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ACL2

ACL2 (A Computational Logic for Applicative Common Lisp) is an automated theorem prover and programming environment designed for specifying, executing, and verifying properties of computational systems, particularly in hardware and software domains. ACL2 originated as part of the Boyer–Moore lineage of theorem provers, tracing its roots to the Pure Lisp Theorem Prover developed in the early 1970s at the University of Edinburgh and the subsequent Nqthm (New Pure Lisp Theorem Prover) from the 1970s and 1980s. The ACL2 project began in 1989 under Robert S. Boyer and J Strother Moore, evolving into its current form through extensive collaboration with Matt Kaufmann, who joined shortly thereafter; since then, it has been primarily developed and maintained by Kaufmann and Moore at the University of Texas at Austin. In recognition of its impact, Boyer, Kaufmann, and Moore received the 2005 ACM Software System Award for the Boyer–Moore family of provers, including ACL2, which has enabled rigorous verification in complex engineering contexts. At its core, ACL2 integrates a first-order logic with a computational subset of Common Lisp, allowing users to define executable functions that can be reasoned about formally through induction, rewriting, and simplification tactics. The system's theorem prover automates proof search using a database of user-supplied rules and lemmas, while supporting interactive guidance for intricate arguments; it emphasizes efficiency, with definitions serving as actual Lisp programs verified under "guards" that enforce domain constraints. A hallmark is its extensibility via "books"—certified libraries of definitions and theorems—that form a communal repository, fostering reuse in large-scale proofs. The current version, 8.6, was released in October 2024 with ongoing updates, distributed freely under a BSD license and supported by a hypertext manual exceeding 5,000 pages. ACL2 has found extensive industrial application since the mid-1990s, particularly in verifying microprocessors, units, and cryptographic protocols at organizations including , , , and . Notable verifications encompass the and processors, cores and subsequent desktop microprocessors, and components of the , demonstrating its scalability for mission-critical systems where exhaustive testing alone is insufficient. The system's annual workshops and growing community continue to advance its role in , bridging theoretical logic with practical .

Introduction

Overview

ACL2 is a that serves as an interactive theorem prover, integrating a programming language, an extensible theory, and an tool to model and verify properties of computer systems. It enables users to specify algorithms and systems in a precise manner while proving about their correctness, behavior, and safety. The primary purposes of ACL2 include in inductive logical theories and of both software and hardware designs, facilitating the detection of subtle errors that might otherwise go unnoticed in traditional testing approaches. At its core, ACL2 is implemented in Common Lisp and emphasizes functional, side-effect-free programming through its applicative subset, which promotes reliable and predictable computations suitable for formal analysis. The high-level architecture seamlessly combines this applicative subset—used for executable specifications—with a quantifier-free that allows for the definition of mathematical objects and the proof of properties via mechanized deduction. This integration supports the development of models that are both runnable as programs and amenable to rigorous logical scrutiny. Released publicly in , ACL2 is distributed as under a three-clause BSD , ensuring broad and encouraging community contributions. It exhibits cross-platform compatibility, running on major operating systems including , macOS, and Windows, which enhances its adoption in diverse development environments. As a mature tool, ACL2 has established itself as a for industrial-strength , building on foundational work from earlier provers like Nqthm and finding notable applications in hardware verification tasks.

Development History

The roots of ACL2 lie in the Edinburgh Pure Lisp Theorem Prover (PLTP), developed by Robert S. Boyer and J. Strother Moore between 1971 and 1973 as the first automated theorem prover for recursive functions and inductive data types in . This system evolved into the Thm prover around 1973–1979, which introduced user-supplied lemmas and improved automation for proving properties like the soundness of tautology checkers. By the early 1980s, it advanced to Nqthm, the Boyer-Moore theorem prover, which added support for , metafunctions, and ordinals, enabling verifications such as the FM9001 design. Development of ACL2 began in August 1989 at Computational Logic, Inc. (CLI), initially led by Boyer and Moore, with Matt Kaufmann joining in 1993 to collaborate with J. Strother Moore on its core implementation. Unlike its predecessors, ACL2 adopted an applicative subset of Common Lisp as its foundational logic, transforming the tool from a pure theorem prover into a comprehensive environment for executable computational logic and automated reasoning. The system saw limited industrial use by 1993, including early verifications at CLI, and became publicly available in the mid-1990s following internal releases starting around 1990. A pivotal occurred in 1995, when ACL2 proved the IEEE compliance of the floating-point division for the , demonstrating its capability for large-scale . In recognition of the Boyer-Moore lineage, including ACL2, Boyer, , and received the 2005 ACM Software System Award for developing influential tools with lasting impact on . ACL2 has progressed through iterative versions, with ongoing enhancements focused on —such as improved proof utilities and metafunction support—and optimizations like hash consing and parallel execution features introduced in later releases. The stable Version 8.6, issued in October 2024, includes new capabilities for handling mutable objects via the attach-stobj mechanism, further boosting efficiency in modeling complex systems.

Language and Logic

Applicative Common Lisp

ACL2's programming language constitutes an applicative, side-effect-free, untyped subset of , designed to support the definition and execution of total recursive functions that always terminate on all inputs. This subset, often referred to as Applicative Common Lisp, forms the executable core of the system, enabling users to model computational systems as pure functions without mutable state or imperative constructs. By restricting to applicative evaluation—where arguments are fully evaluated before —the language ensures predictable, deterministic behavior aligned with mathematical function semantics. Key features of the language include strict applicative-order evaluation, the absence of mutable state to prevent side effects, and the use of as preconditions for . are logical expressions that specify the expected input domains, facilitating proofs of termination, , and correctness; for instance, verifying a guard ensures that a function operates only on valid arguments, avoiding . The language supports efficient execution by compiling definitions to native code through the underlying implementation, while guard verification allows runtime checks to be omitted for certified functions, achieving performance comparable to full . The syntax inherits Lisp's s-expression form, using prefix notation for expressions and supporting familiar constructs like lists, symbols, and numbers. Functions are defined with defun, which includes optional :guard and :measure clauses to specify preconditions and termination measures, respectively; theorems are integrated via defthm, allowing properties to be asserted alongside executable code. For example: 
(defun append (x y)
  (declare (xargs :guard (and (true-listp x) (true-listp y))))
  (if (endp x)
      y
    (cons (car x) (append (cdr x) y))))
This definition declares a guard ensuring both arguments are true lists, promoting both executability and provability. Semantics emphasize totality through guarded definitions: functions must be proven to terminate (via a measure that decreases on recursive calls) and satisfy guards under the logical axioms, which model primitives like cons, car, and cdr as total operations. In contrast to full , ACL2 excludes imperative features such as assignment (setq), mutable variables, destructive list operations (e.g., rplaca), and higher-order functions with side effects, to preserve logical consistency and enable equational reasoning. These restrictions ensure that every expression denotes a pure value, supporting both via execution and rigorous without concerns over state interference.

Logical Foundations

ACL2's logical foundations are rooted in a with equality, which axiomatizes the semantics of applicative functions, ensuring that all reasoning is grounded in a computationally subset of the language. This base theory is quantifier-free, emphasizing equational reasoning over propositional and arithmetic primitives, while supporting definitions and . The logic models the applicative behavior of , treating functions as total and well-defined within specified domains, thereby providing a conservative extension of standard mathematical logics suitable for mechanical verification. Central to this foundation are key axioms that define the core data types and operations. Peano is incorporated for natural numbers, enabling reasoning about integers with up to the ordinal ε₀, which supports termination arguments for recursive functions over potentially domains. Definitions for booleans distinguish true (T) from false (NIL), with axioms such as (not p) = (if p nil t); lists are constructed via pairs with primitives like , satisfying equations like (rest ([cons](/page/Cons) x y)) = y; and primitive operations include , conditional (if), (e.g., binary-+), and ordering relations (<). Ordinals extend the theory to handle descending chains, formalized through well-founded relations like E0-ORD-<, allowing proofs about non-terminating or processes in finite models. The logic's extensibility allows users to build upon this base by introducing new axioms, function definitions, and theorems through events such as and , maintaining conservative extensions that preserve prior theorems. An event introduces constrained functions with witnessing constants and axioms, requiring proofs of existence and uniqueness to ensure the extension does not introduce inconsistencies; for instance, recursive definitions must be admitted only after proving termination via ordinal measures. adds proven lemmas to the theory, enriching the rule base for future proofs without altering the semantics of existing terms. This modular approach enables domain-specific theories while upholding the logic's integrity. Theory invariants ensure the preservation of logical consistency throughout extensions, enforced by single-threaded execution that processes events sequentially and guard verification that checks function applicability before evaluation. Guards specify input constraints (e.g., type or range conditions), and their verification during proof admission prevents undefined behavior, approximating the total functions of the logic in executable models. For non-computable aspects, such as infinite computations, ACL2 approximates them using finite models via choice operators, Skolem functions, and well-founded induction, allowing termination proofs without simulating unbounded processes directly. This framework balances expressiveness with decidability, focusing on finite, ground terms for practical theorem proving.

Theorem Proving System

Proof Process

The proof process in ACL2 relies primarily on automated term rewriting through its simplifier, which applies rewrite rules to transform terms toward provable forms, combined with decision procedures for linear arithmetic and equality to handle specific subgoals efficiently. The simplifier operates by repeatedly rewriting terms using enabled rules, including those derived from previously proved lemmas, until no further changes occur or a contradiction is reached. These decision procedures, such as the built-in linear arithmetic solver, resolve inequalities and equalities without full induction, providing foundational automation for many conjectures. Proof automation centers on the waterfall process, a sequential cascade of tactics that attempts to discharge goals starting with simplification, followed by elimination of irrelevant variables, case splitting, and application of built-in rules or user-supplied hints. If simplification fails to prove a goal outright, the waterfall may invoke lemma induction to generate subgoals based on previously proved lemmas or proceed to metareasoning, where ACL2 reasons about its own proof rules to justify additional inferences. Hints allow users to guide this process by specifying custom rewrite rules, induction instructions, or case analyses at particular waterfall steps, enhancing automation while maintaining soundness. Metareasoning further supports this by enabling proofs about the prover's internal logic, such as verifying the correctness of custom simplifiers. User interaction occurs primarily through the certify-book command, which processes a file of definitions and theorems, attempting automated proofs for each and providing detailed feedback on failures, including counterexamples or stalled subgoals for manual inspection. This interactive development loop allows iterative refinement: users can add hints, lemmas, or adjust definitions based on output, then recertify to verify changes, with the system replaying proofs incrementally. For complex cases, the proof-checker mode offers finer control, enabling step-by-step goal management outside the full waterfall. Induction in ACL2 is handled by automatically generating induction schemes from the termination proofs of recursive function definitions, creating a named induction rule that posits the theorem over the function's measure to justify recursive cases. These schemes are invoked during the waterfall when pattern-matching detects recursive structures, splitting the goal into base and inductive steps, often requiring user hints to select the appropriate scheme among alternatives. Performance optimizations include proof caching, where successful proofs in certified books are stored and reused during subsequent certifications to avoid recomputation, and parallel processing, supported via the set-waterfall-parallelism setting since version 6.1, which distributes independent subgoals across multiple threads for faster execution on multiprocessor systems. This parallelism targets the waterfall's sequential bottlenecks, such as simultaneous simplification of multiple goals, while preserving determinism through controlled scheduling.

Encapsulation and Books

In ACL2, books serve as reusable libraries that encapsulate definitions, theorems, and proofs, allowing users to certify and organize complex specifications as modular units. A book is essentially a file containing a sequence of ACL2 events, which can include both local (temporary) and non-local (permanent) forms, and is certified using the certify-book command to verify its logical consistency with the existing world. Once certified, books can be loaded into a session via the include-book command, which imports only the non-local events while assuming their prior certification, thereby extending the logical world without re-proving contents. This mechanism promotes reusability by enabling the construction of new developments atop compatible libraries, with ACL2 detecting potential name conflicts to maintain integrity. Encapsulation provides a powerful mechanism in ACL2 for abstracting interfaces and hiding implementation details, particularly useful for proving properties about abstract functions without exposing their concrete realizations. The encapsulate event processes a sequence of sub-events in two passes: the first pass admits all events sequentially to check consistency, while the second pass rolls back the world and re-evaluates only non- events, skipping proof obligations that were already verified. Constrained functions are introduced via signatures in the encapsulate form, such as ((foo *) => *), which declare the function's without defining it, allowing theorems about these functions to be proved and exported while their witnesses—temporary definitions used for —are hidden from the final logical . For instance, a definition like (local (defun foo (x) x)) can serve as a to satisfy constraints, enabling proofs of properties such as preservation of data structures, after which only the abstract constraints remain visible. Theory management in ACL2 leverages encapsulation to handle constraints systematically, ensuring that functions adhere to specified properties through verification, which confirms the existence of implementations satisfying the constraints without polluting the . The syntax for encapsulation typically follows (encapsulate (((fname1 arg-types) => return-type) ...) ev1 ... evn), where signatures define constrained functions and embedded events include local proofs and theorems. This approach supports scalable developments by isolating , allowing users to build hierarchical proofs where inner encapsulations provide witnesses for outer ones, thus maintaining logical consistency across modules. The benefits of and encapsulation in ACL2 include enhanced scalability for large-scale projects, as they avoid global namespace pollution by localizing events and enabling certified extensions without redundant proofs. facilitate community-contributed libraries, such as those in the ACL2 community repository, which can be included to bootstrap complex analyses efficiently. Encapsulation further aids by exporting only theorems, reducing the on users and supporting black-box reasoning about components. The workflow for using books and encapsulation emphasizes certification to ensure logical consistency: during book certification, ACL2 processes all events to build a consistent extension, flagging errors if constraints cannot be met; subsequent inclusions or encapsulations then inherit this certified state, preserving the host world's properties and enabling incremental development. This process is crucial for maintaining soundness in distributed proof efforts, where multiple books can be certified independently before integration.

Applications and Case Studies

Industrial Applications

ACL2 has been extensively applied in industrial settings for verifying complex hardware and software systems, particularly in the and sectors, where are essential for ensuring reliability and compliance with standards like for . Since the mid-1990s, companies such as , , , , , and have integrated ACL2 into their workflows to detect subtle , reduce testing time, and certify high-assurance designs. These applications leverage ACL2's ability to model and prove properties of large-scale systems, often involving millions of lines of code or thousands of transistors. A landmark case study is AMD's use of ACL2 in 1995 to verify the floating-point division for the K5 , completing the proof in approximately two months and confirming correctness against IEEE standards, which helped avoid potential post-silicon issues. Similarly, AMD applied ACL2 to verify the floating-point units in the and processors starting in 1998, proving compliance for , , and other operations. At , ACL2 has been routinely used since 2007 to verify x86-compatible processors like the , including a detailed proof of the floating-point adder that spanned 33,700 lines of and 432,322 transistors, ultimately uncovering a bug during the process. employed ACL2 in combination with its SixthSense model checker to verify VHDL circuits, such as multipliers and modular reduction engines in processors, enabling hybrid of arithmetic hardware. has utilized ACL2 for proving properties of transaction protocols and implementations, while has verified floating-point and integer arithmetic units in processors. (formerly ) has applied ACL2 to safety-critical systems, including proofs of security for the AAMP7 processor and correctness of a simulator, supporting certification under standards like for avionic software. has incorporated ACL2 in verifying floating-point designs for , , division, and operations. These industrial verifications demonstrate ACL2's benefits in accelerating design cycles by automating proof reuse upon modifications and detecting errors that simulation might miss, such as the FPU bug, thereby enhancing system reliability without exhaustive testing. Integration with other tools is common; for instance, translates / models into ACL2 using tools like and SVEX for bit-level analysis, allowing nightly regression proofs on multi-core clusters for designs exceeding 700,000 lines. IBM's hybrid approach combines ACL2 theorems with SixthSense for bounded of VHDL signals. At , ACL2 supports hardware/software co-assurance workflows, translating high-level models to verifiable code for cyber-physical systems. The sustained impact of ACL2 in is evident in its in producing reliable chip designs since the 1990s, with adopters reporting scalable proofs that scale to production hardware and contribute to the ACL2 Community Books repository, which contains over 5,000 and more than 170,000 theorems (as of ). This long-term adoption underscores ACL2's maturity for high-stakes applications, from microprocessors to certified avionic components.

Academic and Research Uses

ACL2 has been integrated into various academic curricula focused on and theorem proving, particularly at the graduate and upper-division undergraduate levels. It serves as a practical tool for teaching concepts in , , and , allowing students to model and prove properties of algorithms and systems. For instance, courses at institutions like have employed ACL2 within DrScheme environments to develop skills in and through . Additionally, on formal techniques, such as the Tenth Summer School organized by , introduce ACL2 for proving properties of recursive functions and models, emphasizing its role in hands-on education. The textbook Computer-Aided Reasoning: An Approach by Matt Kaufmann, Panagiotis Manolios, and J Strother Moore provides a foundational resource, outlining methodologies for modeling computing systems and mechanized reasoning, suitable for courses on . In research, ACL2 has advanced through its support for meta-reasoning and , enabling the implementation of custom proof strategies within the system. These capabilities facilitate the of complex algorithms, such as networks via inductive proofs of correctness and like SHA-256, where functional models are certified against security properties. Studies have explored ACL2's efficiency in proof automation, including the development of structures optimized for verification tasks, demonstrating reductions in proof effort for bit-level pipelined machines. Notable academic projects leverage ACL2 in hybrid verification frameworks, combining it with tools like solvers (e.g., Z3) or model checkers (e.g., IBM's ) to enhance for analog-mixed-signal systems and multiplier circuits. These integrations allow ACL2 to handle while delegating propositional checks to specialized engines, improving overall efficiency without compromising soundness. The ACL2 community's publications, including proceedings from annual workshops, document these advancements and extensions of the logic to new domains, such as probabilistic models through reflective . These volumes, published in series like Electronic Proceedings in Theoretical Computer Science (EPTCS), feature peer-reviewed papers on proof techniques and tool integrations, fostering ongoing research collaboration. ACL2's free availability under the BSD license since its initial public release in the mid-1990s has driven widespread adoption, enabling researchers and educators worldwide to experiment with and extend the system without barriers. This has supported its use in over two decades of theorem-proving and theoretical studies, contributing to its status as a benchmark tool in communities.

Community and Extensions

Workshops and Resources

The ACL2 workshops serve as the primary forum for the user community to present research, applications, tutorials, and new features of the theorem prover. The series began with the first workshop in 1999 at the , and has since been held approximately every 18 months, often co-located with conferences such as FMCAD, CAV, or FLoC. Recent workshops include the 2020 event held online via video streaming, the 2022 in , the 2023 in , and the 2025 hybrid in . These events typically feature peer-reviewed papers, invited talks, and discussions on advancements in ACL2's capabilities, with proceedings published in venues like Electronic Proceedings in Theoretical Computer Science (EPTCS); for example, the ACL2-2023 in , included papers on techniques and tool enhancements. Official documentation for ACL2 is hosted by the and includes comprehensive manuals, such as the ACL2 User's Guide and the Introduction to the Theorem Prover, which detail the system's , proof , and programming features. Additional resources encompass tutorial materials like the ACL2 Tutorial section in the manual, which provides step-by-step guidance for modeling and proving properties, and books such as Computer-Aided Reasoning: ACL2 Case Studies derived from early workshops. Online resources from the ACL2 project page offer downloadable versions, certification instructions, and examples for integrating ACL2 into verification workflows. The ACL2 community fosters collaboration through mailing lists for announcements (acl2-announce), discussions (acl2-discuss), and help (acl2-help), as well as a Zulip chat platform for real-time interactions. A key resource is the certification repository of community , maintained on , which contains hundreds of verified libraries and examples contributed by users for domains like , structures, and models. Contribution guidelines emphasize submitting pull requests for and extensions, with reviews ensuring and . Learning paths in ACL2 range from beginner resources, such as the Gentle Introduction to ACL2 Programming in , which covers basic events and definitions, to advanced topics like metareasoning and custom proof strategies outlined in seminar archives and workshop tutorials. Community-contributed libraries, including those for bit-vector operations and symbolic simulation, exemplify practical extensions that users can study and adapt. Maintenance of ACL2 is led by principal developers Matt Kaufmann and J Strother at the University of Texas, with the system distributed as open-source via since version 7.4 in 2016, enabling community contributions to both the core prover and books repository.

Related Tools and Variants

ACL2s, also known as the ACL2 Sedan, is an Eclipse-based (IDE) designed to simplify ACL2 usage for educational purposes. It introduces a simplified syntax, defevaluator support for step-by-step execution, and visualization tools to aid novices in learning theorem proving concepts. Developed initially by Peter Dillinger and Pete Manolios around 2007 and extended through 2013, ACL2s supports progressive user levels from beginner to expert, streamlining the learning curve without altering ACL2's core logic. Other variants and tools enhance ACL2's capabilities in specific areas. The Glucose interface, integrated via the IPASIR framework, enables ACL2 to leverage the for incremental satisfiability solving, improving performance on propositional problems during proof automation. The Certify tool, implemented through the cert.pl command-line utility and the built-in certify-book function, manages book certification by compiling and verifying ACL2 libraries, ensuring modular reusability across projects. ACL2 integrates with hardware description languages and model checkers to extend its verification scope. For instance, the ACL2SIX environment combines ACL2 with IBM's tool to verify circuits, automatically translating hardware models into ACL2 specifications for property checking. Similarly, interfaces allow ACL2 to incorporate results from model checkers like , using external deductions to compose proofs for temporal properties in finite-state systems. Community extensions address domain-specific challenges. The floating-point (FP) library in ACL2's community books provides verified models of IEEE-754 , supporting double-precision computations and enabling formal analysis of FP hardware like multipliers at . For machine learning verification, tools like ACL2(ml) apply techniques to recommend theorems and function definitions, assisting in proofs related to ML algorithms, while broader efforts explore verifying properties within ACL2 models. In comparison to interactive provers like Isabelle and , ACL2 emphasizes automation through its Lisp-based embedding and industrial-scale proof search, reducing user intervention for large verifications, whereas Isabelle and prioritize explicit scripting and higher-order logics for interactive development.