Mechanically-Verified Validation of Satisfiability Solvers
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1 Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler The University of Texas at Austin Dissertation Proposal October 18, 2013
2 Outline Motivation and Proposal Satisfiability and Proofs Task 1: Designing a Proof Format Task 2: Developing an Efficient Checker Task 3: Proving Correctness Timeline and Conclusion Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 2 / 27
3 Motivation Satisfiability solvers are used in amazing ways... - Hardware verification: Centaur x86 verification - Combinatorial problems: van der Waerden numbers [Dransfield, Marek, and Truszczynski, 2004] Gardens of Eden in Conway s Game of Life [Hartman, Heule, Kwekkeboom, and Noels, 2013; Kouril and Paul, 2008] - Unsatisfiability is often more important Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 3 / 27
4 Motivation Satisfiability solvers are used in amazing ways... - Hardware verification: Centaur x86 verification - Combinatorial problems: van der Waerden numbers [Dransfield, Marek, and Truszczynski, 2004] Gardens of Eden in Conway s Game of Life [Hartman, Heule, Kwekkeboom, and Noels, 2013; Kouril and Paul, 2008] - Unsatisfiability is often more important..., but satisfiability solvers have errors. - Documented bugs in SAT, SMT, and QBF solvers [Brummayer and Biere, 2009; Brummayer et al., 2010] - Competition winners have contradictory results (HWMCC winners from 2011 and 2012) - Implementation errors often imply conceptual errors Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 3 / 27
5 Solutions Verify SAT solvers - Requires verification of all crucial search techniques - Delicate balance between efficiency and ease of verification - Verification proofs are specific to each solver - New developments in solving require additional proof effort Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 4 / 27
6 Solutions Verify SAT solvers - Requires verification of all crucial search techniques - Delicate balance between efficiency and ease of verification - Verification proofs are specific to each solver - New developments in solving require additional proof effort Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 4 / 27
7 Solutions Verify SAT solvers - Requires verification of all crucial search techniques - Delicate balance between efficiency and ease of verification - Verification proofs are specific to each solver - New developments in solving require additional proof effort Validate SAT solver output - Emit proof of unsatisfiability from SAT solver - A single proof checker can validate results from many state-of-the-art solvers - Proof checker uses limited number of techniques and can be mechanically verified Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 4 / 27
8 Proposal For my dissertation, I will develop a fast mechanically-verified satisfiability proof checker using ACL2. Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 5 / 27
9 Proposal For my dissertation, I will develop a fast mechanically-verified satisfiability proof checker using ACL2. This project has three tasks: 1. Design a suitable proof format, 2. Implement an efficient proof checker for the format, and 3. Demonstrate a proof of correctness for the proof checker. Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 5 / 27
10 Proof Properties Easy to Emit Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 6 / 27
11 Proof Properties Easy to Emit Compact Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 6 / 27
12 Proof Properties Easy to Emit Compact Checked Efficiently Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 6 / 27
13 Proof Properties Easy to Emit Compact Checked Efficiently Verified Checker Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 6 / 27
14 Proof Properties Easy to Emit Compact Checked Efficiently Verified Checker Expressive Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 6 / 27
15 Proof Properties Easy to Emit Compact Checked Efficiently Resolution Proofs [Zhang and Malik, 2003] [Van Gelder, 2008] [Biere, 2008] with Verified Checker [Weber, 2006, and Amjad, 2009] (Isabelle/HOL) [Darbari et al., 2010] (Coq) [Armand et al., 2011] (Coq) Clausal (RUP) Proofs [Goldberg and Novikov, 2003] [Van Gelder, 2008] Verified Checker Expressive Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 6 / 27
16 Proof Properties Easy to Emit Compact Checked Efficiently Verified Checker Expressive Resolution Proofs [Zhang and Malik, 2003] [Van Gelder, 2008] [Biere, 2008] with Verified Checker [Weber, 2006, and Amjad, 2009] (Isabelle/HOL) [Darbari et al., 2010] (Coq) [Armand et al., 2011] (Coq) Clausal (RUP) Proofs [Goldberg and Novikov, 2003] [Van Gelder, 2008] DRUP (DRUP-Trim) [Heule, Hunt, Jr., and Wetzler, STVR 201X] [Heule, Hunt, Jr., and Wetzler, FMCAD 2013] RAT Proofs [Heule, Hunt, Jr., and Wetzler, CADE 2013] with Verified Checker [Wetzler, Heule, and Hunt, Jr., ITP 2013] (ACL2) Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 6 / 27
17 Proof Properties Easy to Emit Compact Checked Efficiently Verified Checker Expressive Resolution Proofs [Zhang and Malik, 2003] [Van Gelder, 2008] [Biere, 2008] with Verified Checker [Weber, 2006, and Amjad, 2009] (Isabelle/HOL) [Darbari et al., 2010] (Coq) [Armand et al., 2011] (Coq) Clausal (RUP) Proofs [Goldberg and Novikov, 2003] [Van Gelder, 2008] DRUP (DRUP-Trim) [Heule, Hunt, Jr., and Wetzler, STVR 201X] [Heule, Hunt, Jr., and Wetzler, FMCAD 2013] RAT Proofs [Heule, Hunt, Jr., and Wetzler, CADE 2013] with Verified Checker [Wetzler, Heule, and Hunt, Jr., ITP 2013] (ACL2) Proposed Work Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 6 / 27
18 Outline Motivation and Proposal Satisfiability and Proofs Task 1: Designing a Proof Format Task 2: Developing an Efficient Checker Task 3: Proving Correctness Timeline and Conclusion Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 7 / 27
19 Satisfiability Is there an assignment of values to variables such that a given Boolean formula evaluates to TRUE? (x1 x2 x3) ( x1 x2 x3) (x2 x3 x4) ( x2 x3 x4) (x1 x3 x4) ( x1 x3 x4) (x1 x2 x4) ( x1 x2 x4) Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 8 / 27
20 Satisfiability Is there an assignment of values to variables such that a given Boolean formula evaluates to TRUE? Checking a solution is easy. Determining unsatisfiability is more difficult. (x1 x2 x3) ( x1 x2 x3) (x2 x3 x4) ( x2 x3 x4) (x1 x3 x4) ( x1 x3 x4) (x1 x2 x4) ( x1 x2 x4) Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 8 / 27
21 Satisfiability Is there an assignment of values to variables such that a given Boolean formula evaluates to TRUE? Checking a solution is easy. Determining unsatisfiability is more difficult. Formulas are in conjunctivenormal form (CNF). (x1 x2 x3) ( x1 x2 x3) (x2 x3 x4) ( x2 x3 x4) (x1 x3 x4) ( x1 x3 x4) (x1 x2 x4) ( x1 x2 x4) Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 8 / 27
22 Satisfiability Is there an assignment of values to variables such that a given Boolean formula evaluates to TRUE? Checking a solution is easy. Determining unsatisfiability is more difficult. Formulas are in conjunctivenormal form (CNF). CNF Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 8 / 27
23 Proofs and Refutations A clause C is redundant with respect to a formula F if C conjoined with F is satisfiability equivalent, SAT, to F. Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 9 / 27
24 Proofs and Refutations A clause C is redundant with respect to a formula F if C conjoined with F is satisfiability equivalent, SAT, to F. A proof trace is a sequence of clauses that are redundant with respect to a evolving formula. Formula Proof Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 9 / 27
25 Proofs and Refutations A clause C is redundant with respect to a formula F if C conjoined with F is satisfiability equivalent, SAT, to F. A proof trace is a sequence of clauses that are redundant with respect to a evolving formula. Formula SAT Proof Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 9 / 27
26 Proofs and Refutations A clause C is redundant with respect to a formula F if C conjoined with F is satisfiability equivalent, SAT, to F. A proof trace is a sequence of clauses that are redundant with respect to a evolving formula. Formula SAT SAT Proof Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 9 / 27
27 Proofs and Refutations A clause C is redundant with respect to a formula F if C conjoined with F is satisfiability equivalent, SAT, to F. A proof trace is a sequence of clauses that are redundant with respect to a evolving formula. Formula SAT SAT SAT Proof Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 9 / 27
28 Proofs and Refutations A clause C is redundant with respect to a formula F if C conjoined with F is satisfiability equivalent, SAT, to F. A proof trace is a sequence of clauses that are redundant with respect to a evolving formula. Formula SAT SAT SAT Proof A refutation is a proof trace that contains the (unsatisfiable) empty clause,. Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 9 / 27
29 Proofs and Refutations A clause C is redundant with respect to a formula F if C conjoined with F is satisfiability equivalent, SAT, to F. A proof trace is a sequence of clauses that are redundant with respect to a evolving formula. Formula SAT SAT SAT Proof A refutation is a proof trace that contains the (unsatisfiable) empty clause,. Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 9 / 27
30 Outline Motivation and Proposal Satisfiability and Proofs Task 1: Designing a Proof Format Task 2: Developing an Efficient Checker Task 3: Proving Correctness Timeline and Conclusion Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 10 / 27
31 Task 1: Designing a Proof Format Existing proof formats are insufficient. - Resolution proofs are large and hard to emit - Clausal (RUP) proofs are inefficient, but are compact and easy to emit Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 11 / 27
32 Task 1: Designing a Proof Format Existing proof formats are insufficient. - Resolution proofs are large and hard to emit - Clausal (RUP) proofs are inefficient, but are compact and easy to emit Use clausal proofs as a foundation with two extensions: - Add deletion information - Extend equivalence from logical to satisfiability Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 11 / 27
33 Extension 1: Deletion Information Proofs can be extended with clause deletion information. - Solvers remove clauses during search - Remove unnecessary clauses during validation - Emit learning and deletion information - New format called DRUP (Deletion RUP) Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 12 / 27
34 Extension 1: Deletion Information Proofs can be extended with clause deletion information. - Solvers remove clauses during search - Remove unnecessary clauses during validation - Emit learning and deletion information - New format called DRUP (Deletion RUP) CNF RUP DRUP d d d d d d Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 12 / 27
35 Extension 2: Expressiveness Preserve Logical Equivalence Only Preserve Satisfiability Equivalence (D)RUP Resolution CDCL Learning Boolean Constraint Propagation Subsumption RAT Blocked Clause Addition Bounded Variable Addition Extended Resolution Extended Learning Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 13 / 27
36 DRAT Format The DRUP and RAT proof formats can be combined. - How will the two formats interact? - With what frequency are RAT clauses produced? - Will the addition of RAT clauses lead to more deletions? CNF DRUP DRAT d d d d d d d d d d d Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 14 / 27
37 Outline Motivation and Proposal Satisfiability and Proofs Task 1: Designing a Proof Format Task 2: Developing an Efficient Checker Task 3: Proving Correctness Timeline and Conclusion Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 15 / 27
38 Task 2: Developing an Efficient Checker Efficiency is necessary on industrial-scale problems. - Validate proofs in a time similar to solving - Performance without full range of solver techniques Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 16 / 27
39 Task 2: Developing an Efficient Checker Efficiency is necessary on industrial-scale problems. - Validate proofs in a time similar to solving - Performance without full range of solver techniques Several techniques to gain performance. - Proofs can be trimmed before validated - Efficient Boolean constraint propagation - Constant-time, indexed memory access and update Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 16 / 27
40 Proof Trimming Proofs often contain clauses that are unnecessary. Our DRUP-trim tool trims (and checks) proofs. 5,000,000 Input Proof Output Proof 4,000,000 Clauses (#) 3,000,000 2,000,000 1,000, Benchmarks (sorted) Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 17 / 27
41 Proof Trimming Proofs often contain clauses that are unnecessary. Our DRUP-trim tool trims (and checks) proofs. 5,000,000 Input Proof Output Proof Adds optimal deletion information. 4,000,000 Clauses (#) 3,000,000 2,000,000 1,000, Benchmarks (sorted) Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 17 / 27
42 Fast Proof Checking DRUP-trim is able to closely match solving time Trimming and Checking (DRUP-trim) Solving (glucose) 10 2 Time (s) Benchmarks (sorted) Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 18 / 27
43 Fast Proof Checking DRUP-trim is able to closely match solving time Trimming and Checking (DRUP-trim) Solving (glucose) Used to check unsatisfiability results from SAT Competition Time (s) Benchmarks (sorted) Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 18 / 27
44 Fast Boolean Constraint Propagation Clausal proof checkers spend around 95% of their time performing Boolean Constraint Propagation. - Core technique in solvers - Often implemented using a watched-literal data structure Watched-Literal Invariant: All clauses are satisfied or contain at least two unassigned literals. This is just one of many implementation techniques that must be verified. Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 19 / 27
45 Efficiency of ACL2 Code Typical ACL2 list-only data structures are not efficient. - Access and update are linear time operations Instead, one can: - Mimic array-like structures using STOBJs - Disassemble key functions to compare compiled code to a highly optimized version We have implemented a basic RUP proof checker in ACL2 that achieves roughly 60% of a similar proof checker written in C. Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 20 / 27
46 Outline Motivation and Proposal Satisfiability and Proofs Task 1: Designing a Proof Format Task 2: Developing an Efficient Checker Task 3: Proving Correctness Timeline and Conclusion Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 21 / 27
47 Task 3: Proving Correctness Interactive theorem provers assist with verification. - ACL2 combines a programming language, first-order logic, and theorem prover - Proof checker is modeled in ACL2 - Specification for termination and soundness (but not completeness) are formalized - Efficient execution by way of Common LISP compilers Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 22 / 27
48 Task 3: Proving Correctness Interactive theorem provers assist with verification. - ACL2 combines a programming language, first-order logic, and theorem prover - Proof checker is modeled in ACL2 - Specification for termination and soundness (but not completeness) are formalized - Efficient execution by way of Common LISP compilers Incremental approach to proof process: - Prove correctness of proof checkers for different formats - Refine code to resemble C-equivalent Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 22 / 27
49 Verified Proof Checkers Verified SAT solvers and proof checkers using ACL2. - Verified RUP proof checker - Verified IORUP (deletion information) proof checker - Verified RAT proof checker Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 23 / 27
50 Verified Proof Checkers Verified SAT solvers and proof checkers using ACL2. - Verified RUP proof checker - Verified IORUP (deletion information) proof checker - Verified RAT proof checker (defthm main-theorem (implies (and (formulap f) (refutationp r f)) (not (exists-solution f)))) ; Given formula AND ; refutation ; Then formula is unsatisfiable Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 23 / 27
51 Verified Proof Checkers Verified SAT solvers and proof checkers using ACL2. - Verified RUP proof checker - Verified IORUP (deletion information) proof checker - Verified RAT proof checker (defthm main-theorem (implies (and (formulap f) (refutationp r f)) (not (exists-solution f)))) ; Given formula AND ; refutation ; Then formula is unsatisfiable Litany of transformations and refinements eventually resulting in code that corresponds to our C code. Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 23 / 27
52 Outline Motivation and Proposal Satisfiability and Proofs Task 1: Designing a Proof Format Task 2: Developing an Efficient Checker Task 3: Proving Correctness Timeline and Conclusion Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 24 / 27
53 Timeline October 2013 May 2014 Fast RAT Checker in ACL2 Verified Flat RAT Checker in ACL2 Connect Fast and Flat RAT Checkers Design and Testing of DRAT Format Fast DRAT Checker in ACL2 Proof Format Efficient Proof Checker Proving Correctness Verified Flat DRAT Checker Connect Fast and Flat DRAT Checkers Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 25 / 27
54 Conclusion This project has three components: - Design a suitable proof format, - Implement an efficient proof checker for the format, and - Demonstrate a proof of correctness for the proof checker. Easy to Emit Compact Checked Efficiently Verified Checker Expressive Our goal is to increase confidence in all satisfiability solvers by efficiently checking proofs with a mechanically-verified proof checker. Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 26 / 27
55 Recent Work Bridging the Gap Between Easy Generation and Efficient Verification of Unsatisfiability Proofs Marijn J.H. Heule, Warren A. Hunt, Jr., and Nathan Wetzler Accepted: Software Testing, Verification, and Reliability (STVR 201X) Verifying Refutations with Extended Resolution Marijn J.H. Heule, Warren A. Hunt, Jr., and Nathan Wetzler Published: Conference on Automated Deduction (CADE 2013) Mechanical Verification of SAT Refutations with Extended Resolution Nathan Wetzler, Marijn J.H. Heule, and Warren A. Hunt, Jr. Published: Interactive Theorem Proving (ITP 2013) Trimming while Checking Clausal Proofs Marijn J.H. Heule, Warren A. Hunt, Jr., and Nathan Wetzler Published: Formal Methods in Computer-Aided Design (FMCAD 2013) Thank you for your attention! Questions? Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 27 / 27
56 Redundancy Two formulas F1 and F2 are logically equivalent if they have the same set of satisfying assignments. F1 F2 Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 28 / 27
57 Redundancy Two formulas F1 and F2 are logically equivalent if they have the same set of satisfying assignments. F1 F2 Two formulas F1 and F2 are satisfiability equivalent if they are both satisfiable or both unsatisfiable. F1 SAT F2 Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 28 / 27
58 Redundancy Two formulas F1 and F2 are logically equivalent if they have the same set of satisfying assignments. F1 F2 Two formulas F1 and F2 are satisfiability equivalent if they are both satisfiable or both unsatisfiable. F1 SAT F2 A clause C is redundant with respect to a formula F if C conjoined with F is satisfiability equivalent to F. (and F C) SAT F Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 28 / 27
59 Resolution Proof Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 29 / 27
60 Resolution Proof Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 29 / 27
61 Resolution Proof Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 29 / 27
62 Resolution Proof Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 29 / 27
63 Resolution Proof Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 29 / 27
64 Resolution Proof Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 29 / 27
65 Resolution Proof Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 29 / 27
66 RUP Proof Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 30 / 27
67 RUP Proof Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 30 / 27
68 RUP Proof Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 30 / 27
69 RUP Proof Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 30 / 27
70 RUP Proof Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 30 / 27
71 DRUP Proof d d d d d d Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 31 / 27
72 DRUP Proof d d d d d d Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 31 / 27
73 DRUP Proof d d d d d d Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 31 / 27
74 DRUP Proof d d d d d d Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 31 / 27
75 DRUP Proof d d d d d d Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 31 / 27
76 DRUP Proof d d d d d d Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 31 / 27
77 DRUP Proof d d d d d d Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 31 / 27
78 DRUP Proof d d d d d d Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 31 / 27
79 DRUP Proof d d d d d d Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 31 / 27
80 DRUP Proof d d d d d d Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 31 / 27
81 DRUP Proof d d d d d d Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 31 / 27
82 RAT Proof Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 32 / 27
83 RAT Proof Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 32 / 27
84 RAT Proof Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 32 / 27
85 RAT Proof Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 32 / 27
86 RAT Proof Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 32 / 27
87 RAT Proof Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 32 / 27
88 RAT Proof Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 32 / 27
89 RAT Proof Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 32 / 27
90 RAT Proof Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 32 / 27
91 Incremental Approach Efficient code can be difficult to verify. - STOBJs provide array-like memory, but require complex invariants - Abstract STOBJs simplify these invariants by maintaining an equivalence - Currently developing a cons-less model that does not use STOBJs, but organizes data structures in a similar way. - Refinements - Litany of transformations eventually resulting in array-like code Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 33 / 27
92 Proof Properties Easy to Emit Compact Checked Efficiently Verified Checker Expressive Resolution Proofs with Verified Checker Clausal (RUP) Proofs DRUP (DRUP-Trim) RAT Proofs with Verified Checker Proposed Work Mechanically-Verified Validation of Satisfiability Solvers Nathan Wetzler 34 / 27
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