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# Integrating SAT, QBF and SMT Solvers with Interactive Theorem Provers

Department of Information Technology, Uppsala University; 14 September, 2011
by

## Tjark Weber

on 7 October 2013

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#### Transcript of Integrating SAT, QBF and SMT Solvers with Interactive Theorem Provers

Integrating SAT, QBF and SMT Solvers
with Interactive Theorem Provers
Tjark Weber
Department of Information Technology, Uppsala University
14 September, 2011
System Overview
Validation
Automation
SMT
propositional logic
equality
uninterpreted functions
arrays
quantifiers
linear arithmetic
bit-vectors
data types
SAT
QBF
Boolean variables
negation
disjunction
conjunction
propositional logic
Boolean quantifiers
Demo
LCF-style Theorem Proving
Theorems are implemented as an abstract data type.
There is a fixed number of constructor functions - one for each axiom/inference rule of higher-order logic.
More complicated proof procedures must be implemented by composing these functions.
Selected HOL4 Inference Rules
Sequent Clauses
Resolution, LCF-style
Q-resolution is propositional resolution followed by forall-reduction.
Q-Resolution
Propositional Resolution
How can we implement this efficiently on top of existing LCF-style kernels?
Q-Resolution, LCF-style
Example
Forall-Reduction, LCF-style
Z3 uses 34 different inference rules.
Satisfiability Modulo Theories
Proof Structure
Proofs are directed acyclic graphs.
Each node corresponds to an inference step.
Children are premises of inferences. Leaves are axioms/input formulas.
A designated root note concludes .
Proofs can be checked by depth-first postorder traversal.
1. A single primitive inference/theorem instantiation.
2. Combinations of primitive inferences/theorem instantiations.
3. Automated proof procedures (e.g., for linear arithmetic).
4. Combinations of the above.
Combinations of Primitive Inferences
Avoiding Automated Proof Procedures
Combinations of primitive inferences
Instantiation of schematic theorems
Generalization wrt. background theories
Memoization
Speed-ups of up to 3 orders of magnitude!
Evaluation
Integrating SAT, QBF and SMT Solvers
with Interactive Theorem Provers

http://www.cl.cam.ac.uk/~tw333/
SAT
QBF
SMT
LCF-style proof checking is feasible for SAT, QBF and SMT. It scales to large formulas and proofs with millions of inferences.

Naive solutions are slow - an efficient implementation is crucial! Also, some prover kernel optimizations were necessary.

Our integrations provide a high degree of automation in Isabelle and HOL4, enable new applications, and provide high assurance for the external solvers' results.
Conclusions
Evaluation on 69 invalid QBF problems from the 2005 fixed instance and 2006 preliminary QBF-Eval data sets

up to 131 alternating quantiers, 24,562 variables, 35,189 clauses
For QBF validation, HOL4's name-carrying kernel is about 75 times faster (after optimizations) than the kernel that uses de Bruijn indices internally.
QBF certificate validation in HOL4 is about 25 times faster than proof search with Squolem.
All certificates are checked successfully!
Certificate validation in Isabelle/HOL is about 19 times slower than proof search with Z3.
Timeouts are due to theory reasoning; Z3's proofs are not detailed enough.
75% of SMT-LIB problems are checked successfully!
Complex systems almost inevitably contain bugs.
Formal Verification
Program testing can be used to show the presence of bugs, but never to show their absence!
Edsger W. Dijkstra
Interactive Theorem Proving
A theorem prover is a computer program to prove theorems.
Given a precise description of a system, and a formal specification of its intended behavior, we obtain a computer-checked proof that the system meets its specification.
Lemma “ ”
Proof
assume “ ”
from this obtain x where X: “ ” ..
fix y
from X have “ ” ..
then show “ ” ..
qed
Interactive Proof
Notable ITP Projects
L4.verified: a machine-checked proof of the functional correctness of the seL4 microkernel
Flyspeck: a machine-checked proof of the Kepler conjecture
Lemma “ ”
by auto
With 431 million adult victims and at an annual price of \$388 billion, cybercrime costs the world significantly more than the global black market in marijuana, cocaine and heroin combined.
Norton Cybercrime Report, Sept. 2011
Software errors cost the U.S. economy \$60 billion annually.
NIST, 2002
Applications
Tarski-Kleene Algebras
C++ Memory Model
Algebra is helpful for modelling and analysing computing systems.
concise models
abstract semantics
equational reasoning
[JAL 2009]
[ITP 2010, ITP 2011]
[ITP 2010, CPP 2011]
Large TKA repository with over 1000 lemmas
Hierarchy of algebraic structures
High degree of automation (via SMT)
Important models (relations, languages, ...)
Point-free and point-wise reasoning
Automated Algebraic Theory Engineering
in Isabelle/HOL
[RAMiCS 2011, ICFEM 2011]
Solver Integration
... with special thanks to my colleagues and co-authors: Hasan Amjad, Mark Batty, Jasmin Blanchette, Sascha Böhme, Simon Foster, Anthony Fox, Walter Guttmann, Ramana Kumar, Scott Owens, Susmit Sarkar, Peter Sewell, Thomas Sewell, Georg Struth, and many others
[JAL 2009]
[ITP 2010]
[ITP 2010]
A new C++ standard (C++11) was approved on 12 August, 2011.
It contains a memory model that defines how threads interact through shared data.
Example: Store Buffering
atomic_int x = 0;
atomic_int y = 0;
x.store(1, seq_cst);
y.store(1, seq_cst);
Can both loads observe the old value of 0?
The memory model defines various relations over a program's memory actions.
happens-before