# The Mathematical Syntax of Small-step Operational Semantics

This writing was inspired by Chapter 2 of the excellent book Understanding Computation. It is available free online, and is a fine piece of writing. My intent here is to introduce the mathematics that the book leaves as an exercise to the reader, and to further explore the fascinating world of small-step semantics.

Small-step semantics is one way (among many) to rigorously define how a program should execute, including how and when expressions are evaluated, how the environment is manipulated, and so on.

This chapter introduces a fairly standard mathematical notation devised by Gordon Plotkin in 1981 for expressing small-step semantics. It looks scary if you are not a mathematician, but don't worry! We'll work through it in small steps of our own.

A program can be considered as the pairing of an expression and an environment. The idea of small-step semantics is to repeatedly "reduce" a program by applying tiny transformations (rules) that modify either the expression, the environment, or both, until a program is reached that cannot be reduced any further. This is a useful way of describing a programming language in an umambiguous manner.

Let's explore this concept with Ruby. To make things more readable in the code that follows, `[]` is aliased to `new` for all classes.

```class Class
def [](*args); new(*args) end
end```

Now consider a program that consists of a single constant number and an empty environment. This is atomic, and cannot be reduced.

```Number = Struct.new(:value) do
def to_s;       value.to_s end
def inspect;    "«#{self}»" end
def reducible?; false end
end

exp, env = Number, {} # => [«10», {}]```

Not particularly interesting, and so far no math! Let's create an expression type that can be reduced: addition.

```Add = Struct.new(:left, :right) do
def to_s;       "%s + %s" % [left, right] end
def inspect;    "«#{self}»" end
def reducible?; true end
def reduce(env)
end
end

exp, env    = Add[Number, Number], {} # => [«1 + 2», {}]
exp, env, _ = exp.reduce(env)               # => [«3», {}, :addition]```

An addition reduces to a number, also returning the particular transformation step that was used. We can define this more explicitly as a "rule". A rule contains a check for whether it should be applied to the given expression, and the actual logic to actually transform it if so.

```class Add
def self.rules
end

def reduce(env)
rule = self.class.rules.detect {|r| r.apply?(self) }
rule.apply(self, env) << rule
end
end

def apply?(*args); true end
end
end```

More code for the same output, but now a list of rules for the system - our operational semantics - can be printed.

```def format_rule(rule)
rule.class
end

def print_all_rules
puts format_rule(rule)
end
puts
end
print_all_rules```
``AddValues``

These rules are what Plotkin (back in 1981) devised a consistent notation for. The add values rule can be expressed as a transition from one expression and environment to another.

```class Add
def self.labels;    [:x, :y] end
def self.prototype; new(*labels) end
end

def antecedent; '' end
def to_s;   '<%s, σ> → <z, σ>'             % expression_type.prototype end
def clause; 'if z is the sum of %s and %s' % expression_type.labels end
end

def format_rule(rule)
if rule.antecedent.empty?
"%s %s" % [rule, rule.clause]
else
"%s : %s %s" % [rule.antecedent, rule, rule.clause]
end
end

print_all_rules```
``<x + y, σ> → <z, σ> if z is the sum of x and y``

In a sentence: "`x + y` reduces to the sum of `x` and `y` and does not change the environment." (σ = environment because math.)

Now we can continually reduce an expression in tiny steps, printing out the exact mathematical rule that was applied at each step.

```def print_line(exp, env, rule)
puts "%15s | %s" % [exp.inspect + ', ' + env.inspect, rule]
end

def evaluate(exp, env = {})
print_line exp, env, nil
while exp.reducible?
exp, env, rule = exp.reduce(env)
print_line exp, env, format_rule(rule)
end
rescue => e
puts e.message
ensure
puts
end

``````«1 + 2», {} |
«3», {} | <x + y, σ> → <z, σ> if z is the sum of x and y``````

Our operational semantics now contains only one rule: that of addition. What happens in the following case?

`evaluate Add[ Number, Add[ Number, Number ] ]`
``````«1 + 2 + 3», {} |
undefined method `value' for «2 + 3»:Add``````

An error! We have not defined a rule that can handle this case yet. It may seem obvious what the correct behaviour is, but small-step semantics is all about unambiguously defining even the tiniest step in our program, so we need to explictly answer the quest: What should happen if one side of the addition is an expression rather than a number?

```class Add
def self.rules
[
ReduceArgument[self, 0],
ReduceArgument[self, 1],
]
end
end

ReduceArgument = Struct.new(:expression_type, :n) do
def apply?(exp); exp[n].reducible?  end
def apply(exp, env)
r =  exp.each.map.with_index {|x, i|
i == n ? x.reduce(env) : x
}
[exp.class.new(*r), env]
end

def antecedent
label = expression_type.labels[n]
"<%s, σ> → <%s, σ>" % [label, "#{label}'"]
end
def to_s
'<%s, σ> → <%s, σ>' % [
expression_type.prototype,
expression_type.new(*expression_type.labels.map.with_index {|label, i|
i == n ? "#{label}'" : label
})
]
end
def clause; '' end
end

``````«1 + 2 + 3», {} |
«1 + 5», {} | <y, σ> → <y', σ> : <x + y, σ> → <x + y', σ>
«6», {} | <x + y, σ> → <z, σ> if z is the sum of x and y``````

There is now a middle step that evaluates the nested addition. In a sentence, the mathematical notation translates to: "if expression `y` can reduce to `y'` without changing the environment, then `x + y` reduces to `x + y'` and does not change the environment." The "does not change the environment" clause (remember σ = environment) is redundant at the moment, but it will become useful later.

In this case we have chosen to first try and evaluate the left side of the addition, then the right. We will see examples later on that would result in a different end state if the right side was evaluated first - this is why spelling out even tiny assumptions is important!

We have been using, and will continue to use, a condensed horizontal form of the mathematical notation since is much easier to include beside sample executions. In case you come across it in the wild, here is the complete operational semantics for our addition engine, using a more vertical formatter.

```def print_rule_vertical(rule)
puts rule.antecedent.to_s
puts "―" * rule.to_s.length + ' ' + rule.clause
puts rule.to_s
puts
end

``````<x, σ> → <x', σ>
――――――――――――――――――――――――
<x + y, σ> → <x' + y, σ>

<y, σ> → <y', σ>
――――――――――――――――――――――――
<x + y, σ> → <x + y', σ>

――――――――――――――――――― if z is the sum of x and y
<x + y, σ> → <z, σ>``````

In the next section we will introduce environment modification to our semantics and see where that leads us.

h