This is an interpreter for a rule based language. A REPL and web interface are in the works.

Some executable examples are listed at the end of Main.hs.

This document will refer to the Go example, defined in go.arrow and go.graph.

Syntax Elements

A comment starts with the # character and ends at the end of a line.

A name is a sequence of unicode characters not including whitespace or any of the forbidden characters:


An identifier is any name, except the forbidden names ~> and =>. For example:

step α hi-there x+1 間 rel/n →

A symbol literal is a single ' character followed by a name. For example:

'apple 'ripe_banana22

An integer literal is a sequence of digits, possibly preceded by a hyphen to indicate a negative number.

A string literal is a sequence of characters or escape sequences surrounded by double quote (") characters. Multi-line strings are not supported.

The following are special sequences, recognized either as operators or delimiters:

   => ~> , . .. ! _ ( ) = /= < > <= >= + - *

There are no keywords.

Data Model

Programs operate on a database of relations. Each relation has a name called its label and a number called its arity. A relation comprises a set of tuples. The elements of a tuple are called values, and the size of a tuple is its relation’s arity.

In writing, a relation can be described by tag/arity, as in Prolog. For example, a directed graph might use adj-to/2 for the adjacency relation. A tuple is written as

label v1 v2 ... vn

for instance, adj-to 2 7.

There are two types of relations: logical and imperative. The tuples in a logical relation are called facts, and the tuples of an imperative relation are called events. They are interpreted differently and computed by different rule types, described later.

There are four types of values:

Values can be compared with the =, /=, <, >, <=, >= operators. Nodes cannot be represented in a program, but they can be created by imperative rules. Nodes are ordered by creation time.


A rule has two sides, the query and the assertion. They are separated by either the logical arrow ~> or the imperative arrow =>; this marks the rule type. In either case, the query and assertion are both a series of clauses separated by commas.

A query represents a pattern matching procedure. If the procedure can be successfully applied to the database, the rule is said to match, and the assertion indicates a certain transformation to perform. Query matches are computed the same way for either rule type.


Each clause of a query is either a pattern or a constraint.


Pattern clauses bind instances of tuples in the database. When they match, they produce variable bindings in the scope of the rule.

Suppose r/n is some relation. Then

r x1 x2 ... xn,

where x1 through xn are literal values or identifiers, is a pattern which matches all tuples in relation r/n which unify with the pattern. The variables are bound to the values in the matched tuple. If a variable appears at more than one location in a query, it indicates an equality constraint. The special identifier _ matches without binding.


NOTE: given n ≠ m, r/n and r/m are considered distinct relations.

Special Patterns

Negation: !

!r x1 ... xn

When r is a logical relation, !r matches exactly when there are no proofs of the fact present. It requires that the variables be bound elsewhere by the query.

Restriction: .

.event x, p x, r x y, ...

A tuple pattern preceded by . prevents new tuples for other relations in the query from causing a match. In the above example, the rule will only match when a new event tuple is observed.

More semantically: given a match for the dotted clauses of a query, in order to extend it to a full match, only older tuples can be used.

Consumption: ..

..event x, p x, r x y, ...

Marking a tuple pattern with .. makes it a linear pattern. A successful match of a query will consume any tuples bound by linear patterns. They are removed from the database and cannot match any pattern in the future. Linear patterns are only allowed in imperative rules.

example (Go):

dying s, ..stone s l _ => empty l 'black, empty l 'white

This query matches any stone s that has been marked as dying. Once the query matches and the rule is selected for application, it consumes the stone tuple that was matched.


x = y, x /= y, x < y, x <= y, x > y, and x >= y are constraints, with x and y algebraic expressions. An algebraic expression is either a number literal, a variable identifier bound to a number, or a parenthesized arithmetic operation (+, -, and * are allowed).

Variables must be bound elsewhere in the query, and a constraint matches if the inequality holds.

See factorial.arrow for an example.


The effect of a rule match is to change the database by adding or removing certain tuples named by the rule.

Events are tuples that belong to an imperative relation; they can be created or consumed by imperative rules. Each has a unique, hidden timestamp that identifies it.

Facts are tuples that belong to a logical relation. They are actively added and removed so as to form a stable model for the program’s logical rules. A fact is true if its tuple is currently present in a database; otherwise it is false.


Imperative rules are marked with the => arrow.

The right side of an imperative rule constructs new tuples. In the Go example above, two tuples in the empty/2 relation are created. The variable l is bound by the query, and the symbols 'black and 'white are symbol literals.

Variables occurring in an assertion are allowed to be free (not bound in the query).

example (Go):

..make-stone loc color, ..empty loc color => stone s loc color

The variable s on the right hand side is unbound. To create this tuple, the interpreter generates a fresh node value and binds it to that slot.

Fresh nodes are guaranteed to be distinct from every other value in the database.


Logical rules are marked by the ~> arrow. The query of a logical rule defines a precondition which is sufficient to conclude some set of facts specified by its assertion. The facts implied by a particular match of a logical rule are called the consequent of the match. The true facts in a database are expected to form a minimal model for its logical rules at any point in time, so if some tuple used by a match later leaves the database, its consequent may become unsupported.

For example, we can define a relation path-to/2 for nontrivial paths in a graph:

adj-to x y ~> path-to x y
adj-to x y, path-to y z ~> path-to x z

Informally, this says “for all x and y which are adjacent, then there is a path from x to y” and “for all x, y, and z such that x and y are adjacent and a path exists from y to z, then there is a path from x to z.”

Assuming adj-to/2 is an imperative relation, its tuples may be added or removed by other rules of the program, causing the path-to relation to grow or shrink.

All relations appearing on the right hand side of a logical rule are defined to be logical relations. They cannot be modified by imperative rules.

Rule Type Summary

A match for an imperative rule may consume events and create new events. Each new event has a unique identity. They persist unless they are consumed.

A match for a logical rule represents a conditional proof of some fact. If its conditions become false, it is retracted. Multiple proofs of a single fact are combined; patterns can only observe a fact’s truth value. If all proofs of a fact are retracted, the consequent fact is also retracted.

Logical rules are useful for computing dynamic properties of objects.

Imperative rules are useful for interacting with external input, describing state machines, and emulating function evaluation.

Rule Evaluation Order

Rules are not generally required to be confluent: the order they are applied may lead to different results. We assume the rules are linearly ordered, generally with logical rules at a higher precedence. The present interpreter fixes a particular rule evaluation order according to the following algorithm:

At a given point in time, the interpreter maintains one copy of

per rule. This structure is used to ensure that each match is eventually processed exactly once.

Control is always granted to the highest precedence rule with non-empty stack. This rule is allowed to process all of its matches: for each tuple in its stack, all ways of binding it in the rule’s query are attempted. Once a tuple has been considered, it is moved from the stack to the rule’s local old-set. New assertions resulting from it are buffered in an output list. Once all of the rule’s new tuples are processed this way, its output list is appended to the global database of processed tuples and propagated to the new-stacks of other rules.

This process repeats until all new-sets are empty; this is a fixed-point. The algorithm is implemented here.

The algorithm guarantees that a rule is not considered until a fixed-point is reached for the rules of higher precedence.


Each tuple is accompanied by a record of the rule and match that created it. This is used internally as part of the truth maintenance process for logical relations.