Tool Using SMT Solver

. We present an SMT solving based test generation approach for MATLAB Simulink designs, implemented in the HiLiTE tool developed by Honeywell for veriﬁcation of avionic systems. The test requirements for a Simulink model are represented by a set of behavioral equivalence classes for each block in the model, in terms of its input(s) and output. A unique feature of our approach is that the equivalence class deﬁnitions, as well as the upstream subgraph of a block under test, are translated as constraints into SMT expressions. An SMT solver is called at the back-end of HiLiTE to ﬁnd a satisﬁable solution that is further augmented into an end-to-end test case at the model level.


Introduction
As the industry practices engage model-based design increasingly, model-based verification and testing [1] techniques emerge to keep up with the trends. In avionics area, comprehensive testing methods and tools are required to assure that safety-critical systems like flight controls are certified to the guidelines established by standard processes such as the DO-178C [2].
At Honeywell, researchers have developed the Honeywell Integrated Lifecycle Tools & Environment (HiLiTE) suite of tools for the automated verification of avionics applications developed using MATLAB Simulink/Stateflow. HiLiTE performs automatic test generation [3] on Simulink models based upon the lowlevel requirements (LLRs) expressed by the model elements. The tests are then applied to the executable object code generated from the model to verify that the code complies with the LLRs in the design model. HiLiTE has been qualified as a DO-178C verification tool and deployed in several avionics product certifications to deliver significant cost savings in the verification effort.
This paper presents an SMT solving technique to extend the earlier heuristics-based test case generation approaches implemented in HiLiTE, providing improved performance on models with complex constraints or non-linear arithmetic computations. SMT solving is the decision procedure of determining whether a formula in first-order-logic is satisfiable and finding a concrete solution if it is. SMT Solvers, such as Z3 [4], Yices [5], etc., have rapidly matured over the last 5 years and have been used in various areas including automated test case generation [6,7]. In our SMT solving based approach, each LLR equivalence class for a block type is represented by a set of constraints, applied on block-level input(s) and expected output. Meanwhile, test space is also constrained by the subgraph environment that the block under test (BUT) is embedded in. The collection of constraints can be formulated as an SMT problem and expressed in a standard format by HiLiTE in an automatic fashion. SMT solver is then called to generate the satisfiable solution once for all ports in the related subgraph. The solution is merged back to the entire graph for a complete model-level input-tooutput test case. With the integration of heuristics and SMT solving techniques, HiLiTE has been successfully used to generate requirement-based test cases for a great range of large-scale complex constrained avionics models.
Section 2 describes the HiLiTE normal test case generation approach and the need for improvements. Section 3 describes the formalized language of equivalence classes of block's behaviors and SMT solving based test case generation approach. Finally, the conclusion and future work are discussed in Sect. 4.

HiLiTE Test Generation Approach
HiLiTE generates specific tests at the model level to exercise the equivalence classes of the behavior of each block embedded in the model. In the original HiLiTE tool, each equivalence class of a block's behavior is represented by a set of test case templates, each of which uses heuristics to select a specific combination of values for the block under test (BUT) input(s) and output that satisfy this equivalence class. Backward and forward propagation search through the computations of other blocks in the model generates a test vector in terms of model inputs and outputs to ensure controllability of the BUT inputs and observability of the expected BUT output. Figure 1 shows a Simulink model extracted from a complex industrial model to illustrate this. When the product block is the BUT, the test case template (Fig. 2) assigns the two inputs with non-zero values 2 and 4 respectively. After the backward and forward propagation search, the generated test vectors are given in Fig. 2 where blue column heading denotes model input and green denotes model output. When the switch block is the BUT, the equivalence class requires different values at its data inputs (FalseIn, TrueIn) to verify unique impact of an input on the block's output. One test case template assigns 44 to FalseIn and 46 to TrueIn, but this leads to a conflict at the model input AdjustPct after backward computation through the two look-up tables since their data points are in the same range. HiLiTE then further tries several alternative templates based on heuristics, yet all result in search failure. The root cause is that HiLiTE templates heuristics in the equivalence class domain prematurely pick block's local input values, while this problem involves taking into account constraints imposed by the look-up table blocks driven by the same input AdjustPct.

Applying SMT Solving in Test Case Generation
Test generation difficulties such as those noted above can be addressed by an approach that solves computational constraints of the upstream subgraph of BUT in conjunction with the constraints on BUT inputs imposed by the behavior equivalence class. SMT can be thought of the constraint satisfaction problem expressed in Boolean formulas, linear/nonlinear arithmetic in integer/real domain, bit-vectors and so on. In HiLiTE, we added SMT solving based approach that embodies formulating test case generation constraints from both equivalence classes, constraints related to upstream source ports and the subgraph computations upstream of the BUT into an SMT problem. Therefore, constraints can be solved together to find a satisfying solution which excludes any conflicts.

Formal Specification of Equivalence Classes of Block Behaviors
An equivalence class of a block behavior, which represents a test requirement, is now expressed in HiLiTE with formalized rules on the block's input and output ports. These rules are expressed in a language as shown in Table 1 for switch block with rule names in blue. Each rule is automatically translated, based upon formal definitions, into SMT logic formula in a straightforward way. For example, "Exists" for port TrueIn is evaluated to "true" if any value is present, "NEQ(FalseIn, TrueIn)" of port FalseIn is interpreted as "FalseIn = TrueIn", and "Equal(Control, 1)" of port Control is interpreted as "Control = true" since Control has a Boolean type. The overall SMT logic formula for an equivalence class is the conjunction of individual formulas translated from equivalence class rules for each block port. E.g., the equivalence class "Verify TRUE Input" corresponds to "(FalseIn = TrueIn)∧(Control = true)". Note: support for time-dependent blocks (e.g., UnitDelay) also allows us to explore feedback loops in the model for bounded number of steps.

Formulated SMT Problem
HiLiTE explores the upstream subgraph of the BUT to ensure all constraints imposed by the subgraph computations on the test case generation are included. The subgraph exploration uses depth-first search, starting from the inputs of the BUT identified in its equivalence class, all the way back to the model inputs. Once the SMT-available subgraph is obtained, HiLiTE loops through each block in it to collect its SMT logic formulas. SMT logic formulas are further translated into expressions in SMT-LIB 2.0 standard format, recognized by popular SMT solvers like Z3, with actual port names as variables. Additionally, to specify the block connections, each input in the formula is replaced by its source block output. For instance, the SMT expression for the switch block in Fig. 1 is (assert  (and (not (= gainTable 1.O1 gainTable 2.O1)) (= equalTo.O1 true))). Finally, the variables in the SMT expressions are substituted by a short form y i j (time step subscript j is omitted if there are no time-dependent blocks), where each block is assigned with a unique index i as shown in Fig. 3.   Fig. 3. Input file for SMT solver Z3 of test generation for switch block in Fig. 1.
The SMT logic formula is built initially with the number of time steps m determined by the equivalence class of the BUT: if the result of SMT solving is "unsat", the formula is then updated with m ← m + 1. The process is repeated until either SMT solver returns "sat" or a pre-defined time step limit is reached. In the worst case, m may become very large before a value at some point (such as the output of a timer/integrator/counter) of the model is accumulated to satisfy the constraints, in which case SMT solver may break down or return "unknown". To bypass blocks causing over-sized formulas, and certain mathematical blocks (e.g., sin) not supported by SMT solving, HiLiTE identifies those blocks, records them as pending, and explores the neighbor paths, resulting in an incomplete subgraph. The pending blocks and their upstream blocks are excluded from the solution returned by SMT solver. HiLiTE normal method then picks up from here to further propagate the pending values. An improvement can be done if the backward search goes through a switch block, only one data input of which has pending block(s) on its upstream. The values on the branch with pending block(s) do not matter if we force the switch block to disable that branch. This is done by modifying the SMT logic formula of that switch block. Suppose port In 1 of block switch is to be disabled, then the SMT logic formula becomes m−1 j=0 (In 3 j ∧ (Out j = In 2 j )).

Tool Architecture
The SMT solving based test case generation is implemented by HiLiTE as a fully automated process shown in Fig. 4. Test cases needing SMT solving based approach are identified by the complexity of relationships detected during model analysis. For these, the test generation module formulates a collection of SMT expressions and writes them into a .smt2 file as described in Sect. 3. The SMT solver is called as a back-end, generating a solution which is then merged into test generation search space. HiLiTE normal method takes over from here to propagate the switch block output through the forward path to the model output GainAdj via the intervening product block, using a non-zero value for the second input of product block to ensure observability. This process results in valid test cases (Fig. 5) for the switch block in Fig. 2.   Fig. 5. Test vector generated for the switch block in Fig. 2 via SMT solving.

Nonlinear Applications
Modern SMT solvers are capable of solving a great range of non-linear problems used be computational intractable. Figure 6 shows a simple two-variable 2ndorder polynomial model. Z3 returns an answer for this case (as shown in Fig. 6) as in many other nonlinear problems.

Conclusion and Future Work
We extended the HiLiTE test generation capability with an SMT solving based approach for solving certain complex constrained problems. The improved tool combines HiLiTE normal search method and SMT solving, and has been successfully applied on many large-scale industrial models. HiLiTE is also being extended to derive invariant bounds on the number of time steps (e.g., for a timer) that will help bound the array size. We are also applying SMT solving to support such invariant generation, in which each condition-guarded path that captures a certain pattern of model behavior can be validated by SMT solving.
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