automated tactics for separation logic veriml reconstruct z3 proof safe incremental type checker

• Automated tactics for separation logic• VeriML

• Reconstruct Z3 Proof • Safe incremental type checker• Certifying code transformation

• Proof carrying hardware IP

Automated tactics for separation logic• Version 1

• Certified verifier for a fragment of separation logic in Coq• Without predicates ( list, tree … )

• Prover written in SML handle these predicates and output proof trace

• Tactic to reconstruct proof from these proof trace in Coq• Version 2

• Hard coded list and tree into the certified verifier• Use a forked version Coq which supports native arrays to speed up

the verifier• Version 3 ( Ongoing )

• More generic way to support user defined predicates

Reconstruct Z3 Proof• Reconstruct proof in VeriML using output from Z3

Start from SAT Solvers• Decide propositional satisfiablity of sets of clauses:

• Proof Witness from SAT solver:• If satisfiable: assignment of all the variables• If unsatisfiable: proof by resolution of the empty clause

• Resolution rule:

𝑥∨ 𝐴𝑥∨𝐵𝐴∨𝐵

Example• Satisfiable:

• Unsatisfiable:

𝑥1∨¬𝑥2∨𝑥3¬𝑥3𝑥1∨¬𝑥2

𝑥1∨ 𝑥2𝑥1

¬𝑥1∨𝑥3¬ 𝑥3¬𝑥1

∎

SAT Modulo Theories(SMT solver)• Atoms can be formulas from other theories

• Congruence Closure(EUF)• Linear Arithmetic(LIA)• ….

• Proof witness• If satisfiable: assignment of all the variables• If unsatisfiable: proof by resolution of the empty clause while some

leaves are now lemmas from theories

Example• Satisfiable :

• Unsatisfiable:

𝑥≠ 𝑦∨ 𝑓 (𝑥 )= 𝑓 (𝑦 ) 𝑥=𝑦𝑓 (𝑥 )= 𝑓 (𝑦 )

𝑓 (𝑥 )≠ 𝑓 (𝑦 )

∎

Interaction between SAT and SMT

• With boolean abstraction:

• • First try: A:true, B:true, C:true, D:true, E:false

• Unsat by a multi-theory solver, thus we add a theory:

• Repeat until sat or no more model can be found

A B C D E

Reconstruct proof• If sat: apply the assignment and check the result is true

• If unsat: check the resolution tree• Checking resolution step (SAT solver)• Checking theory lemmas (SMT solver)• Combination of theories

What we need• resolution checker • checker for each theory• checker for the resolution tree which calls these two kinds of checkers at each step

Example𝑥1∨¬𝑥2∨𝑥3¬𝑥3

❑ 𝑥1∨ 𝑥2❑

¬𝑥1∨𝑥3¬ 𝑥3❑

❑

Example𝑥1∨¬𝑥2∨𝑥3¬𝑥3

𝑥1∨¬𝑥2𝑥1∨ 𝑥2

❑¬𝑥1∨𝑥3¬ 𝑥3

❑❑

Example𝑥1∨¬𝑥2∨𝑥3¬𝑥3

𝑥1∨¬𝑥2𝑥1∨ 𝑥2

𝑥1¬𝑥1∨𝑥3¬ 𝑥3

❑❑

Example𝑥1∨¬𝑥2∨𝑥3¬𝑥3

𝑥1∨¬𝑥2𝑥1∨ 𝑥2

𝑥1¬𝑥1∨𝑥3¬ 𝑥3

¬𝑥1❑

Example𝑥1∨¬𝑥2∨𝑥3¬𝑥3

𝑥1∨¬𝑥2𝑥1∨ 𝑥2

𝑥1¬𝑥1∨𝑥3¬ 𝑥3

¬𝑥1∎

Encoding in VeriML• Only consider CNF clause• Variable

• Assignment: Nat-> Bool• Literal :

• Clause: List Literal• State: List ( clauseID * Clause )

• get : S -> clauseID -> C set: S -> clauseID -> C -> S• Proof trace: ??• A realistic Checker: State -> Trace -> Bool

Interpretation• Given assignment • Var: v• Literal: • Clause: • Valid state

• Soundness of checker:

Start from SAT solver• Resolution Chain: List Nat• Resolve : Clause->Clause->Clause

• resolution between clause C1 and C2. • Resolution checker: Resolution Chain-> Clause

• for resolution chain [n1,n2,…n] and state S,• R(..(R( S[n1], S[n2]),…),S[n])

• Proof trace: List (clauseID*Resolution Chain)• Reconstruct the proof:

• List.fold (fun (id,rc) => set s (resolution_checker rc) id) trace• After this, we have the empty clause in state which indicates the

state is not valid

Example𝑥1∨¬𝑥2∨𝑥3¬𝑥3

𝑥1∨¬𝑥2𝑥1∨ 𝑥2

𝑥1¬𝑥1∨𝑥3¬ 𝑥3

¬𝑥1∎

Initial State: [ [2,4], [2,5,6], [3,6], [7] ]Resolutions: [ (5,[1,3,0]) , (6,[2,3] ) , (7,[5,6]) ]Resolve two clause: [2,4] [2,5,6] [2,6] ( a modified merge sort )

Naïve framework for SMT solver• Certificate from different theories• Inductive cert:

• | sat_solver sat_cert• | euf_solver euf_cert• | lia_solver lia_cert• ….

• Trace might be like: List (clauseID*cert)• Checkers for each theory• And finally the checker for SMT solver

• checker s trace = • List.fold (fun (id,cert) => set s (cert_checker cert) id

•

Problem Witness

SMT

Resolution

CNF

EUF

LIA

VeriML Checker

Z3 Proof• Example

• 34 Axiom• 19 core rules : mostly are propositional reasoning• 5 equality rules : refl, symm, trans, …• 7 quantifier rules : quant-inst, quant-intro, …• 3 theory rules : rewrite, inconsistent, …

Reconstruction• Overall approach:

• one proof method for every Z3 inference rule• depth-first traversal of Z3 proof

Conclusion• A prototype to reconstruct zChaff(SAT solver) proof• Very slow in performance

• Ongoing with Z3 reconstruction

Safe Incremental Type Checker• Type checking is more and more an interaction between the programmer and the type-checker

• The richer the type system is, the more expensive type checking gets

• Example• Type inference ( unification)• Dependent types • Very large term ( proof term )• Complex language ( C++ )

Goal • Reuse already-computed results• Recheck only the modified part and where it affects

• Example

let f x = x + 1 in (f 2)

let f x = 2 * (x + 1) in (f 2)

let f x = (let y = true in x + 1) in (f 2)

let f x = x > 1 in (f 2)

• Benefit• fast type checking modified code

• this could also be used to tactic language, and refinement?• typed version control as we always have well-typed

program and its type derivations• “undo” for free

• If we simply use memorization• We can only do syntactic comparison• We can’t do

• weakening: let x = 1 and let y = 2 in x let x = 1 and let y = 2 in x

• -equivalence• other possible user extensible conversions

• it’s outside typing rule

Difficulties• A language to describe delta between different versions

• Slice the code into each typing step• A language to store typing derivations

• Higher order terms with meta variables• Build it up

• Repository : store type derivations• Delta: describe difference between two versions of code• Incremental checker: check : repo -> delta -> repo

• try to reduce the complexity to

• A language to store typing derivations• VeriML support contextual terms in computation language, thus

possible to present typing derivations• How to make it generic?

• A language to describe delta between different versions• Got stuck here

Scratch

• Given signature: • t1 = g a

• t2 = g (g a)

Certifying Code Transformation• How to certify code transformation?

• Verified Compiler• Translation Validation

• VeriML approach

Verified Compiler

Translation Validation

Approach with VeriML

Naïve Example

Scratch

• ImplementationLogical language• Heap implementation• Language definition and semantics

• Computation language• Symbolic execution(VCGen)• Tacitc to proof Hoare Triple• Practical Code transformation examples

Conclusion• The thing we need to proof isn't less than translation

validation• we can do it in a single lanuguage which easier to write proof with

and we get more proof• The whole picture is still vague

Proof Carrying Hardware• Current work

• A formalization of Intel 8051 microcontroller in Coq• RC5 hash algorithm • Proof certain ports won’t “leak” secret