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Title
Abstract
Figures
Introduction
Methodology
Reachability Analysis Over a Path
Limitations
Results
Experimental Setup
Related Works
Conclusion
References
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Distributed Systems

SAT-Reach : A Bounded Model Checker for Affine Hybrid Systems

Profile image of rajarshi rayrajarshi ray

ACM Transactions on Embedded Computing Systems

https://doi.org/10.1145/3567425
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36 pages

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Abstract

Bounded model checking (BMC) is well-known to be undecidable even for simple hybrid systems. Existing work targeted for a wide class of non-linear hybrid systems reduces the BMC problem to the satisfiability problem of an SMT formula encoding the hybrid system dynamics. Consequently, the satisfiability of the formula is deduced with a δ -decision procedure. However, the encoded formula can be complex for large automaton and for deep exploration causing the decision procedure to be inefficient. Additionally, a generalized decision procedure can be inefficient for hybrid systems with simple dynamics. In this paper, we propose a BMC algorithm built upon the foundation of the CEGAR technique and targeted for hybrid systems with piecewise affine dynamics, modeled as a hybrid automaton. In particular, our algorithm begins by searching an abstract counterexample in the discrete state-space of the automaton. We check whether a discovered abstract counterexample is spurious or real by a two-...

Figures (13)
Fig. 2. The graph structure of a hybrid automaton is shown for illustration. v; and v17 are the designated initial and the unsafe location respectively in the verification problem. Fig. 1. SAT-assisted reachability and-falsification procedure for BMC of affine hybrid automaton.
Fig. 2. The graph structure of a hybrid automaton is shown for illustration. v; and v17 are the designated initial and the unsafe location respectively in the verification problem. Fig. 1. SAT-assisted reachability and-falsification procedure for BMC of affine hybrid automaton.
along with the smallest infeasible subpath in z’ when z is infeasible. Line 13 checks if the infeasible path z’ is strictly a subpath of , denoted by len(z’) < len(z) to mean that the length of 7’ is shorter than that of z. When found to be a strict subpath, we add 7’ to II for pruning all its extensions. If the length of 7! and 7 are equal, we do not prune the extensions of ! because it is still possible that an unsafe trajectory may be present that follows in an extension of z. For a feasible path z', we call the trajectory splicing algorithm in line 18 to search for a counterexample trajectory along the path. If it successfully finds a counterexample trajectory, the procedure terminates by classifying the verification instance as unsafe together with the concrete trajectory. Line 26 finds
along with the smallest infeasible subpath in z’ when z is infeasible. Line 13 checks if the infeasible path z’ is strictly a subpath of , denoted by len(z’) < len(z) to mean that the length of 7’ is shorter than that of z. When found to be a strict subpath, we add 7’ to II for pruning all its extensions. If the length of 7! and 7 are equal, we do not prune the extensions of ! because it is still possible that an unsafe trajectory may be present that follows in an extension of z. For a feasible path z', we call the trajectory splicing algorithm in line 18 to search for a counterexample trajectory along the path. If it successfully finds a counterexample trajectory, the procedure terminates by classifying the verification instance as unsafe together with the concrete trajectory. Line 26 finds
The computation of R,, is expensive and therefore the overall cost of the proposed path-wise exploration algorithm can be prohibitive in automata with a large number of paths from source to the target location. To address this challenge, we propose an optimization technique based on reusing the computed symbolic states wherever possible based on the observation that paths can have multiple common sub-paths. Note that the classical method of having a closed list.and an open list in search algorithms to store the explored and waiting to be explored symbolic states respectively does not suffice in our algorithm. This is because in addition to checking whether a symbolic state, say S, is present in a closed/waiting list as done in the traditional reachability analysis algorithm, our algorithm also seeks to get the precomputed post,(S) and postg(S) for some transition when S is present in the closed list. This is because the intention is to extract the precomputed reachable states along a path. This requires additional bookkeeping which we propose in the paper. In order to reuse symbolic states computation and hence avoid duplication in computation, we propose to use two map data structures, namely the initial-to-flow and flow-trans-to-init. The initial-to-flow data-structure maps a symbolic state (v, C) to post,((u, C)). The other data-structure flow-trans-to-init maps a pair (F, 5) to a symbolic state (v, C’) = posta(F, 5), where 6 is a transition between locations u and v and F = post,((u, C)). The data structures are populated as well as referenced during the iterative computation of R,, by the rules shown in Figure 3. The rules are expressed as a precondition(s) and the corresponding effects. Rule R; starts the computation from the initial symbolic state (uj,C)), where C, = Init(v,), for a path 2 = vo, e;,U1,..., er, Ue. In the rules, (v;, C;) the largest infeasible sequence of locations L’. If len(L’) < len(L), then L’ is added to the set SL for pruning its extensions, shown in line 27. Algorithm 2 illustrates the path feasibility checking with reachability analysis.
The computation of R,, is expensive and therefore the overall cost of the proposed path-wise exploration algorithm can be prohibitive in automata with a large number of paths from source to the target location. To address this challenge, we propose an optimization technique based on reusing the computed symbolic states wherever possible based on the observation that paths can have multiple common sub-paths. Note that the classical method of having a closed list.and an open list in search algorithms to store the explored and waiting to be explored symbolic states respectively does not suffice in our algorithm. This is because in addition to checking whether a symbolic state, say S, is present in a closed/waiting list as done in the traditional reachability analysis algorithm, our algorithm also seeks to get the precomputed post,(S) and postg(S) for some transition when S is present in the closed list. This is because the intention is to extract the precomputed reachable states along a path. This requires additional bookkeeping which we propose in the paper. In order to reuse symbolic states computation and hence avoid duplication in computation, we propose to use two map data structures, namely the initial-to-flow and flow-trans-to-init. The initial-to-flow data-structure maps a symbolic state (v, C) to post,((u, C)). The other data-structure flow-trans-to-init maps a pair (F, 5) to a symbolic state (v, C’) = posta(F, 5), where 6 is a transition between locations u and v and F = post,((u, C)). The data structures are populated as well as referenced during the iterative computation of R,, by the rules shown in Figure 3. The rules are expressed as a precondition(s) and the corresponding effects. Rule R; starts the computation from the initial symbolic state (uj,C)), where C, = Init(v,), for a path 2 = vo, e;,U1,..., er, Ue. In the rules, (v;, C;) the largest infeasible sequence of locations L’. If len(L’) < len(L), then L’ is added to the set SL for pruning its extensions, shown in line 27. Algorithm 2 illustrates the path feasibility checking with reachability analysis.
Fig. 4. Rae reuses most of the regions (Q9,Q1,Q4, and Q5) which are computed for R71. The symbolic states (vi, Cj) computed by Eqn. 25 is represented by the upper portion (Cj) of the region Q;, and ¥; indicates the region computed by post, ((u;, C;)). For location o;, region Q; is the union of Cj and ¥;. During the computation of R,-2, once ¥3 is computed for C3 and stored in the initial-to-flow map, and (v4,C;) is computed by the Eqn. 25. For region Q4, C4 and ¥4 are already present in the corresponding map. Since Cj is a subset of Cy which is calculated during the computation of R71, the Rz2z computation proceeds with the symbolic state (v4, C4) due to rule Re. mentions that if flow-trans-to-init contains an entry for (Fj, 5), then fetch the data from the map and assign to (Vi+1, Ci41). The rules Ry and R; avoids redundant postg computations across paths. Rule Rg is an optimization to save post, computation on a symbolic state (vj41,Cj41) when there is a symbolic state (vi+1,Cj,,) already present in the init-to-flow map such that C;,, C Cj, ,. The effect of the rule assigns (v;41, Ci41) to (vi41, Cj,,) and stores this in the respective flow-trans-to-init stilry, The computation of R, proceeds with poste( (vis1, Ci.) retrieved from the init-to-flow map. When there are many symbolic states in the map which satisfies the above condition, (v;41, Cis1) is er to that (vi+1, Cj,,) in which C/,, is the tightest superset of C;,,. Expression {(CL 1 ECA) A (CL, BC} Ci)} states that initiar to ftowt map contains C;'_, which is neither superset nor subset of C/,,, while expression lite ¢ CCl.) = (Ci, ¢ cC.)} aiaeantees that if initial-to-flow map has C3, , which is superset of the C;,; then C/,, must be subset of C;;,. This rule trades-off precision for efficiency. The idea behind rule 7 is to store the computed (v;+1, Cj41) to the corresponding data structures for reuse (mentioned in rule R5) only when no superset of Cj; is found in the initial-to-flow map. Rule R; is applied when R, is not applicable. In order to compute R,,, the rules are applied in the order Rj,R2|R3,R4|Rs5,Ro|R7,R2|R3 and so on, where R;|Ri+1 denotes that exactly one of R; or Rj+1 will be applicable. Figure 4 illustrates the path reachability computation for two paths 2; and zg with our rules. During the computations of R,,, our reachability analysis algorithm computes the regions shown invthe figure using the OG a aia ee ede Do Rada thwk-Ehece DO aeeeexnetatetnem wma These wen hewtaw wawd xox eee aw few lmxecitan @ accel fen hx
Fig. 4. Rae reuses most of the regions (Q9,Q1,Q4, and Q5) which are computed for R71. The symbolic states (vi, Cj) computed by Eqn. 25 is represented by the upper portion (Cj) of the region Q;, and ¥; indicates the region computed by post, ((u;, C;)). For location o;, region Q; is the union of Cj and ¥;. During the computation of R,-2, once ¥3 is computed for C3 and stored in the initial-to-flow map, and (v4,C;) is computed by the Eqn. 25. For region Q4, C4 and ¥4 are already present in the corresponding map. Since Cj is a subset of Cy which is calculated during the computation of R71, the Rz2z computation proceeds with the symbolic state (v4, C4) due to rule Re. mentions that if flow-trans-to-init contains an entry for (Fj, 5), then fetch the data from the map and assign to (Vi+1, Ci41). The rules Ry and R; avoids redundant postg computations across paths. Rule Rg is an optimization to save post, computation on a symbolic state (vj41,Cj41) when there is a symbolic state (vi+1,Cj,,) already present in the init-to-flow map such that C;,, C Cj, ,. The effect of the rule assigns (v;41, Ci41) to (vi41, Cj,,) and stores this in the respective flow-trans-to-init stilry, The computation of R, proceeds with poste( (vis1, Ci.) retrieved from the init-to-flow map. When there are many symbolic states in the map which satisfies the above condition, (v;41, Cis1) is er to that (vi+1, Cj,,) in which C/,, is the tightest superset of C;,,. Expression {(CL 1 ECA) A (CL, BC} Ci)} states that initiar to ftowt map contains C;'_, which is neither superset nor subset of C/,,, while expression lite ¢ CCl.) = (Ci, ¢ cC.)} aiaeantees that if initial-to-flow map has C3, , which is superset of the C;,; then C/,, must be subset of C;;,. This rule trades-off precision for efficiency. The idea behind rule 7 is to store the computed (v;+1, Cj41) to the corresponding data structures for reuse (mentioned in rule R5) only when no superset of Cj; is found in the initial-to-flow map. Rule R; is applied when R, is not applicable. In order to compute R,,, the rules are applied in the order Rj,R2|R3,R4|Rs5,Ro|R7,R2|R3 and so on, where R;|Ri+1 denotes that exactly one of R; or Rj+1 will be applicable. Figure 4 illustrates the path reachability computation for two paths 2; and zg with our rules. During the computations of R,,, our reachability analysis algorithm computes the regions shown invthe figure using the OG a aia ee ede Do Rada thwk-Ehece DO aeeeexnetatetnem wma These wen hewtaw wawd xox eee aw few lmxecitan @ accel fen hx
Fig. 6. (a) A counterexample of length 7 for NAV_U2, and (b) length 17 for NAV_U22 navigation benchmark instances. (c) A 3-length counterexample of the F_oscillator_32 instance. (d) A counterexample of length 3 on the Two_tank_U1 benchmark instance. The gray region shows R,. The blue curve shows the counterexample.
Fig. 6. (a) A counterexample of length 7 for NAV_U2, and (b) length 17 for NAV_U22 navigation benchmark instances. (c) A 3-length counterexample of the F_oscillator_32 instance. (d) A counterexample of length 3 on the Two_tank_U1 benchmark instance. The gray region shows R,. The blue curve shows the counterexample.
Table 2. Evaluation of Safe instances. #Locs: the number of discrete locations in the hybrid automaton H. #Dims: The number of system variables in the hybrid automaton H. BMC Bnd: the bound for model checking. Time: The time is taken to decide the safety of the instance. #Paths: the number of explored abstract counterexamples in the discrete state-space. #postc: The number of poste operations by the algorithm. For DREACH, results are with a precision of 5 = 0.001. The gray cells highlight the best algorithm to classify an instance as safe in terms of time taken. Figure: 7 (c) shows the regions computed in SAT-REacu (dark gray) and BFS-REAcH (light gray) for an instance NAV_U17. The exploration in SAT-REAcH is guided towards the forbidden region. These figures show the benefit of path-wise reachability analysis. 1 bh r we: Gera ce a a et — ow 1 — <i lm . a
Table 2. Evaluation of Safe instances. #Locs: the number of discrete locations in the hybrid automaton H. #Dims: The number of system variables in the hybrid automaton H. BMC Bnd: the bound for model checking. Time: The time is taken to decide the safety of the instance. #Paths: the number of explored abstract counterexamples in the discrete state-space. #postc: The number of poste operations by the algorithm. For DREACH, results are with a precision of 5 = 0.001. The gray cells highlight the best algorithm to classify an instance as safe in terms of time taken. Figure: 7 (c) shows the regions computed in SAT-REacu (dark gray) and BFS-REAcH (light gray) for an instance NAV_U17. The exploration in SAT-REAcH is guided towards the forbidden region. These figures show the benefit of path-wise reachability analysis. 1 bh r we: Gera ce a a et — ow 1 — <i lm . a
Table 3. Comparison between SAT-REACH with optimization and SAT-REACH without our proposed path-pruning and opti- mization using memoization. The gray cells highlight the best algorithm in terms of time taken.
Table 3. Comparison between SAT-REACH with optimization and SAT-REACH without our proposed path-pruning and opti- mization using memoization. The gray cells highlight the best algorithm in terms of time taken.
Table 4. Comparison with flow*. #Locs is the number of discrete locations in H. Time and Result in Flow* denotes the time is taken to classify the verification instances and the Result of the tool when Flow* terminates respectively. The gray cells highlight the best algorithm in terms of the time taken for reachability analysis. ACM Trans. Embedd. Comput. Syst.
Table 4. Comparison with flow*. #Locs is the number of discrete locations in H. Time and Result in Flow* denotes the time is taken to classify the verification instances and the Result of the tool when Flow* terminates respectively. The gray cells highlight the best algorithm in terms of the time taken for reachability analysis. ACM Trans. Embedd. Comput. Syst.
Table 5. Comparison with SpAceEx. Time and Result in SPAcEEx denotes the time taken to classify the verification instances and the result of the tool when SpaceEx terminates respectively. The gray cells highlight the best algorithm in terms of the time taken for reachability analysis. ACM Trans. Embedd. Comput. Syst.
Table 5. Comparison with SpAceEx. Time and Result in SPAcEEx denotes the time taken to classify the verification instances and the result of the tool when SpaceEx terminates respectively. The gray cells highlight the best algorithm in terms of the time taken for reachability analysis. ACM Trans. Embedd. Comput. Syst.
Table 6. Comparison with PDB. Time and Result in PDB denotes the time taken to classify the verification instances and the result of the tool when PDB terminates respectively. The gray cells highlight the best algorithm in terms of the time taker for reachability analysis. ACM Trans. Embedd. Comput. Syst.
Table 6. Comparison with PDB. Time and Result in PDB denotes the time taken to classify the verification instances and the result of the tool when PDB terminates respectively. The gray cells highlight the best algorithm in terms of the time taker for reachability analysis. ACM Trans. Embedd. Comput. Syst.

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