1.3 Resolution and theorem proving
4 min read•august 15, 2024
and theorem proving are powerful tools in . They allow us to determine if a set of clauses is satisfiable or if a conclusion logically follows from given premises. This process involves applying the resolution rule to derive new clauses.
The resolution rule combines two clauses with to create a new . By repeatedly applying this rule, we can construct resolution proofs to check or prove theorems. This method forms the basis for automated theorem proving algorithms.
Resolution Rule for Clause Derivation
Applying the Resolution Rule
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- The resolution rule is a single inference rule that produces a new clause (resolvent) from two clauses (parent clauses) containing complementary literals
- Complementary literals are a literal L and its ¬L, which must appear in separate parent clauses for the resolution rule to apply
- The resolution rule combines the remaining literals from both parent clauses, excluding the complementary literals, to form the
- The resolvent clause is a logical consequence of the parent clauses, meaning if the parent clauses are true, the resolvent must also be true
Properties of the Resolution Rule
- The resolution rule is applied to propositional logic clauses, where each clause is a disjunction of literals (propositional symbols or their negations)
- The resolution rule is sound, as it preserves the satisfiability of the original clauses
- If the parent clauses are satisfiable, the resolvent clause will also be satisfiable
- The resolution rule is complete for propositional logic, meaning if a clause is a logical consequence of a set of clauses, it can be derived using the resolution rule
- The resolution rule is the basis for resolution-based theorem proving algorithms, which automate the process of deriving new clauses from a given set of clauses
Resolution Proofs for Satisfiability
Constructing Resolution Proofs
- A is a sequence of applications of the resolution rule to derive new clauses from a given set of clauses until a specific goal is reached
- The goal can be deriving the empty clause (a clause with no literals) or proving the unsatisfiability of the clause set
- To determine the satisfiability of a set of clauses using resolution, the negation of the desired conclusion is added to the clause set
- Resolution is then applied to derive new clauses from the extended clause set
- Resolution proofs can be represented as a graph or tree structure
- Each node represents a clause, and edges connect clauses that have been resolved to produce a new resolvent clause
Interpreting Resolution Proofs
- If the empty clause is derived during the resolution process, it indicates that the original set of clauses, together with the negated conclusion, is unsatisfiable
- This means that the original set of clauses logically entails the desired conclusion
- If the resolution process exhausts all possible applications of the resolution rule without deriving the empty clause, it suggests that the original set of clauses is satisfiable
- In this case, the original set of clauses does not logically entail the negated conclusion
- Resolution proofs provide a systematic way to determine the satisfiability of a set of clauses and to establish logical entailment between clauses
Unification in Resolution
Role of Unification
- is a key concept in resolution, where clauses contain variables and functions in addition to propositional symbols
- Unification is the process of finding a substitution (a mapping from variables to terms) that makes two logical expressions identical
- In the context of resolution, unification is used to determine if two clauses can be resolved by finding a substitution that makes the complementary literals in the clauses identical
Unification Algorithm
- The takes two expressions (literals or terms) as input and returns a substitution that makes the expressions identical, if such a substitution exists
- If no unifying substitution exists, the algorithm indicates that unification has failed
- The (MGU) is a substitution that provides the most general solution for unifying two expressions
- It introduces the fewest constraints on the variables involved
- Unification allows resolution to be applied to first-order logic clauses by finding appropriate substitutions for variables
- This enables the derivation of new clauses that are logically entailed by the original clauses
Resolution-Based Theorem Proving Algorithms
Basic Steps
- Resolution-based theorem proving algorithms automate the process of applying the resolution rule to a set of clauses to determine satisfiability or prove a desired conclusion
- The basic steps of a include:
- Conversion of the given logical formulas into a set of clauses in (CNF)
- Application of the resolution rule to pairs of clauses in the set, generating new resolvent clauses
- Addition of the resolvent clauses to the clause set
- Repetition of steps 2 and 3 until the empty clause is derived (indicating unsatisfiability) or until no more new clauses can be generated (suggesting satisfiability)
Efficiency and Optimizations
- Efficient implementation of resolution-based theorem proving algorithms requires strategies for selecting clauses to resolve
- prioritizes resolving clauses containing a single literal (unit clauses)
- focuses on resolving clauses related to the negated conclusion
- Optimizations and heuristics can be employed to improve the performance of resolution-based theorem provers
- eliminates redundant clauses that are subsumed by more general clauses
- removes clauses that are always true (tautologies)
- Ordering strategies determine the order in which literals are selected for resolution
- Resolution-based theorem provers have been successfully applied to various domains
- Mathematics, formal verification, and artificial intelligence
- They automatically prove theorems and check the consistency of logical theories
Key Terms to Review (22)
Alan Turing: Alan Turing was a British mathematician, logician, and computer scientist who is widely considered the father of modern computing and artificial intelligence. His work laid the foundation for algorithmic processes and computational theory, making significant contributions to logic, including the development of the Turing machine, which is a fundamental concept in the study of computation and theorem proving.
Clause: A clause is a disjunction of literals that represents a logical statement in propositional logic. In the context of formal logic, clauses are critical components used to express logical formulas in a structured way. They can be used in various normal forms, like conjunctive and disjunctive, and play a vital role in resolution and theorem proving, where clauses are manipulated to derive conclusions from premises.
Complementary literals: Complementary literals are pairs of propositional variables where one is the negation of the other. For example, if 'A' is a literal, then its complementary literal is '¬A'. These pairs are crucial in understanding logical resolution because they can be used to simplify logical expressions and prove the unsatisfiability of a set of clauses. Recognizing complementary literals helps in both theorem proving and the unification process during resolution algorithms, allowing for effective deductions in formal logic.
Completeness: Completeness is a property of a logical system that indicates every statement that is semantically true can also be proved syntactically within that system. This concept ensures that if something is true in all models of a theory, there exists a formal proof for it, linking semantics and syntax. Completeness is vital when analyzing how theories are structured and verified, providing a foundation for understanding proofs and logical deductions.
Conjunctive Normal Form: Conjunctive Normal Form (CNF) is a way of structuring logical expressions where a formula is represented as a conjunction of one or more disjunctions of literals. This format is important because it helps in simplifying complex logical expressions and makes them easier to process, especially when applying resolution techniques in proofs and theorem proving. CNF is also closely linked with other normal forms, including disjunctive normal form, and is foundational in converting formulas into simpler equivalents for automated reasoning.
First-order logic: First-order logic (FOL) is a formal system used in mathematics, philosophy, linguistics, and computer science that allows for the expression of statements about objects and their properties using quantifiers, variables, and predicates. It extends propositional logic by enabling the representation of relationships between objects and can express more complex statements involving variables and quantification.
John Robinson: John Robinson was a prominent figure in the development of resolution and theorem proving in formal logic, known for his significant contributions to the field. His work primarily focused on advancing automated reasoning techniques, particularly through the introduction of resolution methods, which became foundational in logical inference systems. His ideas on resolving logical propositions helped shape how problems are approached in computational logic and artificial intelligence.
Most General Unifier: The most general unifier (MGU) is a substitution that makes two expressions identical, while being the most general among all possible substitutions. This means that an MGU can be applied to a broader range of cases than any other unifier, essentially capturing the essential structure of the expressions involved. MGUs are crucial in processes like resolution and theorem proving, substitution mechanisms, and unification algorithms, enabling the systematic derivation of conclusions from logical statements.
Negation: Negation is a fundamental logical operation that transforms a proposition into its opposite truth value. It plays a critical role in understanding and manipulating logical statements, allowing us to express denial or contradiction of a proposition, thus impacting the interpretation of logical expressions and arguments.
Propositional Logic: Propositional logic is a branch of logic that deals with propositions, which are statements that can be either true or false. This area of logic focuses on the relationships between these propositions and how they can be combined using logical connectives such as 'and', 'or', 'not', and 'if...then'. Understanding propositional logic is essential for various processes like resolution and theorem proving, as well as for establishing the foundational principles in artificial intelligence and computer science.
Resolution: Resolution is a rule of inference used in formal logic and automated theorem proving to derive conclusions from a set of premises. It plays a crucial role in simplifying logical expressions, particularly in conjunctive and disjunctive normal forms, and is essential for effectively proving theorems in first-order logic through systematic deductions.
Resolution Proof: A resolution proof is a formal method used in propositional and predicate logic to derive conclusions from a set of premises through a process called resolution. This technique involves refuting a statement by showing that it leads to a contradiction, and it is particularly useful for automated theorem proving, where logical statements are manipulated to achieve a desired outcome.
Resolution-based theorem proving algorithm: A resolution-based theorem proving algorithm is a method for automated reasoning in logic that uses the principle of resolution to derive conclusions from a set of premises. This algorithm works by transforming logical statements into a specific form, typically conjunctive normal form (CNF), and then applying resolution rules to identify contradictions or valid conclusions, thereby proving or disproving theorems systematically.
Resolvent Clause: A resolvent clause is a key component in the resolution method of theorem proving, which is a rule used in propositional logic and first-order logic to derive conclusions from a set of clauses. It results from the process of unifying two clauses that contain complementary literals, thus producing a new clause that is a logical consequence of the original clauses. This new clause plays a critical role in the systematic exploration of logical entailments, allowing for the efficient resolution of logical propositions.
Satisfiability: Satisfiability refers to the property of a logical formula whereby there exists an assignment of truth values to its variables that makes the formula true. This concept is essential in understanding various logical systems, as it helps determine whether certain statements can be made true under specific interpretations and conditions.
Set of Support Strategy: A set of support strategy is a technique used in resolution and theorem proving that focuses on a specific subset of clauses or formulas, which serve as the basis for generating new conclusions or proofs. This approach allows for more efficient problem-solving by limiting the scope of consideration to relevant information, thereby enhancing the effectiveness of the resolution process and improving the chances of successfully finding a proof.
Soundness: Soundness refers to a property of logical systems whereby if a statement can be proven within the system, it is also true in the intended interpretation or model. This ensures that the system does not allow for false conclusions to be derived from true premises, reinforcing the reliability of the logic used.
Subsumption: Subsumption is a method in logic and automated reasoning where one statement or clause is considered to be more general than another, effectively absorbing or including it within its scope. This concept is vital for optimizing proof search in resolution-based systems by reducing the number of clauses that need to be considered, enhancing efficiency and performance in automated theorem proving.
Tautology Elimination: Tautology elimination is a logical principle that allows the removal of tautological expressions from logical formulas without affecting their truth values. This process is crucial in streamlining arguments and simplifying proofs, making it easier to derive conclusions. By identifying tautologies, which are statements that are always true regardless of their variables, one can enhance clarity and efficiency in both resolution and theorem proving.
Unification: Unification is the process of making two or more logical expressions identical by finding a substitution for their variables. This concept is crucial in formal logic, particularly in first-order logic, as it allows for the resolution of statements by transforming them into a common form that can be easily compared and resolved.
Unification Algorithm: The unification algorithm is a systematic method used in logic and computer science to determine whether two logical expressions can be made identical through variable substitutions. It plays a critical role in automated theorem proving and resolution, allowing for the simplification of logical expressions and facilitating the process of finding proofs. This algorithm is essential for effective substitution of variables, ensuring that differing representations of knowledge can be reconciled in various contexts, such as logic programming and artificial intelligence.
Unit Preference Strategy: Unit preference strategy is a method in resolution and theorem proving where the resolution process prioritizes unit clauses—clauses that contain only a single literal. This approach is beneficial because unit clauses can simplify the search space and lead to quicker derivations, making it easier to derive conclusions from a set of premises. By focusing on unit clauses, the process efficiently narrows down potential resolutions, enhancing the overall effectiveness of the proving strategy.