Deep Causality HAFT

HAFT: Higher-Order Abstract Functional Traits

deep_causality_haft is a sub-crate of the deep_causality project, providing traits for Higher-Kinded Types (HKTs) in Rust. This enables writing generic, abstract code that can operate over different container types like Option<T> and Result<T, E>.

What are Higher-Kinded Types?

In Rust, types like Option<T> and Vec<T> are generic over a type T. We can think of Option and Vec as "type constructors": they take a type and produce a new type.

A Higher-Kinded Type is an abstraction over these type constructors. It allows us to write functions that are generic not just over a type, but over the shape or kind of a type constructor. For example, we can write a function that works with any type constructor that can be mapped over (a Functor), without caring if it's an Option, a Result, or something else.

This crate provides the fundamental traits (HKT, HKT2, HKT3, HKT4, HKT5) and functional traits (Functor, Applicative, Monad, Foldable, Traversable) to enable this pattern.

Usage

This crate uses a "witness" pattern to represent HKTs. For each type constructor (like Option), we define a zero-sized "witness" type (like OptionWitness) that implements the HKT trait. These witness types are zero-sized and incur no runtime overhead, making them zero-cost abstractions. This crates also comes with default witness pattern implementations for commonly used Rust types such as:

• Option -> OptionWitness
• Result -> ResultWitness
• Box -> BoxWitness
• Vec -> VecWitness

Each of those withness types implements the following traits and methods:

• Applicative: pure<T>(value: T) and apply<A, B, Func>(f_ab:HKT, f_a:HKT)
• Functor: fmap<A, B, Func>(m_a: HKT, f: Func)
• Foldable: fold<A, B, Func>(fa: HKT, init: B, f: Func)
• Monad: bind<A, B, Func>(m_a:HKT, f: Func)

Witness types that only implement Functor and Fold:

• BTreeMap -> BTreeMapWitness
• HashMap -> HashMapWitness
• VecDeque -> VecDequeWitness

Example: Using Functor and HKT in a generic function

use deep_causality_haft::*;

fn double_value<F>(m_a: F::Type<i32>) -> F::Type<i32>
where
    F: Functor<F> + HKT 
    {
    F::fmap(m_a, |x| x * 2)
}

fn main() {
    // Using double_value with Option
    let opt = Some(5);
    println!("Original Option: {:?}", opt);
    let doubled_opt = double_value::<OptionWitness>(opt);
    println!("Doubled Option: {:?}", doubled_opt);

    // Using double_value with Result
    let res = Ok(5);
    println!("Original Result: {:?}", res);
    let doubled_res = double_value::<ResultWitness<i32>>(res);
    println!("Doubled Result: {:?}", doubled_res);

    // Using double_value with Box
    let b = Box::new(7);
    println!("Original Box: {:?}", b);
    let doubled_box = double_value::<BoxWitness>(b);
    println!("Doubled Box: {:?}", doubled_box);

    // Using double_value with Vec
    let vec = vec![1, 2, 3];
    println!("Original Vec: {:?}", vec);
    let doubled_vec = double_value::<VecWitness>(vec);
    println!("Doubled Vec: {:?}", doubled_vec);
    
    // Using double_value with VecDeque
    let vec_dec = VecDeque::<i32>::from(vec![2, 4, 6]);
    println!("Original VecDec: {:?}", vec_dec);
    let doubled_vec_dec = double_value::<VecDequeWitness>(vec_dec);
    println!("Doubled VecDec: {:?}", doubled_vec_dec);
    assert_eq!(doubled_vec_dec, vec![4, 8, 12]);
}

When you run the example via:

cargo run --example haft_functor_example

You will see:

Original Option: Some(5)
Doubled Option: Some(10)
Original Result: Ok(5)
Doubled Result: Ok(10)
Original Box: 7
Doubled Box: 14
Original Vec: [1, 2, 3]
Doubled Vec: [2, 4, 6]
Original VecDec: [2, 4, 6]
Doubled VecDec: [4, 8, 12]

When combined with the deep_causality_num crate, you can abstract even further:

use deep_causality_haft::{Functor, HKT};
use deep_causality_num::{Float, One}; 
   
fn double_float<F, T>(m_a: F::Type<T>) -> F::Type<T>
   where
     F: Functor<F> + HKT,
     T: Float + One, // T must be a float and support multiplication
     {
     F::fmap(m_a, |x| x * (T::one() + T::one())) // Robust way to multiply by 2.0 for any Float
}

This level of abstraction helps in domains like numerical computing where you often deal with various data structures holding different numeric types. It allows for highly expressive and maintainable code that preserves high performance characteristics.

Example: Using Functor with Option

Here's how you can use the Functor trait with Option via its witness type, OptionWitness.

use deep_causality_haft::{Functor, HKT, OptionWitness};

fn main() {
    let opt_a = Some(5);
    let f = |x| x * 2;

    // Use the fmap function from the Functor implementation in OptionWitness
    let opt_b = OptionWitness::fmap(opt_a, f);
    assert_eq!(opt_b, Some(10));

    let opt_none: Option<i32> = None;
    let opt_none_mapped = OptionWitness::fmap(opt_none, f);
    assert_eq!(opt_none_mapped, None);
}

Example: Using Functor with Result

Here's how you can use the Functor trait with Result<T, E> via its witness type, ResultWitness<E>. HKT2 is used here because Result has two generic parameters, and we are fixing the error type E.

use deep_causality_haft::{Functor, HKT2, ResultWitness};

fn main() {
    let res_a: Result<i32, String> = Ok(5);
    let f = |x| x * 2;

    // Use the fmap function from the Functor implementation in ResultWitness
    let res_b = ResultWitness::fmap(res_a, f);
    assert_eq!(res_b, Ok(10));

    let res_err: Result<i32, String> = Err("Error".to_string());
    let res_err_mapped = ResultWitness::fmap(res_err, f);
    assert_eq!(res_err_mapped, Err("Error".to_string()));
}

Example: Using Foldable with Vec

I Here's how you can use the Foldable trait with Vec via its witness type, VecWitness.

use deep_causality_haft::{Foldable, VecWitness};

fn main() {
    let vec_a = vec![1, 2, 3, 4, 5];

    // Use the fold function from the Foldable implementation in VecWitness to sum elements
    let sum = VecWitness::fold(vec_a, 0, |acc, x| acc + x);
    assert_eq!(sum, 15);

    let vec_empty: Vec<i32> = Vec::new();
    let sum_empty = VecWitness::fold(vec_empty, 0, |acc, x| acc + x);
    assert_eq!(sum_empty, 0);

    let words = vec!["hello".to_string(), "world".to_string()];
    let concatenated = VecWitness::fold(words, String::new(), |mut acc, x| {
        if !acc.is_empty() {
            acc.push(' ');
        }
        acc.push_str(&x);
        acc
    });
    assert_eq!(concatenated, "hello world");
}

Type-Encoded Effect System

use deep_causality_haft::utils_tests::*;
use deep_causality_haft::{Effect5, MonadEffect5, HKT5};

  // 1. Start with a pure value, lifting it into the effect context
    let initial_effect: MyEffectType<i32> = MyMonadEffect5::pure(10);

    // 2. Define a collection of step functions
    // Each function takes an i32 and returns an effectful i32
    let step_functions: Vec<Box<dyn Fn(i32) -> MyEffectType<i32>>> = vec![
        Box::new(|x: i32| {
            MyCustomEffectType5 {
                value: x * 2,
                f1: None,
                f2: vec!["Operation A: Multiplied by 2".to_string()],
                f3: vec![1],
                f4: vec!["Trace: Executing step 1".to_string()],
            }
        }),
        Box::new(|x: i32| {
            MyCustomEffectType5 {
                value: x + 5,
                f1: None,
                f2: vec!["Operation B: Added 5".to_string()],
                f3: vec![1],
                f4: vec!["Trace: Executing step 2".to_string()],
            }
        }),
        Box::new(|x: i32| {
            MyCustomEffectType5 {
                value: x * 3,
                f1: None,
                f2: vec!["Operation C: Multiplied by 3".to_string()],
                f3: vec![1],
                f4: vec!["Trace: Executing step 3".to_string()],
            }
        }),
    ];

    // 3. Execute all step functions in sequence 
    println!("Process Steps: ");
    let mut current_effect = initial_effect;
    for (i, f) in step_functions.into_iter().enumerate() {
        let prev_logs_len = current_effect.f2.len();
        current_effect = MyMonadEffect5::bind(current_effect, f);
        for log_msg in current_effect.f2.iter().skip(prev_logs_len) {
            println!("  Log (Step {}): {}", i + 1, log_msg);
        }
    }

    println!("Sequenced outcome: {:?}", current_effect.value);

When you run the example via:

cargo run --example haft_effect_system_example

You will see:

--- Type-Encoded Effect System Example (Arity 5) ---

Initial effect (pure 10): MyCustomEffectType5 { value: 10, f1: None, f2: [], f3: [], f4: [] }

Process Steps: 
  Log (Step 1): Operation A: Multiplied by 2
  Log (Step 2): Operation B: Added 5
  Log (Step 3): Operation C: Multiplied by 3

Sequenced outcome: 75

... (Truncated)

The Effect3, Effect4, Effect5 and MonadEffect3, MonadEffect4, MonadEffect5 traits provide a powerful mechanism for building a type-encoded effect system. This allows you to manage side-effects (like errors and logging) in a structured, safe, and composable way, which is particularly useful for building complex data processing pipelines. It leverages Rust's powerful type system to ensure that these effects are explicitly handled and tracked throughout your program.

Here's a breakdown of how it works:

1. Effects as Types: Instead of side-effects occurring implicitly, this system represents them explicitly as generic type parameters on a container type. For instance, you might have a custom effect type like MyCustomEffectType<T, E, W>, where:

   • T is the primary value of the computation.
   • E represents an error type.
   • W represents a warning or log type. By making these effects part of the type signature, the presence of potential side-effects becomes explicit and verifiable by the compiler.

2. Higher-Kinded Type (HKT) Witnesses: To make these effect types generic over their primary value T while keeping the effect types (E, W, etc.) fixed, the system utilizes Higher-Kinded Types (HKTs). Traits like Effect3, Effect4, and Effect5 are used to "fix" a certain number of generic parameters of an underlying HKT type (e.g., HKT3, HKT4, HKT5). This allows you to define a "witness" type (e.g., MyEffectHktWitness<E, W>) that represents the shape of your effect container with specific, fixed effect types, leaving one parameter (T) open for the actual value.

3. Monadic Logic for Effects (MonadEffect traits): The core logic for how these effects are handled and combined is defined through MonadEffect traits (e.g., MonadEffect3, MonadEffect4, MonadEffect5). These traits provide:

   • pure: A method to lift a "pure" value (a value without any side-effects) into the effectful context.
   • bind: The central sequencing operation. It allows you to chain computations where each step might produce new effects. The implementation of bind dictates how effects from different steps are combined. For example, in the provided MyCustomEffectType, the bind implementation ensures that if an error occurs at any point, it propagates, and warnings from all steps are accumulated.

4. Specialized Effect Handling (LoggableEffect traits): The system can be extended with specialized traits for specific types of effects. For example, LoggableEffect3, LoggableEffect4, and LoggableEffect5 provide a log function. This function allows you to add a log message (of a specific fixed type, like E::Fixed2 for LoggableEffect3) to the effect container without altering the primary value or causing an error.

5. Compiler-Enforced Safety: A significant advantage of this system is that because effects are part of the type signature, the Rust compiler statically verifies that all effects are handled correctly. This means that if a function is declared to produce a certain type of effect, the compiler ensures that the effect is either explicitly handled or propagated. This prevents common bugs related to unhandled errors or forgotten logging, leading to more robust and predictable code.

Unbound HKTs & Functional Traits (Arity 2-5)

This crate also supports "Unbound" Higher-Kinded Types, where all generic parameters are free to vary. This enables advanced functional patterns from Category Theory that are crucial for complex systems modeling.

Unbound HKT Traits

• HKT2Unbound - HKT5Unbound: Base traits for multi-arity type constructors (e.g., Result<A, B>, (A, B, C)).
• Bifunctor: Maps over both types of a binary constructor simultaneously.
  • Usage: Evolving a coupled system (e.g., (Metric, Plasma)) where both components change type.
• Profunctor: Contravariant input, Covariant output.
  • Usage: Adapters, Optics, and State Machines where you pre-process input and post-process output.
• Adjunction: Defines a dual relationship between two functors ($L \dashv R$).
  • Usage: Conservation laws, optimization (Primal/Dual), and Galois connections.
• ParametricMonad: A Monad where the state type changes (Indexed Monad).
  • Usage: Modeling state transitions (e.g., Solid -> Liquid -> Gas) or protocol state machines.
• Promonad: Models interaction or fusion of contexts.
  • Usage: Tensor products, force calculations (merging fields), and quantum entanglement.
• RiemannMap: Models curvature and scattering (Arity 4).
  • Usage: General Relativity (Curvature Tensor), Particle Physics (Scattering Matrices).
• CyberneticLoop: Models a complete feedback control loop (Arity 5).
  • Usage: Autonomous agents (OODA Loop), Control Theory, and Error Correction.

Example: Bifunctor

use deep_causality_haft::{Bifunctor, HKT2Unbound};

struct ResultWitness;
impl HKT2Unbound for ResultWitness {
    type Type<A, B> = Result<A, B>;
}

impl Bifunctor<ResultWitness> for ResultWitness {
    fn bimap<A, B, C, D, F1, F2>(fab: Result<A, B>, mut f1: F1, mut f2: F2) -> Result<C, D>
    where F1: FnMut(A) -> C, F2: FnMut(B) -> D {
        match fab {
            Ok(a) => Ok(f1(a)),
            Err(b) => Err(f2(b)),
        }
    }
}

// Usage
let res: Result<i32, &str> = Ok(10);
let new_res = ResultWitness::bimap(res, |x| x * 2.0, |e| e.len()); // Result<f64, usize>

non-std support

The deep_causality_haft crate provides support for no-std environments. This is particularly useful for embedded systems or other contexts where the standard library is not available. Note, the std feature is enabled by default thus you need to opt-into non-std via feature flags.

To use this crate in a no-std environment, you need to disable the default std feature. You can optionally enable the alloc feature if you have an allocator available, which enables support for dynamic collections like Vec, Box, BTreeMap, etc.

Cargo Build and Test for no-std

1. Building for no-std with Allocator:

To build the crate for no-std while including support for dynamic collections (via alloc), use the following command:

cargo build --no-default-features --features alloc -p deep_causality_haft

2. Testing for no-std with Allocator:

To run tests in a no-std environment with allocator support, use:

cargo test --no-default-features --features alloc -p deep_causality_haft

3. Building for no-std without Allocator (Core/Algebra only):

If your no-std application does not have an allocator, you can build without the alloc feature. This restricts the crate to core HKT traits and algebraic structures that don't require dynamic memory.

cargo build --no-default-features -p deep_causality_haft

4. Testing for no-std without Allocator:

cargo test --no-default-features -p deep_causality_haft

Bazel Build

For regular (std) builds, run:

   bazel build //deep_causality_haft/...

and

   bazel test //deep_causality_haft/...

for tests. When you want to build for non-std, use

   bazel build --@rules_rust//rust/settings:no_std=alloc //deep_causality_haft/...

and

   bazel test --@rules_rust//rust/settings:no_std=alloc //deep_causality_haft/...

👨‍💻👩‍💻 Contribution

Contributions are welcomed especially related to documentation, example code, and fixes. If unsure where to start, just open an issue and ask.

Unless you explicitly state otherwise, any contribution intentionally submitted for inclusion in deep_causality by you, shall be licensed under the MIT licence, without any additional terms or conditions.

📜 Licence

This project is licensed under the MIT license.

👮️ Security

For details about security, please read the security policy.