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Denotational Semantics

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Title: Denotational Semantics


1
Denotational Semantics
  • COS 441
  • Princeton University
  • Fall 2004

2
Denotational Semantics
  • Describe meaning of syntax by specifying the
    mathematical meaning of syntax tree as
  • Function
  • Function on functions
  • Value, such as natural numbers or strings
  • Sets of values etc
  • defined by each construct

3
Original Motivation for Topic
  • Precision
  • Use mathematics instead of English
  • Avoid details of specific machines
  • Aim to capture pure meaning apart from
    implementation details
  • Basis for program analysis
  • Justify program proof methods
  • Soundness of type system, control flow analysis
  • Proof of compiler correctness
  • Language comparisons

4
Denotational vs. Operational
  • Denotational semantics are more abstract than
    operational approaches
  • Cannot reason about number of steps of a
    computation or algorithmic complexity
  • Specify what the answer should be not how a
    computation takes place

5
Principles of DS
  • Compositionality
  • The meaning of a compound program must be defined
    from the meanings of its parts (not the syntax
    of its parts).
  • Examples
  • P Q
  • composition of two functions, state ?
    state
  • letrec f(x) e1 in e2
  • meaning of e2 where f denotes function ...

6
Notation and Definitions
  • Phrase
  • variable, expression, statement, declarations
  • e , P
  • Syntax tree of a phrase
  • E e , C P
  • Meaning functions (E,C) applied to syntax trees
    of phrases

7
Example Binary Numbers
  • Syntax
  • b 0 1
  • n b nb
  • e n e_1 e2
  • Semantics
  • E 0 ? 0
  • E 1 ? 1
  • E nb ? 2 E n E b
  • E e1 e2 ? E e1 E e2

8
DS of Regular Expressions
  • Syntax
  • a,b 2 ?
  • re a re1 re2 re1 . re2 re
  • Semantics
  • E Regular Exp ! ?

9
Sets of Strings
  • Let a,b 2 ?
  • ? is an unspecified alphabet
  • ? is the set of strings made up from the
    alphabet ?
  • ? 2 ?
  • a 2 ? if a 2 ?
  • s1 . s2 2 ? if s1 2 ? and s2 2 ?

10
Equality on Strings
  • Let s1,s2,s3 2 ?

a a
? . s s
s . ? s
(s1 . s2) . s3 s1 . (s2 . s3)
s1 . s2 s1 . s2 if s1 s1 and s2 s2
11
Operations On Sets of Strings
a is x x a
S1 S2 is x x 2 S1 Ç x 2 S2
S1 . S2 is x s1 2 S1 Æ s2 2 S2 Æ x s1 . s2
S0 is ?
Sn1 is x x 2 S . Sn
S is x 9n. x 2 Sn
12
Some Theorems
Ea b ? a , b
Ea . (b c) ? ab, ac
E(a b) . c ? ac, bc
Ere1 . (re2 re3) ? E(re1 . re2) (re1 . re3)
Ea ? Ea
13
DS for Regular Expressions
Ea ? a
Ere1 re2 ? x x 2 Ere1 Ere2
Ere1 . re2 ? x x 2 Ere1 . Ere2
Ere ? x x 2 Ere
14
Another DS for REs
  • We can given an equivalent semantics by relating
    regular expressions to Non-deterministic Finite
    Automata (NFAs)
  • Equivalent means
  • Any theorem about meanings in one semantics is
    true iff the same theorem is true in the other
    model

15
NFA
  • q0,q1,,qn 2 Q States
  • qinit 2 Q Initial State
  • F ½ Q Final States
  • ? ½ Q (? ?) Q Transition Relation
  • M is (qinit,F,?) NFA

16
NFA Accepting a String
  • ? 2 L(qinit,F,?)
  • if (qinit, ?, qf) 2 ? and qf 2 F
  • a 2 L(qinit,F,?)
  • if (qinit, a, qf) 2 ? and qf 2 F
  • x . s 2 L(qinit,F,?)
  • if (qinit, x, q) 2 ? and s 2 L(q,F,?)

17
Semantics of Regular Expressions
  • NFAa ? (qi, qf, (qi,a,qf))

a
qf
qi
18
Semantics of Regular Expressions
  • NFAre1 re2 ?
  • let q be a unique new state
  • let (qi1, F1, ?1) NFAre1
  • let (qi2, F2, ?2) NFAre2
  • let ? ?1 ?2 (q,?,qi1),(q,?,qi2))
  • (q, F1 F2,?)

?
NFAre1
q
NFAre2
?
19
Semantics of Regular Expressions
  • NFAre1 . re2 ?
  • let (qi1, F1, ?1) NFAre1
  • let (qi2, F2, ?2) NFAre2
  • let ? ?1 ?2 (qf,?,qi2) qf 2 F1
  • (qi1, F2,?)

?
NFAre1
NFAre2
20
Semantics of Regular Expressions
  • NFAre ?
  • let q,qf be new states
  • let (qi, F, ?) NFAre
  • let ? ? (q,?,qi), (q,?,qf)
    (qf,?,qi) qf 2 F
  • (q, qf,?)

?
?
q
qf
NFAre
?
21
Some More Theorems
L(NFAa b) ? a , b
L(NFAa . (b c)) ? ab, ac
L(NFA(a b) . c) ? ac, bc
L(NFAre) ? Ere
22
Ere ? L(NFAre)
  • We can prove the NFA semantics is equivalent to
    the set theoretic semantics via induction on the
    definition of the meaning function E
  • Proof relies on compositional definition of
    meaning function E!

23
Which Semantics?
  • Set of string semantics clearly easier to reason
    about!
  • This is what denotation semantics was designed
    for
  • NFA semantics can be transformed into efficient
    of regular expression matcher
  • Our theorem is a correctness proof about the
    implementation of our NFA conversion function

24
DS for Programming Languages
  • We can build a DS for a programming language
  • Allows us to reason about program equivalence
  • Useful to prove compiler optimizations is safe
  • Provides framework for reason about abstract
    properties of programs statically

25
DS of Imperative Programs
  • Syntax
  • n 2 Numbers
  • x 2 Vars
  • e false true n x e1 e2 e1 e2
  • P x e if e then P1 else P2 P1P2
  • while e do P

26
Semantics for Expression
  • Meaning function for expressions
  • State Vars ! Numbers
  • E Expressions ! State ! Numbers

Efalse(s) ? 0
Etrue(s) ? 1
En(s) ? n
Ex(s) ? s(x)
Ee1 e2(s) ? Ee1(s) Ee2(s)
Ee1 e2(s) ? if Ee1 Ee2 then 1 else 0
27
Semantics for Programs
  • Meaning function for programs
  • Command State ! State
  • C Programs ! Command
  • CPQ(s) ? CQ(CP(s))
  • Cif e then P else Q(s) ?
  • CP(s) if E e(s) 1
  • CQ(s) if E e(s) 0

28
Semantics of Assignment
  • modify State Vars Numbers ? State
  • modify(s,x,a) ?y. if y x then a else s(y)
  • Cx e(s) ? modify(s,x,Ee(s))

29
Semantics of Iteration
  • Cwhile e do P(s) ?
  • The function f such that
  • f(s) s if Ee(s) 0
  • f(s) f(CP(s)) if Ee 1
  • Mathematics of denotational semantics prove that
    there is such a function and that it is uniquely
    determined. Beyond scope of this course.

30
Mathematical Foundations
  • A full DS for imperative programs requires the
    definition of a special class of sets called
    domains
  • From Wikipedia, the free encyclopedia.
  • Dana S. Scott is the incumbent Hillman University
    Professor of Computer Science, Philosophy, and
    Mathematical Logic at Carnegie Mellon University.
    His contributions include early work in automata
    theory, for which he received the ACM Turing
    Award in 1976, and the independence of the
    Boolean prime ideal theorem.
  • Scott is also the founder of domain theory, a
    branch of order theory that is used to model
    computation and approximation, and that provides
    the denotational semantics for the lambda
    calculus.
  • He received his Bachelor's degree from the
    University of California, Berkeley in 1954, and
    his Ph.D. from Princeton University in 1958

31
Abstract Interpretation
  • From Wikipedia, the free encyclopedia.
  • Abstract interpretation is a theory of sound
    approximation of the semantics of computer
    programs, based on monotonic functions over
    ordered sets, especially lattices. It can be
    viewed as a partial execution of a computer
    program which gains information about its
    semantics (e.g. control structure, flow of
    information) without performing all the
    calculations.
  • Abstract interpretation was formalized by Patrick
    Cousot.

32
Abstract vs. Standard Semantics
  • Given a standard semantics abstract
    interpretation approximates the standard
    semantics in a way that guarantees the abstract
    interpretation is correct with respect to the
    original semantics
  • The original DS semantics again is used as part
    of a correctness criterion for the abstract
    semantics

33
The Meaning of Meaning
  • The denotational relates of syntax trees to
    objects in mathematical functions expressed using
    a metalanguage for defining functions
  • Strachey and Scott used the mathematical theory
    of continuous function over partially ordered
    sets (domains) as their target semantics
  • We can explain the meaning of programs in using
    the Strachey and Scott metalanguage

34
DS Scheme
  • Formal semantics of Scheme is defined using the
    Strachey and Scott metalanguage
  • http//www.schemers.org/Documents/Standards/R5RS/r
    5rs.pdf
  • A non-trivial semantics language

35
Abstract Syntax of Scheme
36
Semantic Values
37
Semantic Values (cont.)
38
Semantic Functions
39
Semantic Functions (cont.)
40
Semantic Functions (cont.)
41
SML as a Metalanguage
  • Scheme semantics looks awfully like a complicated
    program written in a obscure language of
    functions which happen to have a precise
    well-understood meaning
  • (If youre a mathematician that is)
  • We could also use SML as a metalanguage to
    express semantics!

42
Semantics of Expressions in SML
  • type var string
  • datatype exp
  • True
  • False
  • N of int
  • V of var
  • Plus of exp exp
  • Minus of exp exp
  • Leq of exp exp

43
Semantics of Expressions in SML
  • type value int
  • type state var -gt value
  • ( val expSem exp -gt state -gt value )
  • fun expSem (True)(s) 1
  • expSem (False)(s) 0
  • expSem (N i)(s) i
  • expSem (V v)(sstate) s v
  • expSem (Plus(e1,e2))(s)
  • expSem(e1)(s) expSem(e2)(s)
  • expSem (Minus(e1,e2))(s)
  • expSem(e1)(s) - expSem(e2)(s)
  • expSem (Leq(e1,e2))(s)
  • if expSem(e1)(s) lt expSem(e2)(s)
  • then 1 else 0

44
Summary
  • Denotational techniques provide give meaning to
    programs by relating syntax to the semantics of
    some well-defined metalanguage
  • DS are abstract semantics that are good to reason
    about correctness of implementations or static
    analysis

45
Next Lecture
  • Using denotationial techniques to specify the
    semantics of resolution independent graphical
    objects
  • Homework for this week turn our semantics into
    SML code that takes a simple picture language
    into an image!

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