- Jared Davis and Magnus O. Myreen.
The reflective Milawa theorem prover is sound (down to the machine code that runs it).
Journal of Automated Reasoning.
June, 2015. Springer.
- Official version (Springer) / Author's pre-print version
This paper presents, we believe, the most comprehensive evidence of
a theorem prover's soundness to date. Our subject is the Milawa
theorem prover. We present evidence of its soundness down to the
Milawa is a theorem prover styled after NQTHM and ACL2. It is based
on an idealised version of ACL2's computational logic and provides
the user with high-level tactics similar to ACL2's. In contrast to
NQTHM and ACL2, Milawa has a small kernel that is somewhat like an
We explain how the Milawa theorem prover is constructed as a
sequence of reflective extensions from its kernel. The kernel
establishes the soundness of these extensions during Milawa's
Going deeper, we explain how we have shown that the Milawa kernel is
sound using the HOL4 theorem prover. In HOL4, we have formalized its
logic, proved the logic sound, and proved that the source code for
the Milawa kernel (1,700 lines of Lisp) faithfully implements
Going even further, we have combined these results with the x86
machine-code level verification of the Lisp runtime Jitawa.
Our top-level theorem states that Milawa can never claim to prove
anything that is false when it is run on this Lisp runtime.
- (See also Milawa's Website and
- Jared Davis, Anna Slobodova, and Sol
Verification—Another Piece of the Microprocessor Verification Puzzle.
Invited talk, ITP 2014. Springer LNCS, 8558. July, 2014.
- Abstract. Despite significant progress in formal hardware verification in the past
decade, little has been published on the verification of microcode. Microcode
is the heart of every microprocessor and is one of the most complex parts of
the design: it is tightly connected to the huge machine state, written in an
assembly-like language that has no support for data or control structures, and
has little documentation and changing semantics. At the same time it plays a
crucial role in the way the processor works.
We describe the method of formal microcode verification we have developed
for an x86-64 microprocessor designed at Centaur Technology. While the previous
work on high and low level code verification is based on an unverified abstract
machine model, our approach is tightly connected with our effort to verify the
register-transfer level implementation of the hardware. The same microoperation
specifications developed to verify implementation of the execution units are
used to define operational semantics for the microcode verification.
While the techniques used in the described verification effort are not
inherently new, to our knowledge, our effort is the first interconnection of
hardware and microcode verification in context of an industrial size
design. Both our hardware and microcode verifications are done within the same
- Jared Davis and Matt Kaufmann.
Industrial Strength Documentation for ACL2.
To appear in ACL2 2014. July, 2014.
EPTCS 152. Pages 9-25.
The ACL2 theorem prover is a complex system. Its libraries are vast. Industrial verification efforts
may extend this base with hundreds of thousands of lines of additional modeling tools, specifications,
and proof scripts. High quality documentation is vital for teams that are working together on projects
of this scale. We have developed XDOC, a flexible, scalable documentation tool for ACL2 that can
incorporate the documentation for ACL2 itself, the Community Books, and an organization's internal
formal verification projects, and which has many features that help to keep the resulting manuals up
to date. Using this tool, we have produced a comprehensive, publicly available ACL2+Books Manual
that brings better documentation to all ACL2 users. We have also developed an extended manual for
use within Centaur Technology that extends the public manual to cover Centaur's internal books. We
expect that other organizations using ACL2 will wish to develop similarly extended manuals.
- (See also the ACL2+Books Manual)
- Magnus O. Myreen and Jared Davis.
The reflective Milawa theorem
prover is sound (down to the machine code that runs it)
In ITP 2014.
July, 2014. Springer, LNCS 8558. Pages 421-436.
- Note: You may prefer the more comprehensive 2015 JAR paper of the same name, see above.
Milawa is a theorem prover styled after ACL2 but with a
small kernel and a powerful reflection mechanism. We have used the
HOL4 theorem prover to formalize the logic of Milawa, prove the logic
sound, and prove that the source code for the Milawa kernel (2,000 lines
of Lisp) is faithful to the logic. Going further, we have combined these
results with our previous verification of an x86 machine-code implementation
of a Lisp runtime. Our top-level HOL4 theorem states that when
Milawa is run on top of our verified Lisp, it will only print theorem
statements that are semantically true. We believe that this top-level
theorem is the most comprehensive formal evidence of a theorem prover's
soundness to date.
- (See also Milawa's Website and
- (See also Slides
from my July 2012 talk at Northeastern University)
- Jared Davis and Sol Swords.
Verified AIG Algorithms in ACL2.
In ACL2 2013. May, 2013.
EPTCS 114. Pages 95-110.
And-Inverter Graphs (AIGs) are a popular way to represent Boolean functions
(like circuits). AIG simplification algorithms can dramatically reduce an
AIG, and play an important role in modern hardware verification tools like
equivalence checkers. In practice, these tricky algorithms are implemented
with optimized C or C++ routines with no guarantee of correctness.
Meanwhile, many interactive theorem provers can now employ SAT or SMT solvers
to automatically solve finite goals, but no theorem prover makes use of these
advanced, AIG-based approaches.
We have developed two ways to represent AIGs within the ACL2 theorem prover.
One representation, Hons-AIGs, is especially convenient to use and reason
about. The other, Aignet, is the opposite; it is styled after modern AIG
packages and allows for efficient algorithms. We have implemented functions
for converting between these representations, random vector simulation,
conversion to CNF, etc., and developed reasoning strategies for verifying
Aside from these contributions towards verifying AIG algorithms, this work
has an immediate, practical benefit for ACL2 users who are using GL to
bit-blast finite ACL2 theorems: they can now optionally trust an
off-the-shelf SAT solver to carry out the proof, instead of using the
built-in BDD package. Looking to the future, it is a first step toward
implementing verified AIG simplification algorithms that might further
improve GL performance.
- Jared Davis.
Embedding ACL2 Models in End-User Applications.
In Do-Form 2013,
AISB 2013, Exeter, UK.
April, 2013. Pages 49-56.
Formal verification, based on mechanical theorem proving, can provide unique
evidence that systems are correct. Unfortunately this promise of correctness
is, for most projects, not enough to justify its high cost. Since formal
models and proof scripts offer few other direct benefits to system developers
and managers, the idea of formal verification is abandoned.
We have developed a way to embed functions from the ACL2 theorem prover into
software that is written in mainstream programming languages. This lets us
reuse formal ACL2 models to develop applications with features like
graphics, networking, databases, etc. For example, we have written a web-based
tool for hardware designers in Ruby on top of a 100,000+ line ACL2 codebase.
This is neat: we can reuse the supporting work needed for formal verification
to create tools that are useful beyond the formal verification team. The value
added by these tools will help to justify the investment in formal
verification, and the project as a whole will benefit from the precision of
formal modeling and analysis.
- Sol Swords and Jared Davis.
Bit-Blasting ACL2 Theorems.
ACL2 2011. November, 2011.
EPTCS 70. Pages 84-102.
- Abstract. Interactive theorem proving requires a lot of human
guidance. Proving a property involves (1) figuring out why it holds, then (2)
coaxing the theorem prover into believing it. Both steps can take a long
time. We explain how to use GL, a framework for proving finite ACL2 theorems
with BDD- or SAT-based reasoning. This approach makes it unnecessary to deeply
understand why a property is true, and automates the process of admitting it as
a theorem. We use GL at Centaur Technology to verify execution units for x86
integer, MMX, SSE, and floating-point arithmetic.
- (See also ACL2 '11 Slides)
- Magnus O. Myreen and Jared Davis. A Verified Runtime for a Verified
Theorem Prover. In Interactive Theorem Proving (ITP 2011). August, 2011,
Nijmegen, The Netherlands. Springer, LNCS 6898. Pages 265-280.
- Abstract. Theorem provers, such as ACL2, HOL, Isabelle and
Coq, rely on the correctness of runtime systems for programming languages like
ML, OCaml or Common Lisp. These runtime systems are complex and critical to the
integrity of the theorem provers.
In this paper, we present a new Lisp runtime which has been formally veried and
can run the Milawa theorem prover. Our runtime consists of 7,500 lines of
machine code and is able to complete a 4 gigabyte Milawa proof effort. When our
runtime is used to carry out Milawa proofs, less unveried code must be trusted
than with any other theorem prover.
Our runtime includes a just-in-time compiler, a copying garbage collector, a
parser and a printer, all of which are HOL4-veried down to the concrete x86
code. We make heavy use of our previously developed tools for machine-code
verication. This work demonstrates that our approach to machine-code verication
scales to non-trivial applications.
- (See also Milawa's Website)
- (See also ACL2 '11 Rump Session Slides)
- Anna Slobadova, Jared Davis, Sol Swords, and Warren A Hunt., Jr. A Flexible Formal Verification Framework
for Industrial Scale Validation. Invited talk, Formal Methods and Models for Codesign (MemoCode
2011). July, 2011. Cambridge, UK. Pages 89-97.
- Abstract. In recent years, leading microprocessor companies
have made huge investments to improve the reliability of their
products. Besides expanding their validation and CAD tools teams, they have
incorporated formal verification methods into their design flows. Formal
verification (FV) engineers require extensive training, and FV tools from CAD
vendors are expensive. At first glance, it may seem that FV teams are not
affordable by smaller companies. We have not found this to be true. This paper
describes the formal verification framework we have built on top of
publicly-available tools. This framework gives us the flexibility to work on
myriad different problems that occur in microprocessor design.
Warren A. Hunt Jr., Sol Swords, Jared Davis, and Anna Slobadova.
Use of Formal Verification at Centaur Technology. In David S. Hardin, editor,
and Verification of Microprocessor Systems for High Assurance Applications.
2010. Springer. Pages 65-88.
- Abstract. We have developed a formal-methods-based hardware
verification toolflow to help ensure the correctness of our X86-compatible
microprocessors. Our toolflow uses the ACL2 theorem-proving system as a
design database and a verification engine. We verify Verilog designs by
first translating them into a formally defined hardware description
language, and then using a variety of automated verification algorithms
controlled by theorem-proving scripts.
In this chapter, we describe our approach to verifying components of VIA
Centaur's 64-bit Nano, X86-compatible microprocessor. We have successfully
verified a number of media-unit operations, such as the packed
addition/subtraction instructions. We have verified the integer multiplication
unit, and we are in the process of verifying microcode sequences that perform
A Self-Verifying Theorem Prover.
The University of Texas at Austin.
Programs have precise semantics, so we can use mathematical proof to establish
their properties. These proofs are often too large to validate with the usual
"social process" of mathematics, so instead we create and check them with
theorem proving software. This software must be advanced enough to make the
proof process tractable, but this very sophistication casts doubt upon the
whole enterprise: who verifies the verifier?
We begin with a simple proof checker, Level 1, that only accepts proofs
composed of the most primitive steps, like Instantiation and Cut. This program
is so straightforward the ordinary, social process can establish its soundness
and the consistency of the logical theory it implements (so we know theorems
are "always true").
Next, we develop a series of increasingly capable proof checkers, Level 2,
Level 3, etc. Each new proof checker accepts new kinds of proof steps which
were not accepted in the previous levels. By taking advantage of these new
proof steps, higherlevel proofs can be written more concisely than lower-level
proofs, and can take less time to construct and check. Our highest-level proof
checker, Level 11, can be thought of as a simplified version of the ACL2 or
NQTHM theorem provers. One contribution of this work is to show how such
systems can be verified.
To establish that the Level 11 proof checker can be trusted, we first use it,
without trusting it, to prove the fidelity of every Level n to Level 1:
whenever Level n accepts a proof of some Phi, there exists a Level 1
proof of Phi. We then mechanically translate the Level 11 proof for each Level
n into a Level n - 1 proof. That is, we create a Level 1 proof of Level
2's fidelity, a Level 2 proof of Level 3's fidelity, and so on. This layering
shows that each level can be trusted, and allows us to manage the sizes of
In this way, our system proves its own fidelity, and trusting Level 11 only
requires us to trust Level 1.
- (See also Milawa's Website)
Memories: Array-like Records for ACL2.
In ACL2 2006.
August, 2006, Seattle, WA, USA.
- Abstract. We have written a new records library for
modelling fixed-size arrays and linear memories. Our implementation provides
fixnum-optimized O(log2 n) reads and writes from addresses 0,
1, ... , n-1. Space is not allocated until locations are used, so large
address spaces can be represented. We do not use single-threaded objects or
ACL2 arrays, which frees the user from syntactic restrictions and slow-array
warnings. Finally, we can prove the same hypothesis-free rewrite rules found
in misc/records for efficient rewriting during theorem proving.
- (See also Memory Library Website)
Reasoning about File Input in ACL2.
In ACL2 2006.
August, 2006, Seattle, WA, USA.
- Abstract. We introduce the logical story behind file input
in ACL2 and discuss the types of theorems that can be proven about filereading
operations. We develop a low level library for reasoning about the primitive
input routines. We then develop a representation for Unicode text, and
implement efficient functions to translate our representation to and from the
UTF-8 encoding scheme. We introduce an efficient function to read UTF-8-encoded
files, and prove that when files are well formed, the function produces valid
Unicode text which corresponds to the contents of the file.
We find exhaustive testing to be a useful technique for proving many theorems
in this work. We show how ACL2 can be directed to prove a theorem by exhaustive
- (See also Input Library Website)
Finite Set Theory based on Fully Ordered Lists.
In ACL2 2004.
November, 2004, Austin, TX, USA.
We present a new finite set theory implementation for ACL2 wherein sets are
implemented as fully ordered lists. This order unifies the notions of set
equality and element equality by creating a unique representation for each set,
which in turn enables nested sets to be trivially supported and eliminates the
need for congruence rules.
We demonstrate that ordered sets can be reasoned about in the traditional style
of membership arguments. Using this technique, we prove the classic properties
of set operations in a natural and effotless manner. We then use the exciting
new MBE feature of ACL2 to provide linear-time implementations of all basic set
operations. These optimizations are made "behind the scenes" and do not
adversely impact reasoning ability.
We finally develop a framework for reasoning about quantification over set
elements. We also begin to provide common higher-order patterns from functional
programming. The net result is an efficient library that is easy to use and
- (See also Sets Library Website)
Gregory L. Wickstrom, Jared Davis, Steve Morrison, Steve Roach, Victor L. Winter.
The SSP: An Example of High-Assurance Systems Engineering.
In High-Assurance Systems Engineering
March, 2004, Tampa, FL, USA. IEEE Computer Society 2004, ISBN 0-7695-2094-4.
The SSP is a high assurance systems engineering effort spanning both hardware
and software. Extensive design review, first principle design, n-version
programming, program transformation, verification, and consistency checking are
the techniques used to provide assurance in the correctness of the resulting