Semantics with Applications

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Semantics with Applications

• Mooly Sagiv
• msagiv_at_post
• Schrirber 317
• 03-640-7606
• html//www.cs.tau.ac.il/msagiv/courses/sem08.htm
  l
• TextbooksWinskel
• The Formal Semantics of Programming
  Languages
• Types and Programming Languages Benjamin C.
  Pierce

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Outline

• Course requirements
• What is semantics
• Who needs semantics
• Forms of semantics
• Tentative Plan
• Trace semantics
• Introduction to operational semantics

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Course Requirements

• Prerequisites
• Compiler Course
• Basic set theory and logic
• A theoretical course
• Forms of induction
• Domain theory
• No algorithms
• Grade
• Course Notes 10
• Assignments 60
• Mostly theoretical with some programming
• Home exam 30

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Modern Programming Languages

• Imperative
• PL/1
• Pascal
• C
• Object Oriented
• C
• Java
• C
• Functional
• Scheme
• ML
• Ocaml
• F
• Haskel
• Logic
• Prolog

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Programming Languages

• Syntax
• Which string is a legal program?
• Usually defined using context free grammar
  contextual constraints
• Semantics
• What does a program mean?
• What is the output of the program on a given run?
• When does a runtime error occur?
• A formal definition

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Benefits of Formal Semantics

• Programming language design
• hard-to-define hard-to-implementhard-to-use
• Avoid design mistakes
• Programming language implementation
• Compiler Correctness
• Correctness of program optimizations
• Design of Static Analysis
• Programming language understanding
• Program correctness
• Type checking
• Program equivalence
• Automatic generation of interpreter
• Techniques used in software engineering

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Desired Features of PL Semantics

• Tractable
• as simple as possible without losing the ability
  to express behavior accurately
• Abstract
• uncluttered by irrelevant detail
• Computational
• an accurate abstraction from runtime behavior
• Compositional
• The meaning of compound language construct is
  defined using the meaning of subconstructs
• Supports modular reasoning

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Alternative Formal Semantics

• Operational Semantics Plotkin, Kahn
• The meaning of the program is described
  operationally
• Trace based Semantics
• Structural Operational Semantics
• Natural Semantics
• Denotational Semantics Strachey, Scott
• The meaning of the program is an input/output
  relation
• Axiomatic Semantics Floyd, Hoare
• The meaning of the program is observed properties
• Proof rules to show that the program is correct
• Complement each other

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Tentative Plan

• A simple programming language IMP
• Natural Semantics of IMP
• Structural operational Semantics of IMP
• Denotational Semantics of IMP
• Axiomatic Semantics
• IMP
• Non-Determinism and Parallelism
• Rely Guarantee Axiomatic Semantics
• Separation Logic
• Type inference/checking

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IMP A Simple Imperative Language

• numbers N
• Positive and negative numbers
• n, m ? N
• truth values Ttrue, false
• locations Loc
• X, Y ? Loc
• arithmetic Aexp
• a ? Aexp
• boolean expressions Bexp
• b ? Bexp
• commands Com
• c ? Com

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Abstract Syntax for IMP

• Aexp
• a n X a0 a1 a0 a1 a0 ? a1
• Bexp
• b true false a0 a1 a0 ? a1 ?b b0
  ?b1 b0 ? b1
• Com
• c skip X a c0 c1 if b then c0
  else c1 while b do c

23?4-5
(2(3?4))-5
((23)?4))-5
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Example Program
Y 1 while ?(X1) do Y Y X X X - 1
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But what about semantics
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Trace Based Semantics

• For every program P define a set potential states
  ?(P)
• Let ? be the set of finite and infinite traces
  over ?
• ? ?(P) ? ?(P)?
• The meaning of P is a set of maximal traces ?P??
  ?

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Example Program
pc?1, x ?2 pc?2, x ?2 pc?3, x ?2 pc?2, x
?1 pc?3, x ?1 pc?2, x ?0 pc?4, x ?0
..
pc?1, x ?-7 pc?2, x ?-7 pc?4, x ?-7
1 while 2(Xgt0) do 3X X 1 4
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Example Program
pc?1, x ?2 pc?2, x ?2 pc?3, x ?2 pc?2, x
?2 pc?3, x ?2 pc?2, x ?2 pc?3, x ?2 ?
..
1 while 2(true) do 3 skip 4
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Limitations of trace based semantics

• The program counter is an implementation detail
• Equivalent programs do not necessarily have the
  same set of traces
• Hard to define semantics by induction on the
  syntax
• Hard to prove properties of the programming
  language

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Chapter 2

• Introduction to
• Operational Semantics

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Expression Evaluation

• States
• Mapping locations to values
• ? - The set of states
• ? Loc ? N
• ?(X) ?Xvalue of X in ?
• ? X ? 5, Y ? 7
• The value of X is 5
• The value of Y is 7
• The value of Z is undefined
• For a ? Exp, ? ??, n ? N,
• lta, ?gt ? n
• a is evaluated in ? to n

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Evaluating (a0 a1) at ?

• Evaluate a0 to get a number n0 at ?
• Evaluate a1 to get a number n1 at ?
• Add n0 and n1

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Expression Evaluation Rules

• Numbers
• ltn, ?gt ? n
• Locations
• ltX, ?gt? ?(X)
• Sums
• Subtractions
• Products

Axioms
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Derivations

• A rule instance
• Instantiating meta variables with corresponding
  values

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Derivation (Tree)

• Axioms in the leafs
• Rule instances at internal nodes

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Computing a derivation

• We write lta, ?gt ? n when there exists a
  derivation tree whose root is lta, ?gt ? n
• Can be computed in a top-down manner
• At every node try all derivations in parallel

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Recap

• Operational Semantics
• The rules can be implemented easily
• Define interpreter
• Natural semantics

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Equivalence of IMP expressions
iff
a0 ? a1
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Boolean Expression Evaluation Rules

• lttrue, ?gt ? true
• ltfalse, ?gt ? false

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Boolean Expression Evaluation Rules(cont)

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Equivalence of Boolean expressions
iff
b0 ?b1
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Extensions

• Shortcut evaluation of Boolean expressions
• Parallel evaluation of Boolean expressions
• Other data types

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The execution of commands

• ltc, ?gt ? ?
• c terminates on ? in a final state ?
• Initial state ?0
• ?0(X)0 for all X
• Handling assignments ltX5, ?gt ? ?

• ltX5, ?gt ? ?5/X

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Rules for commands

• ltskip, ?gt ? ?
• Sequencing
• Conditionals

Atomic
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Rules for commands (while)
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Example Program
Y 1 while ?(X1) do Y Y X X X - 1
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Equivalence of commands
iff
c0 ?c1
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Proposition 2.8
while b do c ? if b then (c while b do c) else
skip
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Small Step Operational Semantics

• The natural semantics defines evaluation in large
  steps
• Abstracts computation time
• It is possible to define a small step operational
  semantics
• lta, ?gt ?1 lta, ?gt
• one step of executing a in a state ? yields a
  in a state ?

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SOS for Additions
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SOS Rules for commands

• ltskip, ?gt ? 1 ?
• Sequencing

Atomic
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SOS Rules for commands

• Conditionals

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SOS rules for while
ltwhile b do c, ?gt ?1 lt if b then (c while b do
c) else skip, ? gt
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Summary

• Operational semantics enables to naturally
  express program behavior
• Can handle
• Non determinism
• Concurrency
• Procedures
• Object oriented
• Pointers and dynamically allocated structures
• But remains very closed to the implementation
• Two programs which compute the same functions are
  not necessarily equivalent