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Push-forth is a light-weight, strongly-typed, stack-based genetic programming language designed by Maarten Keijzer and detailed in his 2013 paper. This project is an implementation of a push-forth interpreter and a compiler for .NET by Shane Celis.


Push-forth is a stack-based language. It uses the following notation to represent a stack: [a b c ...]. A stack has two principle operations:

  • pop([a b c]) -> a, [b c]
  • push(z, [b c]) -> void, [z b c]

Here is an example of a push-forth program that adds one plus one.

[[1 2 +]]

This example works like this. The evaluator pops "1" off the code stack. "1" is not an instruction, so the evaluator pushes the "1" onto the data stack. It does the same thing for the "2". It pops "+" off the code stack. The "+" is an instruction, so the evaluator passes the data stack to it. The "+" instruction pops both "1"s from the data stack, adds them, and pushes the result to the data stack. When no items are left on the code stack, the interpreter halts.

[[1 2 +]]
[[2 +] 1]
[[+] 2 1]
[[] 3]

In general there is a stack which holds the code and the data. The top item of the stack is another stack of code. The rest of the stack is the data. The evaluator pops off the code stack, then it pops an item off the code stack. If it's an instruction, it's executed---pulling its arguments from the data stack and pushing its results to the data stack. If it's not an instruction, it's pushed onto the data stack so it can be used as an argument. Finally the code stack is pushed back onto the data stack. Thus one step of evaluation is completed.

This form of [[code stack] data stack] has some nice properties.

  • The entire state of a program can be captured by its stack. Therefore, it's easy to suspend and resume computations.

  • There's no return stack as in Forth.

  • It's easy to embed and evaluate other programs within it.

[[eval eval eval] [[1 1 +]]]
[[eval eval] [[1 +] 1]]
[[eval] [[+] 1 1]]
[[] [[] 2]]
  • One can write a push-forth interpreter in push-forth.
[[true [eval dup car empty? not] while] [[1 2 -]]] -> [[] [[] -1]]
  • One can write other interpreters in push-forth.
[[true [alt-eval dup car empty? not] while] [[1 2 -]]] -> [[] [[] 1]]


This project uses the dotnet toolchain. To build the library run dotnet build at the command line. To run the unit tests, run dotnet test.

Example Code

To run the one plus two program, here's how one would set it up.

Stack program = "[[1 2 +]]".ToStack();
var interpreter = new NonstrictInterpreter();
program = interpreter.Eval(program); // [[2 +] 1]
Stack result = interpreter.Run(program); // [[] 3]
Console.WriteLine(result.ToRepr()); // Prints "[[] 3]" 


There is a small Read Eval Print Loop (REPL) program one can use to play with the push-forth. Here's an example of a session:

$ cd push-forth-dotnet/repl
$ dotnet run
Welcome to push-forth!
> 1
[[] 1]
> 2
[[2] 1]
[[] 2 1]
> +
[[+] 2 1]
[[] 3]
> 1 2 -
[[1 2 -] 3]
[[2 -] 1 3]
[[-] 2 1 3]
[[] 1 3]


There are actually a handful of interpreters in this project. Most differ only based on their instructions.

The class Interpreter is the base interpreter. It defines the evaluator but does not define any instructions. It's referred to as the "clean" interpreter within the unit tests.

The NonstrictInterpreter is an interpreter that most closely matches Keijzer's description. Its instructions will be applied most forgivingly such that errors caused by too few arguments on the stack will turn into noops, and wrongly typed arguments will be skipped.

The StrictInterpreter is not forgiving. If there are wrongly typed arguments, exceptions are thrown. If there are too few arguments, exceptions are thrown. This interpreter is not intended for genetic programming.

The ReorderInterpreter is a special case that takes a program and applies the forgiving approaches of the NonstrictInterpreter to produce a new program which may be executed by the StrictInterpreter. This kind of "reordering" is put to good use by La Cava, Helmuth, and Spector in their 2015 paper.

Lastly the TypeInterpreter takes a program and returns the types it requires as inputs and the types it produces as outputs.

Instructions Note

Although the instructions differ between each of these interpreters, they are not wholesale re-implementations. For instance, the NonstrictInterpreter uses the same instructions as the StrictInterpreter except they are wrapped with the ReorderWrapper class, which handles too few or wrongly typed arguments in general.


An additional interpreter is called Compiler. It takes a program and generates Intermediate Language (IL) byte-code. Since the IL virtual machine is stack-based, it can mirror many of push-forth's operations directly as IL instructions.

The compiler only accepts strict programs, but the ReorderInterpreter can turn a non-strict program into a strict program.


Any non-strict program can be made strict by reordering arguments and dropping of instructions.

It may be possible that there is a program where reordering arguments dynamically is required to evaluate it. If anyone can find such a counter example program, please do.

This reordering can also be used as an optimization

Implementation Note

The compiler uses the same push-forth evaluator, but its instructions differ and constitute a wholesale reimplementation. It produces CompilationUnit objects on the data stack. Once the program is finished being interpreted, the CompilationUnits left on the data stack are fed an ILGenerator to produce IL.

[[1 1 +]] -> [[] CompilationUnit("ldc.i4 1; ldc.i4 1; add")]

It's helpful sometimes to think of a compiler as an interpreter but with a different resulting data type.

Notable Deviations

There are few deviations from Keijzer's description.

Generalized argument predicates are not supported

Instead of just relying on types, one could use a predicate to enforce type-safety or whatever other constraints. This is easier done in an interpreter than in a compiler. At compile-time we know what the types are. To make the interpreter and compiler compatible, this feature was dropped from the interpreter.

Polymorphic instructions not fully implemented yet

Polymorphic instructions, i.e., a "+" instruction that can add integers, floating-point numbers, or strings have not been implemented yet.

Argument order

Suppose there is a binary-arity function F(x, y) that is bound to the instruction f in push-forth. One has to choose whether [[a b f]] will evaluate as F(a, b) or F(b, a). There is no right choice. It is a matter of convention. This implementation has chosen F(a, b) deviating from Keijzer's presentation but embracing the convention set forth by Forth, Push3, and many other stack-based languages.

Pivot Notation Revisited


Keijzer's pivot notation emphasizes the order of the data stack. For example the program [+ "Hello " "World!" •] evaluates to [• "Hello World!"]. Without pivot notation the code looks less natural [["World!" "Hello " +]]. However, this emphasis of the data stack requires breaking the prevailing argument order convention. In addition the pivot travels left to right which can cause confusion.

Still the pivot notation offers an interesting way of viewing the code and data stack. Perhaps if it preserved the order of the code stack instead of the data stack, it may offer a compelling notation to illustrate execution. Let us introduce a different pivot character for clarity, '⬦' instead of '•'.

revised pivot notation

[⬦ 1 2 -]
[1 ⬦ 2 -]
[1 2 ⬦ -]
[-1 ⬦]

Here the pivot travels left-to-right. The data stack is reversed. The order of the code stack is preserved. And the argument order convention is respected. Let us call Keijzer's pivot notation a "data pivot" and this proposal a "code pivot".

One reason to prefer reversing the data stack is it can be considered an artifact of execution. One would never "see" the data stack while programming Forth, only the code stack. This is less important for a genetic programming language since it's not intended to be written by hand. However, there is an argument to be made for readability since one may want to analyze a program found by a genetic algorithm. (The ability to take a non-strict program and make it strict certainly can help one analyze it too.)

Future Improvements

Since the compiler is an interpreter, it may not be too difficult to implement a mixed-mode compiler that will interpret everything available to it at compile-time. It's like having an C++'s constexpr but automatic. For evolving programs that may provide a worthwhile heuristic to consider: How much code does any computation with input data versus what are effectively constants generated a runtime?

Conventional Interpretation

    [[1 1 +]] -> [[] 2]

Conventional Compilation

    [[1 1 +]] -> [[] CompilationUnit("ldc.i4 1; ldc.i4 1; add")]

Mixed Interpretation and Compilation

    [[1 1 +]] -> [[] 2] -> [[] CompilationUnit("ldc.i4 2")]

If there is an external argument "x", the mixed-mode compiler cannot "precompute" its value.

[[1 1 + x +]] -> [[] [2 x +]] -> [[] CompilationUnit("ldc.i4 2; ldarg.0; add")]

See the file for more details.


  • Sprache is used for parsing.
  • xUnit is used for unit tests.
  • OneOf is used for discriminated unions (sum types). (May have fallen out of use, actually.)
  • dotnet toolchain is used to build and test.


This code is licensed under the MIT License.


Thanks to Keijzer for designing push-forth and providing source code to inspect.

Thanks to Lee Spector et al for designing Push3; it introduced me to stack-based languages for genetic programming.

Thanks to Bill Tozier for blazing a trail and providing a push-forth ruby implementation to inspect.


  • Keijzer, M. (2013). Push-forth: a light-weight, strongly-typed, stack-based genetic programming language. Proceeding of the fifteenth annual conference companion (pp. 1635–1640). New York, New York, USA: ACM.

  • La Cava, W., Helmuth, T., Spector, L., & Danai, K. (2015). Genetic Programming with Epigenetic Local Search (pp. 1055–1062). Presented at the the 2015, New York, New York, USA: ACM Press.


Push-forth is a light-weight, strongly-typed, stack-based genetic programming language for .NET







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