A general proof system for logics of imperfect information

0.25 MB·English

by  Pietro Galliani

A general proof system for logics of imperfect
2 information
1
0
2 Pietro Galliani

n ILLC
a
University of Amsterdam
J
The Netherlands
7
2 ([email protected])

]
January 30, 2012
O
L
.
h Abstract
t
a
Wedevelopasemanticsforlogicsofimperfectinformationwithrespect
m
to general models.
[ Then we build a proof system and prove its soundness and complete-

ness with respect to this semantics.
1
v
1
1 Introduction
1
8
5 Logicsofimperfectinformationareextensionsoffirst-orderlogic(or,sometimes,
. ofotherlogics: seeforexampleTulenheimo[13]andVa¨a¨n¨anen[15])whichallow
1
0 to reason about patterns of dependence and independence between variables.
2
1 Historically, the earliest such logic was branching quantifier logic (Henkin
:
v [5]), which adds to the language of first order logic branching quantifiers such
i as
X
∀x ∃y
φ(x,y,z,w)
r (cid:18) ∀z ∃w (cid:19)
a
whose interpretation, informally speaking, states that the choice of y is not
dependentonthe choiceofz andthechoiceofw is notdependentonthe choice
of x. A significant breakthrough in the study of this class of logics occurred
with the development of independence-friendly logic (Hintikka and Sandu [7]),
through which
1. The syntax of branching quantifier logic was significantly simplified, do-
ingawaywithcomplexstructuresofquantifierssuchastheaboveoneand
introducinginsteadslashedquantifiers (∃x/W)φ,whoseinformalinterpre-
tationis“thereexistsax,notdependentonanyvariablesinW,suchthat
φ”;
1
2. The game-theoretic semantics of logics of imperfect information was de-
fined formally, and its properties were examined in detail.
Thesedevelopmentsmadeitpossibletodefine,in[8],acompositional semantics
forindependence-friendly logicwhichisequivalentto its game-theoreticseman-
tics. This semantics, called team semantics or trump semantics, differs from
Tarski’ssemanticsforfirstorderlogicinthatsatisfactionconditionsofformulas
are predicated not of single assignments, but of sets of assignments1 (which we
will henceforth call Teams, after the terminology of Va¨a¨n¨anen [14]).
This alternate semantics providedone of the main impulses towardsthe de-
velopment of dependence logic [14], which separates the notion of dependence
andindependence fromthe notionofquantificationby doing awaywith slashed
quantifier and introducing instead dependence atoms of the form =(t ...t ),
1 n
where t ...t are terms, which are satisfied by a team X if and only if the
1 n
valueoft isafunctionofthevaluesoft ...t init. This-onlyatfirstsight
n 1 n−1
minor - innovation led to a number of significant advances in the study of the
properties of logics of imperfect information, and, in particular, of their model
theory; apart from the aforementioned [14], we can refer here for example to
the results of (Juha) Kontinen and Va¨a¨n¨anen [10] and (Jarmo) Kontinen [9].
Furthermore,a recentdirection of researchin the field of logics ofimperfect
informationconsistsinthe study ofthe model-theoreticalpropertiesofvariants
of dependence logic obtained by substituting the dependence atoms with other
kinds of non first-order atomic formulas. The earliest work along these lines
wasGr¨adelandVa¨a¨n¨anen[4],whoseindependence logic isexpressivelystronger
than dependence logic and will be the main logical formalism taken in exam in
the rest of this work; furthermore, we have multivalued dependence logic from
Engstr¨om [2] and inclusion logic and exclusion logic from Galliani [3].2
One property common to all these papers is that they are essentially con-
cernedonlywiththesemantics oflogicsofimperfectinformationanditsmodel-
theoretic properties. The corresponding proof theories, instead, are still rela-
tively undeveloped. The recent [11] presents a sound and complete deduction
system for extracting the first-order consequences of a Dependence Logic the-
ory;however,due to the equivalence betweenDependence Logicand existential
second-order logic there exists no hope of extending this system to one for
deducingthe Dependence LogicconsequencesofaDependence Logictheoryac-
cordingtothestandardsemantics. Thepresentpaper,drawinginspirationfrom
Henkin’s treatment of second order logic [6] and from the analysis of branching
quantifiersof[12],maybe seenasadifferentapproachtothe studyofthe proof
1Later,CameronandHodges [1]proved, throughcombinatorial methods,thatnocompo-
sitionalsemanticsforsuchalogicexistsinwhichthesatisfactionconditionsarepredicatedof
singleassignments.
2That paper alsocharacterized precisely the expressive power of independence logic with
respect to open formulas, thus answering an open problem of [4], and proved that inclusion
andexclusionlogicarestrictlyweakerthanindependence logic.
2
theories of logics of imperfect information: instead of restricting our language,
we will weaken the semantics and consider a more general class of models and
then we will develop a proof system capable of extracting all valid formula for
this new semantics.
2 Independence Logic
In this section, we will briefly recall the syntax and the semantics of Indepen-
dence Logic, plus a few of its basic properties. It can be safely skipped by
anyone who is already familiar with the results of [4].
Asisoftendoneinthefieldoflogicsofimperfectinformation,wewillassume
that our expressions are always in Negation Normal Form.
Definition 2.1 (Syntax) Let Σ bea firstorder signature. Then the set NNF
Σ
of the negation normal form formulas of our logic is the smallest set such that
NNF-lit If φ is a first order literal over the signature Σ then φ∈NNF ;
Σ
NNF-ind If ~t , ~t and ~t are tuples of terms with signature Σ then ~t ⊥ ~t
1 2 3 2 ~t1 3
is in NNF ;
Σ
NNF-∨ If φ and ψ are in NNF then φ∨ψ is also in NNF ;
Σ Σ
NNF-∧ If φ and ψ are in NNF then φ∧ψ is also in NNF ;
Σ Σ
NNF-∃ If φ is in NNF and x is a variable then ∃xφ is in NNF ;
Σ Σ
NNF-∀ If φ is in NNF and x is a variable then ∀xφ is in NNF .
Σ Σ
The set Free(φ) of the free variables of a formula φ is defined similarly to the
case of First Order Logic:
Definition 2.2 (Free Variables) Let Σ be a first order signature, and let φ∈
NNF . Then the set Free(φ) of the free variables of φ is defined by structural
Σ
induction on φ as follows:
Free-lit If φ is a first order literal then Free(φ) is the set of all variables oc-
curring in φ;
Free-ind If φ is ~t ⊥ ~t then Free(φ) is the set of all variables occurring in
2 ~t1 3
~t ,~t or ~t ;
1 2 3
Free-∨ If φ is ψ∨θ for some formulas ψ,θ ∈NNF then Free(φ) is Free(ψ)∪
Σ
Free(θ);
Free-∧ If φ is ψ∧θ for some formulas ψ,θ ∈NNF then Free(φ) is Free(ψ)∪
Σ
Free(θ);
Free-∃ If φ is ∃xψ for some variable x and some ψ ∈ NNF then Free(φ) =
Σ
Free(ψ)\{x};
3
Free-∀ If φ is ∀xψ for some variable x and some ψ ∈ NNF then Free(φ) =
Σ
Free(ψ)\{x}.
The following definition is standard:
Definition 2.3 (Team) Let V be a finite set of variables and let M be a first
order model. A team over M with domain V is a set of first order assignments
over M with domain V.
The next definition will be useful to give the semantics for the “lax” (in the
sense of [3]) version of the existential quantifier that we will use:
Definition 2.4 (x-variation) Let M be a first order model, let X be a team
over M, and let x be a variable symbol (not necessarily in Dom(X)). Then
a team X′ of M with domain Dom(X′) = Dom(X)∪{x} is said to be a x-
variation of X, and we write X[x]X′, if and only if the restrictions of X and
X′ to Dom(X)\{x} are the same.
Atthispoint,wehaveallthatweneedinordertodefinetheteamsemantics
for independence logic.
Definition 2.5 (Team Semantics for Independence Logic) LetΣbeafirst
order signature, let M be a first order model of signature Σ, let φ∈NNF and
Σ
let X be a team with domain containing Free(φ). Then we say that X satisfies
φ in M, and we write M |= φ, if and only if
X
TS-lit φ is a first order literal and, for all s ∈ X, M |= φ in the usual first
s
order sense;
TS-ind φis~t ⊥ ~t forsometuplesofterms~t ,~t and~t ,andforalls,s′ ∈X
2 ~t1 3 1 2 3
with ~t hsi = ~t hs′i there exists a s′′ ∈ X with ~t ~t hs′′i = ~t ~t hsi and
1 1 1 2 1 2
~t ~t hs′′i=~t ~t hs′i;
1 3 1 3
TS-∨ φ is ψ ∨ψ for two formulas ψ ,ψ ∈NNF and X =Y ∪Z for some
1 2 1 2 Σ
two teams Y and Z such that M |= ψ and M |= ψ ;
Y 1 Z 2
TS-∧ φ is ψ ∧ψ for two formulas ψ ,ψ ∈NNF , M |= ψ and M |= ψ ;
1 2 1 2 Σ X 1 X 2
TS-∃ φ is ∃xψ for some variable x and some ψ ∈ NNF and there exists a
Σ
team X′ such that X[x]X′ (that is, X′ is a x-variation of X) and such
that M |= ψ;
X′
TS-∀ φ is ∀xψ for some suitable x and M |= ψ, where
X[M/x]
X[M/x]={s[m/x]:s∈X,m∈Dom(M)}.
As[4]shows,thedependenceatom=(t ...t )isequivalenttotheindependence
1 n
atomt ⊥ t . Therefore,DependenceLogiciscontainedinIndependence
n t1...tn−1 n
Logic. The following result is also in [4]:
4
Theorem 2.6 ([4]) Let Σ be a first order signature, let V = {~v} be a finite
set of variables and let φ(~v) ∈ NNF be an Independence Logic formula with
Σ
signature Σ and free variables in V. Then there exists an existential second
order logic formula Φ(R) such that, for all models M with signature Σ and all
teams X over M with domain V,
M |= φ⇔M |=Φ(Rel(X))
X
where Rel(X)={s(~v):s∈X}.
In [3], the converse of this result is proved:
Theorem 2.7 ([3]) Let Σ be a first order signature, let V ={~v} be a finite set
of variables and let Φ(R) be an existential second order formula with signature
Σ and with R as its only free variable, where R is a relational variable of arity
|~v|. Then there exists an independence logic formula φ(~v), over the signature Σ
and with free variables in ~v, such that
M |= φ⇔M |=Φ(Rel(X))
X
for all models M with signature Σ and all nonempty teams X over M with
domain V.
3 General models for independence logic
In this section, we will develop a generalization of team semantics, along the
lines of Henkin’s treatment of second order logic. As we will see, the fact that
Independence Logic correspondsto ExistentialSecond Order Logic (and not to
full Second Order Logic) means that we will be able to restrict ourselves to
consider only a very specific kind of general model.
Definition 3.1 (General Model) Let Σ be a first order signature. A general
model with signature Σ is a pair (M,G), where M is a first order model with
signatureΣ and G is a set of teams over finitedomains, respectingthe condition
• Ifφ(x ...x ,m~,R~)isafirstorderformula, wherem~ is atupleofconstant
1 n
parameters in Dom(M) and where R~ is a tuple of “relation parameters”
corresponding to teams in G, in the sense that each R is of the form
i
R =Rel(X)={s(~z):s∈X }
i i
for some X ∈G, then for
i
kφ(x ...x ,m~,R~)k ={s:Dom(s)={x ...x },M |= φ(x ...x ,m~,R~)}
1 n M 1 n s 1 n
it holds that kφ(x ...x ,m~,R~)k ∈G.
1 n M
5
Lemma 3.2 Let Σ be a first order signature and let (M,G) be a general model
with signature Σ. Then for all X ∈G and all variables y, X[M/y]∈G.
Proof:
Let Dom(X) = {~x}, let R = Rel(X), and consider the formula φ(~x,y) =
∃yR(~x). Then take any assignment s with domain {~x,y}: by construction,
M |= φ(~x,y)⇔∃m s.t. s[m/y] ∈X ⇔s∈X[M/y], as required.3
s |~x
(cid:3)
We can easily adapt the team semantics of the previous section to general
models. We report all the rules here, for ease of reference; but the only differ-
ences between this semantics and the previous one are in the cases PTS-∨ and
PTS-∃.
Definition 3.3 (General Team Semantics for Independence Logic) Let
Σ be a first order signature, let (M,G) be a general model of signature Σ, let
φ ∈ NNF be a formula of Independence Logic and let X ∈ G be a team with
Σ
domain containing Free(φ). Then we say that X satisfies φ in (M,G), and we
write (M,G)|= φ, if and only if
X
GTS-lit φ is a first order literal and, for all s∈X, M |= φ in the usual first
s
order sense;
GTS-ind φ is ~t ⊥ ~t for some tuples of terms ~t , ~t and ~t , and for all
2 ~t1 3 1 2 3
s,s′ ∈ X with ~t hsi =~t hs′i there exists a s′′ ∈X with ~t ~t hs′′i =~t ~t hsi
1 1 1 2 1 2
and~t ~t hs′′i=~t ~t hs′i;
1 3 1 3
GTS-∨ φ is ψ ∨ψ for two formulas ψ ,ψ ∈NNF and X =Y ∪Z for some
1 2 1 2 Σ
two teams Y,Z ∈G such that (M,G)|= ψ and (M,G)|= ψ ;
Y 1 Z 2
GTS-∧ φ is ψ ∧ ψ for two formulas ψ ,ψ ∈ NNF , (M,G) |= ψ and
1 2 1 2 Σ X 1
(M,G)|= ψ ;
X 2
GTS-∃ φ is ∃xψ for some variable x and some ψ ∈ NNF and there exists a
Σ
team X′ ∈G such that X[x]X′ and such that (M,G)|= ψ;
X′
GTS-∀ φ is ∀xψ for some suitable x and (M,G)|= ψ.
X[M/x]
TheusualsemanticsforIndependenceLogicsatisfiesalocalityprinciple: inbrief,
the satisfiability ofa formula φ ina teamdepends only onthe restrictionof the
team to Free(φ). Let us verify that the same holds for entailment semantics:
Lemma 3.4 Let(M,G)beageneralmodel,andletX ∈G besuchthatDom(X)=
~x~y. Then X ={s:Dom(s)=~x,∃m~ s.t. s[m~/~y]∈X} is in G.
|~x
Furthermore, let Y ⊆X be such that Y ∈G. Then the team
|G
X(~x∈Y)={s∈X :s ∈Y}
|~x
3Here by s[m/y]|~x we intend the restriction of s[m/y] to the domain {x1...xn}. If y is
amongx1...xn,thenthisisthesameofs[m/y]itself;otherwise,itissimplys.
6
is in G.
Proof:
By definition, X is kφ(~x,R)k , where φ is ∃~y(R~xy~) and R=Rel(X). There-
|~x M
fore, X ∈G.
|~x
Similarly, X(~x ∈ Y) is kφ(~xy~,R ,R )k , where φ is R ~xy~ ∧R ~x, R is
1 2 M 1 2 1
Rel(X) and R is Rel(Y).
2
(cid:3)
Theorem 3.5 (Locality) Let (M,G) be a general model, let X ∈ G and let φ
be an independence logic formula over the signature of M with Free(φ) = ~z ⊆
Dom(X). Then (M,G)|= φ if and only if (M,G)|= φ.
X X|~z
Proof:
The proof is by structural induction on φ. We present only the passages corre-
sponding to disjunction and existential quantification, as the others are trivial:
• Suppose that (M,G) |= ψ ∨ψ . Then, by definition, there exist teams
X 1 2
Y and Z in G such that X = Y ∪Z, (M,G) |= ψ and M |= ψ . By
Y 1 Z 2
inductionhypothesis,thismeansthat(M,G)|= ψ and(M,G)|= ψ .
Y|~z 1 Z|~z 2
But Y ∪Z =X , and hence (M,G)|= ψ ∨ψ .
|~z |~z |~z X|~z 1 2
Conversely, suppose that (M,G) |= ψ ∨ψ . Then there exist teams
X|~z 1 2
Y′,Z′ in G such that (M,G) |= ψ , (M,G) |= ψ and X = X′∪Y′.
Y′ 1 Z′ 2 |~z
NowletY beX(~z ∈Y′)andZ beX(~z ∈Z′);byconstruction,Y∪Z =X,
and furthermore Y′ =Y and Z′ =Z , and, by the lemma, Y and Z are
|~z |~z
in G. Thus, by induction hypothesis, (M,G) |= ψ and (M,G) |= ψ ,
Y 1 Z 2
and finally (M,G)|= ψ ∨ψ , as required.
X 1 2
• Suppose that (M,G) |= ∃xψ. Then there exists a team Y ∈ G such
X
that X[x]Y and (M,G) |= ψ. By induction hypothesis, this means that
Y
(M,G) |= ψ too; and since X [x]Y , this implies that M |= ∃xψ,
Y|~zx |~z |~zx X|~z
as required.
Conversely, suppose that (M,G) |= ∃xψ. Then there exists a team
X|~z
Y′, with domain ~zx, such that M |= ψ and X [x]Y′. Now let Y be
Y′ |~z
(X[M/x])(~zx ∈ Y′). By the lemma, Y ∈ G; furthermore, Y = Y′, and
|~zx
hence by induction hypothesis (M,G) |= ψ. Finally, X[x]Y: indeed, if
Y
s ∈X then s [m/x] ∈Y′ for some m ∈Dom(M), and hence s[m/x] ∈Y
~z
for the same m, and on the other hand, Y is contained in X[M/x], and
hence if s[m/x]∈Y it follows that s∈X.
Therefore (M,G)|= ∃xψ, as required.
X
(cid:3)
As in the case of Second Order Logic, first-order models can be represented
as a special kind of general model:
7
Definition 3.6 (Full models) Let (M,G) be a general model. Then it is said
to be full if and only if G contains all teams over M.
The following result is then trivial.
Proposition 3.7 Let (M,G) be a full model. Then for all suitable teams X
and formulas φ, (M,G)|= φ in general team semantics if and only if M |= φ
X X
in the usual team semantics.
Proof:
Follows at once by comparing the rules of Team Semantics and General Team
Semantics for the case that G contains all teams.
(cid:3)
How does the satisfaction relation in general team semantics change if we
vary the set G? The following definition and result give us some information
about this:
Definition 3.8 (Refinement) Let (M,G) and (M,G′) be two general models.
Then we say that (M,G′) is a refinement of (M,G), and we write (M,G) ⊆
(M,G′), if and only if G ⊆G′.
Intuitively speaking, a refinement of a general model is another general model
with more teams than it. The following result shows that refinements preserve
satisfaction relations:
Theorem 3.9 Let (M,G) and (M,G′) be two general models with
(M,G) ⊆ (M,G′), let X ∈ G, and let φ be a formula over the signature of M
with Free(φ)⊆Dom(X). Then
(M,G)|= φ⇒(M,G′)|= φ.
X X
Proof:
The proof is an easy induction on φ.
1. If φ is a first order literal, the result is obvious, as the choice of the set of
teams G (orG′)does notenterinto the definitionofsatisfactioncondition
PTS-lit.
2. If φ is an independence atom, the result is also obvious, for the same
reason.
3. If (M,G)|= ψ ∨ψ then there exist two teams Y,Z ∈G such that X =
X 1 2
Y ∪Z,(M,G)|= ψ and(M,G)|= ψ . ButY andZ arealsoinG′,and
Y 1 Z 2
by induction hypothesis we have that (M,G′)|= ψ and (M,G′)|= ψ ,
Y 1 Z 2
and therefore (M,G′)|= ψ ∨ψ .
X 1 2
4. If (M,G) |= ψ ∧ψ then (M,G) |= ψ and (M,G) |= ψ . Then,
X 1 2 X 1 X 2
by induction hypothesis, (M,G′) |= ψ and (M,G′) |= ψ , and finally
X 1 X 2
(M,G′)|= ψ ∧ψ .
X 1 2
8
5. If (M,G) |= ∃xψ then there exists a X′ ∈ G such that X[x]X′ and
X
(M,G) |= ψ. But then X′ is also in G′, and by induction hypothesis
X′
(M,G′)|= ψ, and finally (M,G′)|= ∃xψ.
X′ X
6. If (M,G) |= ∀xψ then (M,G) |= ψ. Then, by induction hypothe-
X X[M/x]
sis, (M,G′)|= ψ, and finally (M,G′)|= ∀xψ.
X[M/x] X
(cid:3)
This result shows us that, as was to be expected from the equivalence between
independence logic and existential second order logic, if we are interested in
formulaswhichholdinall generalmodelsoveracertainfirst-ordermodelweonly
need to pay attention to the smallest (in the sense of the refinement relation)
ones. But do such “least general models” exist? As the following result shows,
this is indeed the case:
Proposition 3.10 Let {(M,G ) : i ∈ I} be a family of general models with
i
signature Σ and over the same first order model M. Then (M, G ) is also
i∈I i
a general model. T
Proof:
Let φ(x ...x ,m~,R~) be a first order formula with parameters, where each R
1 n i
is ofthe formRel(X)for some X ∈∩ G . Thenthe teamkφ(x ...x ,m~,R~)k
i i 1 n M
is in G for all i∈I, and therefore it is in G, as required.
i i∈I
(cid:3)
T
Therefore, it is indeed possible to talk about the least general model over a
first order model.
Definition 3.11 (Least General Model) LetM beafirstordermodel. Then
the least general model over M is the (M,L), where
L= {G :(M,G) is a general model.}
\
Whatisthepurposeofleastgeneralmodels? Theanswercomesasaconsequence
of Theorem 3.9, and can be summarized by the following corollary:
Corollary 3.12 Let Σ be a first order signature, let M be a first order model
over it and let (M,L) be the least general model over it. Then, for all teams
X ∈L and all formulas φ with signature Σ and with free variables in Dom(X),
(M,L)|= φ⇔(M,G)|= φ for all general models (M,G) over M.
X X
Proof:
Suppose that (M,L) |= φ. Then take any general model (M,G): by defini-
X
tion, we have that (M,L) ⊆ (M,G), and hence by Theorem 3.9 we have that
(M,G)|= φ.
X
Conversely, suppose that (M,G) |= φ for all general models (M,G); then
X
in particular (M,L)|= φ, as required.
X
9
(cid:3)
We can also find a more practical characterization of this “least general
model”.
Proposition 3.13 Let M be a first order model. Then the least general model
over it is (M,L), where L is the set of all kφ(~x,m~)k , where φ ranges over all
M
first order formulas and m~ ranges over all tuples of variables of suitable length.
Proof:
If (M,G) is a general model then L ⊆ G by definition; therefore, we only need
to prove that (M,L) is a general model.
Now, let φ(~x,m~,R~) be a first order formula, and let each R be Rel(X )
i i
for some X ∈ L. So for each R , any assignment s and any suitable tuple of
i i
terms t, M |= R~t if and only if M |= ψ (~t,~n ) for some first order formula ψ
s i s i i i
with parameters~n . Now let φ′(~x,m~,~n ,~n ,...) be the expression obtained by
i 1 2
substituting, in φ, each instance of R~t with ψ (~t,~n ); by construction, we have
i i i
that M |= φ(~x,m~,R~) if and only if M |= φ′(~x,m~,~n ,...), and therefore
s s 1
kφ(~x,m~,R~)k =kφ′(~x,m~,~n ,~n ,...)k ∈L
M 1 2 M
as required.
(cid:3)
As long as we are only considering teams in L, studying satisfiability with
respect to the least model (M,L) is the same as considering satisfiability with
respect to all general models over M. This restriction may at first sight seem
a bit unpractical, but it becomes irrelevant when it comes to the problem of
validity:
Definition 3.14 (Validity wrt general models) Let Σ be a first order sig-
nature, let V be a finite set of variables, and let φ ∈ NNF be a formula of
Σ
our language with free variables in V. Then φ is valid with respect to general
models if and only if (M,G)|= φ for all general models (M,G) with signature
X
Σ and for all teams X ∈G with Dom(X)⊇Free(φ). If this is the case, we write
GTS|=φ.
Definition 3.15 (Validity wrt least general models) Let Σ be a first or-
der signature, let V be a finite set of variables, and let φ∈NNF be a formula
Σ
of our language with free variables in V. Then φ is valid with respect to least
general models if and only if (M,L) |= φ for all least general models (M,L)
X
with signature Σ and for all teams X ∈ L with Dom(X) ⊇ Free(φ). If this is
the case, we write LTS|=φ.
Lemma 3.16 Let M be a first order model with signature Σ, and let M′ be
another first order model with signature Σ′ ⊇Σ such that the restriction of M′
to Σ is precisely M. Then for all general models G for M′, for all formulas φ
with signature Σ and for all X ∈G,
(M,G)|= φ⇔(M′,G)|= φ.
X X
10