O + O') @ (EIPiP)) <@ (i -- jc). Image from https://flic.kr/p/ab1bwS, “Spiky balls” Tom Magliery

  36. Open Issues: Quantifier Location Program Definition obs(O: OPred) := mkspec(k P (s: ProcState), (Pltrue) so, O orunsForWithPrefixOfk s o) _ _. Can we hide the quantifiers in the I/O predicate, so that the specification for echo doesn’t have to quantify over streams? Currently, both (* and (* say the wrong thing.

  37. Take Home Messages • We can verify the I/O behavior of simple assembly programs. • Putting effort into Coq automation results in tactics which are: • comparable in speed to manual proofs • capable of push-button verification in well-specified domains

  38. Acknowledgements • Thanks to: • Andrew Kennedy, my host here at MSR, and Nick Benton, who is also on this project. • Georges Gonthier, for help with Coq and SSReflect • Jonas Jensen, Jesper Bengtson, Gregory Malecha, and Edward Z. Yang for discussions about various aspects of I/O specifications, among other things.

  39. Thanks!Questions?Requests for verification demo?

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