(Pre-)Algebras for Linguistics 4. Residuation Carl Pollard - - PowerPoint PPT Presentation

(Pre-)Algebras for Linguistics

• 4. Residuation

Carl Pollard

Linguistics 680: Formal Foundations

Autumn 2010

Carl Pollard (Pre-)Algebras for Linguistics

Inflationary and Deflationary Operations

A unary operation f on a preorder P, ⊑ is called inflationary iff, for all p ∈ P, p ⊑ f(p)

Carl Pollard (Pre-)Algebras for Linguistics

Inflationary and Deflationary Operations

A unary operation f on a preorder P, ⊑ is called inflationary iff, for all p ∈ P, p ⊑ f(p) deflationary iff, for all p ∈ P, f(p) ⊑ p

Carl Pollard (Pre-)Algebras for Linguistics

Closure and Interior Operations

Suppose P, ⊑ is a preorder, and f : P → P a unary

• peration which is both monotonic and idempotent u.t.e.

Then f is called

a closure operation if it is inflationary

Carl Pollard (Pre-)Algebras for Linguistics

Closure and Interior Operations

Suppose P, ⊑ is a preorder, and f : P → P a unary

• peration which is both monotonic and idempotent u.t.e.

Then f is called

a closure operation if it is inflationary an interior (or kernel) operation if it is deflationary

Carl Pollard (Pre-)Algebras for Linguistics

Closure and Interior Operations

Suppose P, ⊑ is a preorder, and f : P → P a unary

• peration which is both monotonic and idempotent u.t.e.

Then f is called

a closure operation if it is inflationary an interior (or kernel) operation if it is deflationary

Examples

topology of the real line

Carl Pollard (Pre-)Algebras for Linguistics

Closure and Interior Operations

Suppose P, ⊑ is a preorder, and f : P → P a unary

• peration which is both monotonic and idempotent u.t.e.

Then f is called

a closure operation if it is inflationary an interior (or kernel) operation if it is deflationary

Examples

topology of the real line modal operators

Carl Pollard (Pre-)Algebras for Linguistics

Closure and Interior Operations

Suppose P, ⊑ is a preorder, and f : P → P a unary

• peration which is both monotonic and idempotent u.t.e.

Then f is called

a closure operation if it is inflationary an interior (or kernel) operation if it is deflationary

Examples

topology of the real line modal operators Kleene closure operation on languages

Carl Pollard (Pre-)Algebras for Linguistics

Residuated Pairs

Suppose P, ⊑ and Q, ≤ are preorders and f : P → Q, g: Q → P functions such that, for all p ∈ P, q ∈ Q, f(p) ≤ q iff p ⊑ g(q)

Carl Pollard (Pre-)Algebras for Linguistics

Residuated Pairs

Suppose P, ⊑ and Q, ≤ are preorders and f : P → Q, g: Q → P functions such that, for all p ∈ P, q ∈ Q, f(p) ≤ q iff p ⊑ g(q) Then we say: f, g is a residuated pair

Carl Pollard (Pre-)Algebras for Linguistics

Residuated Pairs

Suppose P, ⊑ and Q, ≤ are preorders and f : P → Q, g: Q → P functions such that, for all p ∈ P, q ∈ Q, f(p) ≤ q iff p ⊑ g(q) Then we say: f, g is a residuated pair g is a residual of f

Carl Pollard (Pre-)Algebras for Linguistics

Residuated Pairs

Suppose P, ⊑ and Q, ≤ are preorders and f : P → Q, g: Q → P functions such that, for all p ∈ P, q ∈ Q, f(p) ≤ q iff p ⊑ g(q) Then we say: f, g is a residuated pair g is a residual of f f is a residuation of g

Carl Pollard (Pre-)Algebras for Linguistics

Basic Facts about Residuated Pairs

If f, g is a residuated pair, then:

Carl Pollard (Pre-)Algebras for Linguistics

Basic Facts about Residuated Pairs

If f, g is a residuated pair, then: both f and g are monotonic

Carl Pollard (Pre-)Algebras for Linguistics

Basic Facts about Residuated Pairs

If f, g is a residuated pair, then: both f and g are monotonic g(q) is a greatest element of {p ∈ P | f(p) ≤ q}

Carl Pollard (Pre-)Algebras for Linguistics

Basic Facts about Residuated Pairs

If f, g is a residuated pair, then: both f and g are monotonic g(q) is a greatest element of {p ∈ P | f(p) ≤ q} f(p) is a least element of {q ∈ Q | p ⊑ g(q)}

Carl Pollard (Pre-)Algebras for Linguistics

Basic Facts about Residuated Pairs

If f, g is a residuated pair, then: both f and g are monotonic g(q) is a greatest element of {p ∈ P | f(p) ≤ q} f(p) is a least element of {q ∈ Q | p ⊑ g(q)} gf is a closure operation

Carl Pollard (Pre-)Algebras for Linguistics

Basic Facts about Residuated Pairs

If f, g is a residuated pair, then: both f and g are monotonic g(q) is a greatest element of {p ∈ P | f(p) ≤ q} f(p) is a least element of {q ∈ Q | p ⊑ g(q)} gf is a closure operation fg is an interior operation

Carl Pollard (Pre-)Algebras for Linguistics

Residual Operations

Recall that a presemigroup is a preordered algebra P, ⊑, ◦ where ◦ is monotonic in both arguments and associative u.t.e.

Carl Pollard (Pre-)Algebras for Linguistics

Residual Operations

Recall that a presemigroup is a preordered algebra P, ⊑, ◦ where ◦ is monotonic in both arguments and associative u.t.e. Suppose P, ⊑, ◦ is a presemigroup. A binary operation ⊸l (⊸r) on P is called a left (right) residual operation with respect to ◦ iff for all p, q, r ∈ P, p ◦ r ⊑ q iff r ⊑ p ⊸l q (r ◦ p ⊑ q iff r ⊑ p ⊸r q)

Carl Pollard (Pre-)Algebras for Linguistics

Residual Operations

Recall that a presemigroup is a preordered algebra P, ⊑, ◦ where ◦ is monotonic in both arguments and associative u.t.e. Suppose P, ⊑, ◦ is a presemigroup. A binary operation ⊸l (⊸r) on P is called a left (right) residual operation with respect to ◦ iff for all p, q, r ∈ P, p ◦ r ⊑ q iff r ⊑ p ⊸l q (r ◦ p ⊑ q iff r ⊑ p ⊸r q) Left and right residual operations are antitonic in their first argument and monotonic in their second argument.

Carl Pollard (Pre-)Algebras for Linguistics

Residual Operations

Recall that a presemigroup is a preordered algebra P, ⊑, ◦ where ◦ is monotonic in both arguments and associative u.t.e. Suppose P, ⊑, ◦ is a presemigroup. A binary operation ⊸l (⊸r) on P is called a left (right) residual operation with respect to ◦ iff for all p, q, r ∈ P, p ◦ r ⊑ q iff r ⊑ p ⊸l q (r ◦ p ⊑ q iff r ⊑ p ⊸r q) Left and right residual operations are antitonic in their first argument and monotonic in their second argument. If ◦ is commutative u.t.e., then there is no difference between a left residual operation and a right residual

• peration, so we speak simply of a residual operation.

Carl Pollard (Pre-)Algebras for Linguistics

Residuated Presemigroups

A residuated presemigroup is a tuple P, ⊑, ◦, ⊸l, ⊸r where P, ⊑, ◦ is a presemigroup with left and right residual operations ⊸l and ⊸r.

Carl Pollard (Pre-)Algebras for Linguistics

Residuated Presemigroups

A residuated presemigroup is a tuple P, ⊑, ◦, ⊸l, ⊸r where P, ⊑, ◦ is a presemigroup with left and right residual operations ⊸l and ⊸r. These are relevant for understanding the kind of categorial grammar called Lambek calculus.

Carl Pollard (Pre-)Algebras for Linguistics

Residuated Presemigroups

A residuated presemigroup is a tuple P, ⊑, ◦, ⊸l, ⊸r where P, ⊑, ◦ is a presemigroup with left and right residual operations ⊸l and ⊸r. These are relevant for understanding the kind of categorial grammar called Lambek calculus. Example: A∗ with language concatenation as ◦ and the language residuals as the residual operations (see Ch. 6).

Carl Pollard (Pre-)Algebras for Linguistics

Symmetric Residuated Presemigroups

A symmetric residuated presemigroup is a tuple P, ⊑, ◦, ⊸ where P, ⊑, ◦ is a presemigroup, ◦ is commutative u.t.e., and ⊸ is a residual operation.

Carl Pollard (Pre-)Algebras for Linguistics

Symmetric Residuated Presemigroups

A symmetric residuated presemigroup is a tuple P, ⊑, ◦, ⊸ where P, ⊑, ◦ is a presemigroup, ◦ is commutative u.t.e., and ⊸ is a residual operation. These are relevant in linear logic, a kind of propositional logic that underlies certain kinds of categorial grammar, such as abstract categorial grammar and λ-grammar.

Carl Pollard (Pre-)Algebras for Linguistics

Heyting Presemilattices

A heyting presemilattice is a preordered algebra P, ⊑, ⊓, → where

P, ⊑, ⊓ is a lower presemilattice, and

Carl Pollard (Pre-)Algebras for Linguistics

Heyting Presemilattices

A heyting presemilattice is a preordered algebra P, ⊑, ⊓, → where

P, ⊑, ⊓ is a lower presemilattice, and → is a residual operation with respect to ⊓.

Carl Pollard (Pre-)Algebras for Linguistics

Heyting Presemilattices

A heyting presemilattice is a preordered algebra P, ⊑, ⊓, → where

P, ⊑, ⊓ is a lower presemilattice, and → is a residual operation with respect to ⊓.

The residual operation → in a heyting presemilattice is usually called a relative pseudocomplement (rpc)

• peration.

Carl Pollard (Pre-)Algebras for Linguistics

Heyting Presemilattices

A heyting presemilattice is a preordered algebra P, ⊑, ⊓, → where

P, ⊑, ⊓ is a lower presemilattice, and → is a residual operation with respect to ⊓.

The residual operation → in a heyting presemilattice is usually called a relative pseudocomplement (rpc)

• peration.

Example: We can model the propositions preordered by entailment as a heyting presemilattice with:

the meaning of and as the meet operation

Carl Pollard (Pre-)Algebras for Linguistics

Heyting Presemilattices

A heyting presemilattice is a preordered algebra P, ⊑, ⊓, → where

P, ⊑, ⊓ is a lower presemilattice, and → is a residual operation with respect to ⊓.

The residual operation → in a heyting presemilattice is usually called a relative pseudocomplement (rpc)

• peration.

Example: We can model the propositions preordered by entailment as a heyting presemilattice with:

the meaning of and as the meet operation the meaning of if . . . then (or implies) as the rpc operation

Carl Pollard (Pre-)Algebras for Linguistics