What4

Introduction

What is What4?

What4 is a Haskell library developed at Galois that presents a generic interface to SMT solvers (Z3, Yices, etc.). Users of What4 use an embedded DSL to create fresh constants representing unknown values of various types (integer, boolean, etc.), assert various properties about those constants, and ask a locally-installed SMT solver for satisfying instances.

What4 relies heavily on advanced GHC extensions to ensure that solver expressions are type correct. The parameterized-utils library is used throughout What4 as a “standard library” for dependently-typed Haskell.

Quick start

Let’s start with a quick end-to-end tutorial, demonstrating how to create a model for a basic satisfiability problem and ask a solver for a satisfying instance. The code for this quick start may be found in doc/QuickStart.hs, and you can compile and run the quickstart by executing the following line at the command line from the source root of this package.

$ cabal v2-run what4:quickstart

We will be using an example from the first page of Donald Knuth’s The Art Of Computer Programming, Volume 4, Fascicle 6: Satisfiability:

F(p, q, r) = (p | !q) & (q | r) & (!p | !r) & (!p | !q | r)

We will use What4 to:

• generate fresh constants for the three variables p, q, and r
• construct an expression for F
• assert that expression to our backend solver
• ask the solver for a satisfying instance.

We first enable the GADTs extension (necessary for most uses of What4) and pull in a number of modules from What4 and parameterized-utils:

{-# LANGUAGE GADTs #-}
module Main where

import Data.Foldable (forM_)
import System.IO (FilePath)

import Data.Parameterized.Nonce (newIONonceGenerator)
import Data.Parameterized.Some (Some(..))

import What4.Config (extendConfig)
import What4.Expr
         ( ExprBuilder,  FloatModeRepr(..), newExprBuilder
         , BoolExpr, GroundValue, groundEval
		 , EmptyExprBuilderState(..) )
import What4.Interface
         ( BaseTypeRepr(..), getConfiguration
         , freshConstant, safeSymbol
         , notPred, orPred, andPred )
import What4.Solver
         (defaultLogData, z3Options, withZ3, SatResult(..))
import What4.Protocol.SMTLib2
         (assume, sessionWriter, runCheckSat)

We create a trivial data type for the “builder state” (which we won’t need to use for this simple example), and create a top-level constant pointing to our backend solver, which is Z3 in this example. (To run this code, you’ll need Z3 on your path, or edit this path to point to your Z3.)

z3executable :: FilePath
z3executable = "z3"

We’re ready to start our main function:

main :: IO ()
main = do
  Some ng <- newIONonceGenerator
  sym <- newExprBuilder FloatIEEERepr EmptyExprBuilderState ng

Most of the functions in What4.Interface, the module for building up solver expressions, require an explicit sym parameter. This parameter is a handle for a data structure that caches information for sharing common subexpressions and other bookkeeping purposes. What4.Expr.Builder.newExprBuilder creates one of these, and we will use this sym throughout our code.

Before continuing, we will set up some global configuration for Z3. This sets up some configurable options specific to Z3 with default values.

  extendConfig z3Options (getConfiguration sym)

We declare fresh constants for each of our propositional variables.

  p <- freshConstant sym (safeSymbol "p") BaseBoolRepr
  q <- freshConstant sym (safeSymbol "q") BaseBoolRepr
  r <- freshConstant sym (safeSymbol "r") BaseBoolRepr

Next, we create expressions for their negation.

  not_p <- notPred sym p
  not_q <- notPred sym q
  not_r <- notPred sym r

Then, we build up each clause of F individually.

  clause1 <- orPred sym p not_q
  clause2 <- orPred sym q r
  clause3 <- orPred sym not_p not_r
  clause4 <- orPred sym not_p =<< orPred sym not_q r

Finally, we can create F out of the conjunction of these four clauses.

  f <- andPred sym clause1 =<<
       andPred sym clause2 =<<
       andPred sym clause3 clause4

Now we can we assert f to the backend solver (Z3, in this example), and ask for a satisfying instance.

  -- Determine if f is satisfiable, and print the instance if one is found.
  checkModel sym f [ ("p", p)
                   , ("q", q)
                   , ("r", r)
                   ]

(The checkModel function is not a What4 function; its definition is provided below.)

Now, let’s add one more clause to F which will make it unsatisfiable.

  -- Now, let's add one more clause to f.
  clause5 <- orPred sym p =<< orPred sym q not_r
  g <- andPred sym f clause5

Now, when we ask the solver for a satisfying instance, it should report that the formulat is unsatisfiable.

  checkModel sym g [ ("p", p)
                   , ("q", q)
                   , ("r", r)
                   ]

This concludes the definition of our main function. The definition for checkModel is as follows:

-- | Determine whether a predicate is satisfiable, and print out the values of a
-- set of expressions if a satisfying instance is found.
checkModel ::
  ExprBuilder t st fs ->
  BoolExpr t ->
  [(String, BoolExpr t)] ->
  IO ()
checkModel sym f es = do
  -- We will use z3 to determine if f is satisfiable.
  withZ3 sym z3executable defaultLogData $ \session -> do
    -- Assume f is true.
    assume (sessionWriter session) f
    runCheckSat session $ \result ->
      case result of
        Sat (ge, _) -> do
          putStrLn "Satisfiable, with model:"
          forM_ es $ \(nm, e) -> do
            v <- groundEval ge e
            putStrLn $ "  " ++ nm ++ " := " ++ show v
        Unsat _ -> putStrLn "Unsatisfiable."
        Unknown -> putStrLn "Solver failed to find a solution."

When we compile this code and run it, we should get the following output.

Satisfiable, with model:
  p := False
  q := False
  r := True
Unsatisfiable.

Where to go next

The key modules to look at when modeling a problem with What4 are:

• What4.BaseTypes (the datatypes What4 understands)
• What4.Interface (the functions What4 uses to build symbolic expressions)
• What4.Expr.Builder (the implementation of the functions in What4.Interface)

The key modules to look at when interacting with a solver are:

• What4.Protocol.SMTLib2 (the functions to interact with a solver backend)
• What4.Solver (solver-specific implementations of What4.Protocol.SMTLib2)
• What4.Solver.*
• What4.Protocol.Online (interface for online solver connections)
• What4.SatResult and What4.Expr.GroundEval (for analyzing solver output)

Additional implementation and operational documentation can be found in the implementation documentation in doc/implementation.md.

To serialize and deserialize what4 terms, see the following modules:

• What4.Serialize.Printer (to serialize what4 terms into an s-expression format)
• What4.Serialize.Parser (to deserialize what4 terms)
• What4.Serialize.FastSExpr (provides a faster s-expression parser than the default, intended to be used in conjunction with the higher-level parsing in What4.Serialize.Parser)

Formula Construction vs Solving

In what4, building expressions and solving expressions are orthogonal concerns. When you create an ExprBuilder (with newExprBuilder), you are not committing to any particular solver or solving strategy (except insofar as the selected floating point mode might preclude the use of certain solvers). There are two dimensions of solver choice: solver and mode. The supported solvers are listed in What4.Solver.*. There are two modes:

• All solvers can be used in an “offline” mode, where a new solver process is created for each query (e.g., via What4.Solver.solver_adapter_check_sat)
• Many solvers also support an “online” mode, where what4 maintains a persistent connection to the solver and can issue multiple queries to the same solver process (via the interfaces in What4.Protocol.Online)

There are a number of reasons to use solvers in online mode. First, state (i.e., previously defined terms and assumptions) can be shared between queries. For a series of closely related queries that share context, this can be a significant performance benefit. Solvers that support online solving provide the SMT push and pop primitives for maintaining context frames that can be discarded (to define local bindings and assumptions). The canonical use of online solving is symbolic execution, which usually requires reflecting the state of the program at every program point into the solver (in the form of a path condition) and using push and pop to mimic the call and return structure of programs. Second, reusing a single solver instance can save process startup overhead in the presence of many small queries.

While it may always seem advantageous to use the online solving mode, there are advantages to offline solving. As offline solving creates a fresh solver process for each query, it enables parallel solving. Online solving necessarily serializes queries. Additionally, offline solving avoids the need for complex state management to synchronize the solver state with the state of the tool using what4. Additionally, not all solvers that support online interaction support per-goal timeouts; using offline solving trivially allows users of what4 to enforce timeouts for each solved goal.

Known working solver versions

What4 has been tested and is known to work with the following solver versions.

Nearby versions may also work; however, subtle changes in solver behavior from version to version sometimes happen and can cause unexpected results, especially for the more experimental logics that have not been standardized. If you encounter such a situation, please open a ticket, as our goal is to work correctly on as wide a collection of solvers as is reasonable.

• Z3 versions 4.8.7 through 4.8.12
• Yices 2.6.1 and 2.6.2
• CVC4 1.7 and 1.8
• CVC5 1.0.2
• Boolector 3.2.1 and 3.2.2
• STP 2.3.3 (However, note https://github.com/stp/stp/issues/363, which prevents effective retrieval of model values. This should be resolved by the next release)
• dReal v4.20.04.1

Note that the integration with Z3, Yices and CVC4 has undergone significantly more testing than the other solvers.