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Incoming links: Abstract syntax tree
Abstract syntax tree 2026-10-13
An abstract syntax tree is a rooted tree recording the meaningful syntactic pieces of a program and how they fit together. A program identifier such as
x, a literal value such as 2, and a node representing addition have different roles. Their arrangement answers structural questions: which program expression supplies an operand, which name is a variable assignment target, and which construct encloses another. This makes an abstract syntax tree useful to a compiler, an interpreter, and the type system that checks a program.The word "abstract" describes what is omitted. Lexical analysis first recognizes lexical tokens, such as
x, =, 2, +, 3, *, 4. A concrete parse tree can retain grammar derivation structure, including punctuation and intermediate context-free grammar categories such as expression, term and factor. An abstract syntax tree usually keeps the operation and its operands without all those grammar scaffolds. Parentheses can determine its shape without surviving as separate nodes. A parser may construct the abstract syntax tree directly; it need not first allocate a concrete parse tree.An abstract syntax tree records programming language syntax, but does not by itself supply all programming language semantics. In
a+b, the names do not reveal their current values, and + might perform addition of integers or concatenate strings. The programming language rules, name bindings, and operand types determine the meaning. In Python, successfully calling ast.parse also does not guarantee that later compilation will accept the result: parsing return 42 succeeds, but compiling that statement outside a function fails. [ref]ast — Abstract syntax trees, Python 3.14 documentation. ast.parse: parsing performs no scoping checks; the return 42 example
An abstract syntax tree is not machine code. Its nodes describe language constructs. A compiler can check and transform this structure before translating it into executable instructions; an interpreter can evaluate an appropriate tree directly. The little evaluator below demonstrates this distinction without pretending to implement an entire programming language.
Abstract syntax tree precedence example Created 2026-10-13 Updated 2026-10-31
Consider Python programs
x=2+3*4 and x=(2+3)*4. Operator precedence places multiplication inside the right operand of addition in the first abstract syntax tree. The parentheses instead place addition inside the left operand of multiplication in the second. Their right-hand sides evaluate to and . No rule that simply reads all operators left to right would recover both structures. [ref]https://docs.python.org/3.14/reference/expressions.html#operator-precedence. Python language reference, section 6.17, multiplicative operators bind more tightly than additive operators
Two abstract syntax trees for variable assignment: operator precedence gives
. Details. x=2+3*4 the value , while parentheses give x=(2+3)*4 the value 
These are schematic views of Python trees:
Module is omitted, and the Add/Mult and Store markers are written inside their owning nodes. In the actual schema, BinOp has left, op and right fields; Assign has a list of targets and a value. Name(id='x', ctx=Store()) marks the name as a variable assignment target. Reading x in a program expression instead uses Load. A variable assignment is a statement, not an equation asserting that two values are equal. [ref]A restricted recursive evaluator can follow those fields. It accepts only integer literal values and binary addition, subtraction and multiplication; the surrounding variable assignment remains a separate action. Using
type(value) is int excludes Boolean constants, even though Python treats bool as a subclass of int.import ast
OPS = {ast.Add: lambda a, b: a + b,
ast.Sub: lambda a, b: a - b,
ast.Mult: lambda a, b: a * b}
def evaluate(node):
if isinstance(node, ast.Constant) and type(node.value) is int:
return node.value
if isinstance(node, ast.BinOp) and type(node.op) in OPS:
return OPS[type(node.op)](evaluate(node.left), evaluate(node.right))
raise ValueError("outside the integer-expression language")
for source in ("x=2+3*4", "x=(2+3)*4"):
tree = ast.parse(source, optimize=0)
statement = tree.body[0]
print(statement.targets[0].id, evaluate(statement.value))
# x 14
# x 20The children in a rooted tree have ordered roles:
left and right cannot be interchanged. For subtraction, 8-3 gives while 3-8 gives . Operator associativity also matters: 8-3-1 parses as (8-3)-1, giving , whereas 8-(3-1) gives . Here the syntax groups operators to the left; this does not assert the mathematical associativity of subtraction. Operator precedence determines grouping, rather than a general permission to reorder execution. For example, Python evaluates the calls in f()+g()*h() in the order f, g, h, although the multiplication result must be ready before the outer addition. [ref]https://docs.python.org/3.14/reference/expressions.html#evaluation-order. Python language reference, sections 6.16 and 6.17: left-to-right evaluation and left grouping of additive operators
Abstract syntax tree source location 2026-10-13
An abstract syntax tree can retain source locations even while omitting punctuation nodes. Python records one-based line numbers and zero-based UTF-8 byte offsets, with the end offset exclusive. In
x=2+3*4, the multiplication node spans offsets through , and ast.get_source_segment returns 3*4. A copied location on a folded 12 describes the original span, not text newly present in the source. Such metadata supports error messages and editing tools without making an abstract syntax tree a lossless record of formatting. [ref]ast — Abstract syntax trees, Python 3.14 documentation. Node classes: lineno, col_offset, end_lineno and end_col_offset; get_source_segment and copy_location
// Target: computer-science.bigb
Abstract syntax tree transformation 2026-10-13
An abstract syntax tree transformation replaces or rearranges nodes. It can simplify a program expression, insert instrumentation, or translate a language construct into simpler constructs. When intended to preserve programming language semantics, it must preserve the relevant observable behavior, including values, side effects, and failures. A changed tree can be structurally valid while changing the program's meaning.
Abstract syntax tree traversal 2026-10-13
An abstract syntax tree traversal systematically visits its nodes. A depth-first traversal of a tree that processes each child before its parent is called postorder. For the program expression
2+3*4, the schematic postorder is 2, 3, 4, Mult, Add. This order lets an evaluator obtain each operand value before applying its operation. A traversal of the entire Python schema additionally visits marker nodes such as Add and Store; this shorter list concerns only the expression nodes drawn in the figures. Visiting an abstract syntax tree is a structural operation, and is not a general model of execution: a conditional may select only one branch, while a structural visitor examines both. Algorithm W Created 2026-10-14 Updated 2026-10-31
Algorithm W performs type inference by traversing the program's abstract syntax tree, introducing fresh type variables, combining type substitutions obtained by type unification, and generalizing eligible
let bindings. It distinguishes fresh instances of a type scheme from the shared monomorphic types of parameters to a lambda abstraction. Compilation Created 2026-10-13 Updated 2026-10-31
Compilation is the translation process performed by a compiler, which may include checking and transforming an abstract syntax tree. Translation can finish before execution, or happen during execution through just-in-time compilation. Reusing the translated representation can avoid repeated parsing and repeated constant folding when only the inputs change. This does not guarantee a speedup: translation itself costs work, and the choice of representation and execution mechanism matters.
One possible compilation pipeline for
. Details. x*(2+3)-4, with execution from an abstract syntax tree, a stack machine instruction sequence, or machine code
Compilation arithmetic example 2026-10-14
Consider the program expression
x*(2+3)-4 in a deliberately small programming language. Its only program identifier is x, supplied as an integer input. Its literal values are nonnegative decimal numerals, interpreted as integers. Whitespace is ignored between lexical tokens. Its context-free grammar is:expression ::= term (("+" | "-") term)*
term ::= factor ("*" factor)*
factor ::= numeral | "x" | "(" expression ")"
numeral ::= digit digit*
digit ::= "0" | "1" | "2" | "3" | "4"
| "5" | "6" | "7" | "8" | "9"Here
* outside quotation marks means zero or more repetitions. Repeated operators are grouped to the left, specifying operator associativity; term gives multiplication higher operator precedence than addition and subtraction. Parentheses override that grouping. There is no unary minus, variable assignment, function call, or side effect in this language. Evaluation obtains each left operand before its right operand and uses exact integer arithmetic, with no fixed-width overflow.Lexical analysis recognizes
x, *, (, 2, +, 3, ), -, 4. Parsing constructs the concrete abstract syntax tree Sub(Mul(Name(x), Add(Int(2), Int(3))), Int(4)). Parentheses have done their work by making Add a child of Mul; they need not survive as nodes. Constant folding changes this to Sub(Mul(Name(x), Int(5)), Int(4)). It cannot replace Name(x) by a literal value because the input is supplied at execution. The resulting function of the input is .A compiler for our stack machine emits
LOAD x; PUSH 5; MUL; PUSH 4; SUB. LOAD x pushes the current input; PUSH n pushes the integer literal value n. Each binary instruction removes the top item as its right operand, then the next item as its left operand, and pushes the result. The instruction sequence is an intermediate representation that an interpreter can execute; no machine code has been emitted. A compact binary encoding could make it bytecode, but these printed instruction names are only a readable notation.Why does this compilation preserve the result? A structural induction supplies a short proof outline. Translating a literal value or
x appends its value to any initial stack. For a binary abstract syntax tree node, first translating the left child and then the right child appends the two operand values; the matching instruction replaces them by exactly the original operation's result. Previously present stack items remain intact. Constant folding also preserves the result because in the specified integer arithmetic. Together these facts explain agreement for every integer input, rather than merely agreement for one trial. Compiler Created 2026-10-13 Updated 2026-10-14
A compiler translates a program from one representation to another while aiming to preserve its programming language semantics. Its input might be source code or an intermediate representation; its output might be another programming language, bytecode, an intermediate representation, or machine code. Producing machine code is therefore one possible destination, rather than the definition of compilation. For example, the LLVM 20.1.0 tutorial translates an abstract syntax tree into an intermediate representation before introducing execution.[ref]
Compilation can use an abstract syntax tree for checking and transformations such as constant folding before producing machine code. These are useful possibilities, not a compulsory sequence of separate stages for every compiler. The central distinction is between translating a program and executing its operations: a compiler can translate
x*(2+3)-4 without knowing the input value of x, whereas an interpreter needs that value to produce this program expression's result.Different representations expose different questions. Source code preserves what a programmer wrote; an abstract syntax tree exposes grouping; a type system checks permitted combinations; a stack machine makes operand movement explicit. Retaining abstract syntax tree source locations across compilation helps an error message identify the original construct even after constant folding has changed its shape. A malformed
x*(2+)-4 is a programming language syntax error; an unavailable x in the valid example is instead a missing input. Locating the failing rule is more helpful than treating both as an unexplained execution failure. Intermediate representation 2026-10-14
A program representation used between input and final output, convenient for analysis, transformation or execution. An abstract syntax tree and bytecode can each serve this role.
Interpreter arithmetic trace Created 2026-10-14 Updated 2026-10-31
Use the compilation arithmetic example with input . An abstract syntax tree interpreter first reads
x, obtains , forms , and finally computes . Interpreting the folded abstract syntax tree skips the addition but gives the same result. The compiled stack machine instructions instead produce the following trace; the top of the stack is written on the right:instruction stack after instruction
initial []
LOAD x [7]
PUSH 5 [7, 5]
MUL [35]
PUSH 4 [35, 4]
SUB [31]
Execution of
. Details. x*(2+3)-4 at : the original abstract syntax tree and the folded stack machine instructions both give 
The order in
SUB is essential: reversing its two popped operands would give . The same stored instruction sequence works for input , yielding , without repeating parsing or constant folding. An abstract syntax tree interpreter can likewise reuse its already-parsed tree. Reusing a representation is therefore useful to both approaches; it does not itself distinguish compilation from interpretation.The pleasing part is that a nested rooted tree and a flat instruction sequence express the same computation. The structural induction in the compilation arithmetic example connects them by a precise stack property. This tiny language always terminates and has no side effects; richer programming language semantics require a correctness argument that also accounts for failures, externally visible actions, and possible nontermination.
Interpreter (computing) Created 2026-10-13 Updated 2026-10-14
An interpreter executes a program's operations according to its programming language semantics. It can recursively evaluate an abstract syntax tree, or follow an instruction sequence using an execution state such as a stack and an instruction pointer. It need not repeatedly read source code characters, and it need not use an abstract syntax tree at all.
Compilation and interpretation can coexist. In CPython, Python source code is compiled into bytecode for execution by an interpreter; its bytecode is not promised to remain stable between releases.[ref]
Glossary, Python 3.14 documentation. Python 3.14 documentation; Python 3.14 glossary: bytecode and virtual machine entries; compilation, execution and release-specific bytecodes
The Python 3.14 compile function accepts text or an abstract syntax tree and can produce a code object, which eval or exec can execute.[ref]Built-in Functions, Python 3.14 documentation. Python 3.14 documentation; Python 3.14 Built-in Functions: compile, accepted source forms, code objects and execution by eval or exec
These operations make translation and execution separately accessible without requiring them to live in different applications.Just-in-time compilation gives another combination: an interactive implementation can translate a submitted program expression into machine code and then execute it immediately. The LLVM 20.1.0 tutorial demonstrates this by adding an intermediate representation module to its just-in-time compilation facility, looking up the generated function, and calling it.[ref]
4. Kaleidoscope: Adding JIT and Optimizer Support. LLVM 20.1.0 documentation; Implementing a language with LLVM, chapter 4, section 4.4: HandleTopLevelExpression, addModule, lookup and calling the generated native function
Thus calling a programming language permanently "compiled" or "interpreted" hides an implementation choice: ask which representation is translated, which is executed, and when. Machine code 2026-10-13
Encoded instructions executed by a central processing unit. An abstract syntax tree instead describes programming language constructs and their nesting.
Occurs check 2026-10-14
The occurs check tests whether a type variable occurs inside the expression proposed as its replacement during type unification. After discarding a trivial equation such as , solving by a type substitution requires that not occur in . Thus is permitted when and are distinct, but is rejected in our finite simple type grammar.
There is a short reason for the rejection. If a finite replacement for existed, with replacement for , the equation would become . The right side's abstract syntax tree has one new arrow node plus the nodes of and , so it has strictly more nodes than . Equal finite type expressions cannot have different sizes. Recursive types are a different design: they explicitly describe self-reference and specify how it interacts with typing judgments.
Operator precedence 2026-10-14
The syntactic priority that resolves grouping when different operators occur together without parentheses. It determines the abstract syntax tree shape before evaluation.
Parser 2026-10-13
A parser recognizes how text fits a programming language syntax and can construct a parse tree or abstract syntax tree.
Past exam of the computer science course of the University of Oxford 2015 Compilers 1 a Solution 2026-10-31
Use an abstract syntax tree with the following constructors. Here
Expr denotes the separate abstract syntax tree type for a program expression, Name identifies a variable assignment target, and the square brackets denote an ordered list.Stmt = Assign(Name, Expr)
| Seq(Stmt, Stmt)
| Case(Expr, [(Expr, Stmt)])The list in
Case must be nonempty. Its first field is the controlling program expression; each pair contains a label program expression and the corresponding body. Keep the pairs in source order and keep repeated labels: this is an ordered search, with the first successful equality test choosing the body. A label is an abstract syntax tree for a program expression, not a previously evaluated integer or a key in a map data structure. Evaluating or reordering labels during compilation would change the specified programming language semantics.Seq(s,t) executes s before t. The delimiters begin and end need no surviving constructor: the parser uses them to determine the abstract syntax tree's grouping and then returns the enclosed tree. There are no declarations in these blocks that would require a separate binding environment. A list of statements could replace binary Seq; below we choose right-associated binary trees so that the semantic actions are precise.The essential representation is an ordered, nonempty list of expression/body pairs, together with assignment and sequencing nodes. In constructor notation,
Add the abstract syntax tree alternative
Fix Ident Expr. Its syntactic sugar expansion and the semantic equations from part (a) givewhere and . The final step uses left identity from the monad laws and the lookup rule for the newly bound identifier.eval (Fix x e@(Lambda xs body)) rho = result v
where
rho' = extend rho [(x,v)]
v = close e rho'Equivalently, spelling out programming closure construction,The result is a recursive closure, not the result of calling its body. Its self-binding is visible inside that body but does not extend the caller's surrounding semantic environment.
Add
Fixes [Ident] [Expr] to the abstract syntax tree. For the stipulated nonempty, equally long lists of distinct names and lambda abstractions, define all programming closures against one common extended semantic environment:eval (Fixes xs es) rho = result (head vs)
where
rho' = extend rho (zip xs vs)
vs = map (\e -> close e rho') esExpanding the auxiliary function, if ,Every body can therefore call every member of the family, including itself. Constructing the family performs no body computations in the arbitrary computational monad. As in part (c), host lazy evaluation ties the recursive semantic environment; a monadic fixed-point operator is unnecessary.
For example, with two bodies implementing evenness and oddness, respectively, each programming closure sees both names in . Returning the first programming closure still retains access to the second. Building each programming closure in an independently extended old semantic environment would lose this mutual recursion. For , the definition agrees with
Fix. Past exam of the mathematics course of the University of Cambridge 2007 ia Paper 5 10 a Solution Created 2026-09-24 Updated 2026-10-14
Represent the two constructors of the type grammar by subclasses of The nested classes are public for use by clients, while their fields are private;
MLType: a TypeVar node and an Arrow node containing two types. Use immutable fields so a completed type expression cannot acquire cycles through later mutation. The static factory allocates a fresh object and gives it a fresh identifier shared across the whole hierarchy:import java.math.BigInteger;
public abstract class MLType {
private MLType() { }
private static BigInteger next = BigInteger.ONE;
public static synchronized TypeVar createNewTypeVar() {
TypeVar answer = new TypeVar(next);
next = next.add(BigInteger.ONE);
return answer;
}
public static final class TypeVar extends MLType {
private final BigInteger id;
private TypeVar(BigInteger id) { this.id = id; }
}
public static final class Arrow extends MLType {
private final MLType from, to;
public Arrow(MLType from, MLType to) {
if (from == null || to == null)
throw new IllegalArgumentException();
this.from = from; this.to = to;
}
}
}TypeVar can be created only through the factory. Client code uses MLType.TypeVar and MLType.Arrow. A type variable is identified by its unique node, and a function type by its domain and codomain nodes; repeated occurrences of one logical variable share the same TypeVar object.The monotonically increasing
BigInteger identifier avoids fixed-width counter overflow, and synchronization prevents two simultaneous calls from receiving the same identifier. Actual allocation remains subject to available memory. This is an abstract syntax tree representation, possibly sharing subexpressions. Fresh variable nodes and recursive arrow nodes represent every type in the given grammar.