Glossary

5.2 Proof of cut elimination for propositional logic

3 min read•august 7, 2024

is a crucial concept in proof theory, showing that any proof using the can be transformed into a cut-free proof. This process simplifies proofs, making them more direct and easier to analyze.

The proof of cut elimination for propositional logic involves intricate induction principles and transformation steps. By gradually reducing the complexity of cuts, we can ultimately eliminate all cuts, resulting in a more streamlined and constructive proof structure.

Sequent Calculus and Cut Rule

Overview of Sequent Calculus

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  • is a formal system for logical proofs
  • Consists of rules for introducing and eliminating logical connectives
  • Sequents are expressions of the form Γ⊢Δ\Gamma \vdash \Delta, where Γ\Gamma and Δ\Delta are sets of formulas
  • Intuitive meaning: if all formulas in Γ\Gamma are true, then at least one formula in Δ\Delta is true

The Cut Rule and Its Significance

  • Cut rule allows the use of lemmas in proofs
  • Stated as: if Γ⊢Δ,A\Gamma \vdash \Delta, A and Γ,A⊢Δ\Gamma, A \vdash \Delta, then Γ⊢Δ\Gamma \vdash \Delta
  • Principal formula is the formula AA that is "cut out" in the application of the cut rule
  • Cut elimination theorem states that any proof with cuts can be transformed into a cut-free proof
  • are more direct and avoid detours through lemmas

Properties of Cut-Free Proofs

  • Cut-free proofs have the : every formula in the proof is a subformula of the conclusion
  • Cut-free proofs are more constructive and computationally meaningful
  • Cut elimination is a key tool for proving and normalization results

Induction Principles

Induction on Cut-Rank

  • Cut-rank is the complexity of the principal formula in a cut rule application
  • proceeds by showing that cuts on complex formulas can be reduced to cuts on simpler formulas
  • Base case: cuts on atomic formulas can be eliminated directly
  • Inductive step: show how to reduce cuts on A∧BA \land B, A∨BA \lor B, A→BA \to B, etc. to cuts on AA and BB

Induction on Cut-Height

  • Cut-height is the maximum number of cut rule applications along any branch in a proof tree
  • proceeds by showing that proofs with many cuts can be transformed into proofs with fewer cuts
  • Base case: proofs with cut-height 0 are already cut-free
  • Inductive step: show how to reduce the cut-height by eliminating topmost cuts or pushing cuts upwards in the proof tree

Combining Induction Principles

  • Cut elimination proofs often use a double induction on cut-rank and cut-height
  • Outer induction on cut-rank, with inner induction on cut-height for each rank
  • This allows for a gradual simplification of the cuts in a proof until all cuts are eliminated

Transformation Steps

Weakening and Contraction

  • allow formulas to be added to the left or right side of a sequent
  • allow duplicate formulas to be removed from the left or right side of a sequent
  • These rules are often used in cut elimination to adjust the contexts of sequents as needed

Reduction and Permutation Steps

  • are used to reduce the cut-rank by eliminating cuts on complex formulas
  • Example: a cut on A∧BA \land B can be reduced to two cuts on AA and BB
  • are used to push cuts upwards in the proof tree, reducing the cut-height
  • Example: a cut can be permuted above a rule that does not affect the principal formula of the cut

Elimination of Atomic Cuts

  • are cuts on formulas that cannot be further decomposed (propositional variables)
  • These cuts can be eliminated directly by a simple transformation
  • After all non-atomic cuts are eliminated, only atomic cuts remain
  • The final step of cut elimination removes all atomic cuts, resulting in a cut-free proof

Key Terms to Review (21)

Atomic Cuts: Atomic cuts refer to specific types of cut-free proofs in proof theory, where a proof can be split into distinct segments, or cuts, that can be individually evaluated without loss of validity. This concept is important for understanding how certain logical arguments can be simplified and manipulated while still maintaining their correctness. Atomic cuts play a crucial role in the cut elimination process, particularly within propositional logic, by allowing for clearer structures in proofs.
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.
Conjunction: In logic, a conjunction is a compound statement formed by connecting two or more propositions using the logical connective 'and', symbolized as $$\land$$. This operation is fundamental as it defines how multiple statements can be combined to yield a true or false value based on their individual truth values.
Consistency: Consistency refers to the property of a formal system in which it is impossible to derive both a statement and its negation from the system's axioms and inference rules. This ensures that the system does not produce contradictions, making it a crucial aspect of logical frameworks and proof theory.
Contraction Rules: Contraction rules are a set of inference rules in proof theory that allow for the simplification of formulas by eliminating duplicate occurrences of a variable or proposition. These rules are crucial in systems of logic, especially in the context of cut elimination, as they help streamline proofs by reducing complexity without changing the overall meaning or truth of the statements involved.
Cut Elimination: Cut elimination is a fundamental process in proof theory that aims to simplify proofs by removing 'cut' rules, which allow for the introduction of intermediate assertions not originally present in the premises. This concept is crucial for understanding the efficiency and structure of proofs, as it transforms proofs into a more direct form, aligning them closer to intuitionistic principles and providing a clearer representation of derivations.
Cut Rule: The cut rule is a principle in proof theory that allows for the introduction of intermediate formulas in the derivation process, enabling more complex proofs by breaking them down into simpler components. This rule highlights the difference between natural deduction and sequent calculus, as it can be seen as a mechanism for managing how assumptions are applied and discharged within a proof. Understanding the cut rule is essential for grasping the structure of proofs in both propositional and first-order logic.
Cut-free proofs: Cut-free proofs are a type of formal proof in logic that does not utilize the cut rule, which allows for the introduction of intermediate formulas. These proofs are important because they simplify the reasoning process and provide a more direct path from axioms to conclusions, which is crucial in understanding logical systems. The elimination of cuts leads to proofs that are more structured and easier to analyze, and they play a vital role in establishing the consistency and completeness of logical frameworks.
Disjunction: Disjunction is a logical operation that combines two propositions with the word 'or', resulting in a statement that is true if at least one of the propositions is true. This operation plays a key role in formal systems, as it helps construct complex statements, define truth conditions, and facilitate proof strategies.
Gerhard Gentzen: Gerhard Gentzen was a German mathematician and logician known for his groundbreaking contributions to proof theory, particularly in developing natural deduction and sequent calculus. His work laid the foundation for many modern concepts in logic, impacting various aspects of mathematical logic, including soundness, completeness, and proof systems.
Induction on cut-height: Induction on cut-height is a proof technique used in the context of cut elimination for propositional logic, where the complexity of a proof is measured by the height of the cut formulas involved. This method involves proving properties of proofs by showing that if a certain property holds for all proofs with a height less than a certain level, it must also hold for proofs of greater height, allowing for a systematic reduction of proofs to simpler forms without cuts.
Induction on cut-rank: Induction on cut-rank is a technique used in proof theory, particularly to demonstrate the cut elimination theorem for propositional logic. This method involves analyzing proofs based on the complexity of their cuts, measured by cut-rank, which refers to the depth of cuts in a proof tree. By systematically reducing the complexity of these cuts, one can show that any proof with cuts can be transformed into a cut-free proof.
Modus Ponens: Modus Ponens is a fundamental rule of inference in propositional logic that allows one to derive a conclusion from a conditional statement and its antecedent. It asserts that if we have a statement of the form 'If P, then Q' (P → Q) and we know that P is true, we can validly conclude that Q is also true. This principle is crucial for constructing proofs and understanding the flow of logical arguments.
Modus Tollens: Modus tollens is a fundamental rule of inference in propositional logic, stating that if a conditional statement is true and its consequent is false, then the antecedent must also be false. This form of reasoning allows one to draw valid conclusions by denying the consequent of a conditional statement, highlighting its significance in logical deduction and proof systems.
Normal Form: Normal form refers to a standardized representation of a mathematical expression or logical proof that adheres to specific rules or criteria. This concept is crucial as it facilitates simplification, comparison, and analysis of proofs and expressions across various logical systems. Achieving normal form can streamline the process of proving theorems and ensuring consistency in logical deductions.
Permutation steps: Permutation steps refer to the specific transformations applied in a proof system that rearrange or substitute elements in a logical formula while preserving its validity. These steps are crucial for simplifying complex proofs and are often employed during the cut elimination process, ensuring that the logical structure remains intact even as components are reordered.
Reduction: Reduction refers to the process of transforming one proof into another proof, often simplifying the argument while maintaining its validity. This concept is crucial in establishing the relationships between different logical systems and understanding how proofs can be simplified or manipulated without altering their outcomes.
Reduction steps: Reduction steps are the individual actions or operations that simplify a proof in a logical system, aiming to bring it closer to a more canonical or normal form. These steps play a crucial role in proof normalization and cut elimination, as they provide the mechanisms through which complex proofs are transformed into simpler ones, making the logical structure clearer and easier to understand.
Sequent Calculus: Sequent calculus is a formal proof system designed for deriving sequents, which express the validity of implications between formulas in logic. It serves as a key framework for studying proof theory, enabling structured reasoning about logical statements and their relationships, particularly in first-order logic and intuitionistic logic.
Subformula Property: The subformula property is a concept in proof theory that asserts every formula that appears in a proof must be a subformula of the initial assumptions or the conclusion. This property highlights how proofs are constructed in a way that maintains the integrity of logical deductions, ensuring that no extraneous formulas are introduced. This characteristic becomes crucial when comparing different proof systems and understanding how they manage the components of proofs, particularly in relation to cut elimination and its implications for propositional logic.
Weakening rules: Weakening rules are inference rules in proof theory that allow for the introduction of additional assumptions or premises into a proof without affecting its validity. This means that if a statement can be proven from a certain set of premises, it can still be proven even if additional premises are added. Weakening rules are crucial for maintaining flexibility in proofs, especially during cut elimination processes.
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