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SE 2F03 Fall 2005 05 Simple Type Theory Instructor: W. M. Farmer Revised: 22 November 2005 1 Theory-Oriented vs. Practice-Oriented Logics • Most traditional logics are theory oriented: they are designed to be studied and used in theory only. – Examples: First-order logic, Zermelo-Fraenkel set theory. • A practice-oriented logic is intended for actual use in practice by engineers, scientists, mathematicians, and students. – Often are modiﬁcations of traditional logics. – Examples: Versions of Church’s type theory used in the HOL, IMPS, Isabelle, ProofPower, PVS, and TPS systems. • The logic of a mechanized mathematics system needs to be practice oriented. 2 Attributes of a Practice-Oriented Logic • Expressivity • Clarity • Naturalness • Conciseness 3 Issue 1: Conditional Expressions • A conditional expression is an expression whose value depends on speciﬁed conditions. – Example: “If C then v1 else v2.” • Conditional formulas can be expressed using implication and conjunction. – Example: (p(x) ⇒ q(x)) ∧ (¬p(x) ⇒ r(x)). • Conditional terms can be expressed with an if-then-else term constructor or with tables. – Example: if(0 ≤ x, x, −x). – Example: condition value x<0 −x x=0 0 x>0 x 4 Issue 2: Function Abstraction • A function abstraction is an expression that deﬁnes a function from an expression containing variables written formally as λ x . t in computer science and as x → t in mathematics. – Example: λ x . x2. – This is called lambda-notation. • Informal notation for function abstraction in mathematical practice is ambiguous. – Example: f (x) = x2. – Does this example mean the function f is deﬁned as x2 or that the function f applied to x is x2? • Lambda-notation allows functions to be deﬁned directly and unambiguously. – Example: f = (λ x . x2 + x ∗ y + 2 ∗ y). 5 Issue 3: Deﬁnite and Indeﬁnite Description • A deﬁnite description is an expression of the form “the x such that A” written formally as ι x . A. – Example: ι x . 0 ≤ x ∧ x ∗ x = 2. • An indeﬁnite description is an expression of the form “some x such that A” written formally as x . A. – Example: x . x ∗ x = 2. • Deﬁnite descriptions, and to a lesser extent indeﬁnite descriptions, are quite common in mathematical practice, but they often occur in a disguised form. • Improper deﬁnite and indeﬁnite descriptions are undeﬁned. 6 Issue 4: Types • A type is a syntactic object α that denotes a set Dα of values. – α and Dα are often confused with each other. – Types are also called sorts. • Types are used to: – Restrict the scope of variables. – Control the formation of expressions. – Organize values. – Classify expressions by their values. • In mathematical practice, types are informal and used mainly for restricting the scope of variables. • There is a wide range of type systems. • Types are also sometimes used as “mini-speciﬁcations”. 7 Issue 5: Higher-Order Values • A higher-order value is a value constructed from other values. – Examples: Sets, functions, relations, tuples, sequences, and algebras. – A practice-oriented logic needs to support reasoning with high-order values including quantiﬁcation over higher-order values. • A higher-order value may be constructed from values of the same kind. – Examples: A set of sets, a function that maps functions to functions, a relation between relations. • A function f : A × B → C can be represented in three ways: – As a binary function from A and B to C. – As a unary function from A × B to C. – As a curryed function from A to B → C. 8 Issue 6: Undeﬁnedness • A mathematical term is undeﬁned if it has no prescribed meaning or if it denotes a value that does not exist. – Undeﬁned terms are commonplace in mathematics. • Sources of undeﬁnedness: √ 1. Improper function applications: −4. 2. Improper deﬁnite descriptions: “the x such that x2 = 4”. 3. Improper indeﬁnite descriptions: “some x such x2 = −4”. • A practice-oriented logic needs a way of handling undeﬁnedness: – Traditional approach to undeﬁnedness – Unspeciﬁed values – Error values – Ill-formed terms 9 Issue 7: Polymorphism • An operator is polymorphic if it can be applied to expressions of diﬀerent types. • Polymorphic operators are not usually needed in mathematical practice since, by convention, operators can be applied to all expressions (but the applications may be undeﬁned). • A practice-oriented logic needs polymorphic operators in some form: – Type variables. – Macro-abbreviations. – All values are members of a universal class. 10 Issue 8: Syntax as Values • An expression has a two meanings: 1. The value it denotes. 2. Its syntactic construction. • Both meanings are important in mathematics, but the distinction between them is often confused. • A practice-oriented logic needs to be able to reason about both meanings of an expression. 11 What is Simple Type Theory? • A simple, elegant, highly expressive, and practical logic. – Familiar to some computer scientists but not to many mathematicians, engineers, and other scientists. – Also known as higher-order logic. • Most popular form of type theory. – Types are used to classify expressions by value and control the formation of expressions. – Classical: nonconstructive, 2-valued. – Higher order: quantiﬁcation over functions. – Can be viewed as a “function theory”. • Natural extension of ﬁrst-order logic. – Based on the same principles as ﬁrst-order logic. – Includes nth-order logic for all n ≥ 1. 12 Some Versions of Simple Type Theory • Church’s type theory is a version of simple type theory with lambda-notation introduced by A. Church in 1940. • STT is a very simple version of Church’s type theory. – Addresses issues 1,2,3,4,5. • BESTT is a more practical version of Church’s type theory with type variables and support for reasoning with tuples, lists, and sets. – Addresses issues 1,2,3,4,5,6,7. 13 References 1. P. B. Andrews, An Introduction to Mathematical Logic and Type Theory: To Truth through Proof, Second Edition, Kluwer, 2002. 2. A. Church, “A formulation of the simple theory of types”, Journal of Symbolic Logic 5: 56–68, 1940. 3. W. Farmer, “The seven virtues of simple type theory”, SQRL Report No. 18, 30 pp., McMaster University, 2003 (revised 2005). 4. W. Farmer,“A basic extended simple type theory”, SQRL Report No. 14, 12 pp., McMaster University, 2003 (revised 2004). 14 Syntax of STT: Types • A type of STT is deﬁned by the following rules: T1 (Type of individuals) type[ι] T2 (Type of truth values) type[∗] type[α], type[β] T3 (Function type) type[(α → β)] • Let T denote the set of types of STT. 15 Syntax of STT: Symbols • The logical symbols of STT are: – Function application: @ (hidden). – Function abstraction: λ. – Equality: =. – Deﬁnite description: I (capital iota). – An inﬁnite set V of symbols called variables. • A language of STT is a pair L = (C, τ ) where: – C is a set of symbols called constants. – τ : C → T is a total function. 16 Syntax of STT: Expressions • An expression E of type α of a STT language L = (C, τ ) is deﬁned by the following rules: x ∈ V, type[α] E1 (Variable) exprL[(x : α), α] c∈C E2 (Constant) exprL[c, τ (c)] exprL[A, α], exprL[F, (α → β)] E3 (Application) exprL[F (A), β] x ∈ V, type[α], exprL[B, β] E4 (Abstraction) exprL[(λ x : α . B), (α → β)] exprL[E1, α], exprL[E2, α] E5 (Equality) exprL[(E1 = E2), ∗] x ∈ V, type[α], exprL[A, ∗] E6 (Deﬁnite description) exprL[(I x : α . A), α] 17 Syntax of STT: Conventions • Eα denotes an expression E of type α. • Parentheses and the types of variables may be dropped when meaning is not lost. 18 Semantics of STT: Standard Models • A standard model for a language L = (C, τ ) of STT is a triple M = (D, I, e) where: – D = {Dα : α ∈ T } is a set of nonempty domains (sets). – D∗ = {t, f}, the domain of truth values. – Dα→β is the set of all functions from Dα to Dβ . – I maps each c ∈ C to an element of Dτ (c). – e maps each α ∈ T to a member of Dα. • A variable assignment into M is a function that maps each expression (x : α) to an element of Dα. • Given a variable assignment ϕ into M , an expression (x : α), and d ∈ Dα, let ϕ[(x : α) → d] be the variable assignment ϕ into M such that ϕ ((x : α)) = d and ϕ (v) = ϕ(v) for all v = (x : α). 19 Semantics of STT: Valuation Function The valuation function for a standard model M = (D, I, e) for a language L = (C, τ ) of STT is the binary function V M that satisﬁes the following conditions for all variable assignments ϕ into M and all expressions E of L: M 1. Let E is (x : α). Then Vϕ (E) = ϕ((x : α)). M 2. Let E ∈ C. Then Vϕ (E) = I(E). M M M 3. Let E be F (A). Then Vϕ (E) = Vϕ (F )(Vϕ (A)). M 4. Let E be (λ x : α . Bβ ). Then Vϕ (E) is the f : Dα → Dβ M such that, for each d ∈ Dα, f (d) = Vϕ[(x:α)→d](Bβ ). M M M 5. Let E be (E1 = E2). If Vϕ (E1) = Vϕ (E2), then Vϕ (E) = M t; otherwise Vϕ (E) = f. 6. Let E be (I x : α . A). If there is a unique d ∈ Dα such that M M M Vϕ[(x:α)→d](A) = t, then Vϕ (E) = d; otherwise Vϕ (E) = e(α). 20 Abbreviations T means (λ x : ∗ . x) = (λ x : ∗ . x). F means (λ x : ∗ . T) = (λ x : ∗ . x). (¬A∗) means A∗ = F. (Aα = Bα) means ¬(Aα = Bα). (A∗ ∧ B∗) means (λ f : ∗ → (∗ → ∗) . f (T)(T)) = (λ f : ∗ → (∗ → ∗) . f (A∗)(B∗)). (A∗ ∨ B∗) means ¬(¬A∗ ∧ ¬B∗). (A∗ ⇒ B∗) means ¬A∗ ∨ B∗. (A∗ ⇔ B∗) means A∗ = B∗. (∀ x : α . A∗) means (λ x : α . A∗) = (λ x : α . T). (∃ x : α . A∗) means ¬(∀ x : α . ¬A∗). ⊥α means I x : α . x = x. if(A∗, Bα, Cα) means I x : α . (A∗ ⇒ x = Bα) ∧ (¬A∗ ⇒ x = Cα) where x does not occur in A∗, Bα, or Cα. 21 Expressivity • Theorem. There is a faithful interpretation of nth-order logic in STT for all n ≥ 1. • Most mathematical notions can be directly and naturally expressed in STT. • Examples: equiv-rel = λ p : (ι → (ι → ∗)) . ∀ x : ι . p(x)(x) ∧ ∀ x, y : ι . p(x)(y) ⇒ p(y)(x) ∧ ∀ x, y, z : ι . (p(x)(y) ∧ p(y)(z)) ⇒ p(x)(z) compose = λ f : (ι → ι) . λ g : (ι → ι) . λ x : ι . f (g(x)) inv-image = λ f : (ι → ι) . λ s : (ι → ∗) . I s : (ι → ∗) . ∀ x : ι . s (x) ⇔ s(f (x)) 22 Peano Arithmetic • Let PA = (L, Γ) be the theory of STT such that: L = ({0, S}, τ ) where τ (0) = ι and τ (S) = ι → ι. Γ is the set of the following three formulas: 1. 0 has no predecessor: ∀ x : ι . 0 = S(x). 2. S is injective: ∀ x, y : ι . S(x) = S(y) ⇒ x = y. 3. Induction principle: ∀P : ι → ∗ . P (0) ∧ (∀ x : ι . P (x) ⇒ P (S(x))) ⇒ ∀ x : ι . P (x). • Theorem (Dedekind, 1888). PA has (up to isomorphism) a unique standard model M = (D, I, e) where Dι = {0, 1, 2, . . .}. 23 Incompleteness of STT Theorem. There is no sound and complete proof system for STT. Proof. Suppose P is a sound and complete proof system o for STT. By the soundness of P and G¨del’s Incom- pleteness Theorem, there is a sentence A such that (1) M |= A, where M is the unique standard model for PA (up to isomorphism), and (2) PA P A. By the completeness of P, (2) implies PA |= A and hence M |= A since M is the only standard model of PA, which contradicts (1). 2 24 A Proof System for STT (1) • Axioms: A1 (Truth Values) ∀ f : ∗ → ∗ . (f (T∗) ∧ f (F∗)) ⇔ (∀ x : ∗ . f (x)). A2 (Leibniz’ Law) ∀ x, y : α . (x = y) ⇒ (∀ p : α → ∗ . p(x) ⇔ p(y)). A3 (Extensionality) ∀ f, g : α → β . (f = g) = (∀ x : α . f (x) = g(x)). A4 (Beta-Reduction) (λ x : α . Bβ )(Aα) = Bβ [x → Aα] provided Aα is free for x in Bβ . A5 (Proper Deﬁnite Description) (∃ ! x : α . A) ⇒ A[(x : α) → (I x : α . A)]. A6 (Improper Deﬁnite Description) ¬(∃ ! x : α . A) ⇒ (I x : α . A) = ⊥α. 25 A Proof System for STT (2) • Rule of inference: R (Equality Substitution) From Aα = Bα and C∗ infer the result of replacing one occurrence of Aα in C∗ by an occurrence of Bα. • Call this proof system A. – Due to Andrews, 1963. • Theorem (Jensen, 1969). A plus an axiom of inﬁnity is equiconsistent with bounded Zermelo set theory. 26 Ways of Making STT More Practical • Make the logic many-sorted by allowing several types of individuals, e.g., ι1, . . . , ιn. • Add machinery for basic mathematical objects such as sets, tuples, and lists. • Admit polymorphic operators like (λ x : t . x) by introducing type variables. • Enrich the type system of STT with new machinery such as subtypes, dependent types, and user-deﬁned type constructors. • Modify the semantics of STT to include partial functions and undeﬁned expressions. 27 Conclusion • Simple type theory is a logic that is eﬀective for practice as well as theory—unlike ﬁrst-order logic. – More expressive and more convenient. – Closer to mathematical practice. – Based on the same principles as ﬁrst-order logic. – Includes the full machinery of ﬁrst-order logic. – Integrates predicate logic, function theory, and type theory. • We recommend that simple type theory be incorporated into: – Logic courses oﬀered by mathematics departments. – The undergraduate curriculum for computer science and software engineering students. 28

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