formal-verification-guide

Formal Verification Guide

A skill for applying formal methods to verify software and hardware correctness. Covers model checking, interactive theorem proving, specification languages, and practical verification workflows used in systems and programming language research.

Verification Approaches Overview

Methods Comparison

| Approach | Technique | Strengths | Limitations |

|----------|-----------|-----------|-------------|

| Model checking | Exhaustive state exploration | Fully automatic, produces counterexamples | State space explosion |

| Theorem proving | Interactive proof construction | Handles infinite state | Requires expert effort |

| Abstract interpretation | Sound static analysis | Automatic, scales well | May report false positives |

| SMT solving | Constraint satisfiability | Powerful automation | Limited to decidable theories |

| Runtime verification | Execution monitoring | Low barrier, practical | Only checks observed runs |

TLA+ Specification

Specifying Distributed Protocols

TLA+ is the standard specification language for distributed systems:

--------------------------- MODULE TwoPhaseCommit -------------------------
EXTENDS Integers, Sequences, FiniteSets

CONSTANTS RM  \* Set of resource managers

VARIABLES
    rmState,      \* rmState[r] is the state of resource manager r
    tmState,      \* State of the transaction manager
    tmPrepared,   \* Set of RMs that have sent "Prepared"
    msgs          \* Set of messages sent

vars == <<rmState, tmState, tmPrepared, msgs>>

Init ==
    /\ rmState = [r \in RM |-> "working"]
    /\ tmState = "init"
    /\ tmPrepared = {}
    /\ msgs = {}

\* RM r prepares to commit
RMPrepare(r) ==
    /\ rmState[r] = "working"
    /\ rmState' = [rmState EXCEPT ![r] = "prepared"]
    /\ msgs' = msgs \union {[type |-> "Prepared", rm |-> r]}
    /\ UNCHANGED <<tmState, tmPrepared>>

\* TM receives a Prepared message from RM r
TMRcvPrepared(r) ==
    /\ tmState = "init"
    /\ [type |-> "Prepared", rm |-> r] \in msgs
    /\ tmPrepared' = tmPrepared \union {r}
    /\ UNCHANGED <<rmState, tmState, msgs>>

\* TM commits (all RMs have prepared)
TMCommit ==
    /\ tmState = "init"
    /\ tmPrepared = RM
    /\ tmState' = "committed"
    /\ msgs' = msgs \union {[type |-> "Commit"]}
    /\ UNCHANGED <<rmState, tmPrepared>>

\* Safety property: No RM commits unless TM has committed
Consistency ==
    \A r \in RM : rmState[r] = "committed" => tmState = "committed"
========================================================================

Running the TLC Model Checker

# Install TLA+ Toolbox or use command-line TLC
# Define model with specific constants
# RM = {"rm1", "rm2", "rm3"}
java -jar tla2tools.jar -config TwoPhaseCommit.cfg TwoPhaseCommit.tla

# TLC will explore all reachable states and verify:
# - No deadlocks (unless specified)
# - Safety properties (invariants)
# - Liveness properties (temporal formulas)

Interactive Theorem Proving

Coq Proof Assistant

(* Example: Proving properties of a simple functional program *)

(* Define natural number addition *)
Fixpoint add (n m : nat) : nat :=
  match n with
  | O => m
  | S n' => S (add n' m)
  end.

(* Prove: 0 + n = n (left identity) *)
Theorem add_0_l : forall n : nat, add 0 n = n.
Proof.
  intro n.
  simpl.    (* simplification reduces add 0 n to n *)
  reflexivity.
Qed.

(* Prove: n + 0 = n (right identity, requires induction) *)
Theorem add_0_r : forall n : nat, add n 0 = n.
Proof.
  intro n.
  induction n as [| n' IHn'].
  - (* Base case: n = 0 *)
    simpl. reflexivity.
  - (* Inductive step: n = S n' *)
    simpl.               (* add (S n') 0 = S (add n' 0) *)
    rewrite IHn'.        (* apply induction hypothesis *)
    reflexivity.
Qed.

(* Prove associativity of addition *)
Theorem add_assoc : forall a b c : nat,
  add a (add b c) = add (add a b) c.
Proof.
  intros a b c.
  induction a as [| a' IHa'].
  - simpl. reflexivity.
  - simpl. rewrite IHa'. reflexivity.
Qed.

Isabelle/HOL

theory SimpleVerification
  imports Main
begin

(* Define a recursive function *)
fun fib :: "nat => nat" where
  "fib 0 = 0"
| "fib (Suc 0) = 1"
| "fib (Suc (Suc n)) = fib (Suc n) + fib n"

(* Prove a property *)
lemma fib_positive: "0 < fib (Suc n)"
  by (induction n rule: fib.induct) auto

(* Verify a sorting algorithm *)
fun insert :: "nat => nat list => nat list" where
  "insert x [] = [x]"
| "insert x (y # ys) = (if x <= y then x # y # ys else y # insert x ys)"

fun isort :: "nat list => nat list" where
  "isort [] = []"
| "isort (x # xs) = insert x (isort xs)"

(* Prove the output is sorted *)
lemma sorted_insert: "sorted (insert x xs) = sorted xs"
  sorry (* full proof requires additional lemmas *)

end

SMT Solving

Z3 for Program Verification

from z3 import Solver, Int, Bool, And, Or, Not, Implies, ForAll, sat, unsat

def verify_array_bounds():
    """
    Verify that an array access is always within bounds.
    Model a loop: for i = 0 to n-1, access a[i].
    """
    s = Solver()
    n = Int("n")
    i = Int("i")

    # Precondition: n > 0
    s.add(n > 0)

    # Loop invariant: 0 <= i < n at each access
    s.add(i >= 0)
    s.add(i < n)

    # Verify: the access a[i] is within bounds [0, n)
    s.add(Not(And(i >= 0, i < n)))  # try to find a violation

    result = s.check()
    if result == unsat:
        return "VERIFIED: array access is always within bounds"
    else:
        return f"COUNTEREXAMPLE: {s.model()}"

def verify_integer_overflow():
    """
    Check if integer addition can overflow for given constraints.
    """
    from z3 import BitVec, BitVecVal

    s = Solver()
    # 32-bit signed integers
    x = BitVec("x", 32)
    y = BitVec("y", 32)

    # Preconditions: both positive
    s.add(x > 0)
    s.add(y > 0)

    # Check: can x + y wrap around to negative?
    s.add(x + y < 0)

    if s.check() == sat:
        m = s.model()
        return {
            "overflow_possible": True,
            "x": m[x].as_long(),
            "y": m[y].as_long(),
        }
    return {"overflow_possible": False}

Model Checking with SPIN

Promela Specification

/* Mutual exclusion with Peterson's algorithm */
bool flag[2] = false;
byte turn = 0;
byte critical = 0;  /* count of processes in critical section */

active [2] proctype process() {
    byte me = _pid;
    byte other = 1 - _pid;

    do
    :: /* Entry protocol */
       flag[me] = true;
       turn = other;
       (flag[other] == false || turn == me);

       /* Critical section */
       critical++;
       assert(critical == 1);  /* mutual exclusion */
       critical--;

       /* Exit protocol */
       flag[me] = false;
    od
}

/* LTL property: mutual exclusion always holds */
ltl mutex { [] (critical <= 1) }

Verification Workflow

Practical Verification Strategy

1. Specify: Write a formal specification of the desired property
2. Model: Create an abstract model of the system
3. Verify: Run model checker or construct proof
4. Refine: If counterexample found, fix the design or refine the model
5. Extract: Generate verified code from the proof (Coq extraction, Isabelle code generation)

Common Properties to Verify

| Property Type | Example | Specification Pattern |

|--------------|---------|----------------------|

| Safety | "No two processes in critical section" | [] (count <= 1) |

| Liveness | "Every request is eventually served" | [] (request -> <> response) |

| Deadlock freedom | "System always has an enabled transition" | [] <> enabled |

| Termination | "Program always halts" | Well-founded ordering |

Tools and Resources

• TLA+ Toolbox: IDE for TLA+ with integrated TLC model checker
• Coq: Interactive theorem prover with program extraction
• Isabelle/HOL: Higher-order logic prover with Sledgehammer automation
• Z3 / CVC5: SMT solvers for automated reasoning
• SPIN: Model checker for concurrent systems (Promela)
• CBMC: Bounded model checker for C programs
• Dafny: Verification-aware programming language (Microsoft)
• Lean 4: Modern theorem prover and programming language