Reversible Meta-Synthesis

Overview

This project explores program synthesis through the lens of reversible computation, using meta-circular interpreters that can run both forward and backward to induce program structure from input-output examples.

Theoretical Foundation

Reversible Computation

Every computation step must be invertible, preserving information throughout execution. This constraint enables powerful reasoning about program behavior:

;; Reversible cons operation
(define (rcons x y history)
  (values (cons x y) 
          (cons (list 'rcons x y) history)))

;; Its inverse
(define (runcons pair history)
  (values (car pair) 
          (cdr pair)
          (cdr history)))

Meta-Circular Interpretation

The interpreter itself is written in the language it interprets, enabling: - Self-modification during synthesis - Reflection on program structure - Bidirectional execution

Synthesis Algorithm

1. Forward Execution: Run candidate programs on inputs
2. Backward Execution: From outputs, reverse to find programs
3. Meet in the Middle: Combine forward and backward search
4. Program Induction: Extract minimal programs from execution traces

Key Innovations

Reversible Pattern Matching

(define-reversible (append xs ys)
  [(null? xs) ys]
  [(cons? xs) (cons (car xs) 
                    (append (cdr xs) ys))])

Execution Trace Analysis

• Automatic extraction of recursive patterns
• Identification of base cases and recursive cases
• Minimization of synthesized programs

Applications

Educational Tool

• Visualize program synthesis process
• Understand the relationship between examples and programs
• Explore the limits of inductive reasoning

Research Platform

• Test synthesis algorithms
• Explore reversible programming paradigms
• Investigate connections to category theory

Results

The system successfully synthesizes: - List manipulation functions (map, filter, fold) - Tree traversal algorithms - Simple arithmetic functions - Pattern-based string operations

Future Work

• Integration with neural program synthesis
• Probabilistic reversible computing
• Application to quantum computing
• Synthesis of concurrent programs

This work demonstrates that reversibility constraints, rather than limiting expressiveness, can actually guide the synthesis process toward more elegant and understandable programs.