Herbrand-satisfiability of a Quantified Set-theoretical Fragment (Cantone, Longo, Nicolosi CILC2014)

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Set theory Computable Set Theory The Fragment π 0 π 0 as FOL Conclusions and Future Works Herbrand-satisfiability of a Quantified Set-theoretical Fragment Domenico Cantone, Cristiano Longo, Marianna Nicolosi Asmundo Department of Mathematics and Computer Science, University of Catania (Italy) CILC2014

Transcript of Herbrand-satisfiability of a Quantified Set-theoretical Fragment (Cantone, Longo, Nicolosi CILC2014)

Page 1: Herbrand-satisfiability of a Quantified Set-theoretical Fragment (Cantone, Longo, Nicolosi CILC2014)

Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

Herbrand-satisfiability of a QuantifiedSet-theoretical Fragment

Domenico Cantone, Cristiano Longo, Marianna NicolosiAsmundo

Department of Mathematics and Computer Science, University of Catania (Italy)

CILC2014

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

Set Theory

Supporters of set theory strongly believe that every formal theorycan be expressed by this intuitive formalism.

In fact, set theory provided solid and unifying foundations to suchdifferent areas of mathematics:geometry, arithmetic, analysis, . . .

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

Set Theory - Zermelo-Frenkel axiomatization

The universally accepted formalization of set theory is theaxiomatization provided by Zermelo, and extended by Frenkel in alater stage.

Relations: ∈ (membership), =. Variables: x , y , z , . . ..Constants: ∅, a, b, c , . . ..Axioms:

(empty set) ∅ is a set with no members;(couple) {α, β} is a set with α and β as members;(union)

⋃α is the set consisting of the members (sets) of α;

(extensionality) α = β iff they have the same members;(regularity) there are no membership cycles, i.e. if α1 ∈ α2 ∈ . . . ∈ αn,

then α1 6= αn....

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

Set Theory - Example Formulae

Some examples of set-theoretical formulae (unquantified):

x ∈ y , x = y ∪ z ,x = y , x ⊆ y \ z ,x ⊆ y , ∅ ∈ x ∪ y ∪ {z},x = ∅.

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

Set Theory - Pure Sets

Set-theretic formulae are interpreted (usually) in the universe ofpure sets, i.e. sets whose solely members are sets.

Examples of pure sets are: ∅, {∅}, {∅, {∅}}.

The von Neumann standard cumulative hierarchy of sets V is theclass consisting of all the pure sets, recursively defined by

V0 = ∅Vγ+1 = P(Vγ) , for each ordinal γVλ =

⋃µ<λ Vµ , for each limit ordinal λ

V =⋃γ∈On Vγ ,

where P(·) is the powerset operator and On is the class of allordinals.

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

Computable Set Theory

Computable Set Theory is the discipline devoted to study thedecision problem in fragments of set theory.

Decision procedures or undecidability results have been providedfor several fragments of set theory.

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

Computable Set Theory - Unquantified decidable fragmentexamples

Some examples of unquantified fragments of set theory with adecidable satisfiability problem:

MLS x ∈ y , x ⊆ y , x = y , x = y ∪ z , x = y ∩ z , x = y \ z ;

MLSS extends MLSS with x = {y};MLSS×2,m extends MLSS with map variables and operators like

f = x × y , f = g−1, f = gx |y , f = g ∪ h, f = g ∩ h, f = g \ h.

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

Computable Set Theory - ∀π0

Also quantified fragments has been studied. We briefly recall ∀0.

Atomic ∀0-formulae are of the types x ∈ y , x = y , withx , y ∈ Vars ∪ Consts;

Restricted quantified ∀0-formulae have the form

(∀x1 ∈ a1) · · · (∀xn ∈ an)ψ

where x1, . . . , xn ∈ Vars, a1, . . . , an ∈ Consts, and ψ is a booleancombination of atomic formulae;

Finally, ∀0-formulae are boolean combinations of restrictedquantified ∀0-formulae.

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

Computable Set Theory - Examples of ∀0-formulae

Some examples of ∀0-formulae:

x ∈ y ∧ a = x

(∀x ∈ a)(∀y ∈ b)(x ∈ y → x ∈ c)

(∀x ∈ a)(x ∈ b) ∨ (∀x ∈ b)(x ∈ a)

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

Computable Set Theory - ∀π0,2∀π0,2 extends ∀π0 with map constants MConsts = {f , g , h, . . .} andordered pair terms of the form [x , y ].

Atomic ∀π0,2-formulae are of the types x ∈ y , x = y , [x , y ] ∈ f andf = g , with x , y ∈ Vars ∪ Consts and f , g ∈ MConsts;

Restricted quantified ∀π0,2-formulae have the forms

(∀x1 ∈ a1) . . . (∀xh ∈ ah)(∀[xh+1, yh+1] ∈ fh+1) . . . (∀[xn, yn] ∈ fn)δ(∃x1 ∈ a1) . . . (∃xh ∈ ah)(∃[xh+1, yh+1] ∈ xh+1) . . . (∃[xn, yn] ∈ xn)δ

with x1, . . . , xn, y1, . . . , yn ∈ Vars, a1, . . . , ah ∈ Consts,fh+1, . . . , fn ∈ MConsts, and δ is a boolean combination of atomic∀π0,2-formulae.

Finally, ∀π0,2-formulae are boolean combinations of restrictedquantified ∀0-formulae.

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

Computable Set Theory - ∀π0,2 Example

An example of ∀π0,2-formula:

(∀x ∈ a)(∀y ∈ b)([x , y ] ∈ f ) ∨ (∃[x , y ] ∈ f )(x /∈ a ∨ y /∈ b)

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

Computable Set Theory - Applications

During the years, results and algorithms developed in the contextof Computable Set Theory has been applied for

automated proof verification,

program verification,

knowledge representation.

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

Computable Set Theory - Knowledge Representation (1/2)

Description logics are a well-established framework for KnowledgeRepresentation and Semantic Web.

In Description Logics a knowledge domain is represented in termsof individuals, indicating domain elements, concepts, which are setsof domain items, and roles, which represents relations over theknowledge domain.

The Description Logic SROIQ underpins the latest version of theWeb Ontology Language (OWL 2). SROIQ has a N2ExpTimedecision problem.

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

Computable Set Theory - Knowledge Representation (2/2)

DL〈MLSS×2,m〉 is a very expressive description logic with anNP-complete decision problem, as proved by a reduction to theone of MLSS×2,m.

DL〈∀π0,2〉 further extends DL〈MLSS×2,m〉 with several operatorsand the possibility to deal with concepts containing other concepts(metamodeling).

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

The fragment ∀π0

Let us examine in detail the fragment ∀π0 .

∀π0 is a quantified fragment of Set Theory wich comprehendsordered pair members [·, ·] and the non-pair operator π̄(·),indicating the non-pair members of the set given as argument.

MLSS×2,m, ∀0 and ∀π0,2 can be reduced in polinomial time to ∀π0 .

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

The fragment ∀π0- Syntax

Atomic ∀π0 -formulae are of the types x ∈ π̄(y), [x , y ] ∈ z , x = y ,with x , y , z ∈ Vars ∪ Consts.

Prenex ∀π0 -formulae have the following form:

(∀x1 ∈ π̄(a1)) · · · (∀xn ∈ π̄(an))(∀[y1, z1] ∈ b1) · · · (∀[yn, zn] ∈ bn)ψ,

where xi ∈ Vars and ai ∈ Consts, for 1 ≤ i ≤ n, yj , zj ∈ Vars andbj ∈ Consts, for 1 ≤ j ≤ m, and ψ is a boolean combination ofatomic ∀π0 -formulae.

Finally, ∀π0 -formulae are boolean combinations of closed (i.e., withno free variables) prenex ∀π0 -formulae.

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

The fragment ∀π0- Semantics

A binary operation over sets π(·, ·) is a pairing function if and onlyif the following hold:

π(ν, µ) = π(ν ′, µ′) ⇐⇒ ν = ν ′ ∧ µ = µ′.

The semantics for ∀π0 -formulae is given in terms of pair-awareset-theoretical interpretations, i.e., first-order interpretations I withV as domain and such that:

∈ is the membership relation among sets in V;

[·, ·] is interpreted by a pairing function;

Iπ̄(x) = {u ∈ Ix | (∀ν, µ)(I[ν, µ] 6= u)}, for allx ∈ Vars ∪ Consts.

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

The fragment ∀π0- Skeletal Representation

Let us recall the decision procedure for ∀π0 -formulae. We start withsome definitions.A skeletal representation S relative to (V ,T ) (V ∪ T ⊆ Consts,V ∩ T = ∅), is a finite set of ground atomic ∀π0 -formulae suchthat:

1 (regularity) ¬(a ∈+S a) for all a ∈ V ∪ T ;

2 (extensionality1) if a = b and γ are in S, then γ[a→ b] andγ[b → a] are in S as well;

3 (extensionality2) if a = b is not in S, for some a, b ∈ V , thenS |= c ∈ π̄(a) iff S 6|= c ∈ π̄(b) for some c ∈ V ∪ T , orS |= [c , d ] ∈ a iff S 6|= [c , d ] ∈ b for some c , d ∈ V ∪ T ;

4 statements in S of the form a = b can involve only constantsoccurring in V ;

5 Consts(S) ⊆ V ∪ T .

for all a, b ∈ V ∪ T .

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

The fragment ∀π0- Realization

Let V and T = {t1, t2, . . . , tn} be two finite and disjoint sets ofconstants, and let S be a skeletal representation relative to (V ,T ).The realization R of S relative to (V ,T ) is defined as follows:

R [u, v ] =Def πh(u, v)R x =Def {R y | S |= y ∈ π̄(x)} ∪ {R [y , z ] | S |= [y , z ] ∈ x}R ti =Def {R y | S |= y ∈ π̄(ti )} ∪ {R [y , z ] | S |= [y , z ] ∈ ti}∪

{{k + 1, k , i}}

for all u, v ∈ V, x ∈ V , ti ∈ T and with h = |V |+ |T |,k = |V | · (|V |+ |T |+ 3),

π0(u, v) = Def {u, {u, v}}πn+1(u, v) = Def {πn(u, v)} .

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

∀π0 as FOL? - Satisfiability

Theorem

Let ϕ be a ∀π0 -formula, let V = Consts(ϕ) andT = {t1, . . . , tn} ⊆ Consts, with n = 2 · |V |. ϕ is satisfiable if andonly if there exists a skeletal representation S relative to (V ,T )such that the realization R of S relative to (V ,T ) is a pair-awareset-theoretical model for ϕ.

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

The fragment ∀π0- Decision Procedure

A non-deterministic decision procedure for satisfiability of∀π0 -formulae esaily follows. Let ϕ be a ∀π0 -formula withV = Consts(ϕ). Choose T = {t1, . . . , tn} such that n = 2 · |V |.Then

1 guess non-deterministically a skeletal representation S relativeto (V ,T ),

2 checks whether the realization of S relative to (V ,T ) is amodel for ϕ.

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

∀π0 as FOL

Can ∀π0 be regarded as a First Order Language?

It would be nice:

machinery developed in the first order context may be reused;

results for decidable fragments of FOL may be adapted tosubfragments or extensions of ∀π0 (guarded fragment,DATALOG∨,¬, . . . ).

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∀π0 as FOL

Can ∀π0 be regarded as a First Order Language? NO

The interpretation domain is fixed to V for ∀π0 -formulae.

The membership relation must be compliant with theZermelo-Frenkel axioms, in particular extensionality and regularityaxioms.

These difficulties can be circumvented.

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

∀π0 as FOL - Idea

Observe that a skeletal representations relative to (V ,T ) is anHerbrand interpretation with universe (V ,T ) and subject toadditional conditions.

IDEA:

encode these conditions as first-order formulae

prove that the realization of every Herbrand model whichsatisfies these conditions is a model for the formula underconsideration.

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

∀π0 as FOL - encoding axioms (1/)

Let V ,T be two disjoint set of constants, and let

χ(V ,T ) =Def χ1 ∧ χ2 ∧ χ(V ,T )3 ∧ χ(V ,T )

4

NOTE: we will introduce additional predicate symbols.

Lemma

An Herbrand interpretation H is an Herbrand model for χ(V ,T ) iffthe ∀π0 -subset of H is a skeletal represention.

Lemma

Every skeletal representation relative to (V ,T ) can be easilyextended to a Herband model for χ(V ,T ).

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∀π0 as FOL - encoding axioms (1/)

χ1 =Def (∀x , y)(x ∈ π̄(y)→ x∈̂y)∧ (∀x , y)([x , y ] ∈ z → x∈̂z)∧ (∀x , y)([x , y ] ∈ z → y ∈̂z)∧ (∀x , y , z)(x∈̂y ∧ y ∈̂z → x∈̂z)∧ (∀x)¬(x∈̂x)

Constraint: (regularity) ¬(a ∈+S a) for all a ∈ V ∪ T .

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

∀π0 as FOL - encoding axioms (3/)

χ2 =Def (∀x , y)(x = y → y = x)∧ (∀x , y , z)(x = y ∧ y = z → x = z)∧ (∀x , y , z)(x ∈ π̄(y) ∧ x = z → z ∈ π̄(y))∧ (∀x , y , z)(x ∈ π̄(y) ∧ y = z → x ∈ π̄(z))∧ (∀x , y , z , v)([x , y ] ∈ z ∧ x = v → [v , y ] ∈ z)∧ (∀x , y , z , v)([x , y ] ∈ z ∧ y = v → [x , v ] ∈ z)∧ (∀x , y , z , v)([x , y ] ∈ z ∧ z = v → [x , y ] ∈ v)

Constraint: (extensionality1) if a = b and γ are in S, thenγ[a→ b] and γ[b → a] are in S as well.

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

∀π0 as FOL - encoding axioms (4/)

χ(V ,T )3 =Def

∧x,y∈V

(¬(x = y) → dist(x, y) ∨ distπ(x, y)

)∧ (∀x, y)

(dist(x, y) →

∨z∈V∪T

(distBy(x, y, z) ∨ distBy(y, x, z)

))

∧ (∀x, y)

(distπ(x, y) →

∨z,v∈V∪T

(distByπ(x, y, z, v) ∨ distByπ(y, x, z, v)

))∧ (∀x, y, z)

(distBy(x, y, z) → (z ∈ π̄(x) ∧ ¬(z ∈ π̄(y)))

)∧ (∀x, y, z, v)

(distByπ(x, y, z, v) → ([z, v ] ∈ x ∧ ¬([z, v ] ∈ y))

)

Constraint: (extensionality2) if a = b is not in S, for somea, b ∈ V , then S |= c ∈ π̄(a) iff S 6|= c ∈ π̄(b) for some c ∈ V ∪T ,or S |= [c , d ] ∈ a iff S 6|= [c , d ] ∈ b for some c , d ∈ V ∪ T .

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

∀π0 as FOL - encoding axioms (5/)

χ(V ,T )4 =Def

∧x∈V∪T ,t∈T ,x 6=t

¬(x = t).

Constraint: statements in S of the form a = b can involve onlyconstants occurring in V .

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∀π0 as FOL - encoding axioms (6/)

Constraint: Consts(S) ⊆ V ∪ T .

It will directly descend from properties of Herbrand models.

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∀π0 as FOL - Herbrand and Set-Theoretical Models

Lemma

Let ϕ be a ∀π0 -formula, V = Consts(ϕ), T ⊆ Consts, T ∩ V = ∅.Let H be a Herbrand interpretation such that H |= χ(V ,T ) andConsts(H) ⊆ V ∪ T . Let S be the ∀π0 -subset of H, and let R bethe realization of S relative to V , T . Then H is an Herband modelof ϕ iff R is a set-theoretical model of ϕ.

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

∀π0 as FOL - Correspondence

Theorem

Let ϕ be a ∀π0 -formula, let V = Consts(ϕ), and let T be any set ofconstants disjoint from V such that such that |T | = 2 · |V |. Thenϕ is satisfiable if and only if χ(V ,T ) ∧ ϕ is Herbrand-satisfiable.

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

∀π0 as FOL - DATALOG∨,¬

χ(V ,T ) is in DATALOG∨,¬. Thus, if ϕ is a ∀π0 -formula inDATALOG∨,¬, χ(V ,T ) ∧ ϕ is in DATALOG∨,¬.

DATALOG∨,¬ satisfiability checkers can be used to test thesatisfiability of ∀π0 -formulae in DATALOG∨,¬.

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

Conclusions

We identified a correspondence between set-theoretical models andHerbrand models of ∀π0 -formulae.

Extending this correspondence to other decidable fragments ofset-theory should be straightforward, in particular those fragmentswhich can be reduced to ∀π0 , eg. MLSS×2,m and ∀π0,2) and, in turn,their counterparts in description logic.

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

Future Works

Call for Collaborations!

An implementation of the satisfiabity test for the DATALOG∨,¬

subfragment of ∀π0 can be implemented by means of DisjunctiveDatalog reasoners (see DLV).

Mappings to more restricted fragments of Disjunctive Datalog (foresample stratified programs) or other logic programmingformalisms should be studied.

Consequences of our result should be further investigated, forexample applications of answer set programming and queryanswering.

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Set theory Computable Set Theory The Fragment ∀π0 ∀π0 as FOL Conclusions and Future Works

Conclusions and Future Works

Thank you.