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Mathematical Programming Language (MPL)

Core Symbol Set

Mathematical Foundation (LaTeX)

  • Variables: α,β,γ,δ,ε,ζ,η,θ,ι,κ,λ,μ,ν,ξ,ο,π,ρ,σ,τ,υ,φ,χ,ψ,ω
  • Collections: ∅,∪,∩,⊂,⊃,∈,∉,⊆,⊇
  • Logic: ∧,∨,¬,⟹,⟺,∀,∃
  • Operations: +,-,×,÷,∗,∘
  • Relations: =,≠,<,>,≤,≥,≈,∼
  • Functions: f: A → B, λ
  • Assignment: ←
  • Definition: ≜
  • Structure: (),[],{},⟨⟩

Effect Extensions (11 new glyphs)

  • ↯ Raise exception
  • ↴ Handle exception
  • ‖ Parallel composition
  • ⌈⌉ Atomic section/lock
  • ⊕ Allocate resource
  • ⊖ Release resource
  • 𝓜 Module declaration
  • ⇐ Import
  • ⇒ Export
  • ⇀ Send (network)
  • ↽ Receive (network)

Additional Operators

  • ⟨v|e⟩ Choice type (value or error)
  • 🖫 File path prefix
  • ⇡⇣ Stream positioning
  • ⇆ Atomic swap
  • ⟪⟫ Deep update path
  • ⌜⌝ Code quotation
  • ⌞⌟ Code evaluation
  • ? Introspection
  • ⧈ Breakpoint
  • ✎ Trace/log
  • ⏲ Delay
  • ⟳ Periodic task
  • 〔〕 RAII scope

Grammar

Basic Expressions

expr ::= variable | literal | operation | function_call | block

variable ::= α | β | γ | ... | ω
literal ::= number | string | path | list | set
operation ::= expr OP expr
function_call ::= f(expr, ...)
block ::= { statement; ... }

Statements

assignment ::= variable ← expr
definition ::= variable ≜ expr  
conditional ::= condition ⟹ expr
iteration ::= ∀variable∈set: expr
parallel ::= expr ‖ expr
atomic ::= ⌈expr⌉_lock
exception ::= ↯expr | expr ↴ {↯e ⇒ handler}

Types

basic_type ::= ℕ | ℤ | ℚ | ℝ | ℂ | 𝔹
function_type ::= domain → codomain
choice_type ::= ⟨type|type⟩
effect_type ::= type^effect

Example Programs

Hello World

✎"Hello, World!"

Factorial

factorial ≜ λn∈ℕ: n≤1 ⟹ 1 | n×factorial(n-1)
result ← factorial(5)
✎result

File Processing with Error Handling

processFile ≜ λpath: 🖫path ↴ {
    data ← readFile(path)
    result ← transform(data)
    writeFile(result, 🖫"output.txt")
    ⟨"success"|"failed"⟩
} ↴ {↯e ⇒ ⟨⊥|e⟩}

Concurrent Download

downloadAll ≜ λurls: ∀url∈urls: (
    fetchData(url) ‖ processData(url)
) ⟹ mergeResults()

Module Definition

𝓜 Mathematics ⇒ {
    π ≜ 3.14159...
    sin ≜ λx∈ℝ: ...
    cos ≜ λx∈ℝ: ...
}

angle ← π/4
result ← Mathematics‧sin(angle)

Resource Management

databaseQuery ≜ λquery: 〔
    conn ← database ⊕
    ⌈
        result ← execute(conn, query)
        ✎"Query executed"
        result
    ⌉_db_lock
    conn ⊖
〕

Metaprogramming

generateFunction ≜ λname: ⌜
    λx: x × 2
⌝

doubler ← ⌞generateFunction("doubler")⌟
result ← doubler(21)

Real-time System

scheduler ≜ ⟳(
    tasks ← getPendingTasks()
    ∀task∈tasks: execute(task) ‖ monitor(task)
    , 100ms
)

Network Server

server ≜ λport: 〔
    socket ← bind(port) ⊕
    ∀request: (
        data ← ↽_socket request
        response ← processRequest(data)
        ⇀_socket response
    ) ‖ handleNext()
    socket ⊖
〕

Type-safe Database

User ≜ {name: String, age: ℕ∣age>0, email: String}
query ≜ λtable∈Database: ∀row∈table: validateUser(row) ↴ {
    ↯"Invalid user" ⟹ ⊥
}

Critical Implementation Decisions

Lexical Layer

  • Unicode Normalization: NFC on ingest, reject mixed forms
  • Symbol Input: Cross-platform keymap (Ctrl+Alt+g → γ) + ASCII escapes (\gamma → γ)
  • Semicolon handling: Require explicit ; everywhere except before }
  • Comments: -- for single-line comments (to end of line), {- ... -} for multi-line comments (nestable)
  • String Literals:
    • Standard strings: "..." with escape sequences (\n, \t, \\, \", \u{XXXXXX})
    • Raw strings: """...""" for multi-line, no escape processing
  • Path Literals: Both 🖫"path" and \path"path" supported for compatibility
  • Number Literals: Decimal (123, 3.14), hex (0x1A), binary (0b1101), with optional type suffixes later

Operator Precedence Table

Level Operators Associativity
9 function application left
8 ↯ ✎ ? ⧈ ⏲ (prefix) right
7 ∘ left
6 × ÷ ∗ left
5 + - left
4 = ≠ < > ≤ ≥ ≈ ∼ non-assoc
3 ∧ left
2 ∨ left
1 ⟹ right
0 ← right
-1 ‖ left
-2 ; left

Grammar Resolutions

  • Block semantics: Every statement returns value (ML-style)
  • Conditional associativity: Right-associative (a⟹b⟹c = a⟹(b⟹c))
  • Choice type parsing: Different tokens for |value vs |type contexts

Type System Extensions

  • Effect polymorphism: map : (A→ᴱ B) → List A →ᴱ List B
  • Linear resources: Compile-time ⊕/⊖ tracking, 〔〕 = linear region sugar
  • Exception types: f : A →⟨E⟩ B where E is union of raised types

Concurrency Semantics

  • Memory model: Happens-before with acquire/release on ⌈⌉
  • Parallel failure: Fail-fast, cancel siblings, aggregate choice types
  • Deadlock detection: Optional --debug-sync runtime verifier

Module System

  • Naming: 𝓜‧A‧B in source → A/B.mpl on disk
  • Re-export: Vector ⇒ 𝓜‧LinearAlgebra‧Vector
  • Versioning: 𝓜 LinearAlgebra@1.2.0 ⇒ { ... }

Missing Syntax (M1 Requirements)

Pattern Matching

match expr with
| pattern₁ ⟹ expr₁
| pattern₂ ⟹ expr₂
end

pattern ::= _ | literal | variable
          | ⟨Left pattern⟩ | ⟨Right pattern⟩    // choice
          | {field₁ = pattern₁, …}              // records
          | (pattern₁, pattern₂, …)             // tuples

Parametric Types

type_abs ::= ΛT. expr     // type lambda
type_app ::= expr [T]     // application

map ≜ ΛA. ΛB. λf: A→ᴱ B. λxs: List A. …

Foreign Function Interface

𝓜 Crypto@0.1.0 uses "libcrypto.so" {
    foreign digest      : 🖫Path →⟨IOErr⟩ Digest
    foreign randomBytes : ℕ → Bytes
}

Static Semantics Rules

Region Typing

Γ ⊢ e₁ : Resource r          Γ, h:r ⊢ e₂ : α
───────────────────────────────────────────────   (REGION)
Γ ⊢ 〔 x ← e₁ ; e₂ ; x ⊖ 〕 : α

Exception Handling

Γ ⊢ e₁ : α             Γ ⊢ e₂ : β           Γ ⊢ e₃ : β
─────────────────────────────────────────────────────────  (HANDLE)
Γ ⊢ e₁ ↴ { ↯x ⇒ e₂ } : β  ▷  Eff = (Eff(e₁) - {Raise ε}) ∪ Eff(e₂)

M0 Exit Criteria

  1. All spec examples parse and pretty-print round-trip
  2. No shift/reduce conflicts in grammar
  3. 10,000 random token sequences don't crash parser

Implementation Roadmap

  1. M0: ANTLR grammar + 500 LOC test suite
  2. M1: Hindley-Milner + row effects + linearity checker
  3. M2: Stack VM with green threads + deterministic GC
  4. M3: Self-hosting (stdlib + compiler in MPL)
  5. M4: LLVM backend with vectorized math ops
  6. M5: Package manager (fetch/build/run workflow)

This specification provides a complete foundation for implementing a mathematical programming language that maintains cognitive universality while supporting all modern programming paradigms.