many-sorted logic - helsingin yliopisto · many-sorted logic jouko väänänen1;2 1department of...

Many-sorted logic

Jouko Väänänen1,2

1Department of Mathematics and Statistics, University of Helsinki

2Institute for Logic, Language and Computation, University of Amsterdam

January 2014

Jouko Väänänen (Helsinki and Amsterdam) Many-sorted logic January 2014 1 / 114

Introduction


Many-sorted logic MSL

Several “sorts" of variableselements and sequences of elementsvector spaces: scalars and vectorsgeometry: points and linessecond order logic: individuals and subsets.


Many-sorted logic

Can be translated into one-sorted logicMany-sorted version of a theorem is sometimes more powerfulthan the single sorted version.Craig Interpolation Theorem.Applications of the many-sorted interpolation theorem.Applications to “symbiosis" between set theory and model theory.


Single sorted logic


DefinitionAn ordinary single sorted first order structure (model)

M = (M,R1, . . . ,Rm, f1, . . . , fk , c1, . . . , cl)

consists of1 Universe M 6= ∅2 Relations R1, . . . ,Rm between elements of the universe3 Functions f1, . . . , fk between elements of the universe4 Distinguished constants c1, . . . , cl in the universe.


where ...

Ri ⊆ M × ...×M

fi : M × ...×M → M

ci ∈ M


Syntax in single sorted logic

t = t ′

Rt1...tn¬φ∧n φn∨n φn

∃xiφ

∀xiφ

In second order logic add

∃X mi φ where X m

i is m-ary

∀X mi φ where X m

i is m-ary

In monadic second order logic m = 1.Jouko Väänänen (Helsinki and Amsterdam) Many-sorted logic January 2014 8 / 114

Semantics in single sorted logic

Assignments v intoM are functions on voc(L)-variables such thatv(xi) ∈ M for each xi . Modified assignment v(a/xi) as usual.

DefinitionFirst order logic:M |=v ∃xiφ ⇐⇒ M |=v(a/xi ) φ for some a ∈ M.M |=v ∀xiφ ⇐⇒ M |=v(a/xi ) φ for all a ∈ M.

Second order logic:M |=v ∃X m

i φ ⇐⇒ M |=v(A/X mi ) φ for some A ⊆ Mm.

M |=v ∀X mi φ ⇐⇒ M |=v(a/X m

i ) φ for all A ⊆ Mm.


Many sorted logic: its structures


DefinitionA many-sorted structure (model)

M = ({M1, . . . ,Mn},R1, . . . ,Rm, f1, . . . , fk , c1, . . . , cl)

consists of1 Universes (“sorts") M1, . . . ,Mn, each 6= ∅2 Relations R1, . . . ,Rm between elements of the universes3 Functions f1, . . . , fk between elements of the universes4 Distinguished constants c1, . . . , cl in the universes.

We callM n-sorted. M is “strict" if the Mi are disjoint, otherwise “lax".We allow n = 0, i.e. “no sorts" as a special case.


where ...

Ri ⊆ Mi1 × ...×Mis s(Ri) = 〈i1, ..., is〉 ⊆ {1, ...,n}

fi : Mi1 × ...×Mis → Mr s(fi) = 〈i1, ..., is, r〉 ⊆ {1, ...,n}

ci ∈ Mj s(ci) = j ∈ {1, ...,n}

No relation between elements of sort s or sort s′ and elements of sorts′′.


ExampleAn ordinary single-sorted first order structure

(M,R1, . . . ,Rm, f1, . . . , fk , c1, . . . , cl)

resurrects in this notation as a 1-sorted structure

({M},R1, . . . ,Rm, f1, . . . , fk , c1, . . . , cl)


Chu spaces

ExampleA Chu space is a strict 2-sorted structure ({A,X},R), (denoted(A,X ,R)), where R ⊆ A× X . For example:

A X Raxelements sets a ∈ xmodels sentences a |= xsentences models x |= a.


Vector spaces

ExampleA vector space is a strict 2-sorted structure

V = ({F ,V},+F , ·F ,0F ,1F ,+V , ·V ,0V ),

where1 (F ,+F , ·F ,0F ,1F ) is a field2 (V ,+V ,0V ) is an Abelian group3 ·V : F × V → V satisfies

a ·V (u +V w) = a ·V u +V a ·V w(a ·F b) ·V u = a ·V (b ·V u)(a +F b) ·V u = a ·V u +V b ·V u1F ·V u = u


Vector spaces (simpler notation)

ExampleA vector space is a strict 2-sorted structure

V = (F ,V ,+, ·,0),

where1 F is a field2 (V ,+,0) is an Abelian group3 The function (a, v) 7→ av(= a · v) : F × V → V satisfies

a(u + w) = au + aw(ab) · u = a(bu)(a + b)u = au + bu1u = u


Model with a pairing function

ExampleA first order structure with a pairing function is a strict 2-sortedstructure

M = ({N,M}, π, ...),

where π is a pairing function, i.e. a bijection M ×M → N. We can alsolet N = M2 and π(x , y) = (x , y).

Note: If M is finite, π cannot be a function into M, so a new sort isneeded.


Model with finite sequences

ExampleA first order structure with finite sequences is a strict 3-sorted structure

M = ({⋃n

Mn,N,M}, len,0, s, σ...),

where len(a) = n if a ∈ Mn, s is the successor function on N, and

σ(〈a1, . . . ,an〉, i) = ai .


Second order domain

ExampleA first order structure with second order domain is a strict 2-sortedstructure

M = ({P(M),M},∈, ...),

where1 ∈ is the membership relation a ∈ B between a ∈ M and B ⊆ M2 (M, ...) is a first order structure.


Higher order domains

ExampleA first order structure with second and third order domains is a strict3-sorted structure

M = ({P(P(M)),P(M),M},∈1,∈2, ...),

where1 ∈1 is the membership relation a ∈1 B between a ∈ M and B ⊆ M2 ∈2 is the membership relation B ∈2 C between B ⊆ M andC ⊆ P(M)

3 (M, ...) is a first order structure.


Second order structure

ExampleA (full) second order structure is a strict 2-sorted structure

M = ({P(M),M},∈,R, ...),

where1 ∈ is the membership relation a ∈ B between a ∈ M and B ⊆ M2 R, ... are relations, functions and constants between elements of

M ∪ P(M).


Database

ExampleThe database

John male D 2500 MaryMary female E 2600 PaulLaura female B 2500 Mary

is a 4-sorted structure with the sorts

1: Name-sort2: Gender-sort3: Department-sort4: Salary-sort


Database as a many-sorted structure

The database is({M1,M2,M3,M4},R)

whereM1 = {John,Mary ,Laura,Paul}M2 = {female,male}M3 = {B,D,E}M4 = {2500,2600}

andR ⊆ M1 ×M2 ×M3 ×M4 ×M1

is the relation indicated by the database.


ExampleThe two table (relation) relational database

R1:John male D MaryMary female E PaulLaura female B Mary

R2:

A 1000B 1300C 1500D 2500E 2600

is another example of a 4-sorted structure, this time with two relations.


More about many sorted structures


DefinitionA many-sorted structure

M = ({M1, . . . ,Mn},R1, . . . ,Rm, f1, . . . , fk , c1, . . . , cl)

is isomorphic to

M′ = ({M ′1, . . . ,M ′n},R′1, . . . ,R′m, f ′1, . . . , f ′k , c′1, . . . , c′l )

if there is a bijection π :⋃

s Ms →⋃

s M ′s such that1 π � Mi : Mi → M ′i is bijection for i = 1, ...,n2 Ri(a1, ...,as) ⇐⇒ R′i (π(a1), ..., π(as))

3 πfi(a1, ...,as) = f ′i (π(a1), ..., π(as))

4 πci = c′i


DefinitionA many-sorted structure

M = ({M1, . . . ,Mn},R1, . . . ,Rm, f1, . . . , fk , c1, . . . , cl)

is an extension of a many-sorted structure

M′ = ({M ′1, . . . ,M ′n},R′1, . . . ,R′m, f ′1, . . . , f ′k , c′1, . . . , c′l )

if1 M ′i ⊆ Mi for i = 1, ...,n2 R′i (a1, ...,as) ⇐⇒ Ri(a1, ...,as) for a1, ...,as ∈

⋃ni=1 M ′i

3 f ′i (a1, ...,as) = fi(a1, ...,as) for (a1, ...,as) ∈ dom(f ′).4 c′i = ci

ThenM′ is a substructure ofM.


Neither extension nor substructure...

DefinitionA Chu transform between Chu spaces (A,X ,R) and (A′,X ′,R′) is apair h,g of mapping such that

1 h : A→ A′

2 g : X ′ → X3 ∀a ∈ A∀x ′ ∈ X ′(R′(h(a), x ′)↔ R(a,g(x ′)))


Chu transforms

Example

Transforming (monadic) second order logic into first order logic:A sentences of single sorted monadic second order logicX single sorted structures (M, ...)R truthA′ sentences of 2-sorted first order logicX ′ second order domain (P(M),M,∈, ...)R′ truthh translation of A into A′

g the first order partwhere

h(∃xiφ) = ∃x0i h(φ),h(∃X 1

i φ) = ∃x1i h(φ)

g((P(M),M,∈, ...)) = (M, ...)


Chu transforms

ExampleContinuous maps in topology:

A topological spaceX its topologyR elementhoodA′ topological spaceX ′ its topologyR′ elementhoodh continuous mapping A→ A′

g g(U) = {a ∈ A : f (a) ∈ U} i.e. preimageitem Now a ∈ g(U) ⇐⇒ h(a) ∈ U.


Chu transforms

ExampleImportant special case: A ⊆ A′, X ′ ⊆ X , h = g = id .


DefinitionA reduct of a many-sorted structure

M = ({M1, . . . ,Mn},R1, . . . ,Rm, f1, . . . , fk , c1, . . . , cl)

is obtained by leaving out some sorts, relations, functions, andconstants, as in

M′ = ({M1, . . . ,Mn−1},R1, . . . ,Rm′ , f1, . . . , fk ′ , c1, . . . , cl ′).

Relations, functions, and constants which are not meaningful are atthe same time dropped. Respectively then,M is an expansion ofM′.A reduct may also have the same sorts but fewer relations, etc.


ExampleReducts of a vector space (F ,V ,+, ·,0):

The scalar field: F = (F ,+F .·F ,0F ,1F )

The vector addition group: (V ,+,0)


ExampleReducts of a second order domain (P(M),M,∈, ...):

The first order part: (M, ...)

The second order part: (P(M),M,∈)


ExampleA reduct of the database

R:John male IV 2500 MaryMary female V 2600 PaulLaura female II 2500 Mary

is not obtained by forgetting (e.g.) the number-sort:

John male IV MaryMary female V PaulLaura female II Mary

This is a completely new relation, a so-called projection (not reduct) ofthe original. In a reduct the entire relation R would drop out.


ExampleA reduct of the two table (relation) relational database

R1:John male D MaryMary female E PaulLaura female B Mary

R2:

A 1000B 1300C 1500D 2500E 2600

obtained by forgetting the number-sort is

John male D MaryMary female E PaulLaura female B Mary

Relation R2 drops out as it uses the abandoned number-sort.


DefinitionA single sorted reduct of a many sorted L-structure

M = ({Ms : s ∈ sort(L)},R1, . . . ,Rm, f1, . . . , fk , c1, . . . , cl)

is any reduct

M′ = ({Ms},R1, . . . ,Rm′ , f1, . . . , fk ′ , c1, . . . , cl ′)

to an L′ = {s}, where s ∈ sort(L). Relations, functions, and constantswhich are not meaningful are dropped.

Sometimes the single sorted reducts determine the many sortedstructure, but usually not (because the single sorted reducts disregardthe interaction between sorts.


Fundamental translation

DefinitionThe fundamental translation of a relational many-sorted structure

M = ({M1, . . . ,Mn},R1, . . . ,Rm, c1, . . . , cl)

is the single sorted expanded structure

M∗ = (M1 ∪ ... ∪Mn,M1, ...,Mn,R1, . . . ,Rm, c1, . . . , cl)

where M1,...,Mn are new unary predicates.

From the single sortedM∗ the originalM can be readily recovered.We have left function symbols out because there is no satisfactory wayto extend functions from individual sorts to the union of all sorts.Anyway, functions can be transformed to relations and then there is noproblem.


Example({P(M),M},∈, ...)∗ = (P(M) ∪M,P(M),M,∈, ...)M∼=M′ ⇐⇒ M∗ ∼=M′∗

For vector spaces we could define the fundamental translation({F ,V},+,0)∗ = (F ∪ V ,F ,V ,+,0) by letting the value of thefunctions +F , ·F ,+V and ·V to be 0 whenever they are notcanonically defined.Note: M �M∗


Syntax and semantics of many sorted logic


DefinitionThe vocabulary L = voc(M) of a many-sorted structure (model)

M = ({M1, . . . ,Mn},RM1 , . . . ,RMm ,FM1 , . . . ,FMk , cM1 , . . . , cMl )

consists of1 Sort symbols s1, . . . , sn

2 Relation symbols R1, . . . ,Rm

3 Function symbols F1, . . . ,Fk

4 Constant symbols c1, . . . , cl

We write L = {s1, . . . , sn,R1, . . . ,Rm,F1, . . . ,Fk , c1, . . . , cl},sort(L) = {s1, . . . , sn}, rel(L) = {R1, . . . ,Rm}, fun(L) = {F1, . . . ,Fk},con(L) = {c1, . . . , cl}.


DefinitionEach vocabulary L has an arity-function

aL : rel(L) ∪ fun(L)→ N

which tells the arity of each predicate and function symbol, and asort-function sL:

sL(R) ∈ sort(L)aL(R), sL(f ) ∈ sort(L)aL(f ) × sort(L), sL(c) ∈ sort(L).

We do not have symbols for abstract relations between elements ofarbitrary sorts except identity = in lax structures.


DefinitionSuppose L is a many sorted vocabulary. The L-variables overmany-sorted L-structures are denoted:

xs, s ∈ sort(L)

with the intuition (fulfilled in the semantics below) that xs ranges overthe universe Ms of an L-structure

M = ({M1, . . . ,Mn},RM1 , . . . ,RMm ,FM1 , . . . ,FMk , cM1 , . . . , cMl ).

No variables ranging over “everything"! No variables ranging overelements of two different sort s (but this can be arranged by having anew sort for the “union").


DefinitionSuppose L is a many sorted vocabulary. The L-terms are defined asfollows:

1 Constants c in L are L-terms and s(c) is already defined.2 L-variables xs are L-terms and s(xs) = s.3 If t1, ..., tn are L-terms with si = s(ti) and f ∈ L with

s(f ) = 〈s1, . . . , sn, s〉, then ft1...tn (or f (t1, ..., tn)) is an L-term ands(ft1...tn) = s.


Syntax

t = t ′

Rt1...tn¬φ∧n φn∨n φn

∃xsφ∀xsφ

Strict many-sorted logic: we allow t = t ′ only ifs(t) = s(t ′).Lax many-sorted logic: we allow t = t ′ for allterms.


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