13.3 Automated theorem proving and proof assistants
3 min read•august 7, 2024
uses algorithms to prove mathematical statements. It combines techniques like and with SAT and SMT solvers to tackle complex logical problems. These tools automate reasoning and find proofs faster than humans.
Proof assistants like , , and help verify math and software. They offer , where users guide proofs with tactics. This blend of human insight and machine power tackles tricky proofs and boosts confidence in results.
Automated Theorem Proving
Resolution and Unification
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- Resolution is a rule of inference for propositional logic and first-order logic that produces a new clause from two clauses containing complementary literals
- Involves resolving two clauses by assuming one is true and the other is false, then eliminating complementary literals to produce a new clause
- Unification is a key concept in automated theorem proving that involves finding a substitution that makes two logical expressions identical
- Unification algorithms (syntactic unification, semantic unification) find the most general unifier between two expressions, allowing for variables to be replaced with terms
SAT and SMT Solvers
- SAT solvers are programs that determine the of propositional logic formulas, checking if there exists an interpretation that makes the formula true
- Use algorithms like DPLL (Davis-Putnam-Logemann-Loveland) to efficiently search for satisfying assignments
- SMT (Satisfiability Modulo Theories) solvers build upon SAT solvers to decide the satisfiability of formulas in first-order logic with respect to background theories (arithmetic, arrays, uninterpreted functions)
- SMT solvers combine a with specialized decision procedures for the background theories, allowing for reasoning about more expressive formulas
Proof Assistants
Popular Proof Assistants
- Coq is a proof assistant based on the Calculus of Inductive Constructions, a dependent type theory
- Supports writing and proofs, allowing for the verification of complex mathematical theorems and programs
- Isabelle/HOL is a generic proof assistant that supports various logics, with higher-order logic (HOL) being the most widely used
- Offers a rich set of tools for interactive theorem proving and has been used to formalize substantial mathematical theories
- Lean is a newer proof assistant based on dependent type theory, designed for both mathematical reasoning and software verification
- Provides a powerful framework for writing formal proofs and has a growing library of mathematical results
Features and Applications
- Proof assistants offer expressive logics and type systems for specifying mathematical concepts and software systems
- Support the interactive development of formal proofs, with the ability to define custom tactics and automation
- Used in a wide range of applications, including verifying the correctness of algorithms, formalizing mathematical theories, and certifying the safety of critical systems
- Enable the construction of large, machine-checked proofs that can be independently verified, increasing confidence in the correctness of results
Interactive Theorem Proving
Proof Tactics and Interactivity
- are programs that automate common proof steps or strategies in interactive theorem proving
- Examples of tactics include
for introducing hypotheses,intros
for applying a lemma or hypothesis, andapply
for applying a collection of standard tacticsauto - Interactive theorem proving involves a user guiding the construction of a proof by applying tactics and providing input when needed
- Allows for the development of complex proofs that may be difficult to fully automate, while still benefiting from the automation provided by tactics
Proof Scripts and Reproducibility
- are files that contain a sequence of proof commands and tactics used to construct a formal proof in a proof assistant
- Serve as a record of the proof development process and can be replayed to check the correctness of the proof
- Enable the reproducibility of proofs, as other users can examine the proof script to understand the reasoning behind each step
- Can be shared and built upon by the community, allowing for the collaborative development of large formalized theories
- Examples of proof scripts include
files in Coq and.v
files in Isabelle/HOL, which contain the definitions, lemmas, and proofs of a formalization.thy
Key Terms to Review (26)
Automated Theorem Proving: Automated theorem proving refers to the use of computer programs to establish the validity of mathematical statements or logical assertions automatically. This process relies heavily on formal proof systems and algorithms to generate proofs, making it essential in areas like first-order logic, where it can streamline and enhance traditional proof methods. Furthermore, it has profound implications for cut elimination, enabling more efficient reasoning and verification in logical systems, as well as supporting proof assistants that assist users in constructing complex proofs interactively.
Completeness: Completeness refers to a property of a formal system where every statement that is true in all models of the system can be proven within that system. This concept is crucial because it connects syntax and semantics, ensuring that if something is logically valid, there's a way to derive it through formal proofs.
Coq: Coq is an interactive proof assistant that allows users to write mathematical definitions, executable algorithms, and theorems, and then construct proofs for those theorems in a formal way. It combines functional programming with formal verification, making it a vital tool for program verification and formal methods as well as automated theorem proving.
Derivability: Derivability is the concept in logic that refers to the ability to derive a conclusion from a set of premises using a formal proof system. This notion connects closely with the mechanisms of proving statements, where the validity of arguments is systematically explored through rules of inference. In automated theorem proving and proof assistants, derivability plays a crucial role as it determines whether certain logical statements can be formally established or proven within a given system.
DPLL Algorithm: The DPLL algorithm is a complete, backtracking-based search algorithm used for deciding the satisfiability of propositional logic formulas in conjunctive normal form (CNF). It enhances the basic backtracking method by incorporating unit propagation and pure literal elimination to efficiently reduce the search space. This algorithm plays a crucial role in automated theorem proving and has been foundational for modern SAT solvers, which are essential tools in various applications including verification and artificial intelligence.
Formal specifications: Formal specifications are precise, mathematical descriptions of software or systems that define their behavior, properties, and requirements in a formal language. This approach helps ensure that the design meets the necessary standards and allows for automated reasoning about correctness and consistency. By utilizing formal methods, developers can create unambiguous documentation that aids in automated theorem proving and enhances the reliability of proof assistants.
Formal Verification: Formal verification is the process of using mathematical techniques and formal methods to prove the correctness of a system or algorithm with respect to a certain specification or property. This process ensures that systems behave as intended, particularly in critical applications like software, hardware, and protocols. By establishing rigorous proofs, formal verification bridges theoretical logic and practical applications, enhancing reliability and safety in various domains.
Heuristics: Heuristics are mental shortcuts or rules of thumb that simplify decision-making processes and problem-solving. They allow individuals and automated systems to make judgments quickly without exhaustive analysis, which is especially crucial in complex environments like automated theorem proving and proof assistants, where time and resource efficiency is essential.
Interactive theorem proving: Interactive theorem proving is a method of formal verification where users interactively construct proofs with the help of software tools, allowing for a collaborative approach between human intuition and machine assistance. This process typically involves the use of proof assistants that help users ensure the correctness of mathematical proofs and logical arguments through a step-by-step verification mechanism. By combining human creativity with machine rigor, interactive theorem proving enhances the reliability and robustness of proofs in various fields, including mathematics and computer science.
Isabelle/HOL: Isabelle/HOL is an interactive theorem proving system that combines the Isabelle proof assistant with higher-order logic (HOL), allowing for formal verification of mathematical proofs and computer programs. This powerful tool enables users to represent and manipulate logical expressions, automate reasoning processes, and formally verify the correctness of complex systems through rigorous proofs, making it essential in both program verification and automated theorem proving.
Lean: In the context of automated theorem proving and proof assistants, 'lean' refers to a specific proof assistant and programming language designed for formalizing mathematics and creating verified software. It emphasizes simplicity, robustness, and efficiency in both the syntax and underlying architecture, making it accessible for users while supporting powerful proof automation features.
Modal logic: Modal logic is a type of formal logic that extends classical logic to include operators expressing modality, such as necessity and possibility. This allows for reasoning about statements that are not strictly true or false, enabling discussions about what could be, must be, or might have been. Modal logic connects to various areas, enhancing our understanding of semantics, proof structures, and computational applications.
Model checking: Model checking is an automated technique used to verify that a system's design adheres to certain specifications by systematically exploring its possible states. This approach ensures that the system behaves as intended, allowing for the detection of errors and inconsistencies before deployment. By modeling systems and applying logical formulas, model checking provides a rigorous method to validate software and hardware designs, making it integral to ensuring reliability in program verification and formal methods as well as aiding in automated theorem proving and proof assistants.
Natural Deduction: Natural deduction is a proof system used in logic that allows one to derive conclusions from premises using a set of inference rules in a structured and intuitive way. It emphasizes the natural reasoning process, enabling proofs to be constructed through the application of these rules without the need for additional axioms or complex structures, making it particularly useful in various fields of mathematics and philosophy.
Predicate logic: Predicate logic is an extension of propositional logic that incorporates quantifiers and predicates, allowing for more complex statements about objects and their properties. It provides a formal framework for reasoning about relations and functions, enabling the expression of statements involving 'all' or 'some' elements within a domain. This level of expressiveness makes predicate logic foundational in various applications, including completeness, compactness, automated theorem proving, and the development of proof assistants.
Proof normalization: Proof normalization is the process of transforming a proof into a simpler or more canonical form without changing its meaning or validity. This concept is crucial in understanding how proofs can be made more efficient and easier to analyze, particularly in formal systems where clarity and brevity are valued. It connects deeply with the goals of proof theory, the rules of sequent calculus, and the functioning of automated theorem proving and proof assistants.
Proof scripts: Proof scripts are structured sequences of commands and logical steps used to construct formal proofs within automated theorem proving systems or proof assistants. They serve as a bridge between human reasoning and machine verification, enabling users to articulate complex arguments in a format that can be processed by software tools. Proof scripts facilitate the automation of proof-checking, enhancing efficiency in both research and education.
Proof search: Proof search is the process of systematically exploring possible proofs for a given statement or theorem within a formal system. This process often involves generating and evaluating various proof structures, which can be particularly complex in systems like natural deduction. The efficiency and effectiveness of proof search are critical in automated theorem proving and the use of proof assistants, where computational methods help find valid proofs or disprove conjectures.
Proof tactics: Proof tactics are strategic techniques used in formal proofs to manipulate and construct logical arguments effectively. They help streamline the process of theorem proving by breaking down complex statements into manageable parts, guiding the proof in a structured manner. These tactics play a critical role in automated theorem proving and proof assistants, enabling users to interactively create and refine proofs.
Provability: Provability refers to the property of a statement being demonstrable within a formal system based on its axioms and inference rules. It connects deeply with various foundational aspects of mathematical logic, emphasizing the importance of formal proofs in establishing the truth of mathematical propositions.
Resolution: Resolution is a powerful rule of inference used in automated reasoning and logic that allows for the derivation of new clauses from a set of existing clauses. It is based on the principle that if one clause contains a literal and another clause contains the negation of that literal, both clauses can be combined to produce a new clause without that literal. This method plays a critical role in first-order logic proof systems, influencing proof complexity, and is also a foundational technique in automated theorem proving.
Sat solver: A sat solver is a software tool used to determine the satisfiability of propositional logic formulas. It works by evaluating whether there exists an assignment of truth values to variables that makes the entire formula true, which is crucial in automated reasoning and verification tasks.
Satisfiability: Satisfiability refers to the property of a logical formula or set of formulas being true under some interpretation or assignment of truth values to its variables. When a formula is satisfiable, it means there exists at least one model in which the formula evaluates to true, linking it closely to concepts like soundness, completeness, and other foundational principles in logic.
Smt solver: An SMT solver, or Satisfiability Modulo Theories solver, is a tool used in automated reasoning that determines the satisfiability of logical formulas with respect to certain theories. These solvers combine the capabilities of propositional satisfiability (SAT) solvers with support for various theories, such as arithmetic or arrays, enabling them to handle more complex problems that arise in verification and decision-making processes.
Soundness: Soundness is a property of a logical system that ensures if a statement can be proven within that system, then it is true in every model of the system. This concept guarantees that no false statements can be derived from the axioms and rules of inference, connecting the syntactic structure of proofs to their semantic meaning in models.
Unification: Unification is the process of determining a substitution that makes different logical expressions identical, allowing for the resolution of equations or formulas in various logical systems. It plays a crucial role in logic programming and automated theorem proving, as it enables the transformation and matching of terms, making it possible to derive conclusions from a set of premises. By finding a unifying substitution, systems can systematically resolve queries or prove theorems through structured reasoning.