General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

Data Type Declarations
concrete version of abstract data types in SML
- datatype mynat = zero | suc of mynat;
datatype mynat = suc of mynat | zero
this gives us constructor functions zero : mynat and suc : mynat -> mynat.
define functions by (complete) case analysis (abstract procedures)
fun num (zero) = 0 | num (suc(n)) = num(n) + 1;
val num = fn : mynat -> int
fun incomplete (zero) = 0;
stdIn:10.1-10.25 Warning: match nonexhaustive
zero => ...
val incomplete = fn : mynat -> int
fun ic (zero) = 1 | ic(suc(n))=2 | ic(zero)= 3;
stdIn:1.1-2.12 Error: match redundant
zero => ...
suc n => ...
zero => ...
c○: Michael Kohlhase 81
So, we can re-define a type of unary natural numbers in SML, which may seem like a somewhat
pointless exercise, since we have integers already. Let us see what else we can do.
Data Types Example (Enumeration Type)
a type for weekdays (nullary constructors)
datatype day = mon | tue | wed | thu | fri | sat | sun;
use as basis for rule-based procedure (first clause takes precedence)
- fun weekend sat = true
| weekend sun = true
| weekend _ = false
val weekend : day -> bool
this give us
- weekend sun
true : bool
- map weekend [mon, wed, fri, sat, sun]
[false, false, false, true, true] : bool list
nullary constructors describe values, enumeration types finite sets
c○: Michael Kohlhase 82
Somewhat surprisingly, finite enumeration types that are a separate constructs in most programming
languages are a special case of datatype declarations in SML. They are modeled by sets of
base constructors, without any functional ones, so the base cases form the finite possibilities in
this type. Note that if we imagine the Peano Axioms for this set, then they become very simple;
in particular, the induction axiom does not have step cases, and just specifies that the property
P has to hold on all base cases to hold for all members of the type.
Let us now come to a real-world examples for data types in SML. Say we want to supply a library
for talking about mathematical shapes (circles, squares, and triangles for starters), then we can
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