Introduction

oCAS (Open Computer Algebra System) is a high-performance computer algebra system (CAS) written in Rust, released under the LGPL-3.0-or-later license. The project aims to match or exceed Symbolica and SageMath in core symbolic-computation performance, while providing multi-language interfaces through Python (PyO3) and C/C++ (cbindgen) bindings.

oCAS uses a layered crate architecture: 13 workspace members from the low-level arena allocator up to the top-level language bindings, with dependencies strictly pointing downward. All coefficient domains (arbitrary-precision integers/rationals, finite fields, real balls, complex numbers, double-precision floats) are abstracted behind a unified Domain trait, and the polynomial algebra, Gröbner basis, symbolic calculus, and ODE solving modules share the same expression infrastructure.

Version Feature Matrix

The table below lists the key features added in each oCAS release (0.1 → 0.27):

Module Architecture

oCAS consists of 13 crates with the following dependency graph (arrow A → B means A depends on B; ocas-py/ocas-c also depend on ocas-eval, ocas-rewrite and other core crates — only representative edges are shown; dashed edges are optional dependencies):

graph TD
    CORE["ocas-core<br/>Arena, Fuel, thread pool"]
    DOMAIN["ocas-domain<br/>coefficient domains, number theory, algebraic numbers"]
    POLY["ocas-poly<br/>polynomials, matrices, Gröbner bases"]
    ATOM["ocas-atom<br/>expression tree Atom, tensors"]
    PARSE["ocas-parse<br/>parser"]
    REWRITE["ocas-rewrite<br/>pattern matching, simplification"]
    CALC["ocas-calc<br/>calculus, ODE, integration"]
    EVAL["ocas-eval<br/>evaluation, JIT, numeric integration"]
    GPL["ocas-gpl<br/>GPL backend isolation"]
    TOP["ocas<br/>top-level crate + prelude"]
    PY["ocas-py<br/>Python bindings (PyO3)"]
    C["ocas-c<br/>C/C++ bindings (cbindgen)"]
    TESTS["ocas-tests<br/>correctness comparisons, pytest"]

    DOMAIN --> CORE
    ATOM --> CORE
    EVAL --> CORE
    EVAL --> DOMAIN
    EVAL --> ATOM
    PARSE --> ATOM
    REWRITE --> ATOM
    REWRITE --> PARSE
    CALC --> ATOM
    CALC --> DOMAIN
    CALC --> POLY
    CALC --> REWRITE
    POLY --> DOMAIN
    TOP --> DOMAIN
    TOP --> ATOM
    TOP --> POLY
    TOP --> CALC
    TOP --> EVAL
    TOP --> REWRITE
    TOP --> PARSE
    TOP --> CORE
    TESTS --> TOP
    TOP -.-> PY
    TOP -.-> GPL
    PY --> CALC
    C --> CALC

    style CORE fill:#e8f4fd,stroke:#2196F3
    style DOMAIN fill:#e8f4fd,stroke:#2196F3
    style POLY fill:#fff3e0,stroke:#FF9800
    style ATOM fill:#e8f5e9,stroke:#4CAF50
    style CALC fill:#fce4ec,stroke:#E91E63
    style EVAL fill:#f3e5f5,stroke:#9C27B0
    style TOP fill:#fffde7,stroke:#FDD835
    style PY fill:#efebe9,stroke:#795548
    style C fill:#efebe9,stroke:#795548

Crate overview:

Key Features

• 13 crates — layered Rust architecture from the arena runtime to language bindings, with dependencies strictly downward.
• 7 coefficient domains — arbitrary-precision integers/rationals, finite fields $\mathbb{F}_p$, real balls, complex numbers, double-precision floats (DoubleF64 ~31 digits), algebraic number fields $\mathbb{Q}(\alpha)$ / $\mathrm{GF}(p^d)$.
• Polynomial algebra — dense univariate/sparse multivariate polynomials, Karatsuba multiplication, GCD (Brown's modular GCD), multivariate GCD, factorization (Cantor–Zassenhaus + Hensel + Wang EEZ), algebraic number field factorization (Trager).
• Gröbner bases — three algorithms (Buchberger / F4 / F5), FGLM change of order, ideal operations (membership/sum/product/quotient/saturation/intersection/elimination/radical/primary decomposition), Hilbert series.
• Symbolic calculus — differentiation, Taylor expansion, layered integration pipeline (rational → Risch → trigonometric → special functions → heuristic → unevaluated form).
• ODE solving — first-order (separable/linear/Bernoulli/exact/homogeneous/integrating factors), second-order (constant coefficient/Cauchy–Euler/reduction of order/undetermined coefficients/variation of parameters), power-series and Frobenius solutions, Laplace IVP, 2×2 systems.
• Evaluation engine — stack VM interpreter, SIMD batch evaluation (f64x4), Cranelift JIT (f64/f32), streaming evaluation (constant memory).
• Automatic differentiation — forward mode with hyper-dual numbers HyperDual<T>, first- and higher-order partial derivatives.
• Tensor algebra — index slots, explicit contraction, McKay graph-isomorphism canonicalization (1-WL refinement + path invariant pruning), Young projectors.
• Numeric integration — adaptive Monte Carlo (Vegas), supporting high-dimensional definite integrals.
• Number theory — BPSW primality testing, integer factorization (trial division/Pollard rho/p−1/p+1/ECM), discrete logarithms (BSGS + Pohlig–Hellman), Euler φ / Möbius / divisor functions.
• Rewriting and simplification — AC backtracking pattern matching, wildcards (x__/x___), fixpoint simplification, fuel-limited simplification, optional egg e-graph equality saturation.
• Three-language bindings — Rust prelude API, Python (PyO3, 25 classes + 32 functions), C/C++ (cbindgen, 91 exports + RAII wrappers).
• Correctness framework — cross-validation against SymPy, SageMath, and Symbolica across multiple mathematical modules (1550 #[test]/#[tokio::test] annotations as of 0.28.0).
• Optional numeric backends — GMP/MPFR/FLINT behind feature flags, with GPL backends isolated in ocas-gpl.

Quick Navigation

Project Status

oCAS is currently at version 0.28.0. The core symbolic engine, polynomial algebra, Gröbner bases and algebraic geometry, ODE solving, JIT evaluation, three-language bindings, and the correctness comparison framework are feature-complete. The roadmap to a stable 1.0 can be found in the roadmap.