BetLang

A Symbolic Probabilistic Metalanguage / Probabilistic CAS

BetLang is a domain-specific language for symbolic probabilistic computation. It is not a betting language—it is a compositional system for reasoning under uncertainty, built around a minimal ternary core and an extensible tower of uncertainty-aware number systems.

At its heart is a single idea:

Computation is structured choice under uncertainty.

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Core Concept

The fundamental primitive is the ternary form:

(bet A B C)

This represents a probabilistic, lazy choice between three branches.

Unlike conventional probabilistic languages:

• Evaluation is lazy (only the selected branch is computed)

• Choice is first-class and compositional

• The system is symbolic-first, not purely numeric

This makes BetLang closer to a probabilistic computer algebra system (CAS) than a simulation DSL.

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What BetLang Is

• A Symbolic Probabilistic Metalanguage (SPML)

• A Probabilistic CAS for uncertainty-aware computation

• A ternary computation model with lazy semantics

• A hosted language in Racket with formalizable semantics

• A system with a rich uncertainty-aware number tower (14 systems)

What BetLang Is Not

• Not a DeFi or gambling language

• Not just a Monte Carlo scripting tool

• Not limited to numeric probability (supports epistemic uncertainty, intervals, belief functions, etc.)

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v1.0 Architecture

BetLang is a multi-layer system with clearly separated responsibilities:

Racket — Authoritative Frontend / Specification

• #lang betlang defines the language

• syntax-parse + nanopass for IR and transformations

• Lazy ternary semantics are canonical and non-negotiable

• Optional type/refinement layer via Typed Racket / Turnstile+

This is the source of truth for semantics.

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Julia — Compute Kernel

• High-performance execution backend

• Primary path for numerical and statistical workloads

• Integration with:

• Distributions.jl

• StatsBase.jl

• Random.jl

• Planned:

• AbstractAlgebra.jl

• IntervalArithmetic.jl

• Differentiable inference via Zygote / Enzyme

Over time, the full uncertainty number tower migrates here.

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Lean 4 — Proof / Export Layer

• Used for:

• Convergence guarantees

• Safety properties

• Semantic correctness proofs

• Not part of the day-to-day compilation path

This layer ensures BetLang can become formally trustworthy.

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Rust — Optional Tooling Layer

• Status: Paused / Non-core

• Intended for:

• Packaging

• WASM targets

• Native runtime wrappers

Rust is explicitly not part of core semantics.

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Core Features

Ternary Computation

• (bet A B C) — primitive choice

• (bet/weighted …​) — non-uniform probabilities

• (bet_conditional …​) — predicate-driven selection

Lazy Semantics

• Only the chosen branch is evaluated

• Enables symbolic and infinite structures

• Prevents unnecessary computation

Compositionality

• Bets compose like algebraic objects

• Supports chaining, mapping, folding, and higher-order composition

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The Real Moat: Uncertainty-Aware Number Systems

BetLang includes 14 distinct number systems for representing uncertainty:

• Gaussian distributions n- Interval / affine arithmetic

• Fuzzy numbers

• Bayesian numbers

• Risk-based numbers (VaR / CVaR)

• Surreal and hyperreal systems

• p-adic probability systems

• Imprecise probabilities

• Dempster–Shafer belief functions

These are not addons—they are the type system of the language.

See: docs/number-tower.md

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Semantics

BetLang distinguishes between:

• Ternary logic (Kleene-style truth values)

• Ternary probabilistic belief (distributional uncertainty)

This distinction is critical and formalized in:

• docs/ternary-semantics.md

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Tooling Roadmap

Planned unified CLI:

bet check        # static + semantic validation
bet fmt          # canonical formatting
bet lsp          # editor integration
bet run --julia  # execute via Julia backend
bet trace        # execution tracing
bet hash         # reproducibility / identity

See: docs/toolchain-roadmap.md

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Example

(bet 'win 'draw 'lose)

(bet/weighted '(common 7) '(uncommon 2) '(rare 1))

(bet (expensive-computation)
     (cheap-approximation)
     (fallback))

Only one branch is evaluated.

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Use Cases

• Probabilistic programming

• Bayesian inference

• Uncertainty quantification

• Symbolic statistics

• Decision theory

• Risk modeling

• Scientific computing

• Research in probabilistic semantics

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Philosophy

BetLang is built on three principles:

1. Ternary over binary — real-world decisions are rarely yes/no

2. Uncertainty as structure — not noise, but a first-class object

3. Symbolic first — computation should remain inspectable and composable

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Status

• Racket frontend: Authoritative

• Julia backend: Active development

• Lean integration: Planned / partial

• Rust tooling: Optional / paused

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License

BetLang uses PMPL-1.0 (Palimpsest Public License).

See LICENSE for the precise definition and terms.

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Closing

BetLang is not about betting.

It is about making uncertainty programmable.