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# Informal Concept of Syntax

## Surface, Concrete Syntax

How phrase are entered and displayed on computer.

Usually thought of as given by strings of characters from some alphabet.

## Structured, Abstract Syntax

The structure of phrases, specifically how they are composed from other phrases.

# Finite Tree Structure of Syntax

We consider pieces of syntax to be **finite trees** augmented with a means of expressing **binding** and **scope** of identifiers within syntax tree.

We define “**a piece of syntax**“ in 2 stages:

- The definition of
**abstract syntax tree**. Augment abstract syntax tree with means of specifying:

**binding**(declaration)**scope**(range of significance)

of an identifier.

# Abstract Syntax Tree (AST)

An ordered tree whose:

- leaves are
**variables** - interior nodes are
**operators**whose**arguments**are its children.

## Sorts

AST’s are classified into a variety of **sorts** corresponding to **different forms of syntax**.

## Variable

A variable stands for an unspecified, or generic, piece of syntax of a specified sort.

## Operator

Operator has an **arity** specifying:

- The sort of the operator.
- The number and sorts of its arguments.

e.g. $<s_1,\dots, s_n>s$ as an arity for the operator.

## Structural Induction

A method when we want to prove some property $P(a)$ holds for all $a$ AST of a given sort.

- Consider all the methods in which AST $a$ of this sort can be generated.
- Assume $P(t)$ holds for all of $a$’s constituent AST’s.
- We prove that the $P(a)$ holds in each case under the previous assumption.

# AST Precise Definitions

- $S$ is a finite set of
**sorts**. - An
**arity**has the form $(s_1,\dots,s_n)s$, with $\forall s, s_i \in S$ and $n \geq 0$. - $O =\lbrace O_\alpha \rbrace$ is an
**arity-indexed**family of disjoint sets of**operators**$O_\alpha$ of arity $\alpha$. $X = \lbrace X_s\rbrace_{s\in S}$ is a

**sort-indexed**family of disjoint finite sets $X_s$ of**variables**$x$ of sort $s$.- If $x \in X_s$, we say $x$ is of sort $s$.
- If $x \notin X_s$ and $x$ is of sort $s$, then we say $x$
**is fresh for**$X$. - If $x$
**is fresh for**$X$ and $s$ is a sort, then $X, x$ is a family of sets of variables by adding $x$ to $X_s$.

$A[X]$ is a family of

**abstract syntax trees**with the form of $A[X] = \lbrace A[X] \rbrace_{s\in S}$ with each sort $s$ of $A[X]_s$ is the smallest family satisfying- If $x \in X_s$ then $x \in A[X]_s$ $\Rightarrow$ A variable of sort $s$ is an AST of sort $s$.
- If $o$ is of arity $(s_1, \dots, s_n)s$ and if $a_i \in A[X]_{s_i}$, then $o(a_1, \dots, a_n) \in A[X]_s$ $\Rightarrow$ Operators combines AST’s.

**Structural Induction**: to show $P(a)$ holds $\forall a \in A[X]$- If $x \in X_s$, then $P_s(x)$.
- If $o$ is of arity $(s_1, \dots, s_n)s$ and $P_{s_i}(a_i)$, then $P_s(o(a_1, \dots, a_n))$.

**Substitution**gives variables meanings.If $a \in A[X, x]_{s’}$ and $b \in A[X]_s$, then $\lbrack b / x \rbrack a \in A \lbrack X \rbrack _{s’}$.

- $\lbrack b / x \rbrack x = b$ and $\lbrack b / x \rbrack y = y$ if $x \neq y$.
- $\lbrack b / x \rbrack o(a_1, \dots, a_n) = o(\lbrack b / x\rbrack a_1, \dots, \lbrack b / x\rbrack a_n)$.

**Theorem**: $a \in A\lbrack X, x\rbrack$, then $\forall b \in A\lbrack X \rbrack$, there exists**unique**$c \in A[X]$ such that $\lbrack b / x \rbrack a = c$.

# Abstract Binding Tree (ABT)

## Binding and Scope

**Binding** is a method to introduce new **variables and symbols** to enrich ast’s to abt’s.

**Binding** has specified range of significance, called **scope**.

The **scope** of a **binding** is an abt within which the **bound identifier** can be used.

Example: $let$ $x$ $be$ $a_1$ $in$ $a_2$.

The variable $x$ is introduced within expression $a_2$ to stand for $a_1$.

$x$ is bounded by the $let$ expression within $a_2$.

Any use of $x$ within $a_1$ refers to a different variable that happens to have the same time.

## Names of Bound Variables

The **name of bound variables** are not so important to the extent that **they determine the same binding**.

## Abstractor

ABT’s generalize AST’s by allowing an **operator** to **bind any finite number of variables (including zero)** in each argument.

**Argument** to an **operator** is called an **abstractor** with the form $x_1, \dots, x_k.a$.

The sequence of variables $x_1, \dots, x_k$ are bound within the abt $a$.

Example: $let$ $x$ $be$ $a_1$ $in$ $a_2$ can be written in the form of $let(a_1; x.a_2)$.

## Some Shorthand Notations

$\vec{x}$ often stands for $x_1,\dots,x_n$ and $\vec{x}.a$ for $x_1, \dots, x_k.a$.

## Generalized Arity and Valence

Operators are assigned **generalized arities** to account for **binding** with the form $(v_1, \dots, v_n)s$.

It specifies **operator** of sort $s$ with $n$ arguments of **valence** $v_1, \dots, v_n$.

In general, a **valence** has form $s_1, \dots, s_k .s$ specifies the **sort of an argument** as well the **number and sorts** of the variables bound within it.

A sequence $\vec{x}$ of variables is of sort $\vec{s}$.

Example: $let$ is of arity

`(Exp, Exp.Exp) Exp`

.It has 2 arguments:

- The first one is an
`Exp`

.- The second one is an abstractor that binds one
`Exp`

.

# ABT Precise Definitions

- Fix a set $S$ of sorts.
- $O =\lbrace O_\alpha \rbrace$ is an
**arity-indexed**family of disjoint sets of**operators**$O_\alpha$ of generalized arity $\alpha$. - For a given family of
**disjoint sets of variables**$X$, abt’s $B[X]$’s definition is similar, except for the $X$ is not fixed since it may**change as we enter scopes of abstractors**. - If $x \in X_s$, $x \in B\lbrack X\rbrack_s$.
- $\forall o$ with arity $(\vec{s_1}.s_1, \dots, \vec{s_n}.s_n)s$, if $\forall 1 \leq i \leq n$ $a_i \in B[X, \vec{x_i}]_{s_i}$, then $o(\vec{x_1}.a_1, \dots, \vec{x_n}.a_n) \in B[X]_s$.

## Some Flaws and Counter Examples

Check this: $let(a_1; x.let(a_2; x.a_3))$

It is

ill-formedsince the first binding already add $x$ into $X$.Then the second binding cannot add $x$ into $X, x$ since it is not

freshfor $X, x$.

The flaw comes to **renaming of bound variables**.

## Solution: Fresh Renaming

Ensure that each of the arguments is **well-formed** regardless of the choice of the bound variable names.

**Fresh renaming** (with related to $X$) of a **finite sequence of variables** $\vec{x}$ is a **bijection** $\rho: \vec{x} \leftrightarrow \vec{x}’$ between $\vec{x}$ and $\vec{x}’$ where $\vec{x}’$ is fresh for $X$.

**Notation**: $\hat{\rho}(a)$ for the result for replacing each occurrence of $x_i$ in $a$ by $\rho(x_i)$, its **fresh counterpart**.

## Some Minor Changes

$\forall o$ with arity $(\vec{s_1}.s_1, \dots, \vec{s_n}.s_n)s$, if $\forall 1 \leq i \leq n$ and each fresh renaming $\rho_i: \vec{x_i}\leftrightarrow \vec{x_i}’$, we have $\hat{\rho_i}(a_i) \in B[X,\vec{x_i}’]$, then $o(\vec{x_1}.a_1, \dots, \vec{x_n}.a_n) \in B[X]_s$.

# Structural Induction Modulo Fresh Renaming

If we want to show that $P\lbrack X\rbrack (a)$ holds for every $a \in B[X]$, it is necessary to show:

- If $x \in X$, then $P[X]_s(x)$.
- $\forall o$ of arity $(\vec{s_1}.s_1,\dots,\vec{s_n}.s_n)s$, if $\forall 1 \le i \le n$, $P[X, \vec{x_i}’]_{s_i}(\hat{\rho_i}(a_i))$ holds for every $\hat{\rho_i}:\vec{x_i} \leftrightarrow \vec{x_i}’$ with $\vec{x_i}’\notin X$, then $P[X]_s(o(\vec{x_1}.a_1, \dots, \vec{x_n}.a_n))$.

Example: $x \in a$ where $a \in B [X, x]$, which means $x$

occurs freein $a$.Informally, this means $x$ is bound outside of $a$.

So we should apply the

structural induction modulo fresh renaming:

- $x \in x$
- $x \in o(\vec{x_1}.a_1, \dots, \vec{x_n}.a_n)$ if $\forall 1 \le i \le n$ such that for every $\rho:\vec{x_i}\leftrightarrow\vec{z_i}$ we have $x \in\hat{\rho}(a_i)$.

# $\alpha$-Equivalence and Substitution

The relation $a =_\alpha b$ of **$\alpha$-equivalence** means $a$ and $b$ are identical up to the choice of bound variable names.

- $x = _\alpha x$
- $o(\vec{x_1}.a_1,\dots,\vec{x_n}.a_n) = _\alpha o(\vec{x_1}’.a_1’,\dots,\vec{x_n}’.a_n’)$ if $\forall 1 \le i \le n, \hat{\rho_i}(a_i) =_\alpha \hat{\rho_i’}(a_i’)$, $\forall \rho_i:\vec{x_i}\leftrightarrow\vec{z_i}$ and $\forall \rho_i’:\vec{x_i}’\leftrightarrow\vec{z_i}$.

**Substitution** is defined carefully with the following:

- $\lbrack b / x \rbrack x = b$ and $\lbrack b / x \rbrack y = y$ if $x \ne y$.
- $\lbrack b / x \rbrack o (\vec{x_1}.a_1,\dots, \vec{x_n}.a_n) = o(\vec{x_1}.a_1’, \dots, \vec{x_n}.a_n’)$ where $\forall 1 \le i \le n, \vec{x_i} \notin b$ and:
- $a_i’ = \lbrack b / x \rbrack a_i$ if $x \notin \vec{x_i}$
- $a_i’ = a_i$ if $x \in \vec{x_i}$

Example to explain:

$\lbrack b / x \rbrack let(a_1;x.a_2) = let(\lbrack b / x \rbrack a_1; x.a_2)$

If $x$ is bound by an abstractor within $a$, then $x$ doesn’t

occur freein $a$, then it cannot be changed by substitution.In this example, $x$ is not occur free in $x.a_2$.

If $y \in b$ and $x \ne y$, then $\lbrack b / x \rbrack let(a_1; y.a_2)$ is

undefined.More specifically, $\lbrack b / x \rbrack let(num[0], y.plus(x;y))$ is

undefined.Since you are trying to let an unbounded variance be substituted and confuses two different variables called $y$.

This is called

capture of a free variable during substitution.

Some little note about

capture avoidance:

- Although it is an
essential characteristic of substitution, it is justmerely a technical nuisance.- If the names of bound variables have no significance, then it can be avoided by first renaming the
bound variablesin $a$ toavoid the free variable namesin $b$.Example: $\lbrack b / x \rbrack let(num[0], y.plus(x;y))$ with $y \in b$ and $x \ne y$.

We can set $a’ \triangleq let(num[0], y’.plus(x;y’))$ then $\lbrack b / x \rbrack a’$ is

defined.

# Identification Convention in ABT

**ABT’s are always identified up to $\alpha$-equivalence.**

Explain: Substitution can be extended to $\alpha$-equivalence classes of abt’s:

$\rightarrow$ Choosing equivalence classes of $b$ and $a$ in the way of

substitutionis defined.$\rightarrow$ Then we get the

$\alpha$-equivalenceresults.So

substitutionbecomes a well-defined total function.

# Operators and Symbols

It is necessary to consider language whose **abstract syntax** cannot be specified by **a fixed set of operators**, but rather requires the available operators be **sensitive to the context in which they occur**.

We introduce a set of **symbols** that **index families of operators**.

An **indexed operator** $o$ is **a family of operators** indexed by symbols $u$, so $o\lbrack u\rbrack$ is an operator when $u$ is available symbol.

If $U$ is a finite set of symbols, then $B[U; X]$ is the **sorted-index family of abt’s** generated by **operators** and **variables**, admitting all index operator instances by symbols $u \in U$.

The only significance of a symbol is **whether it is the same as or differs from another symbol**, it doesn’t stand for anything.

## Introducing New Symbols

The **set of symbols** is extended by introducing a **new** or **fresh** symbol within a scope using the abstractor $u.a$, binds the symbol $u$ within abt $a$.

The **name** of the **bound symbol** can be varied at will, provided that **no conflicts arise**.

## Symbol and Variable Difference

The only operation on **symbols** is **renaming**, there’s no notion of **substitution** for a symbol.

# Some Notation For Readability

We often **group** and **stage** the **arguments** to an operator:

- using
**round brackets**and**braces**to show**grouping** - regarding
**stages**to progress**from left to right** - All arguments in a group are considered to occur
**at the same stage**. - the successive groups are considered to occur in
**sequential stages**.

Staging and groupingare helpful but not fundamental significant:$o\lbrace a_1;a_2 \rbrace (a_3;x.a_4)$ is the same as $o(a_1; a_2; a_3;x.a_4)$, as would be any other order-preserving

grouping or stagingof its arguments.

# Exercises For Review

## 1.1

Prove by structural induction on ast that $X \subseteq Y$, then $A\lbrack X \rbrack \subseteq A\lbrack Y \rbrack$.

- $\forall v \in X$, $v \in Y$, thus $\forall v \in A\lbrack X\rbrack$, $v \in A\lbrack Y \rbrack$.
- $\forall v_i \in X$, $v_i \in Y$, thus $\forall o(v_1, \dots, v_n) \in A \lbrack X\rbrack$, $\forall o(v_1, \dots, v_n) \in A \lbrack Y\rbrack$.

## 1.2

Prove by structural induction on abt that $X \subseteq Y$, then $B\lbrack X \rbrack \subseteq B\lbrack Y \rbrack$.

First part is the same as ast’s. Jump to second part.

If $\vec{v}.a \in B\lbrack X\rbrack$, then we need to show that $\vec{v}.a\in B\lbrack Y\rbrack$.

Then $\rho:\vec{v} \leftrightarrow \vec{v}’$, we get $\vec{v} \in X$ but $\vec{v}’ \notin X$.

Thus $\hat{\rho}(a) \in B[X, \vec{v}’]$ and it leads to $\hat{\rho}(a) \in B[Y, \vec{v}’]$.

## 1.3

Provided that:

- $a = _\alpha a’$
- $b =_\alpha b’$

Show that:

- $\lbrack b / x \rbrack a$ and $\lbrack b’/x\rbrack a’$ are defined, then $\lbrack b / x \rbrack a =_\alpha \lbrack b’/x\rbrack a’$.

WLOG, we say $\lbrack b / x \rbrack a = _\alpha \lbrack b / x \rbrack a’$ by identification convention of abt.

Then since $b =_\alpha b’$, we can substitute / rename $b$ into $b’$ on RHS. Thus we get the result.

## 1.4

Problem is toooooooooooo long, check the link.

We set a **bound variable occurrence** represented by $bv[i]$, $i \in \mathbb{Z}^+$ for $i^{th}$ enclosing abstractor.

An **abstractor** for **abg** has a notation $.g$ with $”.”$ for introduction of **an (unnamed) bound variable**.

Index **abg**‘s with $n$ enclosing abstractors by notation $G\lbrack X\rbrack_n$.

If $g \in G\lbrack X\rbrack_n$ then $.g\in G\lbrack X\rbrack_{n+1}$.

Also $\forall 1 \le i \le n$, $bv\lbrack i \rbrack \in G\lbrack X\rbrack _n$.