Syzygy Language

Algebraic Programming for Mathematical Computation

Syzygy is a domain-specific language that treats programs as systems of algebraic constraints. It's designed for mathematicians, researchers, and developers working with abstract algebra, category theory, and mathematical structures.

Why Syzygy?

Traditional programming languages struggle with mathematical abstractions. Syzygy bridges this gap by providing native support for:

• Ring Theory - Finite fields, rational numbers, modular arithmetic
• Module Theory - Free modules, generators, algebraic relations
• Category Theory - Morphisms, limits, universal properties
• Constraint Solving - Algebraic relation resolution

Quick Example

// Define algebraic structures
ring Z5 = integers_mod 5
module V = free_module(Z5, 3)

// Declare generators and relations
generators {
    v1 = (1, 0, 0) in V
    v2 = (0, 1, 0) in V
}

relations {
    2*v1 + 3*v2 == 0
}

// Functional programming with pattern matching
define factorial as n .
    case n of {
        0 -> 1;
        _ -> n * factorial(n - 1);
    }

Key Innovations

1. Algebraic-First Design

// Rings are first-class citizens
ring Z7 = integers_mod 7
ring Q = rationals

// Modules over rings
module VectorSpace = free_module(Q, 3)
module MatrixAlgebra = free_module(Z7, 9)

2. Constraint-Based Programming

relations {
    3*x + 2*y == z                    // Linear relations
    kernel(f) == image(g)             // Exact sequences  
    composition(m1, m2) == identity   // Morphism properties
}

3. Mathematical Pattern Matching

case matrix of {
    ((a, b), (c, d)) -> a*d - b*c     // Determinant
    _ -> 0
}

Compiler Architecture

Lexer → Parser → Algebraic Constraint Solver → Code Generator

Built with safety in mind:

• Memory-safe C implementation
• Bounds checking on all operations
• Comprehensive error reporting
• Modular, extensible design

Use Cases

Research Mathematics

• Formal verification of algebraic proofs
• Experimentation with mathematical structures
• Educational tool for abstract algebra

Scientific Computing

• Cryptographic protocol analysis
• Computational algebraic geometry
• Linear algebra over finite fields

Education

• Teaching abstract algebra concepts
• Visualizing mathematical structures
• Interactive mathematical exploration

What Makes Syzygy Unique?

Not Just Another DSL

While other mathematical languages focus on theorem proving or numerical computation, Syzygy specializes in algebraic structure manipulation and constraint resolution.

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Join us in redefining how we compute with mathematical structures!

"Where abstract algebra meets practical computation"