Type-Theoretical Semantics with Coercive Subtyping

Zhaohui LuoDepartment of Computer ScienceRoyal Holloway, University of London

Type theory

Combines computation and logical inference Uniformly in a single language Simple operational semantics + embedded logic

Provides structural mechanisms Natural description power Remark: “types” are natural in math. (cf, “sets” in set

theory) Has nice properties

Eg, Normalisation (decidability + consistency), Subject Reduction, Church-Rosser, … …

Is the basis of implementations of Proof assistants (ALF/Agda, Coq, Lego/Plastic, Matita,

NuPRL, … )

I. Type Theory: an Introduction

Basic judgement of type theory a : A “a is an object of type A.”

Examples 3 : Nat 2, 2+1 : Nat Nat x:Nat.x : Nat Nat

How do we understand “a : A”?

Basics – canonical objects

Examples: A = Nat, a = 3+4, v = 7. A = NatNat, a = (x:Nat.x,x+1)(2), v = 2,3.

Type of natural numbers

Elimination rule explained

“If C holds for all canonical nats, then C holds for every nat.”

General pattern (for all inductive types): C holds for all canonical objects of …

========================================================================

C holds for every object of …

Inductive/dependent types: examples

Finite types (Unit, Bool, ...)Types of lists, vectors, trees, …Types of dependent functionsTypes of dependent pairs (tuples)Types of ordinals, well-orderings, ...... ...

Embedded logic in type theory

Propositions-as-typesformula

type example

A B A B If … then …

x:A.B(x) x:A.B(x)Every man is handsome.

Prop – totality of logical propositions Impredicative type theories such as Simple TT

(Church 40) and UTT (Luo 94) Example – predicates to formalise adjectives/verbs:

handsome : ManPropwalk : ManProp

Combining types with propositions: examples

-types – types of dependent pairs/tuples Intuitively, for A : Type and B(x) : Type [x:A],

(A,B) = { x,y | x :A & y : B(x) }

Examples (when B is an A-indexed proposition):

(Nat, positive) -- type of positive nats(Man, handsome) -- type of

handsome men

Historical remarks

Early development (from early 1900’s) Logical paradoxes (eg, SS if S = { x | x x }?) Ramified type theory (Russell) Simple type theory (Ramsay 1926 & Church 40)

Modern development (since 1970’s) Martin-Löf’s type theory (predicative) (Martin-Löf 84) Impredicative type theories

Polymorphic -calculus – F (Girard 72) Calculus of Constructions – CC (Coquand & Huet 88) Unifying Theory of dependent Types – UTT (Luo 94)

Nice properties: examples

Normalisation: Every computation terminates. Decidability: Given a and A, it is decidable whether

“a : A”. (Compare this with the undecidable “aA” in set theory.)

Logical consistency: “False” is not provable.Other properties

e.g., “Church-Rosser”, “Subject Reduction”, …

Remark: Based on such good properties, we have simple operational semantics can implement type theories as, eg, proof assistants

Applications of type theory

Application example: Interactive theorem proving and proof assistants

ALF/Agda (Sweden) and NuPRL (USA), implementing Martin-Löf’s TT

Coq (France) and Lego/Plastic (UK), implementing CIC and UTT

Formalisation of mathematics (eg, 4-colour Theorem in Coq)

Program verification (eg, security protocols) Dependently-typed programming

Cayenne (Sweden), DML (USA), Epigram (UK), ...

Today: type-theoretical semantics in linguistics

II. Coercive Subtyping

Limitations of type theory? Compare with set theory:a A ------ a : AA B ------ ???

Subtyping A B ! How? Traditional notion of subtyping (“subsumptive

subtyping”) a : A A B===================================================

a : B

Q: Is this OK for type theories with canonical objects? A: No. (Why? Then what?)

Two different views of types

Type assignment Objects exist first – types are then assigned to objects. Overloading -terms, which may reside in different types. Examples: ML-like programming languages. Eg,

x.x : (NatNat, IntInt, AA, … )

Types as collections of canonical objects Types and their objects co-exist (one does not without the

other) Given by introduction/elimination/computation rules Embedded (consistent) logic, unique typing, … Examples: Martin-Löf’s TT and UTT.

Two corresponding views of subtyping

Subsumptive subtyping (for type assignment systems)Subsumption rule:

a : A A B =======================================

a : BDifficulty of subsumption with canonical objects

Q: If a is canonical in A, is it canonical in B?

If so, how would one formulate the elimination rule for B? “Bounded quantification”? (AB C:AType x:A …

Troublesome.) If not, then what? … …

Problematic …

Coercive subtypingResearch on coercive subtyping (Luo 1999 et al) Subtyping as abbreviation

f : B C a : A A c B ===================================================================================

f(a) = f(c(a)) : C a : A c B “a” can be regarded as an object of type

B CB[a] = CB[c(a)], ie, “a” stands for “c(a)”

Example, if we declare 1 as a coercion:

(Man, handsome) 1 Man

Coercive subtyping solves the incompatibility problem of subsumption; is adequate for TTs with canonical objects.

Coercive subtyping generalises/subsumes Injective/subset subtyping: Even N; Man Human. Projective/inheritance subtyping: (Man, handsome)

Man.

Implementations of coercive subtyping: Coq, Lego, Plastic, Matita

Applications, including Proof development Dependently-typed programming ... ...

Today: Coercive subtyping in type-theoretical semantics

Coercive subtyping is

an adequate theory of subtyping for the type theories with canonical

objects.

III. Type-Theoretical Semantics

Formal semantics of NLs Montague grammar (Montague 74) Situation semantics (Barwise-Perry 83) Discourse Representation Theory (Kamp-Reyle 93)

Examples in Montague grammar Common nouns (as functional subsets of entities)

man : CN / 〔 man 〕 : e t Verbs (also as functional subsets over entities)

walk : IV / 〔 walk 〕 : e t Adjectives (as functions from subsets to subsets)

handsome : CN/CN / 〔 handsome 〕 : (et) (et)

Some new developments in, eg, lexical sem Reference transfers (cf, Jackendoff)

“The ham sandwich shouts.” (in a special context) Logical polysemy (cf, Pustejovsky 95)

“burn an interesting book” (“book”: both “physical” and “informational”)

Problem and solution Montague Grammar is not fit for the purpose!

(cf, work by Pustejovsky, Asher, …) Types instead of functional subsets +

coercive subtyping

Type-theoretical semantics: CNs as types

Type theory – “many domains/types” (cf, (Ranta 94))

Common nouns are interpreted as types〔 man 〕 : Type〔 book 〕 : Type

Verbs are interpreted as predicates (functional subsets)〔 walk 〕 : 〔 man 〕 Prop〔 John walks 〕 = 〔 walk 〕 (John) : Prop, where John : Man.

Adjectives are interpreted as predicates (functional subsets)

〔 handsome 〕 : 〔 man 〕 Prop Modified CNs are interpreted by -types

〔 handsome man 〕 = ( 〔 man 〕 , 〔 handsome 〕 )

What about, eg, “A handsome man is a man” ? “Paul walks”, with Paul : 〔 handsome man 〕 ?

Solution: coercive subtyping First projection 1 as coercion: (A,B) 1

A

〔 handsome man 〕 = ( 〔man 〕 , 〔 handsome 〕 ) 1

〔man 〕 〔 Paul walks 〕 = 〔 walk 〕 (Paul) : Prop

because 〔 walk 〕 : 〔man 〕 Prop 〔 handsome man 〕 Prop.

Remark: Subtyping is crucial in type-theoretical semantics.

Examples involving dot-types

Dot-types AB in Generative Lexicon Theory (Pustejovsky 95) Example: PHYINFO PHYINFO PHY and PHYINFO INFO

In type-theoretical semantics,〔 book 〕 PHYINFO 〔 interesting 〕 : INFO Prop

PHYINFO Prop 〔 book 〕 Prop

〔 burn 〕 : Human PHY Prop Human PHYINFO Prop Human 〔 book 〕 Prop Human 〔 interesting book 〕 Prop

〔 John burned an interesting book 〕= b : 〔 interesting book 〕 . 〔 burn 〕 (John,b) : Prop

Another example – copredication (Asher 09)“John picked up and mastered the book.”〔 pick up 〕 : Human PHY Prop

Human PHYINFO Prop Human 〔 book 〕 Prop

〔 master 〕 : Human INFO Prop Human PHYINFO Prop Human 〔 book 〕 Prop

Remarks: Common nouns as types in type-theoretical semantics – so

things work. (But coercive subtyping is crucial!) Problematic if sticking to Montague’s interpretations of CNs

as functional subsets.

Modelling dot-types in type theory

What is AB?Inadequate accounts Intersection type Product type

Representing AB as AB with 1/2 as (implicit) coercions

where the C-condition (disjointness of A/B-components) guarantees that the coercions concerned are coherent.

Coercion contexts

Coercive subtyping only happens in local contexts (eg, reference transfers)Coercion contexts in type theory

Conclusions

Type theories with coercive subtyping provide a logical basis for formal semantics of NLs.Coercive subtyping

is crucial in giving an adequate type-theoretical semantics.

can be used to adequately model many lexical phenomena (eg, by means of dot-types) in lexical semantics.

Selected References

(Asher 09) N. Asher. Lexical Meaning in Context: A Web of Words. (Draft Book to appear in Cambridge Univ Press)

(Church 40) A. Church. A formulation of the simple theory of types. J. Of Symbolic Logic 5(1).

(Luo 94) Z. Luo. Computation and Reasoning: A Type Theory for Computer Science. Oxford Univ Press.

(Luo 99) Z. Luo. Coercive subtyping. J of Logic and Comput. 9(1). (Montague 74) R. Montague. Formal Philosophy. Yale Univ Press. (Pustejovsky 95) J. Pustejovsky. The Generative Lexicon. MIT.(Ranta 94) A. Ranta. Type-Theoretical Grammar. Oxford Univ

Press.