Command-Line Manipulator of 𝜑-Calculus Expressions

This is a command-line normalizer, rewriter, and dataizer of 𝜑-calculus expressions.

First, you write a simple 𝜑-calculus expression in the hello.phi file:

⟦ φ ↦ ⟦ Δ ⤍ 68-65-6C-6C-6F ⟧, t ↦ ξ.k, k ↦ ⟦⟧ ⟧

Installation

Then you can install phino in two ways:

Install Cabal first and then:

cabal update
cabal install --overwrite-policy=always phino-0.0.115
phino --version

Or download binary from the internet using curl or wget:

sudo curl -o /usr/local/bin/phino http://phino.objectionary.com/releases/macos-15/phino-latest
sudo chmod +x /usr/local/bin/phino
phino --version

Download paths are:

• Ubuntu 22.04: http://phino.objectionary.com/releases/ubuntu-22.04/phino-latest
• Ubuntu 24.04: http://phino.objectionary.com/releases/ubuntu-24.04/phino-latest
• MacOS (ARM): http://phino.objectionary.com/releases/macos-15/phino-latest
• MacOS (Intel): http://phino.objectionary.com/releases/macos-14-large/phino-latest
• Windows: http://phino.objectionary.com/releases/windows-2022/phino-latest.exe

Build

To build phino from source, clone this repository:

git clone git@github.com:objectionary/phino.git
cd phino

Then, run the following command (ensure you have Cabal installed):

cabal build all

Next, run this command to install phino system-wide:

sudo cp "$(cabal list-bin phino)" /usr/local/bin/phino

Verify that phino is installed correctly:

$ phino --version
0.0.0

You can ensure scripts are run with a specific version of phino using the --pin global option. It exits with an error when the version supplied doesn't match the installed one:

phino --pin=0.0.0.67 dataize hello.phi

Dataize

Then, you dataize the expression:

$ phino dataize hello.phi
68-65-6C-6C-6F

Every atom fired on the way to the bytes may be recorded in a machine-readable protocol, with the --evaluations option. One firing is one line of three tab-separated fields: the name of the λ function, the formation it was applied to, and the expression it returned:

$ cat sum.phi
⟦
  bytes(data) ↦ ⟦ φ ↦ data ⟧,
  number(as-bytes) ↦ ⟦ φ ↦ as-bytes, plus(x) ↦ ⟦ λ ⤍ L_number_plus ⟧ ⟧,
  φ ↦ 5.plus( 6 )
⟧
$ phino dataize --evaluations=atoms.tsv --quiet --sweet --hide-rho sum.phi
$ cat -T atoms.tsv
L_number_plus^I⟦ x ↦ 6 ⟧^I11

Records follow the syntax of the other options, such as --sweet and --hide-rho, but always stay on one line. The file is truncated at the beginning of every run, and --output=phi is the only output format it works with, since one record must fit into one line.

An atom that cannot fire fails the run: its λ function is unknown to phino, or one of its inputs reaches such an atom. This is what happens when a data input is replaced on purpose by a placeholder formation, such as ⟦ λ ⤍ Sym_arg_0 ⟧. With --partial, dataization becomes partial evaluation instead: what the known inputs decide is computed, the rest survives as the residual program, which is printed in place of the bytes, and the run ends successfully:

$ cat partial.phi
⟦
  bytes(data) ↦ ⟦ φ ↦ data ⟧,
  number(as-bytes) ↦ ⟦
    φ ↦ as-bytes,
    plus(x) ↦ ⟦ λ ⤍ L_number_plus ⟧,
    times(x) ↦ ⟦ λ ⤍ L_number_times ⟧
  ⟧,
  φ ↦ 2.times(3).plus(⟦ λ ⤍ Sym_arg_0 ⟧)
⟧
$ phino dataize --partial --sweet --hide-rho partial.phi
⟦ x ↦ ⟦ λ ⤍ Sym_arg_0 ⟧, λ ⤍ L_number_plus ⟧

Here 2.times(3) was decided by literals, so it was computed (its result, 6, sits in the hidden ρ of the residual program), while plus waits for an x no atom can produce, so it stays in place as a normal-form subterm. Each such stuck site also lands in the --evaluations file, as a record with the first two fields only, since there is no result to report; the inner stuck atom comes first, then the known atom whose input reached it:

$ phino dataize --partial --evaluations=atoms.tsv --quiet \
    --sweet --hide-rho partial.phi
$ cat -T atoms.tsv
L_number_times^I⟦ x ↦ 3 ⟧^I6
Sym_arg_0^I⟦⟧
L_number_plus^I⟦ x ↦ ⟦ λ ⤍ Sym_arg_0 ⟧ ⟧

Evaluation stays demand-driven, as the calculus prescribes: an argument that nothing asked for before the run got stuck is left as it is in the residual program, for the next iteration.

The nested morphing and dataization recursion is bounded by the --max-steps option (default 1000): when the budget is exhausted, the run fails with Dataization did not finish before reaching the limit of steps. This guards against non-terminating terms, which used to loop forever before the bound was introduced:

$ phino dataize --max-steps=50 problem.phi
[ERROR]: Dataization did not finish before reaching the limit of steps: --max-steps=50

Morph

Dataization insists on bytes. Morphing 𝕄 asks a different question: evaluate as far as the object model allows, without demanding data. It resolves Φ against the universe, peels dispatches and applications through normalization, fires whichever atoms sit under a dispatch, and stops at the first formation it reaches, handing that formation back untouched. The morph command runs 𝕄 on its own:

$ cat two.phi
⟦
  bytes(data) ↦ ⟦ φ ↦ data ⟧,
  number(as-bytes) ↦ ⟦ φ ↦ as-bytes, plus(x) ↦ ⟦ λ ⤍ L_number_plus ⟧ ⟧,
  φ ↦ 5.plus( 6 ).plus( 7 )
⟧
$ phino dataize --sweet --hide-rho two.phi
40-32-00-00-00-00-00-00
$ phino morph --locator=Q.φ --sweet --hide-rho two.phi
⟦ x ↦ 7, λ ⤍ L_number_plus ⟧

The inner 5.plus( 6 ) fires, because .plus is dispatched on its result, and 11 lands in the ρ hidden by --hide-rho. The outer application is saturated but bare, so 𝕄 returns it and is finished; firing it is dataization's job and takes dataize on to 18.

The default locator Q morphs the whole top formation, which 𝕄 returns unchanged, so --locator is how one aims 𝕄 at a subterm, exactly as in dataize. Unlike 𝔻, 𝕄 is total: where no formation is reachable the answer is the terminator ⊥, printed rather than reported as a failed run:

$ phino morph --locator=Q.x <<< '⟦ x ↦ ξ ⟧'
⊥

The whole dataize option surface applies unchanged — --sequence, --headers, --steps-dir, --evaluations, --partial, --max-steps, --shuffle/--seed, --output, --focus and the rest.

Rewrite

You can rewrite this expression with the help of rules defined in the my-rule.yml YAML file (here, the !d is a capturing group, similar to regular expressions):

name: My custom rule
pattern: Δ ⤍ !d
result: Δ ⤍ 62-79-65

Then, rewrite:

$ phino rewrite --rule=my-rule.yml hello.phi
⟦ φ ↦ ⟦ Δ ⤍ 62-79-65 ⟧, t ↦ ξ.k, k ↦ ⟦⟧ ⟧

If you want to use many rules, just use --rule as many times as you need:

phino rewrite --rule=rule1.yaml --rule=rule2.yaml ...

You can also use built-in rules, which are designed to normalize expressions:

phino rewrite --normalize hello.phi

Both flags may be combined, so that your own rules are applied alongside the built-in ones, in a single rewriting session:

phino rewrite --normalize --rule=my-rule.yaml hello.phi

Some rules mint fresh synthetic names via the random-string built-in. To keep the output reproducible across runs, phino seeds the random generator deterministically with 0 by default. Use --seed to pick a different seed:

phino rewrite --seed=42 --rule=my-rule.yml hello.phi

If no input file is provided, the 𝜑-expression is taken from stdin:

$ echo '⟦ φ ↦ ⟦ Δ ⤍ 68-65-6C-6C-6F ⟧ ⟧' | phino rewrite --rule=my-rule.yml
⟦ φ ↦ ⟦ Δ ⤍ 62-79-65 ⟧ ⟧

You're able to pass XMIR as input. Use --input=xmir and phino will parse given XMIR from file or stdin and convert it to phi AST.

phino rewrite --rule=my-rule.yaml --input=xmir file.xmir

Also phino supports 𝜑-expressions in ASCII format and with syntax sugar. The rewrite command also allows you to desugar the expression and print it in canonical syntax:

$ echo '[[ @ -> Q.io.stdout("hello") ]]' | phino rewrite
⟦
  φ ↦ Φ.io.stdout(
    α0 ↦ Φ.string(
      α0 ↦ Φ.bytes(
        α0 ↦ ⟦ Δ ⤍ 68-65-6C-6C-6F ⟧
      )
    )
  )
⟧

Merge

You can merge several 𝜑-expressions into a single one by merging their top level formations:

$ cat bytes.phi
⟦ bytes(data) ↦ ⟦ φ ↦ data ⟧ ⟧
$ cat number.phi
⟦
  number(as-bytes) ↦ ⟦
    φ ↦ as-bytes,
    plus(x) ↦ ⟦ λ ⤍ L_number_plus ⟧
  ⟧
⟧
$ cat minus.phi
⟦ number ↦ ⟦ minus(x) ↦ ⟦ λ ⤍ L_number_minus ⟧ ⟧ ⟧
$ phino merge bytes.phi number.phi minus.phi --sweet
⟦
  bytes(data) ↦ ⟦ φ ↦ data ⟧,
  number(as-bytes) ↦ ⟦
    φ ↦ as-bytes,
    plus(x) ↦ ⟦ λ ⤍ L_number_plus ⟧,
    minus(x) ↦ ⟦ λ ⤍ L_number_minus ⟧
  ⟧
⟧

Match

You can test the 𝜑-expression matches against the rule pattern. The result output contains matched substitutions:

$ phino match --pattern='⟦ Δ ⤍ !d, !B ⟧' hello.phi
B >> ⟦ ρ ↦ ∅ ⟧
d >> 68-65-6C-6C-6F

Explain

You can explain the built-in rules by printing them in LaTeX format. Pass exactly one of --normalize, --morph, --dataize or --contextualize for the rewriting, morphing (𝕄), dataization (𝔻) or contextualization (𝒞) rules (or --rule for a custom rule file):

$ phino explain --normalize
\begin{tabular}{rl}
\phinoNormalizationRule{alpha}
  { [[ B_1, \tau -> ?, B_2 ]] ( \phiTerminal{\alpha_{i}} -> e ) }
  { [[ B_1, \tau -> ?, B_2 ]] ( \tau -> e ) }
  { $ i = \vert \overline{ B_1 } \vert $ }
  { }
\phinoNormalizationRule{dc}
  { T ( \tau -> e ) }
  { T }
  { }
  { }
...
\phinoNormalizationRule{stop}
  { [[ B ]] . \tau }
  { T }
  { $ \tau \notin B \;\text{and}\; @ \notin B \;\text{and}\; L \notin B $ }
  { }
\end{tabular}

The morphing and dataization rules are printed the same way:

$ phino explain --morph
\begin{tabular}{rl}
\phinoMorphingRule{mf}
  { \mathbb{M}( [[ B ]], e ) }
  { [[ B ]] }
  { }
  { }
...
\phinoMorphingRule{universe}
  { \mathbb{M}( Q, e ) }
  { \mathbb{M}( \phinoNormalize{ e }, e ) }
  { $ e \not= Q $ }
  { }
\end{tabular}

$ phino explain --dataize
\begin{tabular}{rl}
\phinoDataizationRule{delta}
  { \phinoDataize{ [[ B_1, D> δ, B_2 ]] } }
  { δ }
  { }
  { }
...
\phinoDataizationRule{norm}
  { \phinoDataize{ n } }
  { \phinoDataize{ \mathbb{M}( n, e ) } }
  { }
  { }
\end{tabular}

$ phino explain --contextualize
\begin{phinoContextualizationInference}
  \phinoName{cxi}
  \phinoConclusion{ \phinoContextualize{ \phiTerminal{\xi} }{ k }{ k } }
\end{phinoContextualizationInference}
...
\begin{phinoContextualizationInference}
  \phinoName{cd}
  \phinoPremise{ \phinoContextualize{ n }{ k }{ n_1 } }
  \phinoConclusion{ \phinoContextualize{ n . \tau }{ k }{ n_1 . \tau } }
\end{phinoContextualizationInference}

For more details, use phino [COMMAND] --help option.

Rule structure

This is BNF-like yaml rule structure. Here types ended with apostrophe, like Attribute' are built types from 𝜑-expression AST

Rule:
  name: String
  pattern: String
  result: String
  when: Condition?       # predicate, works with substitutions before extension
  where: [Extension]?    # substitution extensions
  having: Condition?     # predicate, works with substitutions after extension

Condition:
  = and: [Condition]     # logical AND
  | or:  [Condition]     # logical OR
  | not: Condition       # logical NOT
  | eq:                  # compare two comparable objects
      - Comparable
      - Comparable
  | in:                  # check if attributes exist in bindings
      - Attribute'
      - Binding'
  | nf: Expression'      # returns True if given expression in normal form
                         # which means that no more other normalization rules
                         # can be applied
  | absolute: Expression' # returns True if given expression is xi-free, i.e.
                         # there is no ξ outside of a formation: it is Φ, a
                         # formation, a dispatch with a xi-free subject, or an
                         # application with a xi-free subject and argument.
                         # Combined with a normal-form check by the '𝑘'/'!k'
                         # meta variable, which ranges over the absolute
                         # expressions 𝒦 ⊆ 𝒩, used by the Rcopy rule.
  | matches:             # returns True if given expression after dataization
      - String           # matches to given regex
      - Expression
  | part-of:             # returns True if given expression is attached to any
      - Expression'      # attribute in ginve bindings
      - BiMeta'
  | formation:           # returns True if given expression is a formation
      Expression'        # (an abstraction ⟦…⟧); used by morphing 'md'
                         # as 'not (formation 𝑛)', so a non-formation head is
                         # morphed and a formation head is left to 'ml'
  | gt:                  # returns True if the first comparable object is
      - Comparable       # greater than the second one
      - Comparable
  | disjoint:            # returns True if none of the given attributes exists
      - [Attribute']     # in the given bindings
      - Binding'

Comparable:              # comparable object that may be used in 'eq' condition
  = Attribute'
  | Number
  | Expression'

Number:                  # comparable number
  = Integer              # just regular integer
  | IndexMeta'           # 𝑖 (or !i), the index captured by an α𝑖 argument
  | length: BiMeta'      # calculate length of bindings by given meta binding
  | domain: BiMeta'      # calculate number of unique attributes in given
                         # meta binding (excluding 'assets')

Extension:               # substitutions extension used to introduce new meta variables
  meta: [ExtArgument]    # new introduced meta variable
  function: String       # name of the function
  args: [ExtArgument]    # arguments of the function

ExtArgument
  = Bytes'               # !d
  | Binding'             # !B
  | Expression'          # !e
  | Attribute'           # !t

Here's list of functions that are supported for extensions:

• contextualize - function of two arguments, that rewrites given expression depending on provided context according to the contextualization rules
• random-tau - creates attribute with random unique name. Accepts bindings, and attributes. Ensures that created attribute is not present in list of provided attributes and does not exist as attribute in provided bindings.
• dataize - dataizes given expression and returns bytes.
• concat - accepts bytes or dataizable expressions as arguments, concatenates them into single sequence and convert it to expression that can be pretty printed as human readable string: Φ.string(Φ.bytes⟦ Δ ⤍ !d ⟧).
• sed - pattern replacer, works like unix sed function. Accepts two arguments: target expression and pattern. Pattern must start with s/, consists of three parts separated by /, for example, this pattern s/\\s+//g replaces all the spaces with empty string. To escape braces and slashes in pattern and replacement parts - use them with \\, e.g. s/\\(.+\\)//g.
• random-string - accepts dataizable expression or bytes as pattern. Replaces %x and %d formatters with random hex numbers and decimals accordingly. Uniqueness is guaranteed during one execution of phino.
• size - accepts exactly one meta binding and returns size of it and Φ.number.
• tau - accepts Φ.string, dataizes it and converts it to attribute. If dataized string can't be converted to attribute - an error is thrown.
• string - accepts Φ.string or Φ.number or attribute and converts it to Φ.string.
• number - accepts Φ.string and converts it Φ.number
• sum - accepts list of Φ.number or Φ.bytes and returns sum of them as Φ.number
• join - accepts list of bindings and returns list of joined bindings. Duplicated ρ, Δ and λ attributes are ignored, all other duplicated attributes are replaced with unique attributes using random-tau function.

Meta variables

The phino supports meta variables to write 𝜑-expression patterns for capturing attributes, bindings, etc.

This is the list of supported meta variables:

• !t || 𝜏 - attribute
• !i || 𝑖 - the index of a positional (α) application argument, captured by writing α𝑖 (or ~!i)
• !e || 𝑒 - any expression
• !n || 𝑛 - any expression that is already in normal form (behaves like !e/𝑒, but only binds a sub-expression in NF, so no explicit nf: guard is needed)
• !k || 𝑘 - any expression that is absolute, i.e. xi-free and in normal form (ranges over 𝒦 ⊆ 𝒩); behaves like !e/𝑒 but only binds an absolute sub-expression, so no explicit absolute: or nf: guard is needed
• !B || 𝐵 - list of bindings
• !d || δ - bytes in meta delta binding
• !F || 𝑓 - function name in meta lambda binding

Every meta variable may also be used with an integer index, like !B1 or 𝜏0.

A positional (α) application argument is written as α0, ~0 (ASCII), or α𝑖/~!i when its index is captured by an !i/𝑖 meta variable.

Incorrect usage of meta variables in 𝜑-expression patterns leads to parsing errors.

Benchmark

To run performance benchmarks, you need Java 8+ and curl. Maven is downloaded automatically on first run via benchmark/mvnw.

The benchmark uses the compiled Native class from JNA — a large real-world Java class — as its test input. On first run, make bench downloads the class, disassembles it to XMIR via jeo-maven-plugin, converts it to 𝜑 using phino rewrite, and caches the results in benchmark/tmp/. Subsequent runs skip straight to the benchmarks.

make bench

=== parse/phi ===
  warmup:     3 iterations
  batches:    10 x 1
  total:      1781621.932 μs
  avg:        178162.193 μs
  min:        163679.993 μs
  max:        209804.570 μs
  std dev:    17409.000 μs
=== parse/xmir ===
  warmup:     3 iterations
  batches:    10 x 1
  total:      7611171.011 μs
  avg:        761117.101 μs
  min:        679176.096 μs
  max:        899930.605 μs
  std dev:    69464.089 μs
=== rewrite/normalize ===
  warmup:     3 iterations
  batches:    10 x 1
  total:      811837.328 μs
  avg:        81183.733 μs
  min:        67331.161 μs
  max:        92232.373 μs
  std dev:    8117.233 μs
=== print/sweet/multiline ===
  warmup:     3 iterations
  batches:    10 x 1
  total:      4199718.146 μs
  avg:        419971.815 μs
  min:        396063.240 μs
  max:        442595.822 μs
  std dev:    16504.492 μs
=== print/sweet/flat ===
  warmup:     3 iterations
  batches:    10 x 1
  total:      4060839.345 μs
  avg:        406083.934 μs
  min:        387257.807 μs
  max:        417907.724 μs
  std dev:    8861.891 μs
=== print/salty/multiline ===
  warmup:     3 iterations
  batches:    10 x 1
  total:      14257603.693 μs
  avg:        1425760.369 μs
  min:        1405945.748 μs
  max:        1449825.539 μs
  std dev:    11882.320 μs

The results were calculated in this GHA job on 2026-09-07 at 19:51, on Linux with 4 CPUs.

How to Contribute

Fork repository, make changes, then send us a pull request. We will review your changes and apply them to the master branch shortly, provided they don't violate our quality standards. To avoid frustration, before sending us your pull request please make sure all your tests pass:

make all

To generate a local coverage report for development, run:

make coverage

To build a phino executable into the root of the repository, run:

make phino

This produces an executable phino (or phino.exe on Windows) in the project root, which you can run directly for quick local testing:

./phino --version

You will need GHC ≥ 9.6.7 and Cabal ≥ 3.0 (recommended) or Stack ≥ 3.0 installed.