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

Finite-state Verification OK Finite-state system Specification Verification tool  or Error trace Line 5: … Line 12: … Line 15:… Line 21:… Line 25:… Line 27:… … Line 41:… Line 47:…

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

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

Goals of our work … 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

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

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

Definition of Abstractions in BASL abstraction Signs abstracts int begin TOKENS = { NEG, ZERO, POS }; abstract(n) begin n {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 Automatic Generation Forall n1,n2: neg?(n1) and neg?(n2) implies not pos?(n1+n2) 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) Proof obligations submitted to PVS... Example: Start safe, then refine: +(NEG,NEG)={NEG,ZERO,POS}

Compiling BASL Definitions abstraction Signs abstracts int begin TOKENS = { NEG, ZERO, POS }; abstract(n) begin n {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) */ } Compiled

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

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

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

Counter-example guided simulation (?) Use abstract counter-example to guide simulation of concrete program Why it works: –Correspondence between concrete and abstracted program –Unique initial concrete state (Java defines default initial values for all data)

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 to handle 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

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

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

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!

(Possible) Implementation void test(int n) { if (n > 0) { n = n + 1;... } 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; }... } Code PathCondition PC; // =“true” void test(SymVal n) { n = new SymVal(); if(SymOps.gt(n,new SymVal(0)){ n=SymOps.add(n,new SymVal(1));... }

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

Problem: Convergence 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” … Solutions?

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

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

Assumption Generation for Component Verification Problem: Component 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

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

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

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

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

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

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

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