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(One-Path) Reachability Logic

(One-Path) Reachability Logic. Grigore Rosu , Andrei Stefanescu , Brandon Moore University of Illinois at Urbana-Champaign, USA Stefan Ciobaca University Alexadru Ioa n Cuza , Romania. Long-Standing Dream. Deductive program verifier. Parser. Interpreter. Formal Language Definition

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(One-Path) Reachability Logic

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

  2. Long-Standing Dream Deductive program verifier Parser Interpreter Formal Language Definition (Syntax and Semantics) Model checker Compiler Symbolic execution (semantic) Debugger

  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 … 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 Compiler Symbolic execution (semantic) Debugger

  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 Compiler Symbolic execution (semantic) Debugger

  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

  21. 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

  22. 7 Proof Rules for Reachability

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

  24. 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)

  25. 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

  26. 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]

  27. 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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