1. (One-Path) Reachability Logic
Grigore Rosu, Andrei Stefanescu, Brandon Moore
University of Illinois at Urbana-Champaign, USA
Stefan Ciobaca
University Alexadru Ioan Cuza, Romania

2. Long-Standing Dream Deductive program verifier Parser Interpreter
Formal Language Definition
(Syntax and Semantics)
Model checker
Symbolic execution
(semantic) Debugger
One long-standing dream of the programming language community is to have a framework that allows and encourages the language designers to formally define their languages once and for all, using an intuitive and attractive notation, and then obtain essentially for free implementations as well as analysis tools for the defined languages.
Compiler

3. 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

4. C Semantics (in K) C configuration … plus ~1200 user-defined rules
To give an idea what it takes to define a large language, here is, for example, the configuration of C.
It has more than 70 cells!
The heap, which is the subject of so many debates in program verification, is just one of them.
… plus ~1200 user-defined rules
… plus ~1500 automatically generated rules

5. 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!

6. 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.:

7. 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

8. 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

9. Need a means to specify static and dynamic program properties
Deductive program verifier
Parser
Interpreter
Formal Language Definition
(Syntax and Semantics)
Model checker
Symbolic execution
(semantic) Debugger
Compiler

10. 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

11. Configurations
For concreteness, assume configurations having the following syntax:
(matching logic works with any configurations)
Examples of concrete (ground) configurations:

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

13. Matching Logic Reasoning
We can now prove (using [FOL] reasoning) properties about configurations, such as

14. 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]

15. Need a means to specify static and dynamic program properties
Deductive program verifier
Parser
Interpreter
Formal Language Definition
(Syntax and Semantics)
Model checker
Symbolic execution
(semantic) Debugger
Compiler

16. 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

17. Conditional Reachability Rules
The involved patterns can share free variables
Generalize conditional rewrite rules

18. 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

19. Expressivity of Reachability Rules
Capture operational semantics rules:
Capture Hoare Triples:

20. Hoare Triple = Syntactic Sugar
This is a code fragment from a program reversing a singly-linked list verified by MatchC (which we will discuss later). The invariant, states that p points to the part of the list already reversed, and x points to the part of the list yet to be reversed. This Hoare-style invariant is just syntactic sugar for a reachability rule. The LHS combines the code of the loop (shown in red) and the invariant, while in the RHS the code has been executed, and the condition of the loop has been evaluated with the semantics and its negation added as a constraint (shown in blue).

21. Reachability Logic
Language-independent proof system that derives reachability rules from other reachability rules:
The main result of the paper is a language-independent proof system which derives reachability rules specifying program properties from trusted reachability rules. In the beginning the trusted rules are just the operational semantics rules. During the proof one can claim additional reachability rules, which cannot be used right away.
They can be used only after taking at least one step with the trusted rules in A.
// The rules in C are added to those on A only after taking at least one step with the trusted rules in A.
Trusted reachability rules
(starts with operational semantics)
Target reachability rule
Intuitively: symbolic execution with operational semantics + reasoning with cyclic behaviors
Claimed reachability rules

22. 7 Proof Rules for Reachability

25. Traditional Verification vs. Our Approach
Traditional proof systems: language-specific
Our proof system: language-independent

26. 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)

27. 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

28. 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]

29. 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