Automatic Abstraction in Symbolic Trajectory Evaluation
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1 Automatic Abstraction in Symbolic Trajectory Evaluation Sara Adams in collaboration with Magnus Björk, Tom Melham, and Carl-Johan Seger University of Oxford 13 November 2007 Adams, Björk, Melham, Seger (Oxford) Automatic Abstraction in STE 13 November / 16
2 Our Contribution Automatic Discovery of highly non-trivial abstractions that make verification of circuits possible that could not be tackled with STE before Adams, Björk, Melham, Seger (Oxford) Automatic Abstraction in STE 13 November / 16
3 Our Contribution Automatic Discovery of highly non-trivial abstractions that make verification of circuits possible that could not be tackled with STE before Adams, Björk, Melham, Seger (Oxford) Automatic Abstraction in STE 13 November / 16
4 Abstraction in STE Introduction Adams, Björk, Melham, Seger (Oxford) Automatic Abstraction in STE 13 November / 16
5 Symbolic Indexing Introduction Adams, Björk, Melham, Seger (Oxford) Automatic Abstraction in STE 13 November / 16
6 Symbolic Indexing Introduction What s good about it? Powerful abstraction mechanism can transform exponential verification to linear critical enabler for content accessible memory and memory verification What s the problem? Manual derivation tedious discovery of good indexing schemes hard coverage requirement (else: false positives) composition non-trivial Adams, Björk, Melham, Seger (Oxford) Automatic Abstraction in STE 13 November / 16
7 Automatic Re-Indexing Automatic Re-Indexing Melham-Jones Algorithm Input: Output: verification task using abstraction scheme A relation between scheme A and scheme B verification task using abstraction scheme B Special case Start with no abstraction scheme Coverage condition Relation has to guarantee that scheme A and scheme B cover the same cases; usually: cover all possible cases Adams, Björk, Melham, Seger (Oxford) Automatic Abstraction in STE 13 November / 16
8 Automatic Re-Indexing Automatic Re-Indexing What s good about it? Correctness of indexing scheme machine-checkable Compositionality and reasoning of verification What s the problem? Manual derivation of relation tedious Coverage check can be exponential loss of re-indexing profits Automatic Abstraction Generate relation automatically Coverage requirement satisfied by construction Adams, Björk, Melham, Seger (Oxford) Automatic Abstraction in STE 13 November / 16
9 Automatic Re-Indexing Automatic Re-Indexing What s good about it? Correctness of indexing scheme machine-checkable Compositionality and reasoning of verification What s the problem? Manual derivation of relation tedious Coverage check can be exponential loss of re-indexing profits Automatic Abstraction Generate relation automatically Coverage requirement satisfied by construction Adams, Björk, Melham, Seger (Oxford) Automatic Abstraction in STE 13 November / 16
10 Automatic Abstraction Backward Propagation The algorithm Input: Specification (as a circuit) Output: Indexing Relation (needed for Melham-Jones) Why use the specification? Expresses essential properties Uncluttered Adams, Björk, Melham, Seger (Oxford) Automatic Abstraction in STE 13 November / 16
11 Automatic Abstraction Backward Propagation: Using the specification (f, g x) (g, f) (f, g) (f, g) (f, g x) Basic idea On the specification determine: which input combinations force the output to true and false respectively start from output determine which inputs force the output to be true or false resp. when given a choice, introduce an indexing variable Adams, Björk, Melham, Seger (Oxford) Automatic Abstraction in STE 13 November / 16
12 Automatic Abstraction Backward Propagation: Using the specification (f, g x) (g, f) (f, g) (f, g) (f, g x) Basic idea On the specification determine: which input combinations force the output to true and false respectively start from output determine which inputs force the output to be true or false resp. when given a choice, introduce an indexing variable Adams, Björk, Melham, Seger (Oxford) Automatic Abstraction in STE 13 November / 16
13 Automatic Abstraction Example relation for a 3-input AND-gate Adams, Björk, Melham, Seger (Oxford) Automatic Abstraction in STE 13 November / 16
14 Automatic Abstraction Making it work: Encoding Basic algorithm 2-input AND-gates Fresh indexing variables on every choice Efficient algorithm n-input AND-gates, XNOR-gates Reuse indexing variables for better sharing pq: 0XX Adams, Björk, Melham, Seger (Oxford) Automatic Abstraction in STE 13 November / 16
15 Automatic Abstraction Making it work: Over-abstraction Basic algorithm Abstraction dependent on specification only Efficient algorithm Allow declaration of symbolic constants specify which inputs not to abstract Adams, Björk, Melham, Seger (Oxford) Automatic Abstraction in STE 13 November / 16
16 Automatic Abstraction Making it work: Automatic Re-Indexing Melham-Jones General relations expensive quantifications Proof of coverage requirement Modified version Specific structure on relations assumed quantifications eliminated proof in the paper Coverage by construction proof in the paper Adams, Björk, Melham, Seger (Oxford) Automatic Abstraction in STE 13 November / 16
17 Results Content Accessible Memory and Memory Figure: CAM (left) and Memory (right) verification Included: Automatic Abstraction, Re-Indexing, STE run Verification not feasible without symbolic indexing Adams, Björk, Melham, Seger (Oxford) Automatic Abstraction in STE 13 November / 16
18 Scheduler Results Figure: Scheduler verification Specification: retrieve the oldest ready entry Verification not feasible without symbolic indexing Indexing and abstraction highly non-obvious Adams, Björk, Melham, Seger (Oxford) Automatic Abstraction in STE 13 November / 16
19 Future Work Future Work Adams, Björk, Melham, Seger (Oxford) Automatic Abstraction in STE 13 November / 16
20 Future Work Future Work Adams, Björk, Melham, Seger (Oxford) Automatic Abstraction in STE 13 November / 16
21 Future Work Future Work Adams, Björk, Melham, Seger (Oxford) Automatic Abstraction in STE 13 November / 16
22 Future Work Future Work Adams, Björk, Melham, Seger (Oxford) Automatic Abstraction in STE 13 November / 16
23 Future Work Future Work Adams, Björk, Melham, Seger (Oxford) Automatic Abstraction in STE 13 November / 16
24 Future Work Future Work Adams, Björk, Melham, Seger (Oxford) Automatic Abstraction in STE 13 November / 16
25 Backup slides Backwards Propagation in Action
26 Backup slides Backwards Propagation in Action
27 Backup slides Backwards Propagation in Action
28 Backup slides Backwards Propagation in Action
29 Backup slides Backwards Propagation in Action
30 Backup slides Backwards Propagation in Action
31 Backup slides Backwards Propagation in Action
Automatic Abstraction in Symbolic Trajectory Evaluation
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