Denotational Semantics

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Denotational Semantics

• COS 441
• Princeton University
• Fall 2004

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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

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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

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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

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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 ...

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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

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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

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DS of Regular Expressions

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

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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 ?

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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
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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
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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
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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
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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

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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

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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,?)

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Semantics of Regular Expressions

• NFAa ? (qi, qf, (qi,a,qf))

a
qf
qi
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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
?
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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
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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
?
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Some More Theorems
L(NFAa b) ? a , b
L(NFAa . (b c)) ? ab, ac
L(NFA(a b) . c) ? ac, bc
L(NFAre) ? Ere
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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!

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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

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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

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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

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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
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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

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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))

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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.

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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

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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.

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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

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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

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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

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Abstract Syntax of Scheme
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Semantic Values
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Semantic Values (cont.)
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Semantic Functions
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Semantic Functions (cont.)
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Semantic Functions (cont.)
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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!

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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

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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

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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

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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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