Bernays-Sch\"onfinkel-Ramsey with Simple Bounds is NEXPTIME-complete

0.36 MB·English

by  Marco Voigt

Bernays-Sch¨onfinkel-Ramsey with
Simple Bounds is NEXPTIME-complete∗
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1 Marco Voigt
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2 Max Planck Institute for Informatics, Saarbru¨cken, Germany,
Graduate School of Computer Science, Saarland University
n
a
Christoph Weidenbach
J

8 Max Planck Institute for Informatics, Saarbru¨cken, Germany
2

]
O
Abstract
L
. Linear arithmetic extended with free predicate symbols is undecidable, in general. We
s
showthattherestrictionoflineararithmeticinequationstosimpleboundsextendedwiththe
c
[ Bernays-Sch¨onfinkel-Ramseyfreefirst-orderfragmentisdecidableandNEXPTIME-complete.
The result is almost tight because the Bernays-Sch¨onfinkel-Ramsey fragment is undecidable

1 in combination with linear difference inequations, simple additive inequations, quotient in-
v
equationsand multiplicative inequations.
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2 1 Introduction
7
0
. Satisfiability of the Bernays-Scho¨nfinkel-Ramsey (BSR) fragment of first-order logic is decidable
1
and NEXPTIME-complete [Lew80]. The complexity remains if the fragment is restricted to a
0
clause normal form. Only further restrictions on the number of literals per clause enable better
5
1 complexity results [Pla84]. Its extension with linear arithmetic is undecidable. For example,
: Halpern [Hal91] showedthat the combination of PresburgerArithmetic with one unary predicate
v
i yields undecidability. His proof relies on ∀∃ quantifier alternations on the arithmetic part. The
X
naturals can be defined on the basis of linear real arithmetic with the help of one (extra) unary
r predicate. Fietzke and Weidenbach [FW12] showed undecidability for the combination of linear
a
realarithmetic with severalbinary or one ternary predicate. The proofis based on a reduction of
thetwo-countermachinemodel[Min67]tothisfragmentwhereapurelyuniversalquantifierprefix
suffices. Two-counter machine instructions are translated into clauses of the form
(i) x′ =x+1kP(i,x,y)→P(i+1,x′,y)
(ii) x=0kP(i,x,y)→P(j,x′,y)
x>0,x′ =x−1kP(i,x,y)→P(i+1,x′,y)
where P(i,x,y) models the program at instruction i with counter values x, y. Then, clause
(i) models the increment of counter x (analogous for y) and a go to the next instruction; clauses
(ii) model the conditional decrement of x (analogous for y) and, otherwise, a jump to instruction
j. The start state is represented by a clause k→ P(1,n,m) for two positive integer values n, m,
and the halt instruction is represented by an atom P(halt,x,y) and reachability of the halting
state by a clause k P(halt,x,y) → (cid:3). Then, a two-counter machine program halts if and only if
the BSRclausesetoflineararithmetic withoneternarypredicateconstructedoutofthe program
is unsatisfiable. Note that for this reductionit does notmatter whether integeror realarithmetic
is the underlying arithmetic semantics. The first argument of P is always a natural.
∗This work has been partly funded by the German Transregional Collaborative Research Center SFB/TR 14
AVACS.
1
Our first contribution is refinements of the two-counter machine reduction where the arith-
metic constraints are further restricted to linear difference inequations x−y⊳c, simple additive
inequationsx+y⊳c,quotientinequationsx⊳c·yandmultiplicativeinequationsx·y⊳cwherec∈R,
and ⊳ ∈ {<,≤,=,6=,≥,>}. Under all these restrictions, the combination remains undecidable,
respectively (see Section 7).
Onthepositiveside,weprovedecidabilityoftherestrictiontoarithmeticconstraintsconsisting
of simple bounds of the form x⊳c, where ⊳ and c are as above. Underlying the result is the
observation that similar to the finite model property of BSR, only finitely many test points are
sufficient for the arithmetic simple bounds constraints. Our construction is motivated by results
from quantifier elimination [LW93] and hierarchic superposition [BGW94, KW12, FW12]. For
example, consider the two clauses
x 6=5kR(x )→Q(u ,x )
2 1 1 2
y <7,y ≤2k →Q(c,y ),R(y )
1 2 2 1
where u is a free first-order variable, x , y are variables over the reals, and c is a free first-order
1 i i
constant. Our main resultrevealsthatthis clause setis satisfiable if andonly if for everyvariable
atthe firstargumentofQthe constantcissubstituted (Corollary16), forthe secondargumentof
Qtheabstractrealvalues5+εand−∞andforRthevalue−∞(Definitions6,8,Lemma13). The
instantiation does not need to consider the simple bounds y < 7, y ≤ 2, because it is sufficient
1 2
to explore the reals either from −∞ upwards or from +∞ downwards, as is similarly done in
linear quantifier elimination [LW93]. Also instantiation does not need to consider the value 5+ε
for R, motivated by the fact that hierarchic superposition will not derive the respective simple
bound for the first argument of R in any generated clause [BGW94]. This idea can be extended
to free argument positions of predicates and the respective free constants. Every BSR clause set
combined with simple bounds is always sufficiently complete because it does not contain a free
non-constant function symbol.
All abstract values are represented by Skolem constants over the reals, together with defining
axioms. For the example, we introduce the fresh Skolem constants α to represent −∞ and
−∞
α to represent 5+ε together with axioms expressing α < 2 < 5 < α . Eventually, we
5+ε −∞ 5+ε
obtain the ground clause set
α ≥2k →(cid:3)
−∞
5≥α k →(cid:3)
5+ε
α 6=5kR(α )→Q(c,α )
5+ε −∞ 5+ε
α 6=5kR(α )→Q(c,α )
−∞ −∞ −∞
α <7,α ≤2k →Q(c,α ),R(α )
−∞ 5+ε 5+ε −∞
α <7,α ≤2k →Q(c,α ),R(α )
−∞ −∞ −∞ −∞
which has the model αA = 1, αA = 6, RA = {1}, QA = {(c,6),(c,1)}, for instance. Using
−∞ 5+ε
the result that every BSR clause set with simple bounds has a satisfiability-preserving ground
instantiation with respect to finitely many constants, we prove NEXPTIME-completeness of the
fragment in Section 5. For this result, the fine grained instantiation introduced in Section 3 and
explainedabovebyexampleisnotneeded. However,ourfurthergoalistodevelopusefulreasoning
procedures for the fragment for which a smaller set of instances improves efficiency a lot. For the
same reason, we do not restrict our attention to the BS fragment, but consider equality as a first
class citizen of the logic from the very beginning.
Oncea BSRclause setwithsimple bounds is grounded,there area varietyofefficient decision
procedures known, such as SMT solversemploying the Nelson-Oppen[NO79] principle. However,
for a large number of Skolem constants or large clause sets, an a-priori grounding may, due to its
exponentialincreasein size,notbe affordable. We arenotawareofanycalculus thatis actuallya
decisionprocedurefortheBSRfragmentwithsimplebounds,althoughsomeworkinthisdirection
has been done already [BFT08, Ru¨m08, BLdM11, KW12].
The decidability result on simple bounds can be lifted to constraints of the form x⊳s where
x is the only variable and s a ground expression. The lifting is done by the introduction of
2
further Skolem constants for complicated ground terms, following ideas of [KW12] and presented
in Section 6. The paper ends with a conclusion, Section 8.
2 Basic Definitions
Hierarchic combinations of first-order logic with backgroundtheories [BGW94] build upon sorted
logic with equality. A (sorted) signature Σ consists of a set Ξ of sorts, a set Ω of function
symbols together with sort information and a set Π of predicate symbols also equipped with sort
information. For the sake of simplicity, we restrict ourselves to two sorts: the base sort R –
interpretedas the setR ofrealsby allthe interpretationswe shallconsider–andthe free sort S –
interpretedby somefreely selectable nonempty domain. Throughoutthe paper we use convenient
notation such as P : ξ × ... × ξ ∈ Π to address an m-ary predicate symbol with full sort
1 m
information; P/m ∈ Π if there is some predicate symbol P of arity m in Π; P ∈ Π if there is
some m such that P/m ∈ Π. To avoid confusion, we assume that Π contains at most one pair
P : ξ ×...×ξ for every P, and do not allow for multiple occurrences of P with different arity
1 m
or sorting. Similar conventions shall hold for function symbols. In addition, we occasionally use
the notation Ω∩C (where C is an arbitrary set of constant symbols without annotated sorting
information) to denote the set {c|c:ξ ∈Ω for some sort ξ and c∈C}.
We instantiatethe frameworkofhierarchicspecificationfrom[BGW94]bythe hierarchiccom-
binationoftheBSRfragmentoffirst-orderlogicwithabasetheoryallowingfortheformulationof
simple constraintson real-valuedconstantsymbols and variables. Formally,the used specification
of the base theory comprises the base signature Σ := {R}, Ω ∪Ωα ,Π , where
LA LA LA LA
Ω :={r:R|r∈R}∪{c :R,...,c(cid:10) :R}, (cid:11)
LA 1 κ
Ωα :={α :R}∪{α :R|c∈Ω },
LA −∞ c+ε LA
Π :={<:R2, ≤:R2, =:R2, 6=:R2, ≥:R2, >:R2},
LA
together with the class M of models containing one base model M for every possible allocation
of the Skolem constants c ,...,c in Ω and the ones in Ωα to real numbers. Moreover, each
1 κ LA LA
of the base models M ∈ M shall extend the standard model M of linear arithmetic over the
LA
reals, i.e. RM = R and cM = c for every constant symbol c ∈ Ω ∩R and ⊳M = ⊳ for every
LA
⊳ ∈ {<,≤,=,6=,≥,>} – in other words all real numbers serving as constant symbols represent
their canonical value under each M ∈ M and every such M interprets all predicate symbols
<,≤,=,6=,≥,> by their standard meaning on the reals.
The definition of the base theory leaves the exact number κ of additional Skolem constants
open; it may be 0. While adding all reals as constant symbols to the signature serves the aim
of defining M to contain term-generated models only, the Skolem constants c ,...,c serve at
1 κ
leasttwopurposes. Firstly,themodelingcapabilitiesofourlogicallanguageareenhancedbyreal-
valued constantsymbols of whichthe exactvalue is not predetermined. Secondly, in Section6 we
outline a technique which allows us to soften our requirements towards the syntax a bit so that
ground non-constant terms s of the base sort become admissible in addition to constant symbols
and variables. The method has already been described in [KW12] under the name basification
and introduces defining unit clauses such as c = s where c is a fresh Skolem constant and s is a
variable-free term.
Infact,weintroduceevenmoreSkolemconstantsbyaddingthesetΩα tothebasesignature.
LA
For notational convenience, we syntactically distinguish this special kind of base-sort constant
symbols αt, t being of the form −∞ or d+ε for arbitrary base-sort constant symbols d ∈ ΩLA.
These will play a key role when we instantiate base-sort variables later on. We associate an
inherentmeaningwith constantsymbolsαt thatwillbe formalizedbymeans ofspecialofaxioms.
We hierarchically extend the base specification hΣ ,Mi by the free sort S and finite sets
LA
Ω and Π of free constant symbols and free predicate symbols, respectively, each equipped with
appropriate sort information. We use the symbol ≈ to denote the built-in equality predicate on
S. To avoid confusion, we assume each constant or predicate symbol to occur in at most one of
3
the sets Ω , Ωα , Ω, Π , Π and that none of these sets contains the ≈ symbol. In the light of
LA LA LA
sorted terms we consider two disjoint countably infinite sets of variables V – the variables of the
R
base sort, usually denoted x,y,z – and V – the free-sort variables, usually denoted u,w.
S
Definition 1 (BSR Clause Fragment with Simple Bounds).
Let {R,S},Ω,Π be a signature such that Ω exclusively contains constant symbols c of the
free sort and no function symbols of greater arity, and such that for every predicate symbol
(cid:10) (cid:11)
P :ξ ×...×ξ ∈Π and every argument position i≤m it holds ξ ∈{R,S}.
1 m i
Anatomic constraint isoftheformc⊳dorx⊳dorαt⊳dorx=αt orαt =αt′ withc,d∈ΩLA,
x∈VR, αt,αt′ ∈ΩαLA and ⊳∈{<,≤,=,6=,≥,>}.
A free atom A is either of the form s ≈ s′ with s,s′ being free-sort constant symbols in Ω or
free-sortvariablesinV ,respectively,orAisoftheformP(s ,...,s ),whereP :ξ ×...×ξ ∈Π
S 1 m 1 m
is an m-ary predicate symbol. For each i≤m the term s shall be of the sort ξ . If ξ =R, then
i i i
s must be a variable x ∈ V , and in case of ξ = S, s may be a variable u ∈ V or a constant
i R i i S
symbol c:S ∈Ω.
A clause has the form Λ k Γ → ∆, where Λ is a multiset of atomic constraints, and Γ and ∆
are multisets of free atoms. We usually refer to Λ as the constraint part and to Γ→∆ as the free
part of the clause.
WeconvenientlydenotetheunionoftwomultisetsΘandΘ′ byjuxtapositionΘ,Θ′. Moreover,
we often write Θ,A as an abbreviation for the multiset union Θ∪{A}. In our clause notation
empty multisets are usually omitted left of “→” and denoted by (cid:3) right of “→” (where (cid:3) at the
same time stands for falsity).
Intuitively,clausesΛkΓ→∆canbereadas Λ∧ Γ → ∆. Putintowords,themultiset
Λ stands for a conjunction of all atomic constraints it contains, Γ stands for a conjunction of the
(cid:0)V V (cid:1) W
freeatomsinitand∆standsforadisjunctionofthecontainedfreeatoms. Alloccurringvariables
are implicitly universally quantified. Requiring the free part Γ → ∆ of clauses to not contain
any base-sort constant symbols does not pose any restriction to expressiveness. Every base-sort
constant symbol c in the free part can safely be replaced by a fresh base-sort variable x when
c
an atomic constraint x = c is added to the constraint part of the clause (a process known as
c
purification, cf. [BGW94, KW12]).
In the rest of the paper we omit the phrase “over the BSR fragment with simple bounds”
whentalking aboutclauses andclause sets,althoughwe mainly restrictourconsiderationsto this
fragment.
A hierarchic interpretation over the hierarchic specification hΣ ,Mi,h{S,R},Ω,Πi is an
LA
algebra A which extends a model M∈M, i.e. A and M interpret the base sort and all constant
(cid:10) (cid:11)
and predicate symbols in Ω ∪Ωα and Π in exactly the same way. Moreover, A comprises a
LA LA LA
nonempty domain SA, assigns to each constant symbol c:S in Ω a domain element cA ∈SA and
interpretseverypredicatesymbol P:ξ ×...×ξ ∈ΠbyasetPA ⊆ξA×...×ξA. Summing up,
1 m 1 m
Aextendsthestandardmodeloflineararithmeticandadoptsthestandardapproachtosemantics
of (sorted) first-order logics when interpreting the free part of clauses.
Given a hierarchic interpretation A, a variable assignment is a sort-respecting total mapping
β :V ∪V →R∪SA sothatβ(x)∈Rforeveryvariablex∈V andβ(u)∈SA foreveryu∈V .
R S R S
We writeA(β)(s) tomeanthe value of the term s under A with respect to the variable assignment
β. In accordance with the notation used so far, we thus define A(β)(v):=β(v) for every variable
v and A(β)(c):=cA for every constantsymbol c. As usual, we use the symbol |= to denote truth
underahierarchicinterpretationA,possiblywithrespecttoavariableassignmentβ. Indetail,we
have the following for atomic constraints, equational and nonequational free atroms and clauses,
respectively:
• A,β |=s⊳s′ if and only if A(β)(s)⊳A(β)(s′),
• A,β |=s≈s′ if and only if A(β)(s)=A(β)(s′),
• A,β |=P(s ,...,s ) if and only if
1 m
A(β)(s ),...,A(β)(s ) ∈PA,
1 m
(cid:10) (cid:11)
4
• A,β |=ΛkΓ→∆ if and only if
– A,β 6|=s⊳s′ for some atomic constraint (s⊳s′)∈Λ,
– A,β 6|=A for some free atom A∈Γ, or
– A,β |=B for some free atom B ∈∆.
The variables occurring in clauses shall be universally quantified. Therefore, givena clause C, we
call A a hierarchic model of C, denoted A |= C, if and only if A,β |= C holds for every variable
assignment β. For clause sets N, we say A is a hierarchic model of N, also denoted A |= N, if
and only if A is a model of all clauses in N. We call a clause C (a clause set N) satisfiable w.r.t.
M if and only if there exists a hierarchic model A of C (of N). Most of the time, we will omit
the explicit reference to M, although we shall only consider models as satisfying that extend the
standard model of linear real arithmetic.
From now on, we implicitly base all further considerations on a hierarchic specification
hΣ ,Mi,h{S,R},Ω,Πi whoseextensionparth{S,R},Ω,Πifulfillstherequirementsstipulated
LA
inDefinition1andwhichwesimplytakeforgranted. OftenthesetsΩ \R,ΩandΠwillcoincide
LA
(cid:10) (cid:11)
with the constant and predicate symbols that occur in a given clause set, and we thus assume a
proper definition of the hierarchic specification.
Substitutions σshallbedefinedinthestandardwayassort-respectingmappingsfromvariables
to terms over our underlying signature. The restriction of the domain of a substitution σ to a
set V of variables is denoted by σ| and shall be defined so that vσ| := vσ for every v ∈ V
V V
and vσ| = v for every v 6∈ V. While the application of a substitution σ to terms, atoms and
V
multisets thereof can be defined as usual, we need to be more specific for clauses. Consider a
clause C = Λ k Γ → ∆ and let x ,...,x denote all base-sort variables occurring in C for which
1 k
x σ 6= x . We then set Cσ := Λσ,x = x σ, ..., x = x σ k (Γ → ∆)σ| . Simultaneous
i i 1 1 k k VS
substitution of k ≥ 1 distinct variables v ,... v with terms s ,...,s shall be denoted in vector
1 k 1 k
notation v ,...,v s ,...,s (or v¯ s¯ for short), where we require every s to be of the same
1 k 1 k i
sort as v , of course.
i
(cid:2) (cid:14) (cid:3) (cid:2) (cid:14) (cid:3)
Consider a clause C and let σ be an arbitrary substitution. The clause Cσ and variable-
renamed variants thereof are called instances of C. A term s is called ground, if it does not
containanyvariables. AclauseC shallbe calledessentially ground if itdoes notcontainfree-sort
variables and for every base-sortvariable x occurring in C, there is an atomic constraintx=d in
C for some constantsymbol d∈Ω ∪Ωα . A clausesetN is essentially ground if allthe clauses
LA LA
it contains are essentially ground.
We can restrict the syntactic form of clauses even further without limiting expressiveness.
Definition 2 (Normal Form of Clauses and Clause Sets). A clause ΛkΓ→∆ is in normal form
if all base-sort variables which occur in Λ do also occur in Γ→∆. A clause set N over the BSR
fragmentwithsimpleboundsisinnormal form ifallclausesinN areinnormalformandpairwise
variable disjoint.
Let us briefly clarify why the above requirement on clauses does not limit expressiveness.
Any base-sort variable x not fulfilling the stated requirement can be removed from the clause
Λ k Γ → ∆ by existential quantifier elimination methods that transform Λ into an equivalent
constraint Λ′ in which x does not occur.1 Moreover, Λ′ can be constructed in such a way that it
contains only atomic constraints of the form admitted in Definition 1 and so that no variables or
constant symbols other than the ones in Λ are necessary.
Given a clause set N, we will use the following notation: the set of all constant symbols
occurring in N shall be denoted by consts(N). While the set bconsts(N) exclusively contains all
base-sortconstantsymbolsfromΩLA thatoccurinN,allbase-sortconstantsymbolsαt appearing
inN shallbe collectedinthe setαconsts(N):=consts(N)∩Ωα . Thesetofallfree-sortconstant
LA
1Methodsfortheeliminationofexistentiallyquantified realvariablesincludeFourier-Motzkinvariableelimina-
tion [DE73], the Loos-Weispfenning procedure [LW93] and many others, see e.g. Chapter 5 in [KS08] for further
details.
5
symbols in N is called fconsts(N). Altogether, the sets bconsts(N), αconsts(N) and fconsts(N)
together form a partitioning of consts(N) for every clause set N. We moreover denote the set of
all variables occurring in a clause C (clause set N) by vars(C) (vars(N)).
3 Instantiation of Base-Sort Variables
We firstsummarizethe overallapproachdescribedinthis sectioninanintuitive way. To keepthe
informal exposition simple, we pretend that all base-sort constant symbols are taken from R and
thus are interpreted by their canonical value in all hierarchic interpretations. Consequently, we
can speak of real values instead of constant symbols, and even refer to improper values such as
−∞ (a “sufficiently small” real value) and r+ε (a real value “slightly larger than r but not too
large”). A formal treatment with proper definitions will follow.
Given a finite clause set N, we aim at partitioning the reals R into finitely many partitions so
that whenever N is satisfiable, there exists a hierarchic model A of N
(Pr) whose interpretation of predicate symbols does not distinguish between values within the
same partition.
AssoonaswefoundsuchafinitepartitioningP,wepickonerealvaluer ∈pasrepresentativefrom
p
every partition p∈P. The following observation motivates why we can use those representatives
insteadofusinguniversallyquantifiedvariables: givenaclauseC thatcontainsabase-sortvariable
x, and given a set {c ,...,c } of constant symbols such that {cA,...,cA}={r |p∈P} it holds
1 k 1 k p
A|=C ⇐⇒ A|= C x c 1≤i≤k . (1)
i
(cid:8) (cid:2) (cid:14) (cid:3)(cid:12) (cid:9)
The equivalence claims that we can transform univers(cid:12)al quantification over the base domain
into finite conjunction over all representatives of partitions in P. The formal version of this
statement is given in Lemma 13, and we will see that hierarchic models complying with property
(Pr) play a key role in its proof. Since P is supposed to be finite, the resulting set of instances
C x c |1≤i≤k is finite, too.
i
Itturnsoutthatthe notionofelimination sets describedbyLoosandWeispfenning in[LW93]
(cid:8) (cid:2) (cid:14) (cid:3) (cid:9)
(in the context of quantifier elimination for linear arithmetic) can be adapted to yield reasonable
sets of representatives from which we can construct a finite partitioning exhibiting the described
characteristics. In this case the partitions are intervals on the real axis. Intuitively speaking, we
start with the partitioning P consisting of a single partition P = (−∞,+∞) . This initial
0 0
partitionshallbe representedby−∞(the “defaultrepresentative”). We thensuccessivelyextract
(cid:8) (cid:9)
fromthegivenclausesetN largerandlargerpointsrontherealaxisatwhichwehavetointroduce
anewboundary,cutofftheintervalthatiscurrentlyunboundedfromaboveatthatboundaryand
introduce the cut-off part as a new interval to the partitioning. For instance, an interval [r′,+∞)
might be cut into two parts [r′,r] and (r,+∞) for some point r ≥ r′. Both parts become then
new partitions in the partitioning and while the interval[r′,r] will be representedby r′, the other
partition(r,+∞)willhaver+ε asits representative. Atthe endofthis processthe overallresult
will be a finite partitioning of the real numbers with the desired properties.
In fact, we will operate on a more fine-grained level than described so far, as we define such
partitionings independently for certaingroups ofbase-sortvariables. The possible benefit may be
a significant decrease in the number of necessary instances. But there is even more potential for
savings. The complete line ofdefinitions andargumentswillbe laidoutindetailfor one direction
ofinstantiationalongtherealaxis,namelyinthepositivedirectionstartingfrom−∞andgoingon
to largerand largerinstantiation points. However,one could as well proceed in the opposite way,
starting from +∞ and becoming smaller and smaller thereafter. While theoretically this duality
is not worthmuch more than a side note, it appearsto be quite interesting from a practicalpoint
of view. As it turns out, one can choose the direction of instantiation independently for each
base-sort variable that needs to be instantiated. Hence, one could always pick the direction that
resultsinlessinstantiationpoints. Suchastrategymightagainconsiderablycutdownthenumber
6
of instances in the resulting clause set and therefore might lead to shorter processing times when
applying automated reasoning procedures.
Formally,theaforementionedrepresentativesareinducedbyconstantsymbolswheninterpreted
underahierarchicinterpretation. Independentlyofanyinterpretationtheseconstantsymbolswill
serve as symbolic instantiation points, i.e. they will be used to instantiate base-sort variables
similar to the c in equivalence (1). We start off by defining the set of instantiation points that
i
need to be considered for every base-sort variable. This set depends on the constraints affecting
suchavariable. However,wefirstneedtodevelopapropernotionofwhatitmeansforabase-sort
variable to be affected by a constraint. The following example shall illustrate the involved issues.
Example 3. Consider the following clauses:
x 6=−5 k → T(x ), Q(x ,x ) ,
1 1 1 2
y <2, y <0 k → Q(y ,y ) ,
1 2 1 2
z ≥6 k T(z ) → (cid:3) .
1 1
Obviously, the variables x , y , y and z are affected by the constraints in which they occur
1 1 2 1
explicitly. Inthegivenclausesvariablesarejustnamesaddressingargumentpositionsofpredicate
symbols. Thus, it is more suitable to speak of the argument position hT,1i instead of variables
x and z that occur as the first argument of predicate symbol T in the first and third clause.
1 1
Speaking in such terms, argument position hT,1i is directly affected by the constraints x 6= −5
1
andz ≥6,argumentpositionhQ,1iis directlyaffectedbyx 6=−5andy <2,andfinallyhQ,2i
1 1 1
is affected by y <0. As soon as we take logical consequences into account, the notion of “being
2
affected” needs to be extended. The above clause set, for instance, logically entails the clause
x ≥ 6 k → Q(x,y). Hence, although not directly affected by the constraint z ≥ 6 in the clause
1
set, the argument position hQ,1i is still indirectly subject to this constraint. The source of this
effect lies in the first clause as it establishes a connection between argument positions hT,1i and
hQ,1i via the simultaneous occurrence of variable x in both argument positions.
1
One lesson learned from the example is that argument positions can be connected by variable
occurrences. Such links in a clause set N shall be expressed by the relation ⇌ .
N
Definition 4 (Connections between Argument Positions). Let N be a clause set in normalform.
We define the relation ⇌ to be the smallest equivalence relation over pairs in Π × N such
N
that hQ,ji ⇌ hP,ii whenever there is a clause in N containing free atoms Q(...,v,...) and
N
P(...,v,...) in which the variable v occurs at the j-th and i-th argument position, respectively.
(Note that Q=P or j =i is possible.)
The relation ⇌N induces the set [hP,ii]⇌N | P/m ∈ Π,1 ≤ i ≤ m of equivalence classes.
To simplify notation a bit, we write [hP,ii] instead of [hP,ii]⇌ when the set N is clear from the
(cid:8) N (cid:9)
context.
As we have argued earlier, it is more precise to speak of argument positions rather than
variables, since their names are of no particular relevance. Nevertheless, variable names are a
syntactical necessity. The following definition is supposed to provide a means to address the
argument position class a variable stands for.
Definition 5 (Argument Position Class). Let N be a clause set in normal form. Consider a
variable v that occurs at the i-th argumentposition of a free atom P(...,v,...) in N. We denote
theequivalenceclassofargumentpositionsvisrelatedtoinN byapN(v),i.e.apN(v):=[hP,ii]⇌N.
If v is a free-sort variable that exclusively occurs in equations v ≈s in N, we set ap (v):=∅.
N
There is a crucial subtlety in this definition that guarantees well-definedness, namely that the
clausesinN arevariabledisjoint. GivenaclauseC ∈N andavariablev whichoccursindifferent
argumentpositionshQ,jiandhP,iiinC,thedefinitionof⇌ entailshQ,ji⇌ hP,ii. Therefore,
N N
[hQ,ji]and[hP,ii]areidentical. Sincev doesnotoccurinanyotherclauseinN,the classap (v)
N
is well-defined to be [hP,ii].
7
Next, we collect the instantiation points that are necessaryto eliminate base-sortvariables by
means of finite instantiation. In order to do this economically, we rely on the same idea that also
keeps elimination sets in [LW93] comparatively small.
Definition 6 (Instantiation Points for Base-Sort Argument Positions). Let N be a clause set
in normal form and let P : ξ ×...×ξ ∈ Π be a free predicate symbol occurring in N. Let
1 m
J :={i|ξ =R,1≤i≤m} be the indices of P’s base-sortarguments. For every i∈J, we define
i
I to be the smallest set fulfilling
P,i,N
(i) d∈I ifthereexistsaclauseC inN containinganatomP(...,x,...)inwhichxoccurs
P,i,N
as the i-th argument and a constraint x=d or x≥d with d∈Ω appears in C, and
LA
(ii) α ∈ I if there exists a clause C in N containing an atom of the form P(...,x,...),
d+ε P,i,N
in which x is the i-th argument and a constraint of the form x 6= d or x > d or x = α
d+ε
with d∈Ω appears in C.
LA
Themostapparentpeculiarityaboutthisdefinitionisthatatomicconstraintsoftheformx<d
and x≤ d are completely ignored when collecting instantiation points for x’s argument position.
Firstofall,thisisoneoftheaspectsthatmakesthisdefinitioninterestingfromtheefficiencypoint
of view, because the number of instances that we have to consider might decrease considerably
in this way. To develop an intuitive understanding why it is enough to consider constraints x⊳d
with ⊳∈{=,6=,≥,>} when collecting instantiation points, the following example may help.
Example 7. Consider two clauses C = x > 2, x ≤ 5 k → T(x) and D = x < 0 k T(x) → (cid:3).
Recall that we are looking for a finite partitioning P of R so that we can construct a hierarchic
model A of the clause set {C,D} that complies with (Pr), i.e. for every partition p ∈ P and
arbitrary real values r ,r ∈p it shall hold r ∈TA if and only if r ∈TA. Of course, there exist
1 2 1 2
infinitely many candidates for P. A natural one is {(−∞,0),[0,2],(2,5],(5,+∞)} which takes
every atomic constraintin C and D into account. Correspondingly,we find a candidate predicate
for TA, namely the interval (2,5], so that A is a hierarchic model of C and D alike and it obeys
(Pr) with respect to the proposed partitioning P.
But there are other interesting possibilities, too, for instance, the more coarse-grained par-
titioning {(−∞,2],(2,+∞)} together with the predicate TA = (2,+∞). This latter candidate
partitioning completely ignores the constraints x < 0 and x ≤ 5 that constitute upper bounds
on x. Dually, we could have concentrated on the upper bounds instead (completely ignoring the
lower bounds). This would have lead to the partitioning {(−∞,0),[0,5],(5,+∞)} and candidate
predicates TA = [0,5] (or TA = [0,+∞)). Both ways are possible, but the first one induces a
simpler partitioning.
The example has revealed quite some freedom in choosing an appropriate partitioning of the
reals. A larger number of partitions directly leads to a larger number of representatives (one
for each interval). In fact, we use the collected instantiation points as representatives and taken
togethertheyinduce thepartitioning(aswewillseeinthe nextdefinition). Itisduetothisdirect
correspondenceof partitionsandinstantiationpoints that a morefine-grainedpartitioning entails
a larger number of instances that need to be considered. Hence, regarding efficiency, it is of high
interest to keep the partitioning coarse.
Definition 8 (Instantiation Points for Argument Position Classes and Induced Partitioning of
R). Let N be a clause set in normal form and let A be a hierarchic interpretation (over the same
signature). For every equivalence class [hP,ii] induced by ⇌ we define the following:
N
The set I of instantiation points for [hP,ii] is defined by
[hP,ii],N
I :={α }∪ I .
[hP,ii],N −∞ Q,j,N
hQ,ji[∈[hP,ii]
The sequence r ,...,r shall comprise all real values in the set cA c ∈ I ∩Ω or
1 k [hP,ii],N LA
α ∈I ordered so that r <...<r .
c+ε [hP,ii],N 1 k (cid:8) (cid:12)
(cid:12)
(cid:9)
8
Given a real number r, we say I A-covers r if there exists an instantiation point c ∈
[hP,ii],N
I ∩Ω withcA =r;analogously(r+ε)isA-covered byI ifthereisaninstantiation
[hP,ii],N LA [hP,ii],N
point α ∈I with cA =r.
c+ε [hP,ii],N
The partitioning P of the reals into finitely many intervals shall be the smallest
[hP,ii],N,A
partitioning (smallest w.r.t. the number of partitions) fulfilling the following requirements:
(i) Ifr isA-coveredbyI ,then(−∞,r )∈P ;otherwise,(−∞,r ]∈P .
1 [hP,ii],N 1 [hP,ii],N,A 1 [hP,ii],N,A
(ii) Foreveryj,1≤j ≤k,ifr andr +εarebothA-coveredbyI ,then[r ,r ]={r }∈
j j [hP,ii],N j j j
P .
[hP,ii],N,A
(iii) For every ℓ, 1≤ℓ<k,
(iii.i) if r +ε and r are both A-coveredby I , then (r ,r )∈P ,
ℓ ℓ+1 [hP,ii],N ℓ ℓ+1 [hP,ii],N,A
(iii.ii) if r +ε is A-covered by I but r is not, then (r ,r ]∈P ,
ℓ [hP,ii],N ℓ+1 ℓ ℓ+1 [hP,ii],N,A
(iii.iii) if r +ε is not A-coveredby I but r is, then [r ,r )∈P ,
ℓ [hP,ii],N ℓ+1 ℓ ℓ+1 [hP,ii],N,A
(iii.iv) if neither r +ε nor r is A-coveredby I , then [r ,r ]∈P .
ℓ ℓ+1 [hP,ii],N ℓ ℓ+1 [hP,ii],N,A
(iv) If r + ε is A-covered by I , then (r ,+∞) ∈ P ; otherwise [r ,+∞) ∈
k [hP,ii],N k [hP,ii],N,A k
P .
[hP,ii],N,A
Pleasenotethatpartitioningsasdescribedinthedefinitiondoalwaysexist,anddonotcontain
empty partitions. Moreover, it is worth to notice that for all instantiation points c and α in a
c+ε
set IQ,j,N we have c ∈ ΩLA. Hence, the concrete values assigned to constant symbols αt do not
contribute to the respective partitionings of R.
We now fix the semantics of constant symbols αt by giving appropriate axioms that make
precisewhatit means forα to be “smallenough”andfor α to be “alittle largerthanc but
−∞ c+ε
not too large” under any hierarchic model A that satisfies the corresponding axioms.
Definition 9 (Axioms for Instantiation Points αt). Let I be a set of instantiation points and
C ⊆Ω be a set of base-sort constant symbols. We define the set of axioms
LA
Ax := α <c α ∈I and c∈C ∪ d<α α ∈I}
I,C −∞ −∞ d+ε d+ε
(cid:8)∪ d<c (cid:12)→ αd+ε <c αd+ε ∈(cid:9)I an(cid:8)d c∈C\{(cid:12)d}
(cid:12) (cid:12)
∪ (cid:8)d=c → αd+ε =αc(cid:12)+ε αd+ε ∈I and c∈C\(cid:9){d} .
(cid:12)
The axioms we just int(cid:8)roduced are clearly not(cid:12)in the admissible clause fo(cid:9)rm. But they can
(cid:12)
easily be transformed into proper clauses with empty free parts. For convenience, we stick to the
notation given here, but keep in mind that formally the axioms have a form in accordance with
Definition 1.
Thefollowingpropositionshowsthateveryhierarchicinterpretationcanbeturnedintoamodel
of a given set of axioms of the above shape by modifying the values assignedto constant symbols
αt. The propositionrelies on three facts concerning the ordering(R,<): totality of <, the lack of
minimal and maximal elements, and the density of (R,<), i.e. for arbitrary reals r <r there is
1 2
an r′ between them.
Proposition 10. Let I be a nonempty set of instantiation points and let C be a nonempty set
of base-sort constant symbols from Ω . Given an arbitrary hierarchic interpretation A, we can
LA
constructahierarchicmodelBsothat(ii)B |=Ax andBandAdifferonlyintheinterpretation
I,C
of the constant symbols in αconsts(Ax ).
I,C
Proof. ThemodelBcanbederivedfromAbyredefiningtheinterpretationofallconstantsymbols
αt ∈I as follows:
αB :=min{cA |c∈C}−1 ,
−∞
αB := 1 dA+min({cA |c∈C, cA >dA}∪{dA+1}) .
d+ε 2
Everything else is taken over(cid:0)from A. (cid:1)
9
Previously,we have been speaking of representatives of partitions p in a partitioning P of the
reals. As far as we know by now, these are just real values r ∈ p from the specific interval they
p
represent. The next lemma shows that we can find constantsymbols c in the set of instantiation
p
points to address them. Furthermore, we will see where the values of these constant symbols lie
within the respective interval.
In orderto have a compactnotation at handwhen doing case distinctions onintervals, we use
“(·” to stand for “(” as well as for “[”. Analogously, “·)” is used to address the two cases “)” and
“]” at the same time.
Lemma 11. Let N be a clause set in normal form and let A be a hierarchic model of
Ax for an arbitrary argument position pair hP,ii. For every partition p ∈
I[hP,ii],N,bconsts(N)
P we find some constant symbol c ∈I so that cA ∈p and
[hP,ii],N,A [hP,ii],p [hP,ii],N [hP,ii],p
(i) if p=(−∞,r ·) or p=(−∞,+∞), then cA <dA for every d∈bconsts(N),
u [hP,ii],p
(ii) if p=[r ,r ·) or p=[r ,+∞), then cA =r .
ℓ u ℓ [hP,ii],p ℓ
(iii) ifp=(r ,r ·)orp=(r ,+∞),thenr <cA andcA <dA foreveryd∈bconsts(N)
ℓ u ℓ ℓ [hP,ii],p [hP,ii],p
with r <dA.
ℓ
Proof. We proceed by case distinction on the form of the partition p.
Case p = (−∞,r ·) for some real value r . By construction of P , there is a constant
u u [hP,ii],N,A
symbol e ∈ bconsts(N) so that eA = r . Moreover, we find the instantiation point α
u −∞
in I and thus Ax contains the axiom α < d for every constant
[hP,ii],N I[hP,ii],N,bconsts(N) −∞
symbol d∈bconsts(N), in particular α <e. Hence, αA ∈(−∞,r ).
−∞ −∞ u
Case p=(−∞,+∞). We find the instantiation point α in I , and obviously αA lies
−∞ [hP,ii],N −∞
inp. Moreover,Ax containstheaxiomα <dforeveryconstantsymbol
I[hP,ii],N,bconsts(N) −∞
d∈bconsts(N).
Case p=[r ,r ·) for real values r ,r with r ≤r .
ℓ u ℓ u ℓ u
If p is a point interval [r ,r ], then we find a constant symbol e ∈ I ∩Ω so that
ℓ ℓ [hP,ii],N LA
p={eA}.
Ifpisnotapointinterval,i.e.r <r ,thenthedefinitionofP entailstheexistence
ℓ u [hP,ii],N,A
of a constant symbol e ∈ bconsts(N) such that eA = r and either e ∈ I or
ℓ [hP,ii],N
α ∈ I . But since the partition p is not a point interval, r +ε cannot be
e+ε [hP,ii],N ℓ
A-coveredby I , and thus α cannot be in I . Consequently, e must be
[hP,ii],N e+ε [hP,ii],N
in I and we can choose c :=e.
[hP,ii],N [hP,ii],p
The case of p=[r ,+∞) can be argued analogously to the previous case.
ℓ
Case p=(r ,r ·) for real values r ,r with r <r .
ℓ u ℓ u ℓ u
By construction of P , we find some instantiation point α ∈ I so that
[hP,ii],N,A e+ε [hP,ii],N
eA = r . Moreover, there exists a second constant symbol e′ ∈ bconsts(N) for which
ℓ
e′A = r . As a consequence of α ∈ I the set Ax contains the
u e+ε [hP,ii],N I[hP,ii],N,bconsts(N)
axioms e < α and e < d → α < d for every constant symbol d ∈ bconsts(N), in
e+ε e+ε
particular for d = e′. And since A is a model of Ax , requirement (iii) is
I[hP,ii],N,bconsts(N)
satisfied and furthermore αA ∈p follows, since eA <αA <e′A.
e+ε e+ε
Casep=(r ,+∞). Theremustbeaninstantiationpointα inI suchthateA =r . As
ℓ e+ε [hP,ii],N ℓ
wehavealreadydoneinDefinition8,wedenotebyr ,...,r allrealvaluesinascendingorder
1 k
whichAassignstoconstantsymbolsd∈bconsts(N)thateitherthemselvesareinstantiation
points in I or for which α occurs in I . By construction of P , we
[hP,ii],N d+ε [hP,ii],N [hP,ii],N,A
know r = r > r > ... > r , and thus there is no base-sort constant symbol e′ in
ℓ k k−1 1
10