alkahest 2.4.0

A high-performance computer algebra system for Python

Alkahest

A high-performance computer algebra system for Python built for both humans and agents. Symbolic operations run orders of magnitude faster than SymPy and can run on modern accelerated hardware. Every computation produces a derivation log; a meaningful subset can export Lean 4 proofs for independent verification.

Install: the package is published on PyPI; use pip install alkahest (Python 3.9–3.13). See Install below for optional +jit / +full Linux wheels (GitHub Releases or a future extras index) and building from source.

Demo: try the hosted playground (WASM in-browser, or bring your own server/Jupyter URL + token), or run demo-playground/ locally for the full agent and recording stack. See demo-playground/README.md.

Stack: Rust kernel → FLINT/Arb (polynomials, ball arithmetic) → vendored egglog + colored e-graphs (simplification) → Cranelift/LLVM JIT + MLIR (native and GPU codegen) → PyO3 → Python

---

Install

Requirements: Python 3.9–3.13 (PyPI requires-python).

pip install alkahest

For an isolated environment (recommended when juggling versions or building from source):

python3 -m venv .venv && source .venv/bin/activate   # Windows: .venv\Scripts\activate
python -m pip install -U pip
pip install alkahest

Default PyPI wheels include the vendored egglog e-graph backend (egraph feature) and the Gröbner solver (groebner feature — so alkahest.solve, Diophantine, homotopy, and related APIs are available out of the box) but not LLVM JIT, Cranelift, or parallel. Numeric APIs use the tree-walking interpreter fallback. For native LLVM CPU JIT—or JIT plus parallel F4—use a PyTorch-style opt-in wheel (separate artifact / index), not the default PyPI resolver path. From source, add --features cranelift for a pure-Rust fast-compile JIT tier without system LLVM.

Opt-in Linux wheels: +jit and +full (PyTorch-style)

Why a separate index or direct wheel URL: feature-heavy wheels use a PEP 440 local version (for example 2.0.3+jit or 2.0.3+full). Those builds must not be mixed into the main PyPI project’s simple API for the same reason PyTorch publishes CUDA wheels on download.pytorch.org: otherwise pip install alkahest could resolve a +jit / +full build as “newer” than 2.0.3 and pull LLVM (or a much larger binary) when you wanted the default wheel.

There is no pip install alkahest[jit] / alkahest[full] that swaps the native extension: pip extras only add Python dependencies, not alternate binaries for the same wheel slot.

Until a dedicated PEP 503 simple index is published, tagged releases attach Linux linux_x86_64 wheels on GitHub Releases (CI builds them on ubuntu-22.04, not the manylinux image used for default wheels). Pick the .whl whose tags match your Python (cp311, etc.) and linux_x86_64.

Local version	Cargo features	When to use
+jit	egraph groebner jit	LLVM CPU JIT (smaller than +full; groebner/egraph are already in default wheels).
+full	egraph groebner jit parallel	JIT plus parallel F4 S-polynomial reduction (largest wheel; groebner already in default).

Direct-install examples (adjust tag and filename after checking the release assets):

pip install "https://github.com/alkahest-cas/alkahest/releases/download/v2.3.1/alkahest-2.3.1+full-cp311-cp311-linux_x86_64.whl"
pip install "https://github.com/alkahest-cas/alkahest/releases/download/v2.3.1/alkahest-2.3.1+jit-cp311-cp311-linux_x86_64.whl"

These wheels vendor LLVM (for JIT) and related .so files under site-packages/alkahest.libs/. If import alkahest fails with a missing libffi-*.so or libLLVM-*.so, prepend that directory to LD_LIBRARY_PATH (or install matching system packages). Release CI uses the same LD_LIBRARY_PATH step when smoke-testing wheels.

If your client chokes on + in the URL, use percent-encoding (2.3.1%2Bfull in the filename segment).

After installing +jit or +full, alkahest.jit_is_available() should be True. Gröbner-backed APIs such as alkahest.solve are available in all wheels (including the default PyPI wheel) since groebner became a default feature.

macOS and Windows +jit / +full wheels are not produced in CI yet (LLVM / MSYS2 constraints); use building from source there.

Target layout (roadmap): a small extra index URL (PEP 503) hosting only +jit / +full wheels, mirroring PyTorch’s --extra-index-url workflow:

pip install 'alkahest==2.0.3+full' --extra-index-url https://EXAMPLE/alkahest-extras/simple

From source

Required to enable optional features (jit, cuda, parallel) or for development. The groebner and egraph features are already built into default wheels; a source build inherits them automatically. Prerequisites:

• Rust stable ≥ 1.76 and nightly:

  curl --proto '=https' --tlsv1.2 -sSf https://sh.rustup.rs | sh
  rustup toolchain install nightly
  

• uv (recommended Python tool manager): curl -LsSf https://astral.sh/uv/install.sh | sh
• LLVM 15: apt install llvm-15 libllvm15 llvm-15-dev / brew install llvm@15
• FLINT ≥ 2.9 (includes GMP and MPFR): apt install libflint-dev / brew install flint

# Install dev tools (maturin, pytest, ruff, ty, …) without building the Rust extension:
uv sync --no-install-project --group dev
# Build and install the extension into the project venv:
uv run maturin develop --manifest-path alkahest-py/Cargo.toml --release --features "parallel egraph jit groebner"

Without uv, install maturin directly and run the same develop command:

pip install maturin
maturin develop --manifest-path alkahest-py/Cargo.toml --release --features "parallel egraph jit groebner"

Optional Cargo features: parallel (sharded pool + parallel F4 + numpy_eval_par), egraph (vendored egglog backend; default in PyPI wheels), groebner (Gröbner solver + Diophantine + homotopy; default in both the Rust crate and PyPI wheels), cranelift (pure-Rust Tier-1 JIT), jit (LLVM JIT), cuda (NVPTX codegen).

Rust crate

alkahest-cas is also published on crates.io (docs.rs) for use directly from Rust without a Python runtime:

[dependencies]
alkahest-cas = "2"

# groebner is included by default; add other optional features as needed:
# alkahest-cas = { version = "2", features = ["parallel", "egraph"] }

System prerequisites (same libraries as the Python build — must be present before cargo build):

# Debian / Ubuntu
sudo apt-get install -y libflint-dev libgmp-dev libmpfr-dev

# macOS
brew install flint

The jit feature additionally requires LLVM 15 dev headers (apt install llvm-15-dev / brew install llvm@15). A self-contained runnable example is in examples/rust_quickstart/.

---

Quick start

import alkahest as ak

pool = ak.ExprPool()
x = pool.symbol("x")

# Differentiation with derivation log
result = ak.diff(ak.sin(x ** 2), x)
print(result.value)   # 2*x*cos(x^2)
print(result.steps)   # list of rewrite steps

# Integration
r = ak.integrate(ak.exp(x), x)
print(r.value)        # exp(x)

# Simplification
s = ak.simplify(x + pool.integer(0))
print(s.value)        # x

# JIT-compile to native code
f = ak.compile_expr(x ** 2 + pool.integer(1), [x])
print(f([3.0]))       # 10.0

Explicit polynomial representations

import alkahest as ak

pool = ak.ExprPool()
x = pool.symbol("x")
y = pool.symbol("y")

# FLINT-backed univariate polynomial
p = ak.UniPoly.from_symbolic(x ** 3 + pool.integer(-2) * x + pool.integer(1), x)
print(p.degree())        # 3
print(p.coefficients())  # [1, -2, 0, 1]

# GCD
a = ak.UniPoly.from_symbolic(x ** 2 + pool.integer(-1), x)
b = ak.UniPoly.from_symbolic(x + pool.integer(-1), x)
print(a.gcd(b))          # x - 1

# Factorization over ℤ (FLINT — Zassenhaus / van Hoeij)
fac = a.factor_z()
print(int(fac.unit), fac.factor_list())  # unit and list of (UniPoly, exponent)

# Dense univariate mod p (Berlekamp / Cantor–Zassenhaus via FLINT nmod)
fp = ak.factor_univariate_mod_p([1, 0, 1], 2)  # x^2+1 over GF(2)
print(fp.factor_list())

# Rational function with automatic GCD normalization
rf = ak.RationalFunction.from_symbolic(x ** 2 + pool.integer(-1), x + pool.integer(-1), [x])
print(rf)                # x + 1

# Sparse multivariate polynomial
mp = ak.MultiPoly.from_symbolic(x ** 2 * y + x * y ** 2, [x, y])
print(mp.total_degree()) # 3

Sparse multivariate interpolation (Ben-Or/Tiwari, Zippel)

Black-box recovery of sparse polynomials from evaluations, and sparse modular GCD as a substrate for faster exact GCD algorithms.

import alkahest as ak

pool = ak.ExprPool()
x, y = pool.symbol("x"), pool.symbol("y")

# Recover a sparse univariate f ∈ 𝔽ₚ[x] from 2T black-box evaluations
p = 32749  # prime
target = lambda v: (v**5 + 3*v**3 + 7) % p  # hidden: x^5 + 3x^3 + 7
f = ak.sparse_interp_univariate(target, T=3, prime=p)
print(f)   # recovered polynomial

# Recover a sparse multivariate f ∈ 𝔽ₚ[x, y] via Zippel's algorithm
target2 = lambda vals: (vals[0]**3 * vals[1]**2 + vals[0] * vals[1]**4) % p
g = ak.sparse_interp(target2, vars=[x, y], T=2, D=5, prime=p)
print(g)   # recovered MultiPolyFp

# Sparse modular GCD over ℤ[x₁,...,xₙ]
f2 = ak.MultiPoly.from_symbolic((x + y) * (x - y), [x, y])
g2 = ak.MultiPoly.from_symbolic((x + y) * (x + pool.integer(1)), [x, y])
h = ak.gcd_sparse(f2, g2, term_bound=4, degree_bound=4)
print(h)   # x + y

Symbolic summation (Gosper / recurrences)

Indefinite and definite sums for terms whose shift ratio F(k+1)/F(k) is rational in k—typically polynomials multiplied by gamma of a linear expression in k. General multivariate Zeilberger automation is partial; use verify_wz_pair(F, G, n, k) to check a discrete telescoping certificate after simplification.

import alkahest as ak

pool = ak.ExprPool()
k = pool.symbol("k")
n = pool.symbol("n")
term = ak.simplify(k * ak.gamma(k + pool.integer(1))).value
print(ak.sum_indefinite(term, k).value)
print(ak.sum_definite(term, k, pool.integer(0), n).value)

fib = ak.solve_linear_recurrence_homogeneous(
    n, [(-1, 1), (-1, 1), (1, 1)], [pool.integer(0), pool.integer(1)]
)

Difference equations / rsolve

Linear recurrences in one sequence with constant coefficients and a polynomial right-hand side (in the recurrence index n). Write shifts as pool.func("f", [n + integer]), pass the equation as a single expression that simplifies to zero, and optional initials as {n: value} to fix the C0, C1, … symbols.

import alkahest as ak

pool = ak.ExprPool()
n = pool.symbol("n")
f = lambda *a: pool.func("f", list(a))
# f(n) - f(n-1) - 1 == 0  →  general solution n + C0
eq = ak.simplify(f(n) - f(n + pool.integer(-1)) - pool.integer(1)).value
print(ak.rsolve(eq, n, "f", None))
# Fibonacci with f(0)=0, f(1)=1
fib_eq = ak.simplify(
    f(n) - f(n + pool.integer(-1)) - f(n + pool.integer(-2))
).value
print(ak.rsolve(fib_eq, n, "f", {0: pool.integer(0), 1: pool.integer(1)}))

Non-homogeneous order > 2 and sequences with polynomial coefficients in n are not implemented yet (see RsolveError / E-RSOLVE-*).

Symbolic products (∏)

product_definite(term, k, lo, hi) closes $\prod_{i=\text{lo}}^{\text{hi}} \text{term}(i)$ (inclusive) when term simplifies to ℚ(k) whose numerator/denominator factor into ℤ-linear polynomials — the implementation expands each linear factor $\alpha k+\beta$ with $\Gamma$ shifts $\Gamma(\text{hi}+\beta/\alpha+1)/\Gamma(\text{lo}+\beta/\alpha)$ and collects $\alpha^{(\text{hi}-\text{lo}+1)\cdot e}$. product_indefinite returns a Γ/power witness Z(k) with simplify-stable ratio Z(k+1)/Z(k)=term. Product(term, (k, lo, hi)).doit() matches SymPy ergonomics (DerivedResult; use .value). Irreducible quadratics in k, extra symbols besides k, and non-integer powers are rejected (ProductError / E-PROD-*).

import alkahest as ak

pool = ak.ExprPool()
k, n = pool.symbol("k"), pool.symbol("n")
P = ak.Product(k, (k, pool.integer(1), n))
print(ak.simplify(P.doit().value).value)

kp2 = k ** 2
term = ak.simplify(
    ((k + pool.integer(-1)) * (k + pool.integer(1))) / kp2
).value  # (k²-1)/k²

print(ak.simplify(
    ak.product_definite(term, k, pool.integer(2), n).value
).value)

Diophantine equations

Two integer unknowns, equation as a single polynomial = 0: linear families a·x + b·y + c = 0, sum of two squares x² + y² = n (finitely many tuples), and unit Pell x² - D·y² = 1 (fundamental solution (x₀, y₀) via the continued-fraction period of √D). The groebner feature is required and is included in all PyPI wheels since 2.3.1. API: diophantine(equation, [x, y]) → DiophantineSolution with .kind (parametric_linear, finite, pell_fundamental, no_solution) and typed fields.

import alkahest as ak

pool = ak.ExprPool()
x, y = pool.symbol("x"), pool.symbol("y")
sol = ak.diophantine(pool.integer(3) * x + pool.integer(5) * y - pool.integer(1), [x, y])
assert sol.kind == "parametric_linear"
pell = ak.diophantine(x**2 - pool.integer(2) * y**2 - pool.integer(1), [x, y])
assert pell.kind == "pell_fundamental" and int(str(pell.fundamental[0])) == 3

Quadratics with an x·y cross-term, unequal ellipse coefficients, or generalized Pell right-hand sides ≠ 1 are not implemented yet (DiophantineError / E-DIOPH-*).

Integer number theory

Submodule alkahest.number_theory: isprime, factorint, nextprime, totient, jacobi_symbol, nthroot_mod (prime modulus; k=2 or gcd(k,p−1)=1), discrete_log (linear scan for moderate primes), plus quadratic DirichletChi on odd square-free conductors. Implemented via FLINT fmpz in the Rust kernel; raises NumberTheoryError (E-NT-*) on invalid input.

import alkahest as ak
import alkahest.number_theory as nt

assert nt.isprime(2**127 - 1)
assert nt.factorint(2**32 - 1)[65537] == 1
assert nt.discrete_log(13, 3, 17) == 4
assert pow(nt.nthroot_mod(144, 2, 401), 2, 401) == 144 % 401

Noncommutative algebra

Symbols can opt out of multiplicative commutativity: pool.symbol("A", "real", commutative=False). Then A * B and B * A are distinct expressions, and sorting of Mul factors is disabled. The egglog backend automatically falls back to the rule-based simplifier when such symbols appear.

Pauli matrices (names sx, sy, sz) and a minimal orthogonal Clifford pair (cliff_e1, cliff_e2) have built-in rewrite tables; combine default rules with ak.simplify_pauli or ak.simplify_clifford_orthogonal. See examples/noncommutative.py.

Truncated series / Laurent tail

series(expr, var, point, order) builds a symbolic truncation about (var − point) and appends a BigO(⋯) remainder. Smooth functions use repeated differentiation; simple poles such as 1/x at zero take the rational Laurent path. Series.expr is the pooled sum-plus-order expression; ExprPool.big_o(inner) constructs standalone order bounds.

import alkahest as ak

pool = ak.ExprPool()
x = pool.symbol("x")
s_cos = ak.series(ak.cos(x), x, pool.integer(0), 6)
s_inv = ak.series(x ** (-1), x, pool.integer(0), 4)
print(s_cos.expr)

Rigorous interval arithmetic

import alkahest as ak

pool = ak.ExprPool()
x = pool.symbol("x")
result = ak.interval_eval(ak.sin(x), {x: ak.ArbBall(1.0, 1e-10)})
print(result)  # guaranteed enclosure of sin(1 ± 1e-10)

String expressions

import alkahest as ak

pool = ak.ExprPool()
x = pool.symbol("x")

# Parse a string into a symbolic expression
e = ak.parse("x^2 + 2*x + 1", pool, {"x": x})
print(e)                    # (x^2 + (x * 2)) + 1

# Round-trip: parse then pretty-print
expr = ak.parse("sin(x)^2 + cos(x)^2", pool, {"x": x})
print(ak.latex(expr))        # \sin\!\left(x\right)^2 + \cos\!\left(x\right)^2
print(ak.unicode_str(expr))  # sin(x)² + cos(x)²

Lattice reduction and approximate integer relations

Exact LLL reduction on integer bases lives under alkahest.lattice; for floating constants (as float or decimal strings) guess_relation searches for small integer coefficient vectors whose dot product has tiny residual relative to the working precision:

import alkahest as ak

basis = ak.lattice.lll_reduce_rows([[2, 15], [1, 21]])
rel = ak.guess_relation(["1", "2", "3"], precision_bits=256)

The relation finder is an augmented-lattice + LLL heuristic, not Ferguson–Bailey PSLQ; treat results as exploratory unless verified independently.

Regular chains / triangular decomposition

Lex-order Gröbner bases yield triangular sets used by the polynomial solver. The triangularize(equations, vars) API returns one or more RegularChain objects (polynomials as GbPoly tiles), splitting along factored bottom univariates when applicable. The built-in solve() routine retries backsolving from an extracted chain when the full basis is not directly triangular enough.

import alkahest as ak

pool = ak.ExprPool()
x = pool.symbol("x")
y = pool.symbol("y")
eq1 = x**2 + y**2 - pool.integer(1)
eq2 = y - x
chains = ak.triangularize([eq1, eq2], [x, y])
assert len(chains) >= 1

Primary decomposition

Lex-order Gröbner data is used to split ideals via saturations (I : x_i^∞ with (I + (x_i))) and, in the zero-dimensional case, factoring a univariate polynomial in the first Lex variable. primary_decomposition(polys, vars) returns PrimaryComponent objects with .primary() and .associated_prime() Gröbner bases; radical(polys, vars) returns a basis for √I.

import alkahest as ak

pool = ak.ExprPool()
x, y, z = pool.symbol("x"), pool.symbol("y"), pool.symbol("z")
comps = ak.primary_decomposition([x * y, x * z], [x, y, z])
assert len(comps) == 2
r = ak.radical([x**2, x * y], [x, y])
assert r.contains(x)

Differential algebra / Rosenfeld–Gröbner

Polynomial DAEs in implicit form g_i(t, y, y') = 0 can be analysed by prolongation (formal time derivatives of each equation, with the same derivative-state extension rule as Pantelides) and ordinary Gröbner bases over the jet variables. Inconsistent systems yield the unit ideal. Use rosenfeld_groebner(dae, order=..., max_prolong_rounds=...); when Pantelides exhausts its index cap, dae_index_reduce(dae) falls back to this pass.

import alkahest as ak

pool = ak.ExprPool()
t = pool.symbol("t")
y = pool.symbol("y")
dy = pool.symbol("dy/dt")
dae = ak.DAE.new([dy - y, dy - y - pool.integer(1)], [y], [dy], t)
r = ak.rosenfeld_groebner(dae, max_prolong_rounds=2)
assert r.consistent is False

Numerical algebraic geometry / homotopy continuation

Square polynomial systems can be solved numerically with a total-degree homotopy in ℂⁿ ((1-t)·γ·G + t·F), Newton polish on real projections, a conservative Smale-style α heuristic, and ArbBall enclosures attached to each coordinate. Use solve(eqs, vars, method="homotopy") for a list of dict solutions (Expr keys → float). For residuals, certification flags, and enclosures, call solve_numerical(...), which returns CertifiedSolution objects (.coordinates, .smale_certified, .to_dict(), .enclosures(), …).

Ideals whose finite root count in ℂⁿ is strictly below the Bézout bound (often called deficient — e.g. the Katsura family) typically need a polyhedral / mixed-volume start system; only the Bézout start (∏ deg F_i paths) is implemented here.

import alkahest as ak

pool = ak.ExprPool()
x, y = pool.symbol("x"), pool.symbol("y")
neg1 = pool.integer(-1)
sols = ak.solve([x**2 + neg1, y**2 + neg1], [x, y], method="homotopy")
cs = ak.solve_numerical([x**2 + neg1], [x])[0]
print(cs.coordinates, cs.smale_certified, cs.enclosures())

Composable transformations

import alkahest as ak

pool = ak.ExprPool()

@ak.trace(pool)
def f(x):
    return ak.sin(x ** 2)

df = ak.grad(f)          # symbolic gradient
df_fast = ak.jit(df)     # compiled gradient

---

Directory layout

alkahest/
├── alkahest-core/         # Rust kernel (published as the alkahest-cas crate)
│   ├── src/
│   │   ├── kernel/        # hash-consed expression DAG, ExprPool
│   │   ├── algebra/       # noncommutative Pauli / Clifford rules
│   │   ├── parse.rs       # Pratt expression parser (parse / ParseError)
│   │   ├── poly/          # UniPoly, MultiPoly, RationalFunction
│   │   ├── simplify/      # e-graph simplification (egglog)
│   │   ├── diff/          # symbolic differentiation
│   │   ├── integrate/     # symbolic integration
│   │   ├── calculus/      # series / limits
│   │   ├── jit/           # LLVM JIT and interpreter
│   │   ├── ball/          # Arb ball arithmetic
│   │   ├── ode/           # ODE analysis
│   │   ├── dae/           # DAE analysis and index reduction
│   │   ├── diffalg/       # Rosenfeld–Gröbner / differential elimination (groebner)
│   │   ├── solver/        # polynomial solving: Gröbner triangular, regular chains, homotopy
│   │   ├── lean/          # Lean 4 proof certificate export
│   │   └── primitive/     # primitive registration system
│   └── benches/           # criterion benchmarks
├── alkahest-mlir/         # MLIR dialect and lowering passes
├── alkahest-py/           # PyO3 bindings (Rust side)
├── python/alkahest/       # Python package
│   ├── _transform.py      # trace, grad, jit decorators
│   ├── _pytree.py         # JAX-style pytree flattening
│   ├── _context.py        # context manager and defaults
│   └── experimental/      # unstable API surface
├── examples/              # runnable end-to-end examples
│   └── rust_quickstart/   # self-contained Cargo project for alkahest-cas
├── tests/                 # Python test suite (pytest + hypothesis)
├── benchmarks/            # Python benchmarks and competitor comparisons
├── fuzz/                  # AFL++ fuzz targets
├── docs/                  # mdBook and Sphinx documentation
├── website/               # landing page (alkahest-cas.github.io)
│   └── src/               # index.html + styles.css source (deployed via CI)
├── alkahest-skill/        # Skill for AI to use alkahest
├── agent-benchmark/       # benchmark for comparing AI use of alkahest vs other CAS
└── scripts/               # CI helpers (API freeze check, error codes)

---

Expression representations

Type	Description
Expr	Generic hash-consed symbolic expression
UniPoly	Dense univariate polynomial (FLINT-backed)
MultiPoly	Sparse multivariate polynomial over ℤ
MultiPolyFp	Sparse multivariate polynomial over 𝔽ₚ (modular arithmetic)
RationalFunction	Quotient of polynomials with GCD normalization
ArbBall	Real interval with rigorous error bounds (Arb)

Representation types are explicit — no silent performance cliffs. Conversion between them is always an opt-in call (UniPoly.from_symbolic(...), etc.).

---

Result objects

Every top-level operation returns a DerivedResult with:

• .value — the result expression
• .steps — derivation log (list of rewrite rules applied)
• .certificate — Lean 4 proof term, when available

---

Documentation and further reading

• Documentation site — full API reference and user guide
• ROADMAP.md — planned milestones
• CONTRIBUTING.md — Rust vs Python layer guide
• TESTING.md — property-based testing, fuzzing, sanitizers, CI tiers
• BENCHMARKS.md — criterion and Python benchmark suites
• examples/ — runnable end-to-end examples
• LICENSE — Apache 2.0 license

---

Stability

Alkahest follows semantic versioning from 1.0. The stable surface is everything re-exported from alkahest_cas::stable (Rust) and alkahest.__all__ (Python). Experimental APIs live under alkahest_cas::experimental and alkahest.experimental and may change in minor releases.