Abstraction and Modular Reasoning for the Verification of Software

Abstraction and Modular Reasoning for the Verification of Software Corina Pasareanu NASA Ames Research Center

Outline • Bandera Project (Kansas Sate University): • Tool Support for Program Abstraction and Abstract Counter-example Analysis (joint work with the Bandera team) • NASA Ames Projects: • Combining Symbolic Execution with (Explicit State) Model Checking (joint work with Willem Visser) • Assumption Generation for Component Verification (joint work with Dimitra Giannakopoulou and Howard Barringer)

Outline • Bandera Project (Kansas Sate University): • Tool Support for Program Abstraction and Abstract Counter-example Analysis • NASA Ames Projects: • Combining Symbolic Execution with (Explicit State) Model Checking • Assumption Generation for Component Verification

OK Finite-state system Verification tool or Error trace (F W) Line 5: … Line 12: … Line 15:… Line 21:… Line 25:… Line 27:… … Line 41:… Line 47:… Specification Finite-state Verification

Finite-State Verification • Effective for analyzing properties of hardware systems Widespread success and adoption in industry • Recent years have seen many efforts to apply those techniques to software Limited success due to the enormous state spaces associated with most software systems

symbolic state represents a set of states Original system Abstract system abstraction Safety: The set of behaviors of the abstract system over-approximates the set of behaviors of the original system Abstraction: the key to scaling up

Develop multiple forms of tool support for abstraction that is… … applicable to program source code … largely automated … usable by non-experts Evaluate the effectiveness of this tool support through… … implementation in the Bandera toolset … application to real multi-threaded Java programs Goals of our work …

int (n<0) : NEG (n==0): ZERO (n>0) : POS Signs Signs x = ZERO; if (Signs.eq(x,ZERO)) x = Signs.add(x,POS); NEG ZERO POS Data Type Abstraction Collapses data domains via abstract interpretation: Code Data domains int x = 0; if (x == 0) x = x + 1;

Bandera Abstraction Specification Language PVS Concrete Type Abstract Type Inferred Type Abstraction Definition Variable x int y int done bool count int BASL Compiler …. o Object b Buffer Program Abstracted Program Abstract Code Generator Abstraction in Bandera Signs Signs Signs bool Abstraction Library int …. Point Buffer

Automatic Generation Example: Start safe, then refine:+(NEG,NEG)={NEG,ZERO,POS} Forall n1,n2: neg?(n1) and neg?(n2) implies not pos?(n1+n2) Proof obligations submitted to PVS... Forall n1,n2: neg?(n1) and neg?(n2) implies not zero?(n1+n2) Forall n1,n2: neg?(n1) and neg?(n2) implies not neg?(n1+n2) Definition of Abstractions in BASL operator + add begin (NEG , NEG) -> {NEG} ; (NEG , ZERO) -> {NEG} ; (ZERO, NEG) -> {NEG} ; (ZERO, ZERO) -> {ZERO} ; (ZERO, POS) -> {POS} ; (POS , ZERO) -> {POS} ; (POS , POS) -> {POS} ; (_,_) -> {NEG,ZERO,POS}; /* case (POS,NEG),(NEG,POS) */ end abstraction Signs abstracts int begin TOKENS = { NEG, ZERO, POS }; abstract(n) begin n < 0 -> {NEG}; n == 0 -> {ZERO}; n > 0 -> {POS}; end

Compiled Compiling BASL Definitions abstraction Signs abstracts int begin TOKENS = { NEG, ZERO, POS }; abstract(n) begin n < 0 -> {NEG}; n == 0 -> {ZERO}; n > 0 -> {POS}; end operator + add begin (NEG , NEG) -> {NEG} ; (NEG , ZERO) -> {NEG} ; (ZERO, NEG) -> {NEG} ; (ZERO, ZERO) -> {ZERO} ; (ZERO, POS) -> {POS} ; (POS , ZERO) -> {POS} ; (POS , POS) -> {POS} ; (_,_)-> {NEG, ZERO, POS}; /* case (POS,NEG), (NEG,POS) */ end public class Signs { public static final int NEG = 0; // mask 1 public static final int ZERO = 1; // mask 2 public static final int POS = 2; // mask 4 public static int abs(int n) { if (n < 0) return NEG; if (n == 0) return ZERO; if (n > 0) return POS; } public static int add(int arg1, int arg2) { if (arg1==NEG && arg2==NEG) return NEG; if (arg1==NEG && arg2==ZERO) return NEG; if (arg1==ZERO && arg2==NEG) return NEG; if (arg1==ZERO && arg2==ZERO) return ZERO; if (arg1==ZERO && arg2==POS) return POS; if (arg1==POS && arg2==ZERO) return POS; if (arg1==POS && arg2==POS) return POS; return Bandera.choose(7); /* case (POS,NEG), (NEG,POS) */ }

Abstract Counter-example Analysis • For an abstracted program, a counter-example may be infeasible because: • Over-approximation introduced by abstraction • Example: x = -2; if(x + 2 == 0) then ... x = NEG; if(Signs.eq(Signs.add(x,POS),ZERO)) then ... {NEG,ZERO,POS}

Our Solutions • Choice-bounded State Space Search • “on-the-fly”, during model checking • Abstract Counter-example Guided Concrete Simulation • Exploit implementations of abstractions for Java programs • Effective in practice • Implemented in Java PathFinder tool

“Choose”-free state space search • Theorem [Saidi:SAS’00] Every path in the abstracted program where all assignments are deterministic is a path in the concrete program. • Bias the model checker • to look only at paths that do not include instructions that introduce non-determinism • JPF model checker modified • to detect non-deterministic choice (i.e. calls to Bandera.choose()); backtrack from those points

State space searched Undetectable Violation Detectable Violation Choice-bounded Search choose() X X

Case Study: DEOS Kernel (NASA Ames) • Honeywell Dynamic Enforcement Operating System (DEOS) • A real time operating system for integrated modular avionics • Non-trivial concurrent program (1433 lines of code, 20 classes, 6 threads) • Written in C++, translated into Java and Promela • With a known bug • Verification of the system exhausted 4 Gigabytes of memory without completion; abstraction needed • Abstracted using data type abstraction • Checked using JPF and SPIN • Defect detected using choice-bounded search

Conclusion and Future Research Directions • Tool support for abstraction enables verification of real properties of real programs • Extend abstraction support for objects • Heap abstractions tohandle an unbounded number of dynamically allocated objects • Handle recursive procedures, unbounded number of processes • Extend automation • For selection and refinement based on counter-example analysis

Java Path Finder (NASA Ames) • Model checker for Java programs • Built on top of a custom made Java Virtual Machine • Checks for deadlock and violation of assertions; LTL properties • Support for abstraction: • Predicate abstraction • Bandera’s data abstraction • Heuristic search

Symbolic execution tree (PC=“path condition”) n:S PC:true 4 5 2 5 1 3 n:S PC:S<=0 n:S PC:S>0 ... n:S+1 PC:S>0 n:S+1 PC:S>0 & S+1>=3 n:S+1 PC:S>0 & S+1<3 ... ... Symbolic Execution Uses “symbolic names” to represent program inputs Code void test(int n){ [1] if (n > 0) { [2] n = n + 1; [3] if (n < 3) [4] ... } [5] ... }

Symbolic Execution and JPF: Applications • Extends JPF with a new form of abstraction • Test case generation • Abstract counter-example analysis and refinement • Symbolic execution of multithreaded programs • Parameter synthesis …

Implementation in JPF • Easy: • Uses Bandera’s type abstraction • Uses Omega library (Java version) • Manipulates sets of linear constraints over integer variables • Can be used as a “symbolic execution tool with backtracking” • Good for finding counter-examples • No state matching!

PathCondition PC; // =“true” void test(SymVal n) { n = new SymVal(); if(SymOps.gt(n,new SymVal(0)){ n=SymOps.add(n,new SymVal(1)); ... } (Possible) Implementation Code public class SymVal { public SymVal() { ... } public SymVal(int n) { ... } public SymVal(SymVal s1, SymVal s2, String ops) { ... } ... } public class SymOps { public SymVal add(SymVal s1, SymVal s2){ return new SymVal(s1,s2,’+’); } public bool gt(SymVal s1, SymVal s2) { bool result = Verify.chooseBool(); if(result) { // “true” PC.addCondition(s1,s2,’>’); } else { // “false” PC.addCondition(s1,s2,’<=‘); } PC.simplify(); return result; } ... } void test(int n) { if (n > 0) { n = n + 1; ... }

n:S PC:true 4 3 1 2 2 4 3 n:S,x:0 PC:0>=S n:S,x:0 PC:true n:S,x:0 PC:0<S n:S,x:1 PC:0<S n:S,x:1 PC:0<S & 1>=S n:S,x:1 PC:0<S & 1<S .... Problem: Convergence Symbolic execution tree Code void test(int n) { [1] int x = 0; [2] while(x < n) [3] x = x + 1; [4] }

Problem: Convergence Solutions? • Limit the search depth of MC • Unwind loops a fixed number of times (similar to Bounded MC?) • Discover “simple and practical” widening techniques • Acceleration techniques • Heuristics? • Combine with “predicate abstraction” …

Relation to Bounded MC • Extend BMC with symbolic variables? • Widening for C programs? • …

Component Assumption Generation for Component Verification • Problem: Environment Property ? Environment Assumption ? The “weakest” assumption A for component C: for all environments E, E |= A  E || C |= P

Applications • Support for modular verification • Compositional verification • Property decomposition • Run-time monitoring of the environment • Component retrieval • Sub-module construction …

Implementation • In Labeled Transition Systems Analyzer (LTSA) tool - Imperial college • Supports compositional reachability analysis based on software architecture • Incremental system design and verification: • Component abstraction (hiding of internal actions) • Minimization wrt. observational equivalence • Both components and properties expressed as labeled transition systems

Mutex: Mutual Exclusion Property: E.acquire W.acquire E.enterCS W.enterCS W.acquire E.release || E.exitCS W.exitCS Writer: W.enterCS W.exitCS E.enterCS Interface actions W.exitCS E.exitCS W.enterCS E.enterCS E.exitCS W.acquire W.enterCS W.exitCS W.release Example: A System and A Property ||

Property true! (all environments) Step 2: backward reachability with error state Property false! (all environments) Step 3: property extraction (sub-set construction and completion) Assumption Assumption Generation Step 1: composition, hiding of internal actions and minimization

Composite System E.release E.enterCS E.acquire E.exitCS E.release E.exitCS E.enterCS E.release E.enterCS E.enterCS t E.enterCS E.exitCS E.exitCS

Backward Error Propagation (with t) E.release E.enterCS E.acquire E.exitCS E.release E.exitCS E.enterCS E.release E.enterCS E.enterCS t E.enterCS E.exitCS E.exitCS

Backward Error Propagation (with t) E.release E.enterCS E.exitCS E.release E.enterCS E.release E.enterCS E.enterCS E.exitCS E.exitCS

E.acquire, E.release E.enterCS, E.exitCS E.release E.acquire E.acquire Property Extraction E.acquire E.enterCS E.release E.exitCS E.enterCS E.exitCS E.enterCS E.release E.exitCS

Generated Assumption E.acquire, E.release E.enterCS, E.exitCS E.release E.acquire E.acquire E.acquire E.enterCS E.release E.exitCS

Directions for Future Work • Liveness /fairness • Extend to other frameworks • LTL checking (since we are interested only in error behaviors) • Is the sub-set construction needed? • Study other forms of composition …

Abstraction and Modular Reasoning for the Verification of Software