References

We use the following abbreviations for technical reports, proceedings, joumals, etc.:

ACM Association for Computing Machinery

CAAP Proceedings of the Colloquium on Trees in Algebra and Programming

CACM Communications of the ACM

CADE Proceedings of the Conference on Automated Deduction

CRIN Centre de Recherche en Informatique de Nancy, Campus Scientifique, B.P.239,

54506 Vandoeuvre Les Nancy Cedex

ECAI Proceedings of the European Conference on Artificial Intelligence

FGCS Proceedings of the International Conference on Fifth Generation Computer

Systems

GWAI Proceedings of the German Workshop on Artificial Intelligence

ICALP Proceedings of the International Colloquium on Automata, Languages, and Pro-

gramming

ICLP Proceedings of the International Conference on Logic Programming

IEEE The Institute of Electrical and Electronics Engineers

IFB Informatik-Fachberichte, Springer

IFIP Proceedings of the World Computer Congress of the International Federation of

Information Processing Societies

IJCAI Proceedings of the Intemational Joint Conference on Artificial Intelligence

IRIA Institute de Recherche d'Informatique et d'Automatique, Domaine de Voluceau,

Rocquencourt, B.P.10578150, Le Chesnay, France

Journal of the ACM

Journal of Automated Reasoning

Journal of Logic Programming

Journal of Symbolic Computation

Lecture Notes in Computer Science, Springer

Microelectronics and Computer Technology Corporation, 3500 West Balcones

Center Drive, Austin, TX 78759

Proceedings of the Annual ACM Symposium on Principles of Programming Lan-

guages

Proceedings of the International Conference on Rewriting Techniques and Appli-

cations

Society for Industrial and Applied Mathematics

Proceedings of the Symposium on Logic Programming

Journal of Theoretical Computer Science

JACM

JAR

JLP

JSC

LNCS

MCC

POPL

RTA

SIAM

SLP

TCS

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Index

Admissible, 171

Alphabet, 15

Answer substitution, 39

Application of a trivial clause, 179

Arbitrary selections, 77

Arity, 15

Assignment (function), 29

Atom, 17

Axioms of equality, 28

Based on, 28

Body, 26

Canonical, 133

Codomain, 20

Complete lattice, 40

Complete set of EP-unifiers, 72

Complete EP-unification procedure, 74

Complete set of reductions, 133

Completely defined, 159

Complexity, 188

Composition of substitutions, 20

Compression, 220

Computed answer substitution, 46

Conditional term rewriting system, 121

Confluent, 130

Connective, 15

Constant, 15

Constrained selections, 77

Construction, 17

Constructor, 15, 134

Context-free grammar, 162

Continuous function, 40

Correct answer substitution, 39

Critical pair, 132

Decomposable, 113

Decomposition, 176

Defined function symbol, 15, 134

Definition, 27, 134

Depth, 187

Derivation, 46

Derivation tree, 47

Descendant, 76, 187

Difference, 16

Directed set, 40

Directly generates (derives), 162

Disagreement set, 215

Domain, 20, 29

E-falsify, 30

E-interpretation, 29

E-model, 31

E-model intersection property, 34

E-(un)satisfiable, 30

E-valid, 31

Element, 17

Empty clause, 26

Empty substitution, 20

Enumeration problem, 74

Environment, 69, 148

EP-derivation, 78

EP-instance, 68

EP-matcher, 68

EP-matching problem, 68, 161

EP-refutation, 78

EP-resolution, 78

EP-unification problem, 66

EP-unification procedure, 74

EP-unifiable, 66, 75

In~x ~3

EP-unifier, 66

EP-variant, 68

Equality, 16

Equation, 17

Equational clause, 26

Equational logic program, 27

Equational program, 27

Existence problem, 74

F-substitutivity, 28

Failed derivation, 46

Failure, 46

Fair selection function, 64

First order theory, 15

Flat clause, 220

Function symbol, 15

Functional reflexive axioms, 98

Functional term, 17

Goal clause, 26

Greibach Normal Form, 163

Ground, 19

Ground confluent, 117,130

Head, 26

Herbrand --base, 36

Herbrand E-interpretation, 33

Herbrand --interpretation, 36

Herbrand E-model, 33

Herbrand --model, 37

Herbrand =-universe, 36

Herbrand universe, 35

Homogeneous form, 27

Horn clause, 26

Idempotent, 20

Imitation, 179

Immediate descendant, 76, 187

Inference rule, 45

Infinite branch, 47

Initial symbol, 17, 86

Innermost derivation, 145

Innermost narrowing, 148

Innermost redex, 128, 144

Innermost reflection, 148

Input clause, 26

Instance, 21, 99

Instantiation, 99

Instantiation and paramodulation, 100

Instantiation lattice, 68

Instantiation preorder, 67

Interpretation, 29

Ip-success set, 104

Irreducible, 128

Language generated, 162

Lazy narrowing, 177

Lazy paramodulation rules, 180

Lazy resolution, 59

Leftmost generation (derivation), 162

Length, 46

Locally confluent, 130

Logic program, 2 7

Logical E-consequence, 31

LUSH-resolution, 53

Matcher, 123

Minimal complete set of EP-unifiers, 73

Monotonic, 40

More general, 67

More specific, 67

Most general EP-unifier, 73

Most general partial unifier, 215

Most general unifier, 48

Multiset, 16

Name, 26, 134

Narrowing, 136

~4 ~n~x

New variant, 22

Noetherian, 130

Normal form, 128, 138

Normalizable, 138

NRF-resolution, 215

Occur check, 49

Occurrence, 17

P-substitutivity, 28

P-success set, 96

Paramodulation, 94, 178

Predicate symbol, 15

Production, 162

Program clause, 26

Proper subterm, 18

Punctuation symbol, 15

Quantifier, 15

Re.dex, 128, 144, 147

Reduction, 126, 127

Referential transparency, 126

Reflection, 95

Reflexivity, 28

Refutation, 46

Regular, 127

Relative decision problem, 163

Removal of trivial equations, 50

Renaming substitution, 21

Replacement, 18

Representation of a derivation, 148

Resolution, 53

Resolution restricted to equations, 60

Restriction, 20

Rewrite rule, 121

Rewriting, 113, 126, 127, 155

RUE-resolution, 215

S-derivation, 199

S-grammar, 163

SEL-computed answer substitution, 46

Selected subgoal, 45

Selected redex, 128

Selection function, 45

Sentence (start) symbol, 162

Simplify, 199

Skeleton, 148

SLD-derivation, 54

SLD-refutation, 54

SLD-resolution, 53

Strong selection function, 110, 144

Subgoal, 26

Subsumption lattice, 68

Subsumption preorder, 67

Subterm, 18

Substitution, 20

Success branch, 47

Success set, 55

Successful derivation, 46

Superposition algorithm, 132

Surface factor, 220

Symmetry, 28

Term, 17

Term decomposition, 48

Term rewriting system, 121

Terminal symbol, 162

Terminating, 130

Transitivity, 28

Trivial equation, 17

Trivial clause, 133

Trivial program, 133

Trivial rule, 133

Unconditional term rewriting system, 121

Undecidable, 163

In~x ~5

Unifiable, 48, 52

Unification problem, 74

Unification rules, 180

Unifier, 48

Union, 16

Universal matching problem, 161

Universal unification problem, 67

Universal unification procedure, 74

Universe, 29

Value, 29, 127

Variable, 15

Variable elimination, 48, 176

Variable-pure substitution, 22

Variant, 22

Via SEL, 46

The Inference Rules

---)d term decomposition 48

---)dd decomposition of decomposable equations 176

~dn decomposition of non-decomposable equations 176

-----)EP EP-resolution 78

- - - ) f reflection 95,148

----)i instantiation 99

--->if innermost reflection 148

-~im imitation 179

-- ' ) in innermost narrowing 148

---)ip instantiation and paramodulation 100

---~ln lazy narrowing 177

---)lr lazy resolution 59

- - - )n narrowing 136

---)p paramodulation 94

---)pv paramodulation at variable occurrences 178

- - - ) r resolution 53

---)R rewriting 123,126,127,155

~r r resolution restricted to equations 60

---)t removal of trivial equations 50

---~tc application of a trivial clause 179

~ t v removal of trivial variable-equations 50

~ v variable elimination 48

---)vc variable elimination on equations that contain only variables and

constructors 176

---)vf variable elimination on equations that contain a defined function

symbol 176

Examples

AB, 91

APPEND, 132

APPEND', 132

ASSOC, 66

BASIC, 145

CATS, 36

COM, 66

CON, 177

CYCLE, 110

EAX(<EP, LP>), 28

ELEMENT, 62

FUN, 94

FUN', 105

FUN", 142

GM, 121

GS, 124

IDEM, 136

INFINITE, 137

INN, 147

INNER, 139

INTEGERS, 228

LENGTH, 65

MAP-ADD1, 154

MINUS, 135

NC, 85

P-ASSOC, 82

PALINDROM, 203

PLUS-MINUS, 113

RS, 165

SUM, 125

TRIV, 133

Notations

A

a,b,...

B

B_=(<EP, LP>)

[B] C

CFG

Cod(G)

Con(X)

D

D(0)

Dora(G)

E

EAX(<EP, LP>)

EP Ep -1

<EP, LP>

<EP, LP> +

EQ

F

F(EP)

FUN(X)

f,g ....

glb

GNF

H(S)

I

i,j ....

IP-SSEP

LP

lub

ME(<EP, LP>)

M_--(<EP, LP>)

atom

constructors

multiset of atoms

Herbrand =-basis for <EP, LP>

{ [A]IA ~ B }

atom or equation

context free grammar

codomain of G

set of constructions occurring in X

multiset of atoms or equations

depth of 0

domain of G

equation

axioms of equality for <EP, LP>

equational program

{ r---)l~F [ 1---~r~F e EP }

equational logic program

<EP, LP> u EAX(<EP, LP>) without the p-substitutivity axioms

equality symbol

multiset of equations

functional reflexive axioms for EP

set of functional terms occurring in X

defined function symbols

greatest lower bound

Greibach Normal Form

homogeneous form of S

E-interpretation

integers

success set of EP with respect to reflection, instantiation and

paramodulation

logic program

least upper bound

least Herbrand E-model of <EP, LP>

least Herbrand --model of <EP, LP>

16

16

16

36

35

16

162

20

19

16

188

20

16

28

16

85

27

55

15

16

98

19

16

40

163

27

16

16

104

16

40

35

38

Notations 249

mgu

Mod_~--(<EP, LP>)

Occ(X) P-SSEP

P

Q

[Q(tl ..... tn)]

R

r,s~...

RULES

RULES +

S

SEL

SEL +

SLDE

SS (<EP, LP>)

slrcl s lr~t l s =EP t

<S=EPt>

<S<EPt>

TEP

T<EP, LP>

TRANS

T$0 Tl"n

[t] UPc

UPEP

U_=(<EP, LP>)

V

Vat(X) VCod(~)

x,y , . . .

E

A

most general unifier

set of Herbrand --models for <EP, LP>

set of occurrences of X

p-success set of EP

program clause

predicate symbol

abbreviation for Q([tl] ..... [tn])

term rewriting system

terms

set of inference rules

RULES u {---)lr}

set of Horn clauses

selection function

selection function that never selects an equation of the form

EQ(x,y) if it has another choice

special selection function

success-set of <EP, LP>

subterm of s at x

replacement of t in s at n

EQ(s,t) is logical E-consequence of EP

EP-unification problem

EP-matching problem

function assigned to EP

function assigned to <EP, LP>

set of transformations

glb(U)

T(T']'(n- 1))

congruence class that contains t

universal unification procedure

EP-unification procedure

Herbrand ---universe of EP, LP

finite set of variables

set of variables occurring in X

set of variables occurring in the codomain of (r

variables

empty substitution

empty occurrence

48

38

17

96

16

16

35

16

16

46

203

16

45

192

77

55

18

18

32

66

161

88

41

180

41

41

35

74

74

36

16

19

20

16

20

18

250 No:ztS.oos

olv

O" =EP 0 [V]

0 - E P 0 [V]

O)

,-, ,1

A

V

3 V

U

\

=--EP

El

<EP

- E P

< <

#e

ep #r

#s

occurrence

substitutions

restriction of G to V

for all x in V we find ~x =EP 0x

EP-variants with respect to V

first infinite ordinal

not

and

or

implication

equivalence

existential quantifier

universal quantifier

concatenation

(multiset) equality

(multiset) union

(multiset) difference

congruence relation on the set of ground terms generated by EP

finest congruence relation on the set of ground terms generated

by EP

empty clause

instantiation preorder

EP-variants

multiset ordering induced by <

ordering on complexities of refutations

number of equations occurring in a goal clause

number of applications of paramodulation in a derivation

number of lazy resolution steps in a derivation

number of symbols occurring in a goal clause

18

16

20

68

68

41

15

15

15

15

15

15

15

18

16

16

16

30

35

26

67,68

68

188

189

188

188

203

188