SMT formulas over the real numbers can encode a wide range of problems in theorem proving and formal verification. Such formulas are very hard to solve when nonlinear functions are involved. $$\delta$$-Complete decision procedures provided a new general framework for handling nonlinear SMT problems over the reals. We say a decision procedure is $$\delta$$-complete for a set $$S$$ of SMT formulas, where $$\delta$$ is an arbitrary positive rational number, if for any $$\varphi$$ from $$S$$ the procedure returns one of the following answers:

• “unsat”: $$\varphi$$ is unsatisfiable.
• $$\delta$$-sat”: $$\varphi^{\delta}$$ is satisfiable.

Here, $$\varphi^{\delta}$$ is a syntactic variant of $$\varphi$$ that encodes a notion of numerical perturbation on logic formulas. Essentially, we allow such a procedure to give answers with one-sided, $$\delta$$-bounded errors. With this relaxation, $$\delta$$-complete decision procedures can fully exploit the power of scalable numerical algorithms to solve nonlinear problems, and at the same time provide suitable correctness guarantees for many correctness-critical problems. (See references below.)

dReal is an SMT solver for formulas over the reals that can handle various nonlinear elementrary functions in the framework of $$\delta$$-complete decision procedures. It returns “unsat” or “$$\delta$$-sat” on input formulas, where $$\delta$$ can be specified by the user. When the answer is “unsat”, dReal produces a proof of unsatisfiability; when “$$\delta$$-sat”, it provides a solution such that a $$\delta$$-perturbed form of the input formula is satisfied. The tool is based on Interval Constraint Propagation in the DPLL(T) framework to handle nonlinearity, and is designed to be easily extendable with other numerical algorithms.

dReal is built on the following tools: realpaver, opensmt, minisat, and capd.

## Example

Let’s consider the following example which slighly modifies a formula from the Flyspeck project benchmarks:

$\exists^{[3.0,3.14]}x_1. \exists^{[-7.0,5.0]}x_2. 2 \times 3.14159265 - 2 x_1 \arcsin \left(\cos 0.797\times \sin \left(\frac{3.14159265}{x_1}\right)\right) \le - 0.591 - 0.0331 x_2 + 0.506 + 1.0$

### Solving with dReal

To solve the formula using dReal, we first translate it into the following SMT2 formula (172.smt2):

(set-logic QF_NRA)
(declare-fun x1 () Real)
(declare-fun x2 () Real)
(assert (<= 3.0 x1))
(assert (<= x1 3.14))
(assert (<= -7.0 x2))
(assert (<= x2 5.0))
(assert (<= (- (* 2.0 3.14159265) (* 2.0 (* x1 (arcsin (* (cos 0.797) (sin (/ 3.14159265 x1)))))))
(+ (- 0.591 (* 0.0331 x2)) (+ 0.506 1.0))))
(check-sat)
(exit)


Note that we encode the range of $$x_1$$ and $$x_2$$ using four assert commands (assert (<= 3.0 x1), (assert (<= x1 64.0)), (assert (<= -7.0 x2)), and (assert (<= x2 5.0)). Then, we check the $$\delta$$-satisfiability of the formula using dReal:

$dReal 172.smt2 unsat  It takes less than a second to terminate with the unsat result. Recall that this unsat result is exact and does not involve any numerical approximation. In the above example, we did not provide the value of $$\delta$$ and therefore dReal used the default value – $$0.001$$. We do have a command-line argument to specify the delta, which sometimes generate a different result as follows: $ dReal --precision=1.0 172.smt2
delta-sat


To see the detailed decision traces along with the solving process, use --verbose option (the omitted result is in 172.smt.verbose):

$dReal --verbose 172.smt2 ... unsat  ### Proof Checking dReal is also able to generate a proof along with the $$\delta$$-satisfiability result. Using --proof option generates the proof 172.smt2.proof. $ dReal --proof 172.smt
unsat


We also provide a proof checker which can validate the proof for the unsat cases. Our proof-checking process is a semi-algorithm and therefore its termination is not guaranteed. -t option shoud be used to specify the timeout in seconds.

\$ proofcheck.sh -t 30 172.smt2.proof
...
proof verified


172.smt2.proof.output shows that our proof checker solved 4 subproblems in the process of checking and was able to verify the proof within 3 seconds. It saves all the extra information under the directory 172.smt2.proof.extra.