Inductive Definitions

Transcription

1 University of Science and Technology of China (USTC) 09/26/2011

2 Judgments A judgment states that one or more syntactic objects have a property or stand in some relation to one another. The property or relation itself is called a judgment form, and the judgment that an object or objects have that property or stand in that relation is said to be an instance of that judgment form. We write a J for the judgment asserting that J holds on a. Examples: n nat n is a natural number n = n 1 + n 2 n is the sum of n 1 and n 2 τ type τ is a type e : τ expression e has type τ e v expression e has value v

3 Inference Rules An inductive definition of a judgment form consists of a collection of rules of the form J 1... J k J where J and J 1,... J k are all judgments of the form being defined. Premises: judgments above the line. Conclusion: judgment below the line. Premises are sufficient for the conclusion, but may not be necessary. Axioms and proper rules

4 Examples zero nat a nat succ(a) nat Here zero and succ can be viewed as constructors of elements of the data type nat. empty tree a 1 tree a 2 tree node(a 1, a 2 ) tree empty and node are constructors of trees. zero = zero nat a = b nat succ(a) = succ(b) nat

5 Examples zero nat a nat succ(a) nat Here zero and succ can be viewed as constructors of elements of the data type nat. empty tree a 1 tree a 2 tree node(a 1, a 2 ) tree empty and node are constructors of trees. zero = zero nat a = b nat succ(a) = succ(b) nat

6 Examples zero nat a nat succ(a) nat Here zero and succ can be viewed as constructors of elements of the data type nat. empty tree a 1 tree a 2 tree node(a 1, a 2 ) tree empty and node are constructors of trees. zero = zero nat a = b nat succ(a) = succ(b) nat

7 Derivable Judgments A judgment J is derivable iff Either there is an axiom J Or there is a rule J 1... J n J such that J k is derivable, for all 1 k n.

8 Derivations A derivation of a judgment is a finite composition of rules, starting with axioms and ending with that judgment. If J 1... J n J is a rule and i is a derivation of J i, then is a derivation of J n J If n = 0, this is an axiom, also a special derivation. A derivation tree: each node is a rule whose children are derivations (sub-trees) of its premises. A derivation of J is evidence for the validity of the inductively defined judgment J.

9 Constructing a Derivation Tree Backward chaning, or top-down construction: starts with the desired conclusion and works backwards towards the axioms. Forward chaning, or Bottom-up construction: starts with the axioms and works forward towards the desired conclusion. Example: empty tree empty tree node(empty, empty) tree empty tree node(node(empty, empty), empty) tree

10 Rule Induction The set of rules for a judgment is both necessary and sufficient for derivability of judgments. In other words, the rules are an exhaustive description of the judgment. To show that every derivable judgment has a property P, it is enough to show that: For every rule J 1... J n J if J 1... J n have the property P, then J has property P. This is the principle of rule induction.

11 Rule Induction (cont d) Consider the rules for (n nat). We can prove by rule induction that the property P holds over every n such that (n nat). Show that P holds over zero. Assuming (n nat) and P holds over n, show that P holds over succ(n). This is just ordinary mathematical induction.

12 Example: Binary Tree Similarly, we can prove that a property P holds for every t such that (t tree) by showing that P holds over empty; If (a tree), (b tree), and P holds over a and b, then P holds over node(a, b).

13 Example: The Height of a Tree Consider the following definition: hgt(empty) = 0 hgt(node(a, b)) = 1+max(hgt(a), hgt(b)) Lemma: for all t such that (t tree), there exists n such that hgt(t) = n; for all m and n, if hgt(t) = m and hgt(t) = n, then m = n. That is, the above equations define a function (hgt(t) is uniquely defined).

14 Example: The Height of a Tree (cont d) We will prove the lemma by rule induction: If (t tree) is derived by the rule empty tree Then by the first equation we know there exists n = 0 such that hgt(empty) = n, and for all m and n, if hgt(empty) = m, hgt(empty) = n, we know m = n = 0.

15 Example: The Height of a Tree (cont d) If (t tree) is derived by the rule a tree b tree node(a, b) tree Then we may assume (inductive hypothesis): hgt(a) is uniquely defined, hgt(b) is uniquely defined. Therefore there exists a unique n = 1+max(hgt(a), hgt(b)), such that hgt(t) = n.

16 Reading Chapter 2 of Harper.