Inductive Definitions with Inference Rules 1 / 25
Outline Introduction Specifying inductive definitions Inference rules in action Judgments, axioms, and rules Reasoning about inductive definitions Direct proofs Admissibility Rule induction 2 / 25
What are inference rules? Inference rules – a mathematical metalanguage For specifying and formally reasoning about inductive definitions Inductive definition Recursively defines something in terms of itself premises Human ( x ) → Mortal ( x ) Human ( x ) Mortal ( x ) conclusion Introduction 3 / 25
Outline Introduction Specifying inductive definitions Inference rules in action Judgments, axioms, and rules Reasoning about inductive definitions Direct proofs Admissibility Rule induction Specifying inductive definitions 4 / 25
Other metalanguages for inductive definitions Haskell data types Recursive functions in Haskell even :: Nat -> Bool data Nat = Z | S Nat even Z = True data Exp = Add Exp Exp | Neg Exp even (S Z) = False | Lit Nat even (S (S n)) = even n Grammars Can also define all of these with n ∈ Nat ::= Z | S n inference rules ! e ∈ Exp ::= add e e | neg e | n Specifying inductive definitions 5 / 25
Example: defining syntax by inference rules rule schema Grammars n ∈ Nat n ∈ Nat ::= Z | S n Z ∈ Nat S n ∈ Nat e ∈ Exp ::= add e e axiom | neg e (no premises) | n n ∈ Nat e ∈ Exp n ∈ Exp neg e ∈ Exp e 1 ∈ Exp e 2 ∈ Exp add e 1 e 2 ∈ Exp Specifying inductive definitions 6 / 25
Example: defining a predicate Option 1: Constructive judgment Recursive function in Haskell Even ( n ) Even ( Z ) even :: Nat -> Bool Even ( S (S n ) ) even Z = True even (S Z) = False even (S (S n)) = even n Option 2: Relate inputs to outputs Even ( Z , true ) Even ( S Z , false ) Even ( n , b ) Even ( S (S n ) , b ) Specifying inductive definitions 7 / 25
Outline Introduction Specifying inductive definitions Inference rules in action Judgments, axioms, and rules Reasoning about inductive definitions Direct proofs Admissibility Rule induction Specifying inductive definitions 8 / 25
How to define a concept (in general) Three parts of a definition: 1. syntax – how to express the concept 2. type – what kind of information is it? 3. content – the definition itself Example: dictionary definition Example: function definition e · ven | ’ ¯ even :: Nat -> Bool Syntax: ev e n | even Z = True Type: adjective even (S Z) = False Content: (of a number) divisible by even (S (S n)) = even n two without a remainder Specifying inductive definitions 9 / 25
How to define a concept using inference rules 1. Define a judgment form – syntax and type States that one or more values have some property or exist in some relation to each other 2. Write down the rules for the judgment – content • axioms – base cases, only conclusion • proper rules – recursive cases, premises + conclusion Specifying inductive definitions 10 / 25
Judgments 1. Define a judgment form – syntax and type States that one or more values have some property or exist in some relation to each other Syntax Type Property or relation n ∈ Nat AST n is in the syntactic category Nat Even ( n ) Nat n is an even number n 1 < n 2 Nat × Nat n 1 is less than n 2 e : T Exp × Type e has type T Γ ⊢ e : T Env × Exp × Type e has type T in environment Γ Specifying inductive definitions 11 / 25
Set theoretic view of judgments A judgment is (conceptually) a predicate that indicates set membership Example: Even ( n ) ⊆ Nat Even : Nat → B = { ( Z , true ) , ( S Z , false ) , ( S (S Z) , true ) , . . . } ≡ { Z , S (S Z) , S (S (S (S Z))) , . . . } ⊆ Nat Example: n 1 < n 2 ⊆ Nat × Nat < : Nat × Nat → B = { (( 0 , 0 ) , false ) , (( 0 , 1 ) , true ) , . . . (( 5 , 3 ) , false ) , . . . (( 5 , 7 ) , true ) , . . . } ≡ { ( 0 , 1 ) , . . . ( 5 , 7 ) , . . . } ⊆ Nat × Nat Specifying inductive definitions 12 / 25
Giving meaning to a judgment by inference rules Inductively defines the 2. Write down the rules of the judgment – content instances of a judgment • axioms – base cases, only conclusion (i.e. members of its set) • proper rules – recursive cases, premises + conclusion Rules for: Even ( n ) ⊆ Nat Rules for: n 1 < n 2 ⊆ Nat × Nat n 1 < n 2 n 1 < n 2 Even ( n ) Z < S Z Even ( Z ) n 1 < S n 2 S n 1 < S n 2 Even ( S (S n ) ) Specifying inductive definitions 13 / 25
Exercises 1. Define the judgment: Odd ( n ) ⊆ Nat 2. Define the judgment: n 1 + n 2 = n 3 ⊆ Nat × Nat × Nat For reference: Rules for: n 1 < n 2 ⊆ Nat × Nat Rules for: Even ( n ) ⊆ Nat n 1 < n 2 n 1 < n 2 Even ( n ) Z < S Z Even ( Z ) n 1 < S n 2 S n 1 < S n 2 Even ( S (S n ) ) Specifying inductive definitions 14 / 25
Outline Introduction Specifying inductive definitions Inference rules in action Judgments, axioms, and rules Reasoning about inductive definitions Direct proofs Admissibility Rule induction Reasoning about inductive definitions 15 / 25
Expressing claims We can also use inference rules to express claims about judgments Examples Even ( S n ) n 1 < n 2 n 2 < n 3 n 1 + n 2 = n 3 S (S Z) ∈ Nat Odd ( n ) n 1 < n 3 n 2 + n 1 = n 3 How can we prove these claims? Three main techniques: 1. direct proof – derive conclusion from premises using the definition 2. admissibility – derive conclusion from derivations of premises 3. rule induction – reason inductively using the definition Reasoning about inductive definitions 16 / 25
Outline Introduction Specifying inductive definitions Inference rules in action Judgments, axioms, and rules Reasoning about inductive definitions Direct proofs Admissibility Rule induction Reasoning about inductive definitions 17 / 25
Direct proof by derivation Definition: n ∈ Nat Definition: n 1 < n 2 ⊆ Nat × Nat Succ n ∈ Nat n 1 < n 2 n 1 < n 2 S +1 Z ∈ Nat S n ∈ Nat Z < S Z n 1 < S n 2 S n 1 < S n 2 Z ∈ Nat Z < S Z Succ S S Z ∈ Nat Z < S (S Z) Succ +1 S (S Z) ∈ Nat S Z < S (S (S Z)) Reasoning about inductive definitions 18 / 25
Proof trees Definition: e ∈ Exp Axioms: 0 ∈ Nat , 1 ∈ Nat , 2 ∈ Nat , . . . lit n ∈ Nat e ∈ Exp add e 1 ∈ Exp e 2 ∈ Exp neg n ∈ Exp neg e ∈ Exp add e 1 e 2 ∈ Exp 2 ∈ Nat 3 ∈ Nat 4 ∈ Nat lit lit lit 2 ∈ Exp 3 ∈ Exp 4 ∈ Exp add neg add 2 3 ∈ Exp neg 4 ∈ Exp add add (add 2 3) (neg 4) ∈ Exp Reasoning about inductive definitions 19 / 25
Exercises Prove that the following expressions are valid terms in Exp 1. neg (add 5 (neg 2)) 2. add (neg (neg 3)) 4 Definition: e ∈ Exp Axioms: 0 ∈ Nat , 1 ∈ Nat , 2 ∈ Nat , . . . lit n ∈ Nat e ∈ Exp add e 1 ∈ Exp e 2 ∈ Exp neg n ∈ Exp neg e ∈ Exp add e 1 e 2 ∈ Exp Reasoning about inductive definitions 20 / 25
Outline Introduction Specifying inductive definitions Inference rules in action Judgments, axioms, and rules Reasoning about inductive definitions Direct proofs Admissibility Rule induction Reasoning about inductive definitions 21 / 25
Admissibility Construct proofs from assumed derivations of the premises Insights: • If the premise of a claim is satisfied, it must have a derivation • Can use information in the derivations to prove the conclusion Proof technique Show that all possible derivations of premises yield a proof of the conclusion Apply definition rules backwards on the premises, prove for each case! Reasoning about inductive definitions 22 / 25
Super simple example Definition: n ∈ Nat ⊆ AST Bold claim Succ n ∈ Nat S ( S n ) ∈ Nat Z ∈ Nat S n ∈ Nat n ∈ Nat Only possible derivation Proof sketch: • Enumerate derivations of premise Succ n ∈ Nat • Show that each derivation proves S n ∈ Nat Succ the conclusion S ( S n ) ∈ Nat Reasoning about inductive definitions 23 / 25
Outline Introduction Specifying inductive definitions Inference rules in action Judgments, axioms, and rules Reasoning about inductive definitions Direct proofs Admissibility Rule induction Reasoning about inductive definitions 24 / 25
Rule induction Just like structural induction on inductive data types! Definition: e ∈ Exp ⊆ AST n ∈ Nat e ∈ Exp e 1 ∈ Exp e 2 ∈ Exp n ∈ Exp neg e ∈ Exp add e 1 e 2 ∈ Exp Suppose I want to prove property P on all Exp s. Just prove: • ∀ n ∈ Nat , P ( n ) • P ( e ) → P ( neg e ) • P ( e 1 ) → P ( e 2 ) → P ( add e 1 e 2 ) Reasoning about inductive definitions 25 / 25
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