Classical logic, truth tables. Types, Propositions and Problems. Heyting. Brouwer. Conjunction A B A & B T T T T F F F T F F F F.

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1 lassical logic, truth tables Types, Propositions and Problems an introduction to type theoretical ideas engt Nordström omputing Science, halmers and University of Göteborg Types Summer School, Hisingen, 15 ugust 2005 onjunction Disjunction & T T T T F F F T F F F F T T T T F T F T T F F F Implication T T T T F F F T T F F T This assumes that a proposition is either true or false! rouwer Heyting rouwer rejected the idea that the meaning of a mathematical proposition is its truth value. Mathematical propositions do not exist independently of us. We cannot say that a proposition is true without having a proof of it. Heyting was a student of rouwer. He gave the following explanation of the logical constants.

2 Kolmogorov Heyting s explanation of the logical constants (1930) proof of: consists of: & a proof of and a proof of a proof of or a proof of a method which takes any proof of to a proof of a method which takes any proof of to a proof of absurdity has no proof x. an element a in and a proof of [x := a] x. a method, which takes any element x in to a proof of [x := a] Independently of Heyting, Kolmogorov interpreted propositions as problems. Kolmogorov understood the logical constants as problems (1932) The problem: & is solved if we can: solve and solve solve or solve reduce the solution of to the solution of show that there is no solution of has no solution Heyting s and Kolmogorov s explanation proof (solution) of: consists of: & a proof (solution) of and a proof (solution) of a proof (solution) of or a proof (solution) of a method which takes any proof (solution) of to a proof (solution) of a method which takes any proof (solution) of to a proof (solution) of absurdity has no proof (solution) x. an element a in and a proof (solution) of [x := a] x. a method, which takes any element x in to a proof (solution) of [x := a] Question: Is this correct? ould not a proof (solution) of & be obtained by induction, for instance?

3 Direct and indirect proofs Examples of indirect proofs When we say that we have a proof of a proposition, then we mean that we have a method which when computed yields a direct proof of it. ompare this with mathematics and programming: When we say that and fst(< 45 2, 9 >) are natural numbers, then we mean that they can be computed to a natural number. Terminology: proofs: objects: direct vs. indirect proof value vs. expression canonical vs. non-canonical proof canonical vs. noncanonical element introduction vs. elimination proof nd-elimination & If we have a proof of &, then we can compute it to a direct proof. This always consists of a proof of and a proof of. Hence we may always obtain a proof of from a proof of &. Mathematical induction n N P(0) ( n N)P(n) P(succ(n)) P(i) What is a proposition (problem)? To summarize Heyting s and Kolmogorov s explanations: What does it mean to understand a proposition? I understand a proposition when I understand what a direct proof of it is. This looks very similar to: What does it mean to understand a set? I understand a set when I understand what a canonical element of it is. Propositions and sets proof (element) of: consists of: & a proof (solution) of and a proof (solution) of an element in and an element in a proof (solution) of or a proof (solution) of + an element in or an element in a method which takes any proof (solution) of to a proof (solution) of a method which takes any element in to an element in has no proof (solution) has no elements x. an element a in and a proof (solution) of [x := a] x. a method, which takes any element x in to a proof (solution) of [x := a]

4 urry s contribution This similarity leads to the urry-howard isomorphism & = = + = = = urry noticed the formal similarity between the axioms of positive implicational logic: ( ) ( ) and the type of the basic combinators: K S ( ) ( ) and modus ponens corresponds to the typing rule for application: f a f a Proofs as Programs in a functional programming language onstructors are introduction rules direct consists of: s a type: proof of: a proof of or data Or = Ori1 Ori2 ; a proof of & a proof of and data nd = ndi ; a proof of a method taking a proof of data Implies = Impi ; to a proof of Falsity data Falsity = ; & [] Ori1 Ori2 ndi & Impli ( )

5 Elimination rules can be defined Proof checking = Type checking orel ( ) ( ) orel (Ori1 a) f g = f a orel (Ori2 b) f g = g b andel & ( ) andel (ndi a b) f = f a b implel implel (Impli f ) a = f a & [] [, ] [] orel andel implel In this way we can prove propositional formulas in a typed functional programming language. The problem of proving for instance ( & ) ( & ) is then the problem of finding a program in this type. The type checker will check if the proof is correct. In this case, we can use implel the(impli following f ) a program: = f a Impli (λx.ndi (andel x λy.λz.z) (andel x λy.λz.y)) What about the quantifiers? Overview of Martin Löf s type theory Propositions and sets proof (element) of: x. Σx. x. Πx. consists of: an element a in and a proof (solution) of [x := a] an element a in and an element in [x := a] a method, which takes any element x in to a proof (solution) of [x := a] a method, which takes any element x in to an element in [x := a] Type theory is a small typed functional language with one basic type and two type forming operation. It is a framework for defining logics. new logic is introduced by definitions.

6 What types are there? What programs are there? Set is a type El() is a type, if Set. (x ) is a type, if is a type and a family of types for x. Programs are formed from variables and constants using abstraction and application: pplication bstraction c (x ) a c a [x := a] b [x ] [x]b (x ) constants are either primitive or defined onstants There are two kinds of constants: primitive: (not defined) have a type but no definiens (RHS): identifier Type defined: have a type and a definiens: identifier = expr Type There are two kinds of defined constants: explicitly defined implicitly defined Primitive constants computes to themselves (i.e. are values). constructors in functional languages. introduction rules and formation rules in logic postulates Examples: N Set 0 N s N N & Set Set Set &I ( Set) ( Set) & Π ( Set) ( Set) Set λ ( Set) ( Set) ((x ) (x)) Π(, )

7 Explicitly defined constants have a type and a definiens (RHS). the definiens is a welltyped expression abbreviation derived rule in logic. names for proofs and theorems in math. Examples: 2 N ( Set)( Set) Set +(x N)(y N) N = succ(succ 0) = Π ( Set)( Set) Set = natrec [x]n x y [u, v](succ v) Implicitly defined constants The definiens (RHS) may contain pattern matching and may contain occurrences of the constant itself. The correctness of the definition must in general be decided outside the system Recursively defined programs Elimination rules (the step from the definiendum to the definiens is the contraction rule). Examples: add(x N)(y N) N add 0 y = y add (succ u) y = succ (add u y) &E( Set)( Set)( Set) (f (x ) (y ) (&I x y)) (z & ) (z) &E f (&I a b) = f a b = Π [x] Two approaches to the usage of implicit constants: The editing process The conservative approach: Use them only to define induction principles for sets (elimination rules). These are functions, which for an inductively defined set produces a function in (x ) ( z) for a family of sets Set. The liberal approach: Use them when they are convenient. The idea is to build expressions from incomplete expressions with holes (placeholders). Each editing step replaces a place holder with another incomplete expression

8 Place holders To construct an object We start to give the name of the object to define, and the computer responds with We use the notation 1,..., n c 1 c = 2 for place holders (holes). Each place holder has an expected type and a local context (variables which may be used to fill in the hole). We must first give the type of c by refining 1. We can either enter text from the keyboard, or do it stepwise, replace it by (x 3 ) 4 Set, or 3... n Refinement of an object Refinement of an object: application When we have constructed the type of the constant c, we can start to define it: c c = 0 Here, the expected type of 0 is. In general, we are in a situation like c = where we know the expected type of the place holders. To refine a place holder 0 with a constant c (or a variable) is to replace it by c 1... n where 1 1,..., n n. The system computes the expected types of the new place holders and some constraints from the condition that the type of c 1... n must be equal to. We have reduced the problem to the subproblems 1,... n using the rule c.

9 Refinement of an object: abstraction Hence: to prove is to build a proof object To refine a place holder 0 with an abstraction is to replace it by [x] 1 The system checks that is a functional type (x ) and the expected type of 1 is and the local context for it will contain the assumption x. To apply a rule c is to construct an application of the constant c. To assume is to construct an abstraction of a variable of type. To refer to an assumption of is to use a variable of type. We have reduced the problem (x ) to the problem by using the assumption x. Summary: Type Theory Types: Programs: T ::= Set El(e) (x T ) T e ::= e e [x]e x c onstants: Primitive (without a definition): c T Explicitly defined: Implicitly defined: c = e T c p 1... p n = e. c p 1... p n = e