Functional Semantics for Non-prenex QBF

Igor St´ephan

LERIA, Universit d’Angers, 2 Boulevard Lavoisier, 49045, Angers, Cedex 01, France

Keywords:

QBF, Semantics, Non-prenex.

Abstract:

Quantiﬁed Boolean Formulae (or QBF) are suitable to represent ﬁnite two-player games. Current techniques

to solve QBF are for prenex QBF and knowledge representation is rarely in this form. We propose in this

article a functional semantics for non-prenex QBF. The proposed formalism is symmetrical for validity and

non-validity and allows to give different interpretations to the quantiﬁers. With our formalism, the solution of

a non-prenex QBF is consistent with the speciﬁcation, directly readable by the designer of the QBF and the

locality of the knolewge is preserved.

1 INTRODUCTION

Quantiﬁed Boolean Formula (or QBF) is a generaliza-

tion of satisﬁability in which propositional symbols

may be universally and existentially quantiﬁed. Many

important problems in Artiﬁcial Intelligence may be

speciﬁed in QBF. The satisﬁability problem (SAT) of

propositional logic is nothing more than the validity

problem for QBF constituted of a propositional for-

mula embedded in existential quantiﬁers associated

with their propositional symbols. Hence, most of the

more recent decision procedures for the validity prob-

lem of QBF are based on the (propositional version

of the) search-based algorithm of Davis, Logemann

and Loveland (DLL) for SAT which is a direct conse-

quence of the semantics of the existential quantiﬁer.

The semantics of QBF is usually presented ei-

ther in its decision form (Stockmeyer, 1977) either

in a functional form but only for prenex QBF es-

sentially thanks to Skolem functions (Kleine B¨uning

et al., 2007; Benedetti, 2005a)

which may be ex-

pressed by policies (Coste-Marquis et al., 2006) or

strategies (Bordeaux and Monfroy, 2002).

The QBF semantics presented in its decision form

is very suitable for theoretical problems but does not

allow to extract solutions of the QBF. QBF are also

suitable to represent ﬁnite two-player games. The va-

lidity of a QBF ensures to the existential player that

there exists a strategy to win whatever plays the uni-

versal player. But in this case the decision semantics

Boolean functions associated to the existentially quan-

tiﬁed propositional symbols which depend on the univer-

sally quantiﬁed propositional symbols which precede them

is no more sufﬁcient to help the existential player.

The QBF semantics presented in its functional

form is restricted to prenex QBF

but this restric-

tion is a major drawback: knowledge representation

is rarely in prenex form. There exists a prenexing pro-

cess which preserves validity but this prenexing pro-

cess has many drawbacks:

• It is a non-deterministic process and the chosen

prenexing strategy impacts the time complexity of

the obtained QBF (Egly et al., 2003) (even the

impact may be reduce by so-called dependency

schemes (Lonsing and Biere, 2010)).

• The elimination of biconditionals leads to an ex-

ponential growth of the size of the formula (and

of its search space, see (Da Mota et al., 2009)

for a discussion about the translation of Plaisted-

Greenbaum (Plaisted and Greenbaum, 1986) for

QBF).

• The loss of locality of the quantiﬁed propositional

symbols introduces an increase of the size of the

search space (anyway many systems add ab ini-

tio miniscoping associated to quantiﬁer trees in

order to recover the lost scopes of the quanti-

ﬁers (Benedetti, 2005b; Giunchiglia et al., 2006)).

• The choice of a total order induced by the par-

tial order deﬁned by the quantiﬁers introduces

new dependencies between existentially quanti-

ﬁed propositional symbols and universally quan-

tiﬁed propositional symbols which does not ex-

ist from the QBF designer point of view and are,

hence, interpreted in the solution with difﬁculty.

i.e. nested quantiﬁers are forbidden

358

Stéphan I..

Functional Semantics for Non-prenex QBF.

DOI: 10.5220/0004760303580365

In Proceedings of the 6th International Conference on Agents and Artiﬁcial Intelligence (ICAART-2014), pages 358-365

ISBN: 978-989-758-015-4

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

• Parts of a solution may have no meaning at all for

the QBF designer: for example in a two-player

game over a space containing n moves, the height

of the tree representing a winning strategy is nec-

essarily n even in the subtrees where the victory

conditions are fulﬁlled before the game space is

completely ﬁlled

These different drawbacks lead us to propose new

procedures for QBF with a richer expressivity. But

we face then to two issues:

• the lack of a functional semantics for non prenex

QBF and

• the lack of techniques to verify the results of those

new procedures.

In this article, we focus on the ﬁrst issue. In or-

der to explicit our motivations, we need some pre-

liminaries (Section 2) which present some basic el-

ements about propositional logic (§ 2), QBF syntax

(§ 2) and decision and functional semantics for QBF

(§ 2). We ﬁrst present in Section 3 an example which

give a more technical presentation of our issue (§ 3.1)

then our proposition about a non-prenex QBF seman-

tics (§ 3.2).

2 PRELIMINARIES

Propositional Logic. Boolean values are denoted

t (for true) and f (for false) and the set of Boolean

values is denoted BOOL. The set of propositional

symbols is denoted P S . Symbols ⊤ and ⊥ stand

for Boolean constants. Symbol ∧ stands for conjunc-

tion, ∨ for disjunction, ¬ for negation, → for impli-

cation, ↔ for biconditional. The set of binary oper-

ators {∧, ∨, →, ↔} is denoted O. The set of propo-

sitional formulae PROP is deﬁned inductively as fol-

lows: propositional symbols or constants are elements

of PROP, if F is an element of PROP then ¬F is

also an element of PROP, if F and G are elements of

PROP and ◦ an element of O then (F ◦ G) is an el-

ement of PROP. A literal is a propositional symbol

or the negation of a propositional symbol. A cube is a

conjunction of literals. A clause is a disjunction of lit-

erals. A valuation v is a function from P S to BOOL

and the set of valuations is denoted VAL

PROP.

The semantics of the propositional formulae uses

the semantics of propositional constants and op-

erators which is deﬁned as usual: To each con-

stant and operator (resp. ⊤, ⊥, ¬, ∧, ∨, →,

In fact, moves after victory are any and rules are not

necessarily respected otherwise victory may be invalidated

after the party is over.

↔) is associated its semantics as a Boolean func-

tion (resp. i

⊤

, i

⊥

:→ BOOL, i

: BOOL → BOOL,

∧

, i

∨

, i

→

, i

↔

: BOOL× BOOL → BOOL). The se-

mantics of the propositional formulae is deﬁned in-

ductively for any valuation v as follows: v

∗

(⊥) =

⊥

= f, v

∗

(⊤) = i

⊤

= t, v

∗

(x) = v(x) if x ∈ P S ,

∗

((F ◦ G)) = i

◦

∗

(F), v

∗

(G)) if F, G ∈ PROP and

◦ ∈ O and v

∗

(¬F) = i

∗

(F)) if F ∈ PROP. A

propositional formula F is a tautology if for every val-

uation v, v

∗

(F) = t. A Boolean function f of arity n

(i.e. a function from BOOL

to BOOL) is associ-

ated to a propositional formula µ( f) on the proposi-

tional symbols {x

, . . . , x

} such that v

∗

(µ( f)) = t if

and only if f(v(x

), . . . , v(x

)) = t for every valuation

Syntax of Quantiﬁed Boolean Formulae. Symbol

∃ stands for existential quantiﬁer, ∀ stands for uni-

versal quantiﬁer and q stands for any quantiﬁer. The

set QBF of quantiﬁed Boolean formulae is deﬁned

inductively as follows: if F is an element of PROP

then it is also an element of QBF, if F is an element

of QBF and x is a propositional symbol then (∃x F)

and (∀x F) are elements of QBF, if F is an element

of QBF then ¬F is an element of QBF, if F and G

are elements of QBF and ◦ is an element of O then

(F ◦ G) is an element of QBF. An occurrence of a

propositional symbol x is free if it does not appeared

into the scope of ∃x or ∀x. A QBF is closed if it con-

tains no free occurrence of propositional symbol. A

substitution is a function from the set of propositional

symbols to the set of formulae. We deﬁne a substi-

tution of x by F in G, denoted [x ← F](G), as the

formula obtained from G by replacing all the occur-

rences of x by F except for the occurrences of x under

the scope of a quantiﬁer associated to x. A binder is a

character string q

. . . q

with x

, . . . , x

some sep-

arate propositionalsymbols and q

, . . . , q

some quan-

tiﬁers. A QBF QM is under prenex form if Q is a

binder and M is a Boolean formula. We deﬁne the

function

(.) which inverts the quantiﬁers and is such

that (

∃x) = ∀x and (∀x) = ∃x ; this function is ex-

tended classically to the binder.

Quantiﬁed Boolean Formula Semantics. The

QBF semantics [[.]] : VAL

PROP → BOOL pre-

sented uses the semantics of the Boolean operators

(and constants) of propositional logic and is deﬁned

inductively by:

FunctionalSemanticsforNon-prenexQBF

359

[[⊥]](v) = i

⊥

[[⊤]](v) = i

⊤

[[x]](v) = v(x) if x ∈ P S

[[(F ◦ G)]](v) = i

◦

([[F]](v), [[G]](v))

if F, G ∈ QBF and ◦ ∈ O

[[¬F]](v) = i

([[F]](v)) if F ∈ QBF

[[(∃x F)]](v) =

∨

([[F]](v[x := t]), [[F]](v[x := f]))

if F ∈ QBF

[[(∀x F)]](v) =

∧

([[F]](v[x := t]), [[F]](v[x := f]))

if F ∈ QBF

A closed QBF F is valid if [[F]](v) = t for every

valuation v. For example the QBF ∃a∃b∀c((a∨ b) ↔

c) is not valid while the QBF ∀c∃a∃b((a ∨ b) ↔ c)

is. This example shows that the order of quantiﬁers is

crucial to decide the validity of a QBF.

As in the propositional case, an equivalence re-

lation denoted ≡ is deﬁned for the QBF by F ≡ G

if [[F]](v) = [[G]](v) for every valuation v. In con-

nection with the above example, ∃x∃yF ≡ ∃y∃xF,

∀x∀yF ≡ ∀y∀xF, ¬∃xF ≡ ∀x¬F and ¬∀xF ≡ ∃x¬F

but ∃a∃b∀c((a∨ b) ↔ c) 6≡ ∀c∃a∃b((a∨ b) ↔ c).

The decision semantics of QBF is extended to a

functional semantics of prenex QBF thanks to the

notion of functional valuation: a partial function

sk of the set of propositional symbols to the set of

the Boolean functions is a functional valuation for

a prenex QBF if for every existentially quantiﬁed

propositional symbol x there exists a unique pair (x 7→

ˆx) ∈ sk such that the Boolean function ˆx has for ar-

ity the number of universally quantiﬁed propositional

symbols which precede x in the binder

. The set of

functional valuations is denoted VAL

FONC. The

decision semantics of QBF is extended to a func-

tional semantics [[.]] : VAL

PROP× VAL FONC →

BOOL for prenex QBF:

[[F]](v, sk) = v

∗

(F) if F ∈ PROP

[[(∃x F)]](v, sk) = [[F]](v[x := ˆx], sk)

if (x 7→ ˆx) ∈ sk and F ∈ QBF

[[(∃x F)]](v, sk) =

∨

([[F]](v[x := t], sk), [[F]](v[x := f], sk))

if (x 7→ ˆx) 6∈ sk and F ∈ QBF

[[(∀x F)]](v, sk)

= i

∧

([[F]](v[x := t], sk(t)),

[[F]](v[x := f], sk(f)))

if F ∈ QBF.

A functional valuation sk is a QBF model for a

prenex closed QBF F if [[F]](v, sk) = t, for any valu-

ation v. A closed prenex QBF is valid if and only if it

admits (at least) a QBF model.

We recall that the decision problem of the satis-

ﬁability of a Boolean formula (SAT) is NP-complete

A functional valuation is a set of Skolem functions.

while the decision problem of the validity of a QBF is

PSPACE-complete (Stockmeyer, 1977).

3 NON-PRENEX QBF

SEMANTICS

The introduction shows that a functional semantics

for non-prenex QBF is useful for QBF designer in or-

der to preserve the expected meaning of quantiﬁers

and locality of knowledge. In what follows we give a

more technical presentation of our issue on an exam-

ple (§ 3.1), then our proposition about a non-prenex

QBF semantics (§ 3.2).

3.1 Motivations

Let ρ = (∀t (t ↔ ((∃x φ(x, t)) ∨ ¬(∃y ψ(y, t))))) be a

QBF with φ(x, t) and ψ(y, t) also two QBF.

If we linearize this QBF (minimizing the depen-

dencies), we obtain the following prenex QBF:

ρ ≡ ∀t((t → ((∃x φ(x, t)) ∨ ¬(∃y ψ(y, t))))

∧(((∃x

′

φ(x

′

, t)) ∨ ¬(∃y

′

ψ(y

′

, t))) → t))

≡ ∀t∃x∃y

′

∀x

′

∀y((t → (φ(x, t) ∨ ¬ψ(y, t)))

∧((φ(x

′

, t) ∨ ¬ψ(y

′

, t)) → t))

The designer of the QBF who chooses to model

its problem by ¬(∃y ψ(y, t)) and not by (∀y ¬ψ(y, t))

waits for the existentially quantiﬁed propositional

symbols x and y Skolem functions with parameter

t. But the propositional symbol y is now univer-

sally quantiﬁed and is one of the parameters of the

model associated to ψ(y, t). Moreover, a new existen-

tially quantiﬁed propositional symbol y

′

has appeared

which has no meaning for the designer.

We have developped a QCSP search-based solver

which is validity oriented: validity check QF ≡ ⊤ is

treated unchanged but non validity check QF ≡ ⊥ is

replaced by the equivalent check

Q¬F ≡ ⊤. If we

look at the execution of a constraint-based validity-

oriented solver, for the check (ρ ≡ ⊤) we obtain the

following propagations:

• If t is substituted by ⊤, then necessarily

((∃x φ(x, ⊤)) ∨ ¬(∃y ψ(y, ⊤))) ≡ ⊤ and then ei-

ther (∃x φ(x, ⊤)) ≡ ⊤ or (∃y ψ(y, ⊤)) ≡ ⊥.

• If t is substituted by ⊥, then necessar-

ily ((∃x φ(x, ⊥)) ∨ ¬(∃y ψ(y, ⊥))) ≡ ⊥ and then

(∃x φ(x, ⊥)) ≡ ⊥ and (∃y ψ(y, ⊥)) ≡ ⊤.

For a validity-oriented solver, (∃y ψ(y, ⊤)) ≡ ⊥ is

treated as (∀y ¬ψ(y, ⊤)) ≡ ⊤ and (∃x φ(x, ⊥)) ≡ ⊥

is treated as (∀x ¬φ(x, ⊥)) ≡ ⊤.

A model for a non-prenex QBF will have the fol-

lowing shape (if (∃y ψ(y, ⊤)) ≡ ⊥):

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360

...

⊤

{

...

⊥

while a semantic certiﬁcate

for a validity-

oriented solver will have the shape:

≡ ⊤ : ∀t

≡ ⊥ : ∀y

...

⊤

{

≡ ⊥ : ∀x

...

≡ ⊤ : ∃y

...

⊥

In the model, the binders are positively inter-

preted, i.e. respecting the modeling, while in the cer-

tiﬁcate, the binders are either positively interpreted

(≡ ⊤), as for example for the QBF ρ itself and

(∃y ψ(y, ⊥)), or negatively (≡ ⊥), as for example for

the QBF (∃y ψ(y, ⊤)) and (∃x φ(x, ⊥)) depending on

whether the QBF is valid or not.

3.2 Functional Semantics for

Non-prenex QBF

To be able to deﬁne models for non-prenex QBF, we

need a new deﬁnition of QBF allowing easier access

to nested binders.

Deﬁnition 1 (QBF). Let D be a set of deﬁnition

symbols such that D ∩ P S =

0. A quantiﬁed Boolean

formula (or QBF) is a set of triplets

def := QΣ

(def ∈ D, Q a binder, Σ a propositional formula de-

ﬁned on D ∪ P S and QΣ a fragment) associated to a

partial order over the deﬁnition symbols with a least

element, the deﬁnition symbol root, such that

• every deﬁnition symbol appears only once in the

left-hand side;

• the deﬁnition symbol root only appears once in the

right-hand side;

• any other deﬁnition symbol than the root appears

only once in the right-hand side;

• every deﬁnition symbol in the left-hand side is

smaller than the deﬁnition symbol which appears

in the right-hand side;

• any binder except the one associated with the root,

is empty.

The function (†) is such that, for every triplet

def := QΣ of a QBF, def

†

= QΣ.

A certiﬁcate is any piece of information that provides

self-supporting evidence of the correctness of an execution.

The four ﬁrst items deﬁne a tree structure while

the last item allows to express easily the semantics.

If someone substitutes the deﬁnition symbols (ex-

cept the root) of a QBF in their right-hand sides of

the deﬁnitions, a “classical” QBF (i.e. in the meaning

of the preliminary section) is obtained. In what fol-

lows, we makes no distinction between the QBF, its

root and its “classical” deﬁnition.

Example 1. We deﬁne a QBF with root ψ :

ψ := ∃x∀y∃z(¬ψ

∨ ω)

:= ∀t∃w(ψ

0.0

∨ ψ

0.1

)

0.0

:= ∃uξ

0.1

:= ∀sγ

with ω = (x∧ y∧ z), ξ = ((u∨ y) ∧ ¬t) and γ = ((s∨

w) ∧t). This QBF is none other, by substitution, than

the “classical” non-prenex QBF:

ψ = ∃x∀y∃z(¬∀t∃w(∃uξ∨ ∀sγ) ∨ ω)

whose representation as a tree is:

∃x∀y∃z

∨

∀t∃w

∨

∃u

∀s

To deﬁne the semantics of our deﬁnition of QBF,

we associate ﬁrst to the propositional symbols of the

binder Boolean functions; such a function is either

positive if the binder is considered unchanged or neg-

ative if the binder is considered in its reversed polar-

ity: universal quantiﬁers are existantially interpreted

and reciprocally.

Deﬁnition 2 (Local Valuation). A positive local valu-

ation (resp. negative local valuation) of a QBF, whose

root deﬁnition has a binder Q, is a partial function

vl from the set of propositional symbols to the set

of Boolean functions such that for every existentially

(resp. universally) quantiﬁed propositional symbol x

of the binder Q, there exists a unique pair (x 7→ ˆx) ∈ vl

such that the Boolean function ˆx has for arity the

number of universally (resp. existentially) quantiﬁed

propositional symbols which precede x in the binder

Q. The set of local valuations is denoted VAL

LOC.

Example 2 (example 1 continued)

• The partial function vl

= {(u 7→ (7→ t))} is a pos-

itive local valuation for the QBF of root ψ

0.0

FunctionalSemanticsforNon-prenexQBF

361

• the partial function vl

= {(s 7→ (7→ f))} is a neg-

ative local valuation for the QBF of root ψ

0.1

• the partial function

0 is a positive local valuation

for the QBF of root ψ

0.1

• the partial function vl

= {(w 7→ {(t 7→ t), (f 7→

t)})} is a positive local valuation for the QBF of

root ψ

• the partial function vl

= {(y 7→ {(t 7→ f), (f 7→

f)})} is a negative local valuation for the QBF

of root ψ,

• ﬁnally, the partial function {(x 7→ (7→ f)), (z 7→

{(t 7→ f), (f 7→ t)})} is a positive local valuation

for the QBF of root ψ.

A deﬁnition symbol def of a triplet def := QΣ, if

it is interpreted positively, has for semantics a deﬁni-

tion valuation which may contain

• not only a positive local valuation (i.e. a function

on the combinations over the values for the uni-

versally quantiﬁed propositional symbols of the

binder Q) to give a semantics to the existentially

quantiﬁed propositional symbols of the binder Q,

• but also a semantics to every deﬁnition symbol of

the right-hand side thanks to a function which as-

sociates to every combination over the values for

the universally quantiﬁed propositional symbols a

deﬁnition valuation.

Deﬁnition 3 (Deﬁnition Valuation and Deﬁnition

Boolean Function). A deﬁnition valuation of a QBF

is a partial function vd from the set of deﬁnition sym-

bols to the set of (p, vl, fbd) constiting of a polar-

ity p ∈ {+, −}, a local valuation vl and a deﬁnition

Boolean function fbd under the following contraint:

for every (d 7→ (p, vl, fbd)) ∈ vd if the polarity p is +

then the local valuation vl and the deﬁnition Boolean

function fbd are positive for the QBF with root deﬁ-

nition symbol d otherwise the polarity is − and the

local valuation and the deﬁnition Boolean function

are negative. A deﬁnition association is a quadruplet

(d 7→ (p, vl, fbd)). The set of QBF valuations is de-

noted VAL

DEF.

A positive deﬁnition Boolean function (resp. neg-

ative) of a QBF, whose root deﬁnition has a binder

Q, is a partial Boolean function fbd from the set of

deﬁnition valuations such that f bd has for arity the

number of universally (resp. existentially) quantiﬁed

propositional symbols of the binder Q. The set of def-

inition Boolean functions is denoted FBD.

Example 3 (example 2 continued)

• The quadruplet ad

0.0

= (ψ

0.0

7→ (+, vl

0)) is a

deﬁnition association for the QBF of root ψ

0.0

(vl

is a positive local valuation);

• the quadruplet ad

0.1

= (ψ

0.1

7→ (−, vl

0)) is a

deﬁnition association for the QBF of root ψ

0.1

(vl

is negative local valuation);

• the quadruplet ad

0.0

= (ψ

0.1

7→ (+,

0)) is a def-

inition association for the QBF of root ψ

0.1

Thus, the function vd

= {ad

0.0

, ad

0.1

} is a deﬁnition

valuation.

The partial function

fbd

= {(t 7→ vd

), (f 7→ {ad

0.0

})}

is a positive deﬁnition Boolean function for the

QBF of root ψ

and the quadruplet ad

= (ψ

7→

(+, vl

, fbd

)) is its deﬁnition association.

Finally, the partial function

fbd

= {((t, t) 7→

0), ((t, f) 7→ {ad

}),

((f, t) 7→ {ad

}), ((f, f) 7→

0)}

is a negative deﬁnition Boolean function for the QBF

of root ψ and the quadruplet (ψ 7→ (+,

0, fbd

)) is its

deﬁnition association.

Deﬁnition 4 (QBF Valuation). A QBF valuation is

a pair consisting of a propositional valuation and a

deﬁnition valuation. The set of QBF valuations is de-

noted VAL

QBF.

Deﬁnition 5. The semantics of a QBF F is deﬁnied

by a function

[[.]] : VAL

QBF → BOOL

associated with two auxiliary functions

[[.]]

, [[.]]

−

VAL

PROP× VAL LOC× FBD → BOOL

all inductively deﬁned as follows.

for the propositional symbols:

(P S ) [[x]](v, vd) = v(x) if x ∈ P S ;

for the propositional logical connectors:

(⊥) [[⊥]](v, vd) = f ;

(⊤) [[⊤]](v, vd) = t ;

(◦) [[(G◦ H)]](v, vd) = i

◦

([[G]](v, vd), [[H]](v, vd))

if G, H are extended propositional formulas and

◦ ∈ {∧, ∨, →, ↔, ⊕} ;

(¬) [[¬G]](v, vd) = i

([[G]](v, vd))

if G is an extended propositional formula.

for the deﬁnition symbols:

) [[d]](v, vd) = [[d

†

]]

(v, vl, fbd) if d ∈ D, and

vd(d) = (+, vl, fbd) ;

) [[d]](v, vd) = i

([[d

†

]]

−

(v, vl, fbd)) if d ∈ D, and

vd(d) = (−, vl, fbd) ;

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362

for the quantiﬁers:

(∃

) [[(∃x G)]]

(v, vl, fbd) =

[[G]]

(v[x := ˆx], vl, f bd)

if G is an extended propositional formula, x ∈ P S

and (x 7→ (7→ ˆx)) ∈ vl ;

(∃

) [[(∃x G)]]

(v, vl, fbd) =

∨

([[G]]

(v[x := t], vl, fbd),

[[G]]

(v[x := f], vl, fbd))

if G is an extended propositional formula, x ∈ P S

and there is no pair (x 7→ ˆx) ∈ vl (ˆx a Boolean func-

tion) ;

(∃

) [[(∃x G)]]

−

(v, vl, fbd) =

∧

([[G]]

−

(v[x := t], vl(t), fbd(t)),

[[G]]

−

(v[x := f], vl(f), fbd(f)))

if G is an extended propositional formula and x ∈

P S .

(∀

) [[(∀x G)]]

−

(v, vl, fbd) =

[[G]]

−

(v[x := ˆx], vl, f bd)

if G is an extended propositional formula, x ∈ P S

and (x 7→ (7→ ˆx)) ∈ vl ;

(∀

) [[(∀x G)]]

−

(v, vl, fbd) =

∨

([[G]]

−

(v[x := t], vl, fbd),

[[G]]

−

(v[x := f], vl, fbd))

if G is an extended propositional formula, x ∈ P S

and there is no pair (x 7→ ˆx) ∈ vl (ˆx Boolean func-

tion).

(∀

) [[(∀x G)]]

(v, vl, fbd) =

∧

([[G]]

(v[x := t], vl(t), fbd(t)),

[[G]]

(v[x := f], vl(f), fbd(f)))

if G is an extended propositional formula and x ∈

P S .

) [[G]]

(v, vl, {(7→ vd)}) = [[G]](v, vd)

if G is an extended propositional formula.

) [[G]]

−

(v, vl, {(7→ vd)}) = i

([[G]](v, vd))

if G is an extended propositional formula.

If we restrict Deﬁnition 5 to the rules (⊤), (⊥),

(◦), (¬) and (P S ) (neglecting the arguments for the

local valuation and the deﬁnition Boolean function

and replacing [[.]]

by [[.]]), the propositional seman-

tics is obtained. If we add the rules (∃

) and (∀

)

the semantics of decision procedure for non-prenex

QBF is then obtained. The rule (D

) interprets the

deﬁnition symbol and begins the interpretation of the

binder, of the fragment associated to the symbol, pos-

itively; the rule (P

) ends the interpretation of the

binder and evaluates the extended propositional for-

mula of the fragment. The rule (D

) interprets the

deﬁnition symbol and begins the interpretation of the

binder, of the fragment associated to the symbol, neg-

atively ; the rule (P

) ends the interpretation of the

binder and evaluates the extended propositional for-

mula of the fragment ; both combinated for a fragment

QΣ the rules apply the equivalence QΣ ≡ ¬

Q¬Σ.

Example 4 (example 3 continued). We show in this

example how the semantics is applied on the fragment

∃x∀y∃z(¬ψ

∨ ω) which is negatively interpreted

∃x∀y∃z(¬ψ

∨ ω) ≡ ¬∀x∃y∀z¬(¬ψ

∨ ω) (1)

by:

1. the application of the rule (D

) which introduces

the ﬁrst negation, the one before the binder;

2. the elimination of the quantiﬁers with a reversed

interpretation according to the rules (∃

) and

(∀

3. the application of the rule (P

) which introduces

the seconde negation, the one after the binder.

Let us compute [[ψ]](

0, vd) with vd = {(ψ 7→

(−,

0, fbd

))}. Let σ = (¬ψ

∨ ω).

[[ψ]](

0, {(ψ 7→ (−,

0, fbd

))})

= i

([[ψ

†

]]

−

(

0, fbd

))

†

= i

([[∃x∀y∃zσ]]

−

(

0, fbd

))

∃

= i

∧

( [[∀y∃zσ]]

−

([x := t],

0, fbd

(t)),

[[∀y∃zσ]]

−

([x := f],

0, fbd

(f))))

The existential quantiﬁer is negatively interpreted.

Let us denote v

= [x := t] and let us compute in

details:

[[∀y∃zσ]]

−

0, fbd

(t))

∀

= i

∨

( [[∃zσ]]

−

[y := t],

0, fbd

(t)),

[[∃zσ]]

−

[y := f],

0, fbd

(t)))

The universal quantiﬁer is negatively interpreted

and in its deﬁnitional version of its semantics (since

the local valuation is empty).

Let us denote v

= v

[y := t] and let us compute in

details:

[[∃zσ]]

−

0, fbd

(t))

∃

= i

∧

( [[σ]]

−

[z := t],

0, fbd

(t, t)),

[[σ]]

−

[z := f],

0, fbd

(t, f)))

Let us denote v

= v

[z := f], fbd = f bd

(t, f) and

let us compute in details:

[[σ]]

−

[z := f],

0, fbd)

= i

([[σ]](v

, fbd))

◦

= i

∨

([[¬ψ

]](v

, fbd), [[ω]](v

, fbd)))

= i

∨

([[ψ

]](v

, fbd)), [[ω]](v

, fbd)))

By the rule (P

), the negative interpretation of

the end of the binder introduces the seconde negation

of (1).

FunctionalSemanticsforNon-prenexQBF

363

Since the formula ω is propositional formula, let

us denote b

= [[ω]](v

, fbd) = (v

)

∗

(ω).

[[σ]]

−

[z := f],

0, fbd)

= i

∨

([[ψ

]](v

, fbd)), b

))

If we add the rules (∃

) to the rules (⊤), (⊥), (◦),

(¬), (P S ), (∃

) and (∀

) (ignoring the argument for

the deﬁnition Boolean function) then the functional

semantics of prenex QBF is obtained.

Example 5 (example 4 continued). This example

shows the positive interpretation of a binder accord-

ing to rules (D

), (∀

), (∃

) and (P

) and the inter-

pretation of the existential quantiﬁer in its functional

semantics and not decision one.

We recall that ψ

:= ∀t∃w(ψ

0.0

∨ ψ

0.1

), vl

{(w 7→ {(t7→ t), (f 7→ t)})}, fbd

= {(t7→ vd

), (f 7→

{ad

0.0

})} and vd

= {(ψ

0.0

7→ (+, vl

0)), (ψ

0.1

7→

(−, vl

0))}.

Let us denote σ

= (ψ

0.0

∨ ψ

0.1

) and let us com-

pute in details:

[[ψ

]](v

, (ψ

7→ (+, vl

, fbd

)))

)

= [[ψ

†

]]

, vl

, fbd

)

†

= [[∀t∃wσ

]]

, vl

, fbd

)

(∀

)

= i

∧

([[∃wσ

]]

[t := t], vl

(t), fbd

(t)),

[[∃wσ

]]

[t := f], vl

(f), fbd

(f)))

We have vl

(t) = {(w 7→ (7→ t))} and let us denote

= v

[t := t]. Let us compute in details:

[[∃wσ

]]

, vl

(t), fbd

(t))

= [[∃wσ

]]

, {(w 7→ (7→ t))}, fbd

(t))

(∃

)

= [[σ

]]

[w := t], vl

(t), fbd

(t))

The existential quantiﬁer is not interpreted with its

decision semantics but the value of the propositional

symbol is obtained from the local valuation.

We have fbd

(t) = {(7→ vd

)} and let us denote

= v

[w := t].

[[(ψ

0.0

∨ ψ

0.1

)]]

, vl

(t), {(7→ vd

)})

(◦)

= i

∨

( [[ψ

0.0

]]

, vl

(t), {(7→ vd

)}),

[[ψ

0.1

]]

, vl

(t), {(7→ vd

)}))

)

= i

∨

([[ψ

0.0

]](v

, vd

), [[ψ

0.1

]](v

, vd

))

The propositional formula of a fragment may

contain many deﬁnition symbols and the associated

binders are not necessarily interpreted in the same

manner.

Example 6 (example 5 continued). This example

shows the access to two deﬁnition associations and

the application of the rules (D

) and (D

) to inter-

pret the deﬁnition symbol and its associated binder.

We recall that

= {(ψ

0.0

7→ (+, vl

0)), (ψ

0.1

7→ (−, vl

0))}

Hence vd

(ψ

0.0

) = (+, vl

0) and vd

(ψ

0.1

) =

(−, vl

0). Thus

∨

([[ψ

0.0

]](v

, vd

), [[ψ

0.1

]](v

, vd

))

)&(D

)

= i

∨

([[ψ

†

0.0

]]

, vl

0),

[[ψ

†

0.1

]]

−

, vl

0))

We give without proof (but the arguments are clear

from above) soundness and completeness theorems

w.r.t. the decision semantics of the “classical” QBF.

Theorem 1 (Soundness). Let σ be a QBF, v a propo-

sitional valuation and vd deﬁnition valuation. If

[[σ]](v, vd) = t then v

∗

(σ) = t.

Theorem 2 (Completeness). Let σ be a QBF and

v a propositional valuation. If v

∗

(σ) = t then there

exists (at least) a deﬁnition valuation vd such that

[[σ]](v, vd) = t.

4 CONCLUSIONS

We have proposed in this article a functional seman-

tics for non-prenex quantiﬁed Boolean formulas. The

proposed formalism is symmetrical w.r.t. the validity

or the non-validity and allows to associate different

interpretations to quantiﬁers. In particular, it allows

to follow the choice of the designer of the QBF w.r.t.

the quantiﬁers. In case of QBF solvers based on a

quantiﬁed search algorithm, the extraction of the so-

lution is easy since our functional semantics follows

the inductive structure of the non-prenex QBF. Since

the prenexing process is not applied, dependencies

between propositional symbols are kept and the com-

puted QBF valuation may be directly interpreted by

the designer as a solution of its problem. Moreover,

a QBF valuation representing a winning strategy for

a ﬁnite two-player game develops no deﬁnition valua-

tion for the propositional symbols beyond the fulﬁlled

victory conditions.

Our formalism is also enough ﬂexible to allow to

deﬁne the notion of certiﬁcate for search procedure

for non-prenex QBF. The way the certiﬁcate certiﬁes

the soundness of the result is not developped due to

lack of space but is independent of the speciﬁcations

of the solver.

This work is implemented in Prolog

and is being

implemented for a quantiﬁed search algorithm based

This work is accessible at the URL: http://www.info.

univ-angers.fr/pub/stephan/Research/Download.html

ICAART2014-InternationalConferenceonAgentsandArtificialIntelligence

364

on the Gecode system

and will be available soon. In

our implementation with the Gecode system, if the in-

put format of QBF is “classical”, the internal structure

follows our deﬁnition of QBF.

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The solver is in fact a QCSP solver with a constraint

approach for QBF solving.

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365