Logic and Computation Zena M. Ariola University Of Oregon
Outline of Lectures • Lecture 1: λ - calculus • Minimal Propositional Logic and simply typed λ - calculus • as deduction system and as a computation engine • Hilbert axioms and Combinatory Logic • Lecture 2: • Second-order Propositional Logic and System F • Classical Logic and Control effects • Lecture 3 : Sequent calculus as a logic and as a PL • Lecture 4: Applications in compilations, execution, design and reasoning of program � 2
The foundational crisis of mathematics At the beginning of the 19th center set theory was born This theory was very promising because it offered a common foundation to all the fields of mathematics. ℛ = { x ∣ x ∉ x } ℛ ∈ ℛ ⟺ ℛ ∉ ℛ Georg Cantor (1845-1918) � 3
Need of logic to define mathematics Let’s devise a complete and finite set of axioms from which all true statements can be derived. Gentzen introduced a formal system which modeled mathematical reasoning quite directly David Hilbert (1862-1943) Gerhard Gentzen (1909-1945) Gödel in 1935 published his incompleteness result. Kurt Gödel (1906-1978) � 4
Natural Deduction
Natural Deduction - Gentzen (1935) Formulae: A , B ::= X ∣ A ⊃ B ∣ A ∧ B ∣ A ∨ B Γ ⊢ A Judgement: If all assumptions in Γ are true then A is true Introduction and Elimination rules � 6
Natural deduction Γ , A ` A Ax Γ ` A Γ ` B Γ ` A ^ B Γ ` A ^ B ^ I ^ E 1 ^ E 2 Γ ` A ^ B Γ ` A Γ ` B Γ , A ` B Γ ` A � B Γ ` A Γ ` A � B � I � E Γ ` B Γ ` A _ B Γ , A ` C Γ , B ` C Γ ` A Γ ` B Γ ` A _ B _ I 1 Γ ` A _ B _ I 2 _ E Γ ` C � 7
Harmony (Dummett 1976) Local Soundness shows that the elimination rules are not too strong. Γ ` A _ B Γ ` B Local Completeness show that the elimination rules are not too weak Γ ` A ^ B Γ ` A � 8
Local Soundness Elimination rules applied to the result of an introduction rule do not produce new information. D E B E Γ ` A Γ ` B D ⟹ ^ I Γ ` A ^ B Γ ` A ^ E Γ ` A Local Reductions � 9
Local Soundness Elimination rules applied to the result of an introduction rule do not produce new information. D I Γ , A ` B E A B ⟹ D [ E / A ] Γ ` A � B � I ` Γ ` A Γ ` B � E Γ ` B Local Reductions � 10
Local Soundness D Γ , A ` B Γ ` A � B � I E E D [ E / A ] E = ) Γ ` A Γ ` B � E Γ ` B A � A , B ` A � A A � A ` B � ( A � A ) � I A ` A B , A ` A ` ( A � A ) � B � ( A � A ) � I ` A � A � I B ` A � A � I � E = ` B � ( A � A ) � I ` B � ( A � A ) ) � 11
Local Completeness The elimination rules are strong enough to get all the pieces out of a structured proposition so that they can be recomposed. A proof can always end with an introduction rule. Γ ` A ^ B Γ ` A ^ B ^ E ^ E A B = ) Γ ` A ^ B Γ ` A Γ ` B ^ I Γ ` A ^ B Local Expansion � 12
Local completeness The elimination rules are strong enough to get all the pieces out of a structured proposition so that they can be recomposed. A proof can always end with an introduction rule. Γ , A ` A � B Γ , A ` A � E Γ , A ` B Γ ` A � B A B = ) Γ ` A � B � I A Local Expansion � 13
Caution with variables Γ ` 8 x . P ( x ) x Γ ` P ( x ) I Γ ` P ( M ) 8 E Γ ` 8 x . P ( x ) 8 I P x ===== ` 8 x . 9 y . x 6 = y x = 0 ` x = 0 x 0 x = 0 ` 8 x . x = 0 ` 9 y . y 6 = y x x 0 Z 1 n n n x 2 dx [ Y X n !1 n 2 a i a i a i lim 0 i = 1 i = 1 i = 1 � 14
Lambda calculus
First [1932] version of lambda-calculus Church. Postulates for the Foundations of Logic, Annals of Mathematics, 33(2), 1932 Church. Postulates for the Foundations of Logic, Annals of Mathematics, 34(4), 1934. λ x . e Church introduced the notation for the function x x e with parameter and body .Variable is bound and Alonzo Church can be renamed without changing the meaning of the (1903-1995) function. Church proposes to encode all these different notations as operators which take a function as parameter: ∀ x . P ( x ) = ∀ ( λ x . P ( x )) ∃ x . P ( x ) = ∃ ( λ x . P ( x )) There is only one mechanism to introduce a variable - Higher-order abstract syntax � 16
Lambda calculus M , N ::= x variable λ x . M abstraction M N application &( M , N ) M and N ¬ ( M ) not M ι ( M ) an integer x such that M x Y ( M , N ) forall x. if M(x) then N(x) X ( M ) ∃ x , Mx A ( M , N ) abstraction from N with respect to M 37 axioms (36 in the revision) and 5 inference rules λ x . M = α λ z . M [ z / x ] ( λ x . M ) N = β M [ N / x ] � 17
Propositions as Terms Predicates and propositions are lambda-calculus terms A ⊃ ( A & A ) Similar to logic programming today (e.g. Prolog) Predicates takes predicates in input (higher-order logic) Sets are identified with their characteristic function: = x ∈ A A x { x | P ( x ) } = λ x . P x { x } = λ y . x = y = A ∪ B λ x . A x ∨ B x λ x . ¬( x x ) The set of all x's such that x does not belong to x: � 18
Lambda calculus is inconsistent R = λ x . ¬( x x ) Does R contains itself? Given = R 2 R R R = ( λ x . ¬ ( x x ))( λ x . ¬ ( x x )) = ¬ ( R R ) = R 62 R R ∈ R ⟺ R ∉ R How did Church react? This is not a paradox because he did not accept excluded middle (every proposition is either true or false). R is neither � 19
Curry paradox (1942) Let’s focus on implication: ( X � ( X � Y )) � ( X � Y ) ( 1 ) ( MP ) X � Y , X ` Y X , X = Y ` Y ( Eq ) We define the term/proposition: term/proposition: X = X � ( X � Z ) = X � ( X � Z ) ( 4 ) X = ( X � ( X � Z )) � ( X � Z ) ( 5 ) � � � � ( 6 ) ( X � ( X � Z )) � ( X � Z ) ( 1 ) provable: It ( 7 ) X � ( X � Z ) ( 5 ) and ( Eq ) ( 8 ) ( 4 ) and ( Eq ) X ( 9 ) ( MP ) and ( 7 ) , ( 8 ) X � Z ( 10 ) ( MP ) and ( 8 ) , ( 9 ) Z cannot use -calculus as a logic, if we can define the formula X ! For it is Everything is provable � 20
Fixed point theorem X = X ⊃ ( X ⊃ Z ) Can we define the formula Fixed point theorem. For every F there exists an X such that F X = X. D = λ x . F ( x x ) Proof. Let X = D D Take X = D D = F ( D D ) = F ( X ) We were looking for a proposition: X = X ⊃ ( X ⊃ Z ) X = ( λ x . x ⊃ ( x ⊃ Z )) X X λ x . x ⊃ ( x ⊃ Z ) becomes the fixed point of � 21
How to solve the problem? Let’s get rid off the logical connectives Computation Recognize that not all terms represent propositions Type system � 22
Pure Lambda-calculus Church with his students Kleene and Rosser (1935) studied the pure lambda- calculus : M ::= x ∣ λ x . M ∣ M M ( λ x . M ) N → M [ N / x ] They orient the beta-axiom: M → M 1 → M 2 → ⋯ → N → Terms represent computation: Theorem (Church-Rosser). beta-reduction is confluent A normal form (i.e. the answer of a program) if it exists is unique � 23
Lambda-calculus as a computational model Church (1936), it is undecidable if a term has a normal form Hilbert program cannot be realized Kleene (1935) and Turing (1937) show the correspondence of the following computational models: - Partial recursive functions - Turing machines - Functions defined by lambda-terms f : N → N is lambda-definable if there exists a term A function F ∀ n ∈ N . F ⌜ n ⌝ = ⌜ f n ⌝ such that � 24
Lambda calculus as the prototypical Functional programming language Functional language = lambda-calculus + evaluation strategy + data structures John McCarthy designed LISP (LISt Processor) in 1958 as an implementation of lambda-calculus, but implemented substitution incorrectly (1927-2011) Peter Landin described ISWIM (If you See What I Mean) in The Next 700 Programming Languages, published in the Communications of the ACM in 1966. (1930-2009) � 25
Simple Theory of Types (Church, 1940) Types σ , τ ::= ι ∣ o ∣ σ → τ ⊃ : o → o → o ∀ : ( ι → o ) → o M ::= x ∣ λ x : σ . M ∣ M M Terms Static semantics Γ , x : σ ` x : σ Γ , x : σ ` M : τ Γ ` M : σ ! τ Γ ` N : σ Γ ` M N : τ Γ ` λ x : σ . M : σ ! τ ⊢ a ⊃ a : o ⊢ λ x . σ . λ y . τ . x : σ → τ → σ ⊢ λ x : σ → τ . λ y : τ → ρ . λ z : σ . y ( xz ) : ( σ → τ ) → ( τ → ρ ) → σ → ρ ⊢ λ x : σ . λ y : ( τ → σ ) → σ . y ( λ z : τ . x ) : σ → (( τ → σ ) → σ ) → σ � 26
Properties Not every type has an inhabitant (i.e. a closed term of that type): ( σ → σ ) → σ Theorem (Subject reduction) Γ ⊢ N : τ Γ ⊢ M : τ M → → N and then . Theorem (Strong Normalization) there is no infinite reduction Γ ⊢ M : τ sequence starting from M � 27
Formulae as Types A , B ::= X ∣ A ⊃ B Formulae σ , τ ::= b ∣ σ → τ Types Change of perspective: formulae are not terms � 28
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