Inductive Definitions (our meta-language for specifications)

1
Inductive Definitions(our meta-language for
specifications)

• Examples
• Syntax
• Semantics
• Proof Trees
• Rule set as a function
• Proof by induction
• Function ( relation) definition by induction
• Generalizing rule form

2
Examples

• An inductive definition by
• one axiom, one proper rule

3

• Questions
• Do we know which elements are in S?
• (at least, can we enumerate them?)
• 2. Given c, can we answer is c a member of S?
• 3. Which is the underlying universe here ?
• A any of N, Z, Q, .
• It is common to omit the universe, U, since
• It is often known from the context
• The rules tell us what it is
• Even if there are several candidates, the
  defined set is often the same in them

4

• 4. Is there just one set in N that satisfies
  (r1), (r2)?
• If not, which one is the one being
  defined?
• A The intention is
• the least set that satisfies the
  rules
• But, least is often omitted
• Note difference between minimal, least (in
  a collection)
• There is an abuse of language. It should be
• Easier to use S for both defined and a set
  variable

5

• Can we enumerate?
• Solve membership?
• Is the defined set independent of the universe?

6

• A new problem goal resolution
• Given c, can we find all y s.t. S(c,y)?
• c a goal (on 1st position)
• y, such that S(c,y) a solution
• Can we solve when the goal is on 2nd position?

7

• Q is the defined set independent of universe?
• What about goal resolution on either column?

8
Syntax

• Inductive definition a set R of rules
• A rule conjunction of premises (?????)
• implies conclusion
• or body implies head
• Our style
  (no commas!)

9

• Empty body (0 premises) an axiom
• Otherwise, a proper rule
• 1st Example
• Here, x, y are individual variables

10
Semantics

• Variable assignment
• maps individual variables to constants
  (in U)
• Applied to a rule ? a ground (var-free) rule
  instance

S satisfies rule r if for each r-instance,
if body is true in S then head is true
in S (Individual vars are
universally quantified)
S satisfies R, if it satisfies each r in R
11

• If U (the universe)
• contains the constants in R,
• is closed under the operations in R,
• then it satisfies R satisfying set exists
• Intersection property
• If each set satisfies
  R,
• so does
• Hence, a (unique) least set that satisfies R
  exists
• This set is the semantics of R indR

12
Extensions

• Define several sets simultaneously
• Can use in rules, for each defined
• Can assume certain sets are given
  (known)
• Can use in premises
• (This is not a real extension, can use these
  in conditions)
• These enable construction of definition
  hierarchies.

13
Proof Trees

• Example of a proof of membership in an inductive
  set
• Each leaf an axiom instance
• Each internal node inferred from its children
• a rule instance

14

• We abbreviate to
• Element a is inferred from by
  rule r if ...
• Q What when r is an axiom?

15
Proof Tree

• A tree in which each node is
• labeled by
• a is the element at the node
• Associated with a rule r such that
• is inferred from the children of
  the node by r
• (for a leaf --- associated rule is an
  axiom)
• (Labels of a node its children rule instance)
• Proof trees contain only membership
  formulas
• Proof tree a composition of inferences

16

• Fact a is in indR iff it has a proof
  tree
• Proof tree is a proof of membership for its
  root
• How do we prove this?
• An arbitrary satisfying set contains every
  element that has a proof tree --- induction on
  depth of tree
• The set of elements that have proof trees is a
  satisfying set

17
Rule set as a function
Fact closed under r/R satisfies r/R Fact

(same for )

• Generalize
• For example 1

18

• Define Q is closed under r/R if it is closed
  under
• Fact closed under r/R satisfies r/R
• Fact
  
• same for

19

• Thm Knaster-Tarski, 1955
• A monotone function f on sets has the
• intersection property for its closed sets
• ? if some set is closed under f, a least closed
  set exists
• ? for any R, a least set closed under R exists
• (for a large enough universe)
• (remember closed under satisfies)

20
Proof by induction

• Prove for each axiom that P(e) holds
• Prove for each rule
  
• that if (and ) then
  P(e) holds
• (actually, prove it for each variable
  assignment)
• proof by rule induction
• Example prove for Example 1 that each element
  in S is divisible by 4

21
Function ( relation) definition by induction

• For each axiom , define f(e)
• For each rule
• define f(e) in terms of
• Example
• fact(0) 1
• fact(n1) (n1)fact(n)
• Why is it called definition by induction?
• Hint What is the domain, and the rules that
  define it?
• and what is the form of the function definition

22

• Let rfact be the graph of fact , a binary
  relation.
• Compare
• To fact(0) 1
• fact(n1)
  (n1)fact(n)

• Inductive has a double meaning
• the domain is inductively defined
• The function is inductively defined (in a special
  syntax)
• and the two rules sets correspond

23

• Function def. by induction
• function defined by clauses, one for each rule
  defining the domain
• Significance
• One-to-one correspondence ? fact is total
• Sometimes, we need a partial function
• we provide clauses for only some of the rules
• that define the domain
• Example head and tail on lists

24

• Consider
• Define
• We have
• What went wrong? (think of proof trees)
• Function is total, but ill-defined!

25
Function relation definitions, summary

• Over an inductively defined domain
• Inductive definition of relation is always ok.
• Inductive function definition
• Is always total
• may fail to be well-defined
• a sufficient condition for well-defined-ness
• each value of domain has a unique proof tree

26
Generalizing rule form

• It is allowed to combine membership formulas by
• Conjunction in head
• Disjunction in body
• But,
• Disjunction in head, negation in body/head are
• forbidden
• Why?
• As for quantifiers --- see reading material