CLASSICAL LOGICS FOR ATTRIBUTE-VALUE LANGUAGES 
J iirgen Wcdekind 
Xerox Palo Alto Research Center 
and 
C.S.L.I.- Stanford University 
USA 
Abstract 
This paper describes a classical logic for attribute-value (or fea- 
ture description) languages which ate used in urfification gram- 
mar to describe a certain kind of linguistic object commonly 
called attribute-value structure (or feature structure). Tile al- 
gorithm which is used for deciding satisfiability of a feature 
description is based on a restricted deductive closure construc- 
tion for sets of literals (atomic formulas and negated atomic 
formulas). In contrast to the Kasper/Rounds approach (cf. 
\[Kasper/Rounds 90\]), we can handle cyclicity, without the need 
for the introduction of complexity norms, as in \[Johnson 88J 
and \[Beierle/Pletat 88\]. The deductive closure construction is 
the direct proof-theoretic correlate of the congruence closure 
algorithm (cf. \[Nelson/Oppen 80\]), if it were used in attribute- 
value languages for testing satisfiability of finite sets of literals. 
1 Introduction 
This paper describes a classical logic for attribute-value (or fea- 
ture description) languages which are used in unification gram- 
mar to describe a certain kind of linguistic object commonly 
called attribute-value structure (or fcz~ture structure). From a 
logical point of view an attribute-vMue structure like e.g. tile 
following (in matrix notation) 
PRED 'PROMISE' 
TENSE PAST 
suBJ Pl  :i) 'JOliN'\] 
XCOMP \[ SUBJ m \] PRED 'COME' 
can be regarded as a graphical representation of a mini- 
mal model of a satisfiable feature description. If we assume 
that the attributes (in the example: PRED, TENSE, SUB J, 
XCOMP) are unary partial function symbols and the values 
(a, 'PROMISE', PAST, 'JOIIN', 'COME') are constants then 
the given feature structure represents graphically e.g. the min- 
imal model of the following description: 
'PRED SUBJa ~ 'JOIIN' &TENSEa ~, PAST & 
PREDa ~ 'PROMISE' & SUBJa ~ SUBJ XCOMPa & 
PRED XCOMPa ~ 'COME') 
I Note that the terms arc h)rnlcd without using brackets. (Since 
all function symbols are unary, the introduction of brackets would 
So, in the following attribute-value languages are regarded & 
quantifier-free sublanguages of classical first order language~ 
with equality whose (nonlogical) symbols are given by a set o" 
unary partial function symbols (attributes) and a set of con- 
stants (atomic and complex values). The logical vocabulary 
includes all propositional connectives; negation is interpreted 
(:lassically. 2 
For quantifier-free attribute-value languages L we give an ax- 
iomatic or IIilbert type system ll°v which simply results from 
an ordinary first order system (with partial function symbols), 
if its language were restricted to the vocabulary of L. Accord- 
ing to requirements of tile applications, axioms for the constant- 
consistency, constant/complex-consistency and acyclicity can 
be added to force these properties for the feature structures 
(models). 
For deciding consistency (or satisfiability) of a feature descrip- 
tion, we assume .first, that the conjunction of the formulas 
ill,the feature dc'scription is converted to disjunctive normal 
form. Since a formula in disjunctive normal form is consis. 
tent, ill" at least one of its disjuncts is consistent, we only need 
all algorithm for.deciding consistency of finite sets of literals 
(atomic formulas or negated atomic formulas) S. In contrast 
to the reduction algorithms which normalize a set S accord. 
ing to a complexity norm in a sequence of norm decreasing 
rewrite steps 3 wc use a restricted deductive closure algorithm 
for deciding the consistency of sets of literMs. 4 The restric- 
tion results from the fact that it is sufficient for deciding the 
consistency of S to consider proofs of equations from ,.q with 
a certain subterm property. For tile closure construction only 
those equations are derived from S whose terms are subterms of 
the terms occurring in the formulas of S. This guarantees that 
the construction terminates with a finite set of literals. The ad- 
equacy of this subterm property restriction, which was already 
shown for the number theoretic calculus K in \[Kreisel/Tait 61\] 
by \[Statman 74\], is a necessary condition for the development 
of more efficient Cut-free Gentzen type systems for attribute- 
not improve tile readability essentially.) Therefore we write e.g. 
PRED SUBJa instead of PRED(SUBJ(a)). 
2For intuitionistic negation cf. e.g. \[Dawar/Vijay-Shanker 90\] 
and \[Langholm 89\]. 
aCf. e.g. \[Kreisel/Tait 61\], \[Knuth/Bendix 70\], and ap- 
plied to attrlhute-value languages \[Johnson 88\], \[Beierle/Plntat 88\], 
\[Smolka 89\]. 
4Since we allow cyclicity, unrestricted deductive closure algo- 
rithms (cf. e.g. \[Kasper/Rounds 86\] and \[Kasper/nounds OO\]) can- 
not be applied. 
- 204 - 
value languages) 
Moreover, this closure construction is the direct prooI. 
theoretic correlate of the congruence closure algorithm (cf. 
\[Nelaon/Oppen 80\]), if it were used for testing satisffability of 
finite sets of literals in HOt,. As it is shown there, the congru- 
ence closure algorithm can bc used to test consistency if the 
terms of the equations are represented as labeled graphs and 
the equations as a relation on the nodes of that graph. 
O~ the basis of the algorithm for deciding satlsfiability of finite 
sets o| formulas we then show the completeness and decidability of//~t,. 
2 Attribute-Value Languages 
In this section we define the type of lauguagc wc want to con- 
sider i~nd introduce some additional notation. 
2.t Syntax 
2.1. DEFINITION. A quantifier-free attribute-value language 
(L.:%~) consists of the Jogical connective~ ± (false), ~ (nega- 
tion), :) (implication), the equality symbol ,~ and the paren- 
theses (,). The nonlogical vocabulary is given by a finite set of 
constants C and a finite set of unary partial Junction symbols 
r; (¢nr~ =~). 
2.2. DEFINITION. The class of terms (7") of L is recursively 
defined as follows: each constant is a term; if f is a function 
symbol and r is a term, then fr is a term. 
2.3. DEFINITION. The set of atomic formulas: of L is 
!n ~ "~ I r,, r~+7,} u {±}. 
2.4. DEFINITION. The formulas of L are the atomic formulas 
4nd, whenever ~ and ~b are formulas, then so are (+ ~b) and 
~.5. DEFINITION. If ~ is a well-formed expressio n (term or 
formula), then a\[r~/r~\] is used to designate an expression ob- 
tained from a by replacing some (possibly all or none) occur- 
r¢nces Of r~ in ~ by r~. 
We assume that the connectives V (disjunction), ~:(conjunc- 
tion) and ~ (equivalence) are introduced by their usual defi- 
nitions, Furthermore, we write sometimes ri ~ rz ;instead of 
-,, ~'~ ~ r2 and drop the parentheses according tolthe usual 
conventions, e 
2~2 Semantics 
A model|or L consists of a nonempty universclt anti an inter. 
pre~a|ion function 9. Since not every term denotes an element 
In M if the function symbols are interi)reted as unary partial 
functions, we generalize the partiality of the denotation by as- 
stltl~l~Ig that ~) itself is a partiM function. Thus in general not 
tCf. also \[Statmml 77\]. 
sWe drop the outermost brackets, assume that the connectives 
h~ty e the precedence ,~> & > v >:), _-- and are left associative. 
all of the constants and function symbols are interpreted by ~). 
Redundancies which result from the fact that non-interpreted 
function symbols and function symbols interpreted as empty 
functions are then regarded as distinct are removed by requiring 
these partial funct~ions to be nonempty. Suppose \[X ,-, Y~(p) 
designates the set of all (partial) functions from X to Y~ then 
a model is defined as follows: 
2.6. DEFINITION. A model for L is a pair M = (//, ~)), cpn- 
sisting of a nonempty set U and an interpretation function 
9 = 9c U ~Fi, such that 
(i) 9~\[c ~ u\]~ 
(iii) Vf~F,(I~Dom(9) --, 9(f) # ~). 
The (partial) denotation function for terms ~ (~;¢\[T ~-*/at\] e) 
induced by 9 is defined as follows: 7 
2.7. DEFINITION. For every ceC anti freT" (feFl), 
~(c) = (9(c) if ceDom(9) 
undefined otherwise 
{ 9(f)(~(r)) if feDom(9) A~(r) definedA 
~(fr) = ~(r)eDom(9(f)) 
undefined otherwise. 
2.8. DEFINITION. The satisfaction relation between models 
M and formulas ~b (~M ~b, read: M satisfies ~, M is a model 
of ~b, ~ is true in M) is defined recursively: 
V=M ± 
~M r ~. r' ~ 9(r),9(r') defined Ag(r) = ~(r') 
J=u,/,3x .-. l=M,/,-.l=~x. 
A formula ~b is valid (\[= ~), iff ~b is true in all models. A 
formula ~b is satisfiable, iff it has at least one model. Given 
a set of formulas F, we say that M satisfies r (~ r), iff M 
satisfies each formula ~b in F. F is satisfiable, iff there is a model 
that satisfies each formula in F. ~ is logical consequen¢~ of F 
(F ~ ¢), iff every model that satisfies F is a model of ~. 
3 The System H°v 
? 
In this section we describe an axiomatic or Hilbert type system 
H°v for quantifier-free attribute-value languages L. We give a 
decision procedure for the saris|lability of finite sets of formulas 
and show the completeness and decidability of H~v on the b~mis 
of that procedure. 
3.1 Axioms and Inference Rules 
If L is a fixed attribute-value language, then the system consiSts 
of a traditional axiomatic propositional calculus for L ud two 
additional equality axioms. For any formulas ~,~b,X , terms 
71n the text following tile definition we drop the overllne. 
- 205 - 
r, r', and every sequence of functors a (aeF;) of L the form,las 
under A1 - A4 are propositionalaxioms s and the formulas under 
El and E2 are equality axioms. ° The Modus Ponens (MP) is 
the only inlerence rule) ° 
AI )--~ _L 
A2 k ~b D (¢ D ~) 
A3 b (~ :9 (~b :3 X)) 2) ((~b 2) ~b) 2) (# D X)) 
A4 ~ (~ ¢ 2)~ ¢) 2) (¢ 2) ~) 
E1 t-ar~r'Dr~r 
E2 k r ,~ r' :3 (¢ 2) ¢\[r/r'\]) 
MP ~b 2) ¢^4 b ~b 
A formula ff is derivable from a set of formulas F (I" b ~,), iff 
there is a finite sequence of formulas ff~...qL, such that ft, = q~ 
and every ~i is an axiom, one of the formulas in U or follows by 
MP from two previous formulas of the sequence, ff is a theorem 
(F ~), iff ~ is derivable from the empty set. A is derivable from 
F (r I- A), iff each formula of A is derivable from P. F and A 
are deductively equivalent (I" -U- A), iff r I- A and A F I'. 
The system is sound: n 
3.1. THEOREM. For every Jormula c~: l/k" c~, then ~- qb. 
Beside this weak version also the stro.g soundness theorem is 
provable for H°Av: 
3.2. THEOREM. For every set oJ Jormulas \[' and every for- 
mula ~: If r t- c~, then r ~ c~. 
3.2 Satisfiability 
We now prove 
3.3. TIIEOREM. The satisfiability of a fi.ite set oJ formulas 
F is decidable. 
by providing a terminating procedure: First the conjunctio, of 
all formulas in F (denoted by A F) is converted into disjunctive 
normal form (DNF) using the well-known standard techniques. 
Then A F is equivalent with a DNF 
= (4,&4~&...&¢k,) v (4~&...&4~,~) v ... v ~v-, ... v,k., 
where the conjuncts 4i (i = 1 .... n; j = 1 .... ki) are either 
atomic formulas or negations of atomic formulas, henceforth 
called iiterals. By the definition of the satisfiability we get: 
scf. e.g. \[Church 56\]. 
9Axlom El restricts the reflexivity of identity to denoting terms: 
if a term denotes, then also its suhterms do (cf. the definition of ~). 
Thus equality is not a reflexive, but only a subterm reflexive relation. 
1°If (i.) constant-consistency and (li.) constant/complex- 
consistency are to be guaranteed for a set Of atomic values V (V C_ C), 
for each a, beV (a # b) and leFt, axiomsof the form (i.) F a ~ b and 
(ii.) b fa ~ Ja have to be added (a finite set). I\[ also acyclicity has 
to be ensured, axioms of the form (iii.) bar ~ ~', with ¢eFI + , veT, 
have to be added. Although this set is i,finite, we only need a finite 
subset for the satisfiability test and for deci,lal,illty (see below). 
II F'or the propositional calculus of. the sta,dard proofs, l"or ax- 
ioms E1 and It,2 cf. \[Johnson 88\]. 
3.4. LEMMA. Let A St v A Sav ... v A s" be a DNF d/A r 
consisting of conjunctions A Si of the literals in S i, then A r 
is satisfiable, iff at least one disjuncl A Si is satisfiablel 
We complete the proof of Theorem 3.3 by an algorithm that 
converts a finite set of literals S i into a deductively equivalent 
set of literals in normal form S i which is satisfiable iff it is not 
equM to {.L}. 
3.2.1 A Normal Form for Sets of Literals 
The normal form is constructed by closing S deductively by 
those equations whose terms are subterms of the terms occur- 
ring in S. For the construction we use the following derived 
rules: 
R1 or ~ r' I- r ~ r Subterm Reflexivity 
R2 r ~ r'A4l- 4\[r/r'\] Substitutivity 
R3 r .~ r' I- r' ~ r Symmetry. 
We get RI and R2 from E1 and E2 by the deduction theorem. 
R3 is derivable from R1 and R2, since we get from r ~ r' first 
r ~ r by R1 and then r' ~, z by R2. 
If Ts denotes the set of terms occurring in the formulas of S (Ts 
= {r, r' I (~)r .~ r'cS}), and SUB(Ts) denotes the set of 
all subterms of the terms in "Is n 
SUB(7"s) = {~ I ~,,~r~, with aeFl*}, 
then the normal form is constructed according to the following 
inductive definition. 
3.5. I)EFIN1TION. For a given set of literals S we define a 
sequence of sets Si (i >_ O) by induction: 
With S~= S U {r' ..~ r \[ r ,~ r'eS}, 
f { l} if/cS; otherwise 
So = <\[s~u{~=~l~=~,,g} 
f {.L} if :lq~(Si(,., #(Si); otherwise 
S,÷l = ~ /S~ u ..... ,.In ~ r2,r ~ r'~&A 1 
tin ~ r2)trl r J\[~- - ...... ? • \[ 
Since Si C Si+l, for Si÷l # {l}, tile construction terminates 
oil tile basis of the subterm condition either with a finite.set of 
literals or with {l}. If each term of the equations in Si+, is a 
subterm of tile terms in Ts, no term of the equations in $~+1 
can be longer than the longest term in Ts. 
EXAMPLE 1. Assume that L consists of the constants a, b, c, e 
and the function symbols f,g, h,m, n,p. Then, for the set of 
literals 
ga = ha, a .~ If a, ngffa ~ e 
the following sequence of sets is constructed. We represenL 
the equations of a set Si by tile system of sets of equivalent 
terms ind.ced hy S,. I.e.: If O is a set of terms under Si and 
12T s C_ SUB('Ts) holds by definition. 
- 206 - 
r,r'rO, then r ~ r'cSi. Furthermore, we mark by an arrow 
that a set under Si is also induced (without modifications) by 
the equations in Si+l. 
So St $2 = S~ 
ngf fa .'# e .....* "-4 
{e,~e} ~ -~ 
{b} --* --, 
{e,a} ) 
{a, ffa} {c, a, ffa, ffc} "--* 
{ffe} 
{ge,pmb} --* --* 
{rob, rig/f c} {rob, ngffc, ngffa} ---* 
{fc} % ) {fc, fa} -... 
{fa} ¢ 
Dffc} {gffc,~ffa} \ Df fc,~f Ia,~a,h,q 
Da, ~a} D~, ~a, ~fla} / 
3.6. DEFINITION. Let S, = S,; with t = min{i I S, = S,÷~}. 
3.7. LEMMA. For Sv holds: S -iF- Sv. 
PROOF. If Sv # {.I-}, then S and Su are deductively equiva- 
lent, since S is a subset of Sv and Sv only contains formulas 
derivable from S. For Sv = {.1_} the same holds for S~_t. Since 
S~_~ is inconsistent, S is deductively equivalent with {.1.}. 
Note that for each equation in Si (Si # {a_}) there is a proof 
from S with the anbterm property, as defined below. This fol- 
lows from the subterm condition in the inductive construction. 
3.8. DEFINITION. A proof of an equation from S has the 
subterm property, iff each term occurring in the equations of 
that proof is a subterm of the terms in Ts, i.e. an element of 
su~(7-s). 
So, if S is not trivially inconsistent (£ not in S), the con- 
struction terminates with {_1.}, since there exists a proof of an 
equation from S with the subterm property, whose negation is 
in $. 
EXAMPLE 2. For the inconsistent set 
S' = S o {gmme ~ pnh f f a} the constructi'on terminates after 
4 steps (S~ = {.L}), sittce there is a proof of gmme m, pnhffa 
from S' with the subterm property of depth 3. 
e~me e~me mb~.ngJJc cma ~_amha amJJa 
9empmb e~mme mb'~ngfJa 9\]JamhJJa 
gmme = pmb mfi m nh f f a 
9mine ~ pnh f f a 
; 
The deductive closure construction restricted by the subterm 
property is a proof-theoretic simulation of the congruence clo- 
sure algorithm (cf. \[Nelson/Oppen 80\]t3), if used for testing 
satisfiability of finite sets of literals in H°v. Strictly speaking, 
if 
i. the congruence closure algorithm is weakened for partial 
functions, 
ii. S is not trivially inconsistent (.1_ not in S), and 
iii. the failure in the induction step of 3.5. is overruled, 
tZCL also \[Gallier 87\]. 
then r ,.mr' is in Sv iff the nodes which represent the terms r 
and r' in the graph constructed for S are congruentfl t More- 
over, for unary partial functions the algorithm is simpler, since 
the arity does not have to be controlled. 
3.9. LEMMA. The set ol all equations in S~ is closed under 
subterm reflexivity, symmetry and transitivity. 
PROOF. For S~ = {.!_} trivial. If S~ # {.L}, then Sv is 
closed under subterm reflexivity and symmetry, since these 
properties are inherited from So to its successor sets. Sv is 
closed under transitivity, since we first get ra~SUB(Ts) from 
rl ~ r2, r~ ~ rsESu and then according to the construction also 
7"1 ~ r2\[r2/rs\]~Sv+l = Sv, with r2\[ra/rs\] = rs. \[3 
3.2.2 Satisfiability of Sets of Literals 
For the proof that the satisfiability of a finite set of fiterals is 
decidable we first show that a set of literals in normal form is 
satisfiable, iff the set is not equal to {.L}. For Sv = {.L} we get 
trivially: 
3.10. LEMMA. Sv = {.1.} ~ "~3M(J=M Sv). 
Otherwise we can show the satisfiability of Sv by the construc- 
tion of a canonical model that satisfies S~. 
Let Ev be the set of all (nonnegated) equations in Sv, TE~ the 
set of terms occurring in Ev and mEv the relation induced by 
E~ on T~ ({(r,r') \[ r ~ r'eE~}). Then, we choose as the 
universe of the canonical model M~ = (Uv,~v) the set of all 
equivalence classes of ~ on TE~, if T~ #- g. By Lemma 3.9 
this set exists. If Sv contains no (unnegated) equation, we set 
Uv = {fl}, sittce the universe has to be nonempty. 
3.11. DEFINITION. For a set of iiterals S~ in normal form, the 
canonical term model for Sv is given by the pair My - (Uv, ~lv}, 
consisting of the universe 
llv = {0} otherwise 
attd the interpretation function ~v, which is defined for c¢C, 
feFt and \[r\]d4v by: Is 
f \[c\] if ccT~ ~c(c) 
= ~, undefined otherwise 
\[It'\] if r'e\[r\] and fr'eT~,, 
~Ft(f)(\[r\]) = undefined otherwise. 
It follows from the definition that ~ is a partial function. Sup- 
pose further for ~)Ft(f) that \[rl\] = \[r2\] and that ~Ft(f)(\[rt\]) 
is defined. Then 
~F, (f)(fn\]) = ~F~ (f)(fr2\]). 
For this, suppose ~F,(f)(\[rl\]) -- \[frq, with r'e\[rl\]. Since 
~E~ is an equivalence relation we get r'e\[r~\] and thus 
~, (f)(\[~\]) = \[fr'\]. 
t4CL \[Wedekind 90\]. 
lSWe drop the ~E~-index of the equivalence classes. 
- 207 - 
EXAMPLE 3. Tile canonical model for S of Example I which 
is constructed using ,.(;2 = Sv is given by: 
{ {e,.,e},{b}, {e,a,.f\]u,/.fe}, 
l/v = ~ {ge, pmb}, {rob, ngf fc, rig/f a}, 
{, {/e, fa}, {Pile, of/s, gu, ha} J 
~(e) = \[el ~(e) 
~(b) = \[b\] ~.(a). = \[el 
f (\[a\],{m, t (\[Iul, IIial)J 9din) = ~, (\[bl, tmbl) I \[ ~.(f) = 
/(\[e\],\[~e\]), ~(~) = \[ (\[a\], \[~a\]) } 
~.(.) = { (\[~u\], \[n~lle))} 
~(h) = {(\[a\], \[ha\])} 
D.(p) = {(\[.,b\], L~-,b\])}. 
For each term r in Tg~ it follows from tile definition of ~c and 
~,: ~(r) = \[d. 
By the following lemma we show in addition that the domain 
of £rv restricted to Ts~ is equal to TE~. 
3.12. LEMMA. For each term r in Ts~: 11 ~ is defined for 
r, then ~,(r) = \[r\], with retd,. 
PROOF. (By induction on the length of r.) Suppose first that 
~v is defined for r. For every coustant c it follows from the 
definition of ~)c that i~c(c) = \[c\], with c(7"E~. Assume for fr by 
inductive hypothesis ~v(r) = \[r\], with roTEs, then it follows 
from the definition of ~F~(f) that ~rt(f)(\[r\]) ----- \[fr~\]~ witlt 
frtcTF.~ and r'(\[r\]. Since r' is a subterm of \]r', wc first get 
r'eT-i;~ and by Lemma 3.9 fr' .~ fr',r' "~ r~S~. Because of 
fr(SUB(Ts), then also fr m \]r(Sv. So, fr must also be in 
Tg~ and hence c~, (f)(\[r\]) = \[fr\]. \[3 
Next we show for the model My: 
3.13. LEMMA. S~ # {,L} -.I=M~ S~. 
PROOF. (We prove I=~ @, for every ¢, i, S~ hy induction oil 
the structure of @.) 
L is not element of S~. If 1 were in S~, we would get by the 
definition of S~ S~ = {a.} which contradicts our assumption. 
For @ =~ ,L, ~=MJ" £ holds trivially. 
Suppose ~ = r ~ r', then r,r' are in T~, ~ is defined for 
r and r ~, and ~v(r) = \[r\], ~(r') = \[r'\]. Because of r 
r'(S~, it follows that \[r\] = \[r'\]. So ~v(r) = ~(r') and hence 
~M,, 7" ,~, r t. 
Assume that @ is ~ (r ~ r'). If r .m r' were satisfied by M~, 
~(r) would be equal to ~,,(r'). By Lemma 3.12 we would 
then get $~(r) = \[r\] and ~v(r') = \[r'\], with r, r'(Tg~. Since 
~g, is an equivalence relation on 7"g~, r ~ r'¢Su would follow 
from \[r\] = Jr'l, and, contradicting the assumption, we would 
get S~ = {'L} by tile defipition of S~. n 
It can be easily shown that Mv is a unique (up to isomorphism) 
minimal model for Sv. :s Strictly speaking, if M is & model for 
16It can be verified very easily by using this fact that we need to 
add to a set of literals S only a finite number of axioms to ensure the 
=cycllcity. All axioms of the form ~" ~ ~ (¢~¢Ft, ~'e'T), with la'r~ _~ 
ISUB(T~)I, are e.g. more than enough, since from a consistent but 
cyclic set of literals S must follow an equation ar ~ ~ (aeFi + ,~'eT), 
with I~1 < I~1, and I~1 _< ISUB(TE)I holds by the construction of 
S~ homomorl~hic to My, then every minimal submodel of M 
tl, al, satisfies c~, is isomorphic to My. 
From the two leuinlata above it follows first that tile sails\]la- 
bility of sets of formulas in normal form is decidable: 
Since S, and S are deductively equivalent, we can establish by 
the following lemma that the satisfiability of arbitrary finite 
sets of literals S is decidable. 
3.14. LEMMA. S~ # {_L} ~ 3M(~M S). 
PROOF. (--,) If Sv # {,L}, we know by Lemma 3.13 that My 
is a model for S~. Then, by the soundness Su i- S "--* VM(~M 
Sv --*~M S). Since S is derivable from Sv, it follows ~M, S 
and thus S~ # {.L} ---, :IM(~M S). 
(,-) If S~ = {.L}, then for each model M V=M S~. From the 
soundness we get S I- Sv --* VM(~M S "-*~M Sv). Since S=. 
is derivable from S, it follows VM(~M Sv "*~=M S) amd hence 
S~ = {.l_} -- VM(~M S). O 
3.3 Completeness and Decidability 
Using tile procedure for deciding satisfiability we can easily 
show the completeness and decidability of lt°A v . 
3.15. TIIEOREM. For euery finite set of formulas P, and\]or 
each formula ~: 1I F ~ q~, then r b ~. 
PROOF. By definition @ is a logical consequence of F, iff 
F O {N @} is unsatisfiable. Using the equivalences of Theorem 
3.3, wc first get: 
r o {~ ¢} + {A(r u {~ ¢})}. 
S,,l,l,OSe, that A S' v,..vAs" is a DNFofA(Fu{~ @}), then 
ru{~ ~} + {A s' v...v AS"} 
and by tile decision procedure 
V= ru {~ ~b} ,-., s~ = {_L} A...A Sv n = {.L}. 
If r U {.-, @) is unsatisfiable, it follows that £ U {,,, @} -iF 
{2.}, since each S i is deductively equivalent with {.L}. 
From £ U {.~ @} k -L it follows by the deduction theorem first 
FI-,,.~D.L and thus Ft-,-, -L D ~. From I'F~ / D ~ and 
F I-~ -L by MP then r I- ~. 13 
3.16. COROLLARY. For every finite set o\] \]ormulas F and 
each \]ormula ~, F ~" ~ is decidable. 
PROOF. By the completeness and soundness we know F I- @ .-. 
I' ~ ~. Since @ is a logical consequence of r, iff ~ r u {,., ~}, 
we can decide r I-- ¢~ by tile procedure for deciding ~= FU{,., ~}. 
13 
Acknowledgments 
The author has been supported during tile writing of the sub- 
mitted draft version of this paper by the EEC Esprit project 
- 208 - 
DYANA at the Institut fiir maschinelle Sprachverarbeitung, 
Universit~t Stuttgart. The author would like to thank Jochen 
D6rre, Mark Johnson, liana Kaml,, It(,n Kal,lau , Paul King, 
John Maxwell and Stefan Momma as well as all anonymous 
reviewer for their comments on earlier versions of this paper. 
All remaining errors are of course lily own. 
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