Reachability Analysis Using Octagons
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1 Reachabilit Analsis Using Octagons Andrew N. Fisher and Chris J. Mers Department of Electrical and Computer Engineering Universit of Utah FAC 0 Jul 9, 0
2 Digitall Intensive Analog Circuits Digitall intensive analog circuits attempt to replace analog components with digital ones whenever possible. DLF DAC VCO (LC) out REFCLK DPD TDC Verilog (snthesizable) Verilog (non snthesizable) Result is optimized power efficienc and performance as well as improved robustness to process variabilit. These circuits though further complicate the verification problem.
3 Simulation-Based Verification Digital verification tpicall uses switch or RTL-level simulations. AMS verification uses detailed transistor-level (SPICE) simulations. SPICE simulation of a PLL can take weeks or even months. Long simulation time makes sstem-level simulation difficult. Functional bugs can be missed resulting in catastrophic failures.
4 Analog Verification If the digital designers did verification the wa analog designers do verification, no chip would ever tape out. (DACezine, Januar 008) Sandipan Bhanot CEO of Knowlent
5 Model Checking Model checking uses non-determinism and state eploration to formall verif designs over all possible behaviors. Has had tremendous success for verifing of both digital hardware and software sstems (now routinel used at Intel, IBM, Microsoft, etc.). For AMS circuits, it is a promising mechanism to validate designs in the face of noise and uncertain parameters and initial conditions. AMS verification is complicated b the need to: Construct abstract formal models of the AMS circuits. Specif formal properties that are to be verified. Represent continuous variables efficientl (voltages, currents, and time).
6 Model Checking Model checking uses non-determinism and state eploration to formall verif designs over all possible behaviors. Has had tremendous success for verifing of both digital hardware and software sstems (now routinel used at Intel, IBM, Microsoft, etc.). For AMS circuits, it is a promising mechanism to validate designs in the face of noise and uncertain parameters and initial conditions. AMS verification is complicated b the need to: Construct abstract formal models of the AMS circuits. (FAC 0) Specif formal properties that are to be verified. (FAC 0) Represent continuous variables efficientl (voltages, currents, and time).
7 Zones Used for formal verification of timed automata and time(d) Petri nets. Simple geometric polhedra formed b the intersection of hper-planes representing inequalities of the form c. Implies polhedra with onl 0, 90, and positive angles. For timed sstems, all variables evolve at a rate of, and zone evolves along a positive angle. Algorithms to restrict, project, and advance time are fast and simple. Can use Flod s all pairs shortest-path algorithm to construct a canonical maimall tight representation. Convenientl represented using a difference bound matri (DBM).
8 Zones M M m m b b 0 M M m 0 b m b 0
9 Zones
10 Zone Warping To verif AMS circuits, need variables that evolve at non-unit rates. Zones can be used with a variable substitution. Replace variable v with non-zero rate r with a variable v r. The new variable v r evolves at a rate of. Resultant polhedra is no longer a zone. Warping creates the smallest zone that contains it.
11 Positive Zone Warping
12 Positive Zone Warping
13 Positive Zone Warping
14 Negative Zone Warping
15 Negative Zone Warping
16 Negative Zone Warping
17 Negative Zone Warping: False Negative = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
18 Negative Zone Warping: False Negative = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
19 Negative Zone Warping: False Negative = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
20 Negative Zone Warping: False Negative = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
21 Negative Zone Warping: False Negative = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
22 Negative Zone Warping: False Negative = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
23 Negative Zone Warping: False Negative = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
24 Negative Zone Warping: False Negative = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
25 Negative Zone Warping: False Negative = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
26 Negative Zone Warping: False Negative = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
27 Negative Zone Warping: False Negative = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > fail t {( >= ) ( >= )} p t < := > fail t
28 Octagons Etension of zones that allow negative degree angles.
29 Octagon DBM Can be represented using a DBM (Mine, 00) and manipulated with efficient algorithms. b b b b + M + m + M m m M m M 0
30 Octagon DBM Can be represented using a DBM (Mine, 00) and manipulated with efficient algorithms. b + + b b b b b + + b + b b + b + b + b m b b M 0 b b + b b 0 m b b M 0
31 Reachabilit Analsis Using Octagons Utilized for software checking, and efficient restriction, projection, and constraint tightening algorithms have been developed. New algorithms are needed to add new continuous variables, advance time, and warp the octagon.
32 Adding Variables to Octagons Adding new continuous variables and clocks is simpl a matter of re-interpreting the algorithms for zones in the language for octagons. When adding a continuous variable v with rate r, the maimum and minimum values for v are divided b r and added to the DBM (after multipling b ). Relational entries are set to infinit, indicating no relationship.
33 Octagon Time Advancement Etend the octagon along the lines. For zones, to advance time, simpl set the upper bounds for all the variables to the maimum allowed value before an event occurs. For octagons, line slicing the upper right hand corner has a limiting effect on the upper bounds of the two variables involved. Entries associated with inequalities + c must also be set to their maimum allowed value in relation to the maimums of and.
34 Octagon Warping Again replace ever variable v b v r where r is the rate of v. Replace resulting polhedra with smallest octagon that contains it. Accomplished b using a few algebraic equations that determine where the new ais intercepts are in terms of the old intercept values. 8
35 Octagon Eample = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
36 Octagon Eample = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
37 Octagon Eample = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
38 Octagon Eample = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
39 Octagon Eample = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
40 Octagon Eample = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
41 Octagon Eample = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
42 Octagon Eample = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
43 Octagon Eample = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
44 Octagon Eample = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
45 Comparison with Zones = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
46 Comparison with Zones = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
47 Comparison with Zones = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
48 Comparison with Zones = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
49 False Negatives Octagons do not eliminate the possibilit of false negatives even in the case where rates are onl ±. Time advancement also introduces a degree of over-approimation, related to the negative lines. Advancement in three dimensions of one of these negative line segments belongs to a plane of the form a + b + cz = d. The bounding hper-planes are of the form ±v i ± v j c and not able to capture this plane produced b advancing time.
50 False Negative Eample
51 False Negative Eample
52 LEMA: LPN Embedded Mied-Signal Analzer Transistor Level Design SPICE Traditional Analog Circuit Verification Kulkarni et al., FDL (0) Fisher et al., LDMT (0) Verification Propert (LAMP) Simulation Traces Model Generator Kulkarni, MS Thesis (0) Kulkarni et al., VW EDA (0) Batchu, MS Thesis (00) Little et al., IJFCS (00) Little at al., IEEE TCAD (0) Walter et al., IEEE TCAD (008) Little, PhD (008) Walter, PhD (00) Labeled Petri Net (LPN) Model Checker Pass or Fail + Error Trace SstemVerilog Model Simulation Engine Assertion Pass/Fail RTL for Digital Components Fisher et al., MWSCAS (0)
53 Acknowledgements Satish Batchu (Qualcomm) Andrew Fisher (Utah) Kevin Jones (Aberdeen) Dhanashree Kulkarni (Intel) Scott Little (Intel) David Walter (Virginia State) Supported b SRC Contracts 00-TJ-0, 00-TJ-, 008-TJ-8, NSF Grant CCF-, and b Intel Corporation. Fisher / Mers (U. of Utah) Octagons FAC 0 / Jul 9, 0
54 Octagon DBM = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
55 Octagon DBM = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
56 Octagon DBM = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
57 Octagon DBM = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
58 Octagon DBM = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
59 Octagon DBM = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
60 Octagon DBM = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t t + t + + t t
61 Octagon DBM = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t t + t + + t t
62 Octagon DBM = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t t + t + + t t
63 Octagon DBM = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t t + t + + t t
64 Octagon DBM = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t t + t + + t t
65 Octagon DBM = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t t + t + + t t
66 Octagon DBM = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t t + t + + t t
67 Octagon DBM = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t t + t + + t t
68 Octagon DBM = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t t + t + + t t
69 Octagon DBM = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t t + t + + t t
70 Octagon DBM = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t t + t + + t t
71 Octagon DBM = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t t + t + + t t
72 Octagon DBM = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
73 Octagon DBM = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
74 Octagon DBM = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
75 Octagon DBM = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
76 Octagon DBM = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
77 Octagon DBM = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
78 Octagon DBM = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t t + t + + t t
79 Octagon DBM = p0 = = = t { ( >= ) ( >= )} < := 0, := 0 > [0, ] < := > t {( >= ) ( >= )} p t < := > t
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