checking reachability using matching logic

Checking Reachability using Matching Logic

Grigore Rosu and Andrei StefanescuUniversity of Illinois, USA

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Main GoalLanguage-independent program verification

frameworkDerive program properties from operational semanticsQuestions:

Is it possible? Is it practical?

Answers: Sound and complete proof system, so YES, it is possible! Efficient automated verifier MatchC, so YES, it is practical!

Overview

State-of-the-art in Certifiable Verification

Our ApproachSpecifying Reachability PropertiesReasoning about Reachability

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Operational SemanticsEasy to define and understand

Can be regarded as formal “implementations”Require little mathematical knowledge

Great introductory topics in PL coursesScale up well

C (>1000 rules), Java, Scheme, Verilog, …, definedExecutable, so testable

C semantics tested against real benchmarks

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Operational SemanticsSample rule (may require a configuration

context)

Define languages only with rules of the form

l, r are configuration termsb is a Boolean side condition

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Unfortunately …Operational semantics considered inappropriate

for program verification; proofs are low-level and tedious:Formalization of and working with transition systemTypically by induction

on the structure of the programon the number of execution stepsetc.

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Axiomatic Semantics(Hoare Logic)

Emphasis on program verificationProgramming language captured as a formal proof

system deriving Hoare triples

precondition postcondition

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Axiomatic SemanticsNot easy to define and understand, error-prone

Not executable, hard to testRequire program transformations, behavior loss

Write e = 1 and you’ve got a wrong semantics!

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State-of-the-art inCertifiable Verification

Define an operational semantics: trusted language model

Define an axiomatic semantics: for verification purposesProve axiomatic semantics sound for operational

semanticsNow we have trusted verification … BUT

Requires two semantics of the same language C operational semantics took more than 2 years!

Must be done individually for each language

Overview

State-of-the-art in Certifiable Verification

Our ApproachSpecifying Reachability PropertiesReasoning about Reachability

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Our ApproachUnderlying belief: one semantics for each

language! Executable (testable), easy to define and understand Suitable for program verification, “as is”

Approach: language-independent proof system Takes operational semantics unchanged Derives program properties Both operational semantics rules and program

specifications stated as reachability rules

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Reachability RulesPairs of configuration predicates

Reachability: Any concrete configuration satisfying and terminating reaches a configuration satisfying , in the transition system induced by the operational semantics .

Overview

State-of-the-art in Certifiable Verification

Our ApproachSpecifying Reachability

Properties Reasoning about Reachability

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Reachability Rules- Operational + Axiomatic

-Operational flavor

Axiomatic flavor

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Hoare Triple = Syntactic Sugar

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Matching LogicState static properties of program configurations

Parametric in a model of configurationsExtends first-order logic with patterns

Special predicates which are configuration termsConfigurations satisfy patterns iff they match them

C Configurations Extra 70 cells

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Model of Configurations- Properties -

Configuration abstraction (list)“Separation” achieved at term level

Operations (reverse)

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Separation Logic =Matching Logic Instance

Separation logic: popular logic for heap propertiesMechanical translation to matching logic (see

paper)Configuration:Separation encoded using different sub-terms

No expressiveness loss from using matching logicMatching logic gives “structural separation”

anywhere in the configuration, not only in the heap

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Operational and Axiomatic Semantics Rules as Reachability Rules

Reachability rules generalizeOperational semantics rulesHoare triples

Operational semantics rule is syntactic sugar for reachability rule

Hoare triple encoded in a reachability rule with the empty code in the right-hand-side (see FM’12)

Overview

State-of-the-art in Certifiable Verification

Our ApproachSpecifying Reachability PropertiesReasoning about Reachability

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Reasoning about Reachability

The main result of our paper is a proof system deriving reachability rules from reachability rules:

Trusted reachability rules(starts with operational

semantics)

Target reachability rule

Claimed reachability rules

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Reachability Proof System- 8 Rules -

Symbolic execution (multiple steps) Symbolic execution (one step)Code with circular behavior

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Circular behaviorsCircularity and Transitivity proof rules

Hoare logic rule for while loops

Language-independent

Language-specific

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Soundness

Theorem: If is derivable by the proof system, then is semantically valid.

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Relative Completeness

RelativityValidity oracle for static configuration properties

Language-independent result, unlike Hoare logics

Theorem: If is semantically valid, then is derivable by the proof system, with the operational semantics of a language.

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MatchCProof-of-concept verifier for a C fragmentDerives program specifications from the

operational semantics (in K framework) using the proof systemNo Hoare/separation logic, no WP, no VC generation

Automated, user only provideSpecifications for recursive functions and loops

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MatchC Snapshot

List reverse: code + invariant

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ImplementationHeuristics for applying the proof system

(forward) symbolic executionMatching logic reasoning

Maude: efficient structure matching and rearrangingmatching a list the heap, …

SMTs (CVC3, Z3): simplifying constraintssmall queries (milliseconds each)

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Preliminary EvaluationProgram Time (s)Buffered read-write 0.15Stack inspection 0.24Insertion sort 0.41Merge sort 0.47Quicksort 1.97AVL find 0.15AVL insert 43.5AVL delete 133.58Schorr-Waite (tree) 0.28Schorr-Waite (graph) 1.73… …

Dozens more programs at matching-logic.org

Only annotated main functions (insert/delete).

Inlined auxiliary functions (balance, rotate, …).

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ConclusionsMatching logic reachability proof system

Sound and (relatively) completePractical

MatchC, an automated verifierExpressiveEfficient

Operational semantics based verification is viable!

matching-logic.org