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# Reconstruction of Z3's Bit-Vector Proofs in HOL4 and Isabelle/HOL

CPP 2011; Kenting, Taiwan; 8 December, 2011
by

## Tjark Weber

on 3 October 2013

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#### Transcript of Reconstruction of Z3's Bit-Vector Proofs in HOL4 and Isabelle/HOL

Tjark Weber
CPP 2011
8 December, 2011
System Overview
Validation
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.
Z3 uses about 34 different inference rules.
Z3: Inference Rules
Z3: Proof Structure
Proofs are
directed acyclic graphs
.
Each node corresponds to an inference step.
Children are premises of inferences. Leaves are axioms/assumptions.
A designated root note concludes .
Proofs can be checked by depth-first
postorder traversal
.
Evaluation
Reconstruction of Z3's Bit-Vector Proofs in HOL4 and Isabelle/HOL
SMT-LIB
Conclusions
Proofs
... should be easy (and fast) to generate.
... should be easy (and fast) to check.
Reconstruction of Z3's Bit-Vector Proofs in
HOL4 and Isabelle/HOL
Sascha Böhme
Anthony Fox
Thomas Sewell
http://user.it.uu.se/~tjawe125/
Automation
Motivation
Related Work
Arrays
Bit-Vectors
Core
Ints
Reals
Reals_Ints
SMT-LIB's Theory of Fixed-Width Bit-Vectors
sorts
BitVec m
, for every m > 0

bit-vector constants, e.g.,
#b0

logical operations (
bvnot
,
bvand
,
bvor
, ...)

arithmetic operations (
bvneg
,
,
bvmul
, ...)

concatenation, extraction, shift operations, etc.
Bit-Vectors in Higher-Order Logic
SMT-LIB
HOL
Translating bit-vector
terms
between SMT-LIB and HOL is relatively straightforward.
bit-vector sorts
bit-vector constants
logical operations
arithmetic operations
...
bit-vector types
bit-vector literals
logical operations
arithmetic operations
...
Only 2 rules involve bit-vector reasoning.
rewrite
th-lemma-bv
Proof Reconstruction for rewrite and th-lemma-bv
Schematic theorems
arbitrary theory-specific lemmas
74% of SMT-LIB problems are checked successfully!

Proof reconstruction is an order of magnitude faster than LCF-style bit-blasting.

Proof generation with Z3 is up to two orders of magnitude faster than proof reconstruction.

Timeouts are due to theory reasoning; Z3's proofs are not detailed enough.
Theorem memoization
Strategy selection
rewrite
th-lemma-bv
Z3 timeout/error
Error
QF_AUFBV
Timeout
Success
QF_BV
QF_UFBV
Benchmarks
Profiling
th-lemma-bv
rewrite
other
% runtime
QF_AUFBV
QF_BV
QF_UFBV
Full transcript