CmSc 315 Programming languages
Chapter 7 . Semantics
1. Introduction
Semantics: What is the meaning of a program?
Motivation:
To provide an authoritative definition of the meaning of all language constructs for:
1. Programmers
2. Compiler writers
3. Standards developers
A programming language is complete only when its syntax, type system, and semantics
are well-defined.
Several approaches to semantics
 Grammatical models: grammar rules are paired with semantic rules.
Resulting grammars are called attribute grammars
Rule
EE+T
ET
TTxP
TP
PI
P  (E)
Attribute
value(E1) = value(E2) + value(T)
value(E) = value(T)
value(T1) = value(T2) x value(P)
value(T) = value(P)
value(P) = value of number I
value(P) = value(E)
Attribute grammars, that associate with each non-terminal in the grammar a set of
attributes, are one of the earliest semantic models (Knuth, Donald E., Semantics of
context-free languages, Mathematical Systems Theory 2 2 :127-145, 1968.) and they are
still in use. Their most beneficial feature is that they can be used for efficient automatic
translation. However, attribute grammars are not sufficiently powerful to represent the
entire semantics of programming languages - they are too tightly coupled with parse
trees.

Operational models
Describe the meaning of the language constructs in terms of machine states, i.e. memory
and register contents
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
Denotational semantics
The meaning of programs is described in terms of mathematical functions on programs and
program components. Programs are translated into functions about which properties can be
proved using the standard mathematical theory of functions. Based on Lambda calculus.
(Scott, D.S. and Strachey, C., Towards a mathematical semantics for computer languages,
in Proc. Symp. Computers and Automata , pages 19-46. Polytechnic Press, NY, 1971.)

Axiomatic models
Describe the meaning as pre-conditions and post-conditions
Used in program verifications

Specification models
Describe the relationship among various functions that implement a program.
e.g. pop(push(S,x)) = S
2. Operational semantics
Sequence control : the control of the order of execution of the operations both primitive
and user defined.
Implicit: determined by the order of the statements in the source program or by
the built-in execution model
Explicit: the programmer uses statements to change the order of execution (e.g.
uses If statement)
Levels of sequence control
Expressions: How data are manipulated using precedence rules and parentheses.
Statements: conditional and iteration statements change the sequential execution.
Declarative programming: an execution model that does not
depend on the order of the statements in the source program.
Subprograms: transfer control from one program to another.
3. Sequencing with expressions
The issue: given a set of operations and an expression involving these operations,
what is the sequence of performing the operations?
How is the sequence defined, and how is it represented?
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An operation is defined in terms of an operator and operands.
The number of operands determines the arity of the operator.
Basic sequence-control mechanism: functional composition
Given an operation with its operands, the operands may be:
 Constants
 Data objects
 Other expressions
Example:
3 * (var1 + 5)
operation - multiplication, operator: *, arity - 2
operand 1: constant (3)
operand 2: operation addition
operand1: data object (var1)
operand 2: constant (5)
Functional compositions imposes a tree structure on the expression, where we
have one main operation, decomposable into an operator and operands.
In a parenthesized expression the main operation is clearly indicated. However we
may have expressions without parentheses.
Example 2: 3* var1 +5
Question: is the example equivalent to the above one?
Example 3: 3 + var1 +5
Question: is this equivalent to (3 + var1) + 5, or to 3 + (var1 + 5) ?
In order to answer the questions we need to know:
 Operator's precedence
 Operator's associativity
Precedence concerns the order of applying operations, associativity deals with the
order of operations of same precedence.
Precedence and associativity are defined when the language is defined - within the
semantic rules for expressions.
3.1. Arithmetic operations / expressions
In arithmetic expressions the standard precedence and associativity of operations
is applied to obtain the tree structure of the expression.
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Linear representation of the expression tree:



Prefix notation (- ( + a b) (* c d))
Postfix notation a b + c d * Infix notation a + b – c * d
Postfix notation is parentheses-free. Problems with unary operators
Cambridge prefix notation: operator followed by operands. The expression is
enclosed in parentheses. Used in LISP and Scheme
There are algorithms to evaluate prefix and postfix expressions and algorithms to
convert an infix expression into prefix/postfix notation, according to the operators'
precedence and associativity.
History:
In the 1920's, Jan Lukasiewicz (1878-1956) developed a formal logic system which
allowed mathematical expressions to be specified without parentheses by placing the
operators before (prefix notation) or after (postfix notation) the operands. For example
the (infix notation) expression
(4 + 5) × 6
could be expressed in prefix notation as
× 6 + 4 5
or
× + 4 5 6
and could be expressed in postfix notation as
4 5 + 6 ×
or
6 4 5 + ×
Prefix notation also came to be known as Polish Notation in honor of Lukasiewicz.
Postfix notation became known as RPN: Reversed Polish Notation
From these ideas, Charles Hamblin (Australian philosopher and computer scientist)
developed a postfix notation for use in computers. Hamblin's work on postfix notation
was in the mid-1950's. Calculators, notably those from Hewlett-Packard, used various
postfix formats beginning in the 1960s.
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3.2. Other expressions
Languages may have some specific operations, e.g. for processing arrays and
vectors, built-in or user defined. Precedence and associativity still need to be
defined - explicitly in the language definition or implicitly in the language
implementation.
3.3. Execution-time representation of expressions
Machine code sequence
Tree structures - software simulation
Prefix or postfix form - requires stack, executed by an interpreter.
3.4. Evaluation of tree representation
Eager evaluation - evaluate all operands before applying operators.
Lazy evaluation
Problems:
Side effects - some operations may change operands of other operations.
Error conditions - may depend on the evaluation strategy (eager or lazy
evaluation)
Boolean expressions - results may differ depending on the evaluation strategy.
Example if(a < b < c)
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