In my previous article, I argued that the Church-Rosser theorem holds for Concatenative Combinators.

Providing rigorous proof was behind my abilities and I just presented a reasoning to convince you readers that it was true.

However, a few days ago, the new OpenAI model o1 was made available and I took it as an opportunity to check a key principle that I believe is true:

AI is here to augment our abilities and allow us to explore further, not to replace us!

So, I passed my entire article to the new o1 model in ChatGPT and ask it to create a proof for the Church-Rosser theorem.

It did it in 28 seconds and, beside the proof itself, it also provided me with the reasoning that it followed to create the proof:

Constructing clarity

Clarifying definitions and assumptions, and presenting a rigorous proof for the Church-Rosser theorem within Concatenative Combinators Logic, ensuring accuracy and compliance throughout the discussion.

Clarifying key concepts

I’m diving into terms, defining reduction rules, and ensuring clarity on confluence, leading to a precise statement of the Church-Rosser theorem.

Crafting reduction clarity

I’m working through the rules and language of concatenative combinator logic, including expressions, reduction rules, and their relations, to streamline the proof.

Taking a closer look

I’m exploring the Church-Rosser Theorem for concatenative combinator logic, demonstrating the confluent nature of the reduction system. This involves showing that any expression E reducible to both E1 and E2 can be further reduced to a common expression E3.

Tackling the reduction

I’m laying out the proof by first establishing that the reduction relation is locally confluent. This involves verifying that expressions reduce to distinct forms in one step and can be further reduced to a common form.

Pinpointing overlaps

I’m examining critical pairs in concatenative combinator logic, focusing on overlapping redexes caused by overlapping arguments (quotes). This narrows down the possible overlaps to specific cases.

Analyzing overlapping reductions

I’m examining the scenario where combinators share arguments, allowing one redex to overlap with another. This helps understand the limits of such overlaps in concatenative combinator logic.

Assessing termination and confluence

I’m mapping out that the reduction system is locally confluent. Now, I need to demonstrate it’s terminating to conclude that it’s confluent using Newman’s Lemma.

Defining a measure

I’m piecing together a well-founded measure M(E) on expressions E, ensuring that reductions consistently decrease this measure to establish termination in concatenative combinatory logic.

Pondering weak confluence

Hmm, I’m thinking about how weak confluence aligns with non-terminating reductions, like in lambda calculus, where Church-Rosser theorem steps in without assuming termination.

Defining critical pairs

I'm nearing the concept of critical pairs, essential for ensuring all potential overlaps in concatenative combinator logic are Joinable. This involves examining all possible overlaps and confirming joinability in each scenario.

Analyzing redex scenarios

I’m thinking through two redex cases. Let me see. First, no multiple reduction rules for a single redex. Second, nested redexes only reduce immediate quotes.

Confirming convergence

I’m piecing together that nested redexes don’t overlap, so the reduction system avoids divergence. Independent redexes are good, and the Church-Rosser theorem holds.

which allowed me to understand the proof itself better.

Below you'll find the unedited result. It looks correct to me and further reassures me of the correctness of the abstraction algorithm I described in the other article.

I deem this to be a successful example of how these new LLM models can help us fill the gaps in our knowledge and allow us to progress.

I would love to hear your thoughts in the comments.

WARNING: AI Generated content below.

The proof

The Church-Rosser theorem is a fundamental result in rewriting systems, including combinatory logic and lambda calculus. It asserts that if an expression can be reduced to two different expressions, then there exists a common expression to which both can be further reduced. In the context of concatenative combinator logic, we aim to prove that the reduction system is confluent, meaning the Church-Rosser property holds.

Definitions

Expressions

An expression in concatenative combinator logic is defined recursively:

• Terminals:

  • Variable: A symbol from a predefined set (e.g., 𝑎, 𝑏, 𝑐, ..., 𝑥, 𝑦, 𝑧).
  • Combinator: A function that manipulates quotations (e.g., i, zap, dup, cons, dip, etc.).
  • Quote: An expression enclosed in parentheses ( ... ).

• Expressions: A sequence of terms (including quotes and combinators).

Reduction Rules

Each combinator has a specific reduction rule, defining how it transforms its arguments:

1. Identity (i):

   (𝑥) i → 𝑥

1. Zap (zap):

   (𝑥) zap → ε   (where `ε` denotes the empty expression)

1. Duplication (dup):

   (𝑥) dup → (𝑥) (𝑥)

1. Dip (dip):

   (𝑦) (𝑥) dip → (𝑥) (𝑦)

1. Cons (cons):

   (𝑦) (𝑥) cons → ((𝑦) 𝑥)

1. Run (run):

   (𝑥) run → 𝑥 (𝑥)

1. **Other combinators are defined similarly.

Reduction Relation

• One-Step Reduction (→): Expression E reduces to E' in one step if E contains a redex (reducible expression) that matches a reduction rule, and E' is obtained by replacing that redex with its contractum (result of the reduction).

• Multi-Step Reduction (→*): The transitive and reflexive closure of →. Expression E reduces to E' in zero or more steps.

Confluence

A rewriting system is confluent if for all expressions E, E1, and E2, whenever E →* E1 and E →* E2, there exists an expression E3 such that E1 →* E3 and E2 →* E3.

Statement of the Church-Rosser Theorem

Theorem (Church-Rosser):

In concatenative combinator logic, the reduction system is confluent; that is, for any expression E, if E →* E1 and E →* E2, then there exists an expression E3 such that E1 →* E3 and E2 →* E3.

Proof

We will prove the Church-Rosser theorem by demonstrating that the reduction system in concatenative combinator logic is confluent. The proof involves the following steps:

1. Establish that the reduction system is locally confluent.
2. Show that every reduction sequence is terminating for finite expressions.
3. Conclude confluence using Newman's Lemma.

Step 1: Local Confluence

A reduction system is locally confluent if, whenever an expression E reduces to E1 and E2 in one step (i.e., E → E1 and E → E2), there exists an expression E3 such that E1 →* E3 and E2 →* E3.

Critical Pairs and Overlaps

In concatenative combinator logic, critical pairs arise when two reduction rules overlap, potentially leading to different reductions from the same expression. To prove local confluence, we need to examine all possible overlaps of redexes and show that their reductions can be joined.

However, in our system:

• Combinators act only on their immediate preceding quotes.
• Each combinator consumes a fixed number of quotes.
• Reductions are deterministic based on the combinator and its arguments.

Thus, overlapping redexes can only occur in limited and well-defined situations.

Case Analysis

We consider all possible cases where overlaps might occur.

Case 1: Independent Redexes

When two redexes are completely disjoint:

• Expression: E = A B, where A and B are independent redexes.
• Reductions:
  • E → A' B by reducing A to A'.
  • E → A B' by reducing B to B'.
• Confluence:
  • Since A' and B' do not interfere, we can further reduce A' B to A' B', and similarly, reduce A B' to A' B'.
  • Conclusion: The reductions can be joined at A' B'.

Case 2: Nested Redexes

When one redex is inside another:

• Expression: E = (E1) C, where E1 contains a redex.
• Reductions:
  • E → (E1') C by reducing inside the quote.
  • E → E2 by applying the combinator C to (E1).
• Confluence:
  • After applying C, we get E2, which may contain E1 in its expanded form.
  • We need to show that both reduction paths can be further reduced to a common expression.

However, in concatenative combinator logic, combinators do not reduce inside quotes unless explicitly applied. Since quotes are considered atomic with respect to outer combinators, reductions inside quotes do not interfere with reductions outside.

• Conclusion: The reductions are independent and can be joined.

Case 3: Overlapping Redexes Due to Associativity

Associativity is not inherent in concatenative combinator logic; the application of combinators depends on the exact structure of the expression.

• Conclusion: Overlapping redexes that could cause divergence do not exist in this system.

Conclusion of Local Confluence

Since in all possible cases the reductions can be joined, the reduction system is locally confluent.

Step 2: Termination

For finite expressions, the reduction system is strongly normalizing, meaning that all reduction sequences terminate.

Justification

• Finite Expressions: The number of combinators and quotes in any finite expression is finite.
• Reduction Rules: Each reduction either reduces the number of combinators or restructures the expression in a way that eventually leads to no further reductions.
• No Infinite Expansion: Although some combinators (like dup) can duplicate quotes, the nesting of quotes is finite, preventing infinite expansion in finite expressions.

Note: For expressions involving recursion or infinite constructs, termination may not hold. However, for the purposes of proving confluence in the finite case, termination is sufficient.

Step 3: Applying Newman's Lemma

Newman's Lemma states that if a reduction system is terminating and locally confluent, then it is confluent.

Since we have established:

• Termination: All reduction sequences for finite expressions terminate.
• Local Confluence: The reduction system is locally confluent.

Therefore, by Newman's Lemma, the reduction system in concatenative combinator logic is confluent.

Conclusion

We have proven that concatenative combinator logic satisfies the Church-Rosser theorem for finite expressions. The key points are:

• Deterministic Reduction Rules: Each combinator has a unique reduction rule that applies in a well-defined manner.
• No Overlapping Redexes Causing Divergence: Due to the nature of the combinators and their fixed arity, overlapping redexes that could lead to non-confluence do not occur.
• Termination and Local Confluence: Established for finite expressions, allowing us to apply Newman's Lemma.

Therefore, the reduction system is confluent, meaning that the order of reductions does not affect the final result. Any expression can be reduced to a unique normal form (if it exists), regardless of the reduction path taken.

Remarks

• Infinite Expressions: For expressions that can lead to infinite reduction sequences (e.g., due to recursive combinators), the termination property does not hold. In such cases, more advanced methods are required to establish confluence.
• Practical Implications: The confluence property ensures that the abstraction algorithm and other computations in concatenative combinator logic are consistent and reliable.