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### (One-Path) Reachability Logic

Grigore Rosu, Andrei Stefanescu, Brandon Moore

University of Illinois at Urbana-Champaign, USA

Stefan Ciobaca

University AlexadruIoanCuza, Romania

Long-Standing Dream

Deductive program verifier

Parser

Interpreter

Formal Language Definition

(Syntax and Semantics)

Model checker

Compiler

Symbolic execution

(semantic) Debugger

Language Frameworks

- PLT-Redex/Racket (Findler et al.)
- OTT (Sewell et al.)
- PLanComps (Mosses et al.)
- Raskal (Klint et al.)
- RLS-Maude (Meseguer et al.)
- K (Rosu et al.)
- …
- All based on operational semantics
- Defined semantics serve as language reference models of languages, but are close to useless for verification
- Takes 1-2 years to define a language

C Semantics (in K)

C configuration

… plus ~1200 user-defined rules

… plus ~1500 automatically generated rules

Operational Semantics

- Virtually all operational semantics can be defined with rewrite rules of the form
- We would like to reason about programs using precisely such operational semantics!

State-of-the-Art

Many different program logics for “state” properties: FOL, HOL, Separation logic…

- Redefine the language using a different semantic approach (Hoare/separation/dynamic logic)
- Very language specific, error-prone; e.g.:

State-of-the-Art

- Thus, these semantics need to be proved sound, sometimes also relatively complete, wrt trusted, operational semantics of the language
- Verification tools developed using them
- So we have an inherent gap between trusted, operational semantics, and the semantics currently used for program verification

Our Proposal

- Use directly the trusted operational semantics!
- Has been done before (ACL2), but proofs are low-level (induction on the structure of program or on steps in transition system) and language-specific
- We give a language-independent proof system
- Takes unchanged operational semantics as axioms
- Derives reachability rules
- Both operational semantics rules and program properties stated as reachability rules
- Is sound(partially correct) and relatively complete

Need a means to specify static and dynamic program properties

Deductive program verifier

Parser

Interpreter

Formal Language Definition

(Syntax and Semantics)

Model checker

Compiler

Symbolic execution

(semantic) Debugger

Matching Logic

[Rosu, Ellison, Schulte 2010]

- Logic for specifying static properties about program configurations and reason with them
- Key insight:
- Configuration terms with variables are allowed to be used as predicates, called patterns
- Semantically, their satisfaction means matching
- Matching logic is parametric in a (first-order) configuration model: typically the underlying model of the operational semantics

Configurations

- For concreteness, assume configurations having the following syntax:

(matching logic works with any configurations)

- Examples of concrete (ground) configurations:

Patterns

- Concrete configurations are already patterns, but very simple ones, ground patterns
- Example of more complex pattern
- Thus, patterns generalize both terms and [FOL]

Matching Logic Reasoning

- We can now prove (using [FOL] reasoning) properties about configurations, such as

Matching Logic vs. Separation Logic

- Matching logic achieves separation through matching at the structural (term) level, not through special logical connectives (*).
- Separation logic = Matching logic [heap]

SL:

ML:

- Matching logic realizes separation at all levels of the configuration, not only in the heap
- the heap was only 1 out of the 75 cells in C’s def.

[OOPSLA’12]

Need a means to specify static and dynamic program properties

Deductive program verifier

Parser

Interpreter

Formal Language Definition

(Syntax and Semantics)

Model checker

Compiler

Symbolic execution

(semantic) Debugger

Reachability Rules - Syntax

- “Rewrite” rules over matching logic patterns:
- Since patterns generalize terms, matching logic reachability rules capture term rewriting rules
- Moreover, deals naturally with side conditions:

turn into

Conditional Reachability Rules

- The involved patterns can share free variables
- Generalize conditional rewrite rules

Reachability Rules - Semantics

- In the transition system generated by the operational semantics on the configuration model, any terminating configuration that matches reaches a configuration that matches (patterns can share free variables)
- That is, partial correctness

Expressivity of Reachability Rules

- Capture operational semantics rules:
- Capture Hoare Triples:

Reachability Logic

Language-independent proof system that derives reachability rules from other reachability rules:

Trusted reachability rules

(starts with operational semantics)

Target reachability rule

Intuitively: symbolic execution with operational semantics + reasoning with cyclic behaviors

Claimed reachability rules

Traditional Verification vs. Our Approach

Traditional proof systems: language-specific

Our proof system: language-independent

Results

- Soundness (partial correctness): Under weak well-definedness conditions on (see paper)
- Mechanized in Coq, for verification certificates
- Relative completeness: Under weak assumptions on the configuration model (e.g., it can express Godel’s beta predicate)

Implementation

- Being implemented within the K framework
- Symbolic execution using the operational semantic rules; custom solver for the matching part + Z3 solver for the model reasoning part (for the Consequence rule)
- Circularity steps given by user (via pre/post/inv annotations), everything else automatic
- Online interface available for fragment of C at

http://matching-logic.org

Related Work and Limitations

- Hoare logic: already explained
- Dynamic logic: need to redefine language semantics (invariant rules, etc.), but more expressive:
- CTL*: expressive, but not clear how to integrate with operational semantics; maybe CTL* over ML patterns?
- Currently we only support one-path reachability for conditional rules. We have a similar proof system for all-path reachability, but only with unconditional rules
- Previous one-path attempts: [ICALP’12] , [OOPSLA’12]

Conclusion

- Program verification using the language operational semantics is possible and feasible
- Language-independent 7-rule reachability proof system, which is sound and complete
- Circularity generalizes the invariant rules
- Being implemented in the K programming language design framework

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