A Proof-Theoretic Analysis of Goal-Directed Provability

Abstract

Uniform proofs have been presented as a basis for logic programming, and it is known that by restricting the class of formulae
it is possible to guarantee that uniform proofs are complete with respect to provability in intuitionistic logic. In this
paper we explore the relationship between uniform proofs and classes of formulae more deeply. Firstly we show that uniform
proofs arise naturally as a normal form for proofs in first-order intuitionistic sequent calculus. Next we show that the class
of formulae known as hereditary Harrop formulae are intimately related to uniform proofs, and that we may extract such formulae
from uniform proofs in two different ways. We also give results which may be interpreted as showing that hereditary Harrop
formulae are the largest class of formulae for which uniform proofs are guaranteed to be complete. Finally we briefly discuss
some possibilites for a slightly more general approach using intermediate and infinitary logics.

Full text

A PROOF-THEORETIC ANALYSIS OF GOAL-DIRECTED
PROVABILITY
JAMES HARLAND
Department of Computer Science, University of Melbourne
Parkville, 3052, Victoria, Australia
Abstract
One of the distinguishing features of logic programming seems to be the
notion of
goal-directed
provability, i.e. that the structure of the goal is used
to determine the next step in the proof search process. It is known that by
restricting the class of formulae it is possible to guarantee that a certain class
of proofs, known as
uniform proofs
, are complete with respect to provability in
intuitionistic logic. In this paper we explore the relationship b etween uniform
proofs and classes of formulae more deeply. Firstly we show that uniform proofs
arise naturally as a normal form for proofs in rst-order intuitionistic sequent
calculus. Next we show that the class of formulae known as hereditary Harrop
formulae are intimately related to uniform proofs, and that we may extract such
formulae from uniform proofs in two dierent ways. We also give results which
may be interpreted as showing that hereditary Harrop formulae are the largest
class of formulae for which uniform proofs are guaranteed to be complete, along
the lines of an interpolation theorem.
1 Introduction
It has long been known that there are ecient implementation techniques which make
Horn clauses, a particular fragment of rst-order logic, able to be used as a program-
ming language
12)
, and that this class of formulae forms the semantic basis for the
programming language Prolog
2
;
19)
. It has also been shown that this computational
paradigm is as powerful as that of Turing machines
20)
. Thus we may think of Horn
clauses as incorporating some form of algorithmic knowledge. As Horn clauses are
not a particularly large fragment of rst-order logic, it is perhaps not surprising that
this class of formulae has such a relatively strong property. There have been vari-
ous schemes proposed for logic programming languages which are extensions of Horn
clauses
4
;
13
;
14
;
16
;
17
;
18)
. Given these various extensions, it seems natural to ask whether
there is a maximal class of formulae which may be used as a programming language.
Moreover, there does not seem to b e a universally agreed criterion which may be used
to determine what constitutes a logic programming language, without which any such
notion of maximality would seem premature. A criterion of this nature has been pro-
posed by Miller et al.
16)
, in that they identify various rst-order and higher-order
fragments as logic programming languages by showing that these fragments satisfy
a completeness property for a certain class of proofs. However it would seem that a
general criterion should be strong enough not only to verify that certain fragments

may be used as programming languages, but also to discover such fragments in the
rst place. Thus it seems natural to use the criterion of
16)
, namely the completeness
of
goal-directed provability
, to investigate this question of maximality for (rst-order)
logic programming languages.
A useful notion this context is that of a
uniform proof
16
. A uniform proof is one
in which the principal connective of the formula is introduced in the last step of the
proof; in other words, when searching for a proof of a given formula, we need only
consider the immediate subformulae of the desired conclusion. Hence we may think of
uniform proofs as
goal-directed
, in that when searching for a uniform proof of a given
goal, we may use the structure of the goal to determine the structure of the proof.
We will denote uniform provability by
`
u
. Such pro ofs lead to an identication of
formulae with operations in a search space, and hence have a natural interpretation as
instructions, thus establishing a direct relationship between proof and computation.
This restriction also allows a more feasible implementation of the proof search process
than may be done in the case of arbitrary (intuitionistic) proofs.
Uniform proofs may be used as a basis for logic programming
16
;
15)
, and there are
several interesting investigations along these lines. However, it is not the case that all
intuitionistic proofs are uniform. Hence, one of the features of this approach is that
the formulae involved are restricted to a class for which uniform proofs are complete
with respect to intuitionistic logic, i.e.
F
1
`
I
F
2
i
F
1
`
u
F
2
when
F
1
and
F
2
belong
to a certain class of formulae. One such restriction is that the formulae which may be
used as assertions (i.e. those which may appear on the left of
`
) are Horn clauses, and
the goals (i.e. those formulae which may app ear on the right of
`
) are conjunctions
of atoms, so that when
F
1
is a set of Horn clauses and
F
2
is a conjunction of atoms,
then
F
1
`
I
F
2
i
F
1
`
u
F
2
. Larger classes of formulae for which the existence of
uniform pro ofs is guaranteed may also be given
16)
, and the largest class of rst-order
formulae for which this property has been established is known as
hereditary Harrop
formulae
. Intuitively, these formulae may be thought of as those which contain no
negative occurrences of
9
or
_
.
In this paper we examine the relationship between uniform proofs and hereditary
Harrop formulae, and we give several results which may b e interpreted as establishing
the maximality of this class of formulae. As uniformity is a property of proofs rather
than formulae, it is not strictly possible to establish that a given class of formulae
is the largest one for which uniform proofs are complete. For example, given that
F
1
`
u
F
2
, where
F
1
and
F
2
satisfy some restriction, for any formula
F
whatsoever,
F
1
; F
`
u
F
2
. However, as we shall see, there is a natural relationship between uniform
proofs and hereditary Harrop formulae. Essentially this is that whilst
F
1
; F
`
u
F
2
,
there is a hereditary Harrop formula
D
such that
D
`
u
F
2
, and
D
is related to
F
and
F
1
in such a way that
D
is the formula \doing the work" in the uniform pro of. The
relationship between the formulae is made precise in later sections.
An important insight which arises from this analysis is that the class of hereditary
Harrop formulae arises naturally out of the
permutation properties
of the rules of

intuitionistic logic
3
;
11)
This occurs by determining when it is possible to permute
certain combinations of inference rules so that an arbitrary pro of may be converted
into a uniform proof. Thus we may identify hereditary Harrop formulae as a logic
programming language purely from the notion of a uniform pro of and the pro of theory
of intuitionistic logic; no prior knowledge of logic programming languages
per se
is
needed. This suggests that the strategy of studying p ermutation rules in order to
investigate the completeness of goal-directed provability may b e used to identify logic
programming languages independently of the logic in use; such a strategy has been
used to identify logic programming languages in linear logic
8
;
9)
In this way we may
think of the permutation properties of the proof theory of the logic in question (in
conjunction with the notion of goal-directed search) as determining what fragments
of the logic may be used as a logic programming language.
2 Preliminaries
First we dene hereditary Harrop formulae
16)
. We assume the existence of a nite
set of constant and function symbols, and a countable set of variables. We refer to
the set of all ground terms as the
Herbrand universe
, denoted by
U
.
Denition 2.1
D
and
G
formulae are given by the grammar
D
::=
A
j 8
xD
j
D
1
^
D
2
j
G

A
G
::=
A
j 9
xG
j 8
xG
j
G
1
^
G
2
j
G
1
_
G
2
j
D

G
where
A
is an atom.
We refer to
D
formulae as
denite
formulae, and to
G
formulae as
goal
formulae.
The set of all denite formulae will be referred to as
D
, and the set of all goal formulae
as
G
.
A
program
is a set of closed denite formulae, and a goal is any closed goal
formula.
We will often refer to the ab ove classes of formulae as hereditary Harrop formulae.
Note that we do not allow negations here. We will refer to the formulae which do not
contain any negations as
negation-free
formulae.
It was shown in
16)
that an operational notion of proof
`
o
may be given for the
above class of formulae in such a way that for a program
P
and a goal
G
,
P
`
o
G
i
P
`
I
G
where
`
I
denotes intuitionistic provability, so that
P
`
o
G
i there is a
proof in intuitionistic logic of the sequent
P
?!
G
. Below we give a slightly dierent
denition, which we will denote as
`
u
. The rules for the standard sequent calculus
for intuitionistic logic are given in the Appendix. We will often refer to the rules
^
-L,
_
-L,
9
-L,
8
-L and

-L as left rules, and the rules
^
-R,
_
-R,
9
-R,
8
-R and

-R
as right rules. The
?
-R rule will be of little interest, as we will not be dealing with
formulae which may contain
?
.

Denition 2.2
We dene the
uniform rule
for a formula
F
as fol lows:

The uniform rule for an atom is

-
L

The uniform rule for
F
1
^
F
2
is
^
-
R

The uniform rule for
F
1
_
F
2
is
_
-
R

The uniform rule for
9
xF
is
9
-
R

The uniform rule for
8
xF
is
8
-
R

The uniform rule for
F
1

F
2
is

-
R
We say that a formula
F
is
compound
if
F
is not an atom.
Denition 2.3
A proof

is
uniform
if for each non-initial sequent
?
?!
F
in

where
F
is a compound formula, the rule used to derive
?
?!
F
is the uniform rule
for
F
.
It should be clear that the following prop osition holds.
Proposition 2.1
Let
F
be a formula containing no negations, and let
?
be a set of
such formulae.
Then

?
`
u
F
1
_
F
2
i
?
`
u
F
1
or
?
`
u
F
2

?
`
u
F
1
^
F
2
i
?
`
u
F
1
and
?
`
u
F
2

?
`
u
9
xF
i
?
`
u
F
[
t=x
]
for some
t
2 U

?
`
u
8
xF
i
?
`
u
F
[
y=x
]
where
y
is not free in
?
or
F

?
`
u
F
1

F
2
i
?
; F
1
`
u
F
2
It is not hard to show that
P
`
o
G
i
P
`
u
G
; for more details, the reader is
referred to
16
;
6
).
Our interest in hereditary Harrop formulae is due to the fact that uniform pro ofs
are complete with respect to intuitionistic logic for this class of formulae, rather than
due to a desire to implement a particular style of theorem prover for intuitionis-
tic logic. The notion of uniform proof is a stronger requirement than intuitionistic
proof; for example,
9
xp
(
x
)
`
I
9
xp
(
x
), but there is no uniform pro of of the sequent
9
xp
(
x
)
?! 9
xp
(
x
). In this way we are more interested in the strength of our conclu-
sions than a particular proof system.

3 Uniform Provability and Deniteness
Whilst the restriction to hereditary Harrop formulae is sucient to guarantee the
existence of uniform proofs, a natural question to ask is whether this restriction is
necessary. As mentioned above, it was shown in
16)
that if the antecedent is a set
of denite formulae and the consequent a goal formula, then the sequent has a proof
i it has a uniform proof. The converse to this result is not (strictly) true. For
example,
p
(
a
)
_
p
(
b
)
;
(
9
xp
(
x
)

q
)
`
u
q
, but the antecedent is not a set of denite
formulae. Similarly,
9
xp
(
x
)
;
8
x
(
p
(
x
)

q
)
`
u
q
, but the antecedent is not a set of
denite formulae.
Hence, it is not strictly true that for
F
1
`
u
F
2
to hold we must have that
F
1
is a denite formula. However it seems that the rst uniform proof above relies on
the fact that
p
(
a
)
;
(
9
xp
(
x
))

q
`
u
q
and
p
(
b
)
;
(
9
xp
(
x
))

q
`
u
q
, in which both
the antecedents are denite formulae. Similarly the second uniform proof above is
dependent on the fact that the universally quantied variable may be replaced by any
term, and hence the pro of may be thought of as a template for a number of proofs
of sequents of the form
p
(
t
)
;
8
x
(
p
(
x
)

q
)
?!
q
for any term
t
. In this way there
seems to be a more subtle relationship between uniform proofs and hereditary Harrop
formulae. Indeed, as mentioned above, it is not possible to give a strict classication
of the largest class of formulae for which uniform proofs are complete, but it does seem
that there is a relationship b etween uniform proofs and hereditary Harrop formulae
which may be elucidated.
A result reported in
15)
is that for sequents of the form ?
?!
G
where ? is a
set of denite formulae, there are no occurrences of the
9
-L or
_
-L rules. Hence,
if there is an intuitionistic proof of a sequent in which the antecedent is a denite
formula and the consequent a goal formula, then there are no occurrences of the
9
-L
and
_
-L rules and the sequent has a uniform proof. Thus we may conject that if a
uniform proof of
F
1
?!
F
2
contains no occurrences of either of these rules, then
F
1
is a denite formula and
F
2
is a goal formula. This again is not true, as there may
be parts of the formula
F
1
which ensure that
F
1
is not a denite formula, but are
not used in the proof. For example,
9
xq
(
x
)
; p
(
a
)
`
u
p
(
a
)
_
p
(
b
), due to the fact that
p
(
a
)
`
u
p
(
a
)
_
p
(
b
), and hence
F; p
(
a
)
`
u
p
(
a
)
_
p
(
b
) for any formula
F
. This means
that the relationship between a sequent
F
1
?!
F
2
and some \equivalent" sequent
D
?!
G
will require more investigation. In particular, the role of the rules
9
-L and
_
-L need examination.
Note that apart from

-L, the left rules may be thought of as converting the
antecedent into a desired form so that the appropriate right rules may be used. Hence,
from the point of view of goal-directed provability, it will often be useful to perform
these manipulations before starting the \main" proof, as it were. This will be the
case if we can interchange the order of the rules when a right rule precedes a left one.
It turns out that the nature of the
9
-L and
_
-L rules may make this dicult, and so
it may not always b e p ossible to re-arrange a given pro of so that all the manipulation

of the assertions can b e done prior to the proof search process. However, there are
some conditions under which this can be done.
For these reasons we introduce below the concept of a
denite
proof.
Denition 3.1
A proof

is
denite
if

contains no occurrences of either the
9
-L
rule or the
_
-L rule. We denote denite provability by
`
d
.
For this reason we will sometimes refer to the
9
-L and
_
-L rules as
indenite
rules.
As mentioned above, it was shown in
15)
that denite proofs are complete with
respect to intuitionistic provability for a large fragment of hereditary Harrop formulae.
Below we state the generalisation of this result for hereditary Harrop formulae.
Proposition 3.1
Let
?
be a set of denite formulae, and let
G
be a goal. Then any
proof

of
?
?!
G
is denite.
Note that it is not true that ?
`
u
F
)
?
`
d
F
, as when the succedent is just
an atom we may use either
_
-L or
9
-L without violating the uniformity property.
However, the converse is true, i.e. that if ?
`
d
F
, then ?
`
u
F
. In other words, a
sequent with a denite proof has a uniform proof, but a uniform proof need not be
denite.
Theorem 3.2
Let
F
be an negation-free formula, and let
?
be a set of negation-free
formulae. Then
?
`
d
F
)
?
`
u
F
This result may be established by using the permutation properties of intuitionistic
logic, as determined by Kleene
11)
; space prevents us from giving the pro of here.
The above theorem may be thought of as showing that if we ignore the
9
-L and
_
-L rules, then we need only consider uniform proofs. Note also the strength of the
contrapositive of the theorem, i.e. that if ?
?!
F
has a proof but no uniform proof,
then all proofs of ?
?!
F
contain an occurrence of an indenite rule. Thus an
obvious way to ensure the completeness of uniform proofs is to restrict the class of
formulae so that the indenite rules become redundant.
One such class of formulae are denite formulae, and the redundance of the in-
denite rules for denite formulae is precisely why denite formulae are interesting.
Denite formulae seem very apt in this context, as they force the programmer to
present his or her knowledge in a relatively strong way. We may think of an indenite
formula as conveying less information than a denite one. For example, the formula
9
xp
(
x
) carries less information than the formula
p
(
t
), which may be used to derive
the former one. Indeed, if we may imagine an intuitionistic programmer asserting
that
9
xp
(
x
) is true, we may expect him to be able to construct a term
t
such that
p
(
t
) is true. In fact this is a requirement if we insist upon goal-directed provability,
as
9
xp
(
x
)
`
I
9
xp
(
x
), but we cannot derive the truth of any instance of
p
(
x
). Hence

we may imagine a compiler taking as input a set of formulae, and retaining only the
denite parts of the formulae, as the indenite parts do not provide us with enough
information to make them useful.
In this way it seems that there is a strong connection between denite proofs and
denite formulae, which is that given a denite proof of ?
?!
F
, we may extract a
set of denite formulae and a goal formula from the sequent, in the manner briey
described above. A more precise description is given below. We denote by
>
the
formula \true".
Denition 3.2
Let
F
be an negation-free formula. Then we dene
def(
A
) =
A
goal(
A
) =
A
def(
F
1
^
F
2
)
=
def(
F
1
)
^
def(
F
2
) goal(
F
1
^
F
2
) = goal (
F
1
)
^
goal(
F
2
)
def(
F
1
_
F
2
)
=
>
goal(
F
1
_
F
2
) = goal (
F
1
)
_
goal(
F
2
)
def(
9
xF
)
=
>
goal(
9
xF
) =
9
x
goal(
F
)
def(
8
xF
)
=
8
x
def(
F
) goal(
8
xF
) =
8
x
goal(
F
)
def(
F
1

F
2
)
=
goal(
F
1
)

def (
F
2
) goal(
F
1

F
2
) = def (
F
1
)

goal(
F
2
)
We also dene def
(
f
F
1
;:::F
n
g
) =
S
n
i
=1
f
def(
F
i
)
g
.
Note that def(
F
) is either
>
or a denite formula, and that goal(
F
) is a goal
formula. Note also that goal(
F
) can never be
>
. We thus arrive at the following
useful lemma.
Lemma 3.3
Let
F
be an negation-free formula. Then
1.
F
`
I
def (
F
)
2.
goal(
F
)
`
I
F
Hence we see that def(
F
) and goal(
F
) preserve certain information, in that any-
thing deducible from def(
F
) is deducible from
F
, and that anything deducible from
F
is deducible from goal(
F
). In addition, as shown below, the converse relationships
hold for uniform provability.
Proposition 3.4
Let
F
be an negation-free formula, and let
?
be a set of negation-
free formulae.
If
?
`
d
F
, then there is a set of denite formula
?
0
and a goal formula
G
such
that
1.
?
`
I
V
?
0
2.
G
`
I
F
3.
?
0
`
u
G

Thus if ?
?!
F
has a denite proof, then not only do es the same sequent have a
uniform proof, but also we may extract a set of denite formulae ?
0
from ? such that
?
`
I
?
0
and ?
0
`
u
F
, and a goal formula
G
from
F
such that
G
`
I
F
and ?
`
u
G
. In
this way we may think of this result as a version of Craig's Interpolation theorem
1)
,
in that given a proof of ?
?!
F
, then provided that there are no occurrences of
9
-L
or
_
-L in the pro of, then we can interpolate a denite formula
D
such that ?
`
I
D
and
D
`
I
F
. Thus given ?, we can derive a denite formula which is provable from
? and has the same consequences, provided that we consider only denite proofs.
Hence, denite formulae arise naturally out of consideration of denite proofs, which
in turn arise naturally out of consideration of the permutability of the left and right
rules in intuitionistic logic.
4 Maximality of Information and Deniteness
The result above may be interpreted as showing what eciencies we can make in the
process of searching for a pro of provided that we restrict our attention to denite
proofs. As described ab ove, we may think of this in a similar manner to the Inter-
polation theorem. A criticism which may be made of this approach is that whilst
indenite formulae may contain less information than denite ones, that information
is lost when the indenite parts of the formulae are ignored. Also, the requirement
that the proof b e denite is a stronger one than merely requiring the proof to be
uniform. Hence it may b e interesting to examine what may be done to preserve (or
strengthen) the original information rather than weakening it, and to see if uniform
proofs are still sucient in these circumstances.
An obvious alternative approach to extracting denite information from a proof
is to nd a denite formula of which the premise is a consequence, rather than a
denite formula which is a consequence of the premise. We may think of this ap-
proach as attempting to supply sucient information in order to make the formula
denite, rather than ignoring the indenite parts of the formula, and hence we will be
suggesting hypotheses which will make the formula true. This leads us to the concept
of a
denite condition
and a
denite consequence
.
Denition 4.1
A
denite condition
of a formula
F
is a formula which is the same
as
F
except that
1. Every positive occurrence of a subformula
9
xF
0
in
F
is replaced by
F
0
[
t=x
]
for
some term
t
in which all variables of
t
appear universal ly quantied elsewhere
in
F
outside the scope of
9
x
.
2. Every positive occurrence of a subformula
F
1
_
F
2
in
F
is replaced by one of
F
i
,
i
= 1
;
2
We denote by defprem
(
F
)
the set of all denite conditions of
F
.

If
?
is a set of formulae, then
D
is a denite condition of
?
if
D
is a conjunction
of denite conditions of each element of
?
.
Denition 4.2
A
denite consequence
of a formula
F
is a formula which is the same
as
F
except that
1. Every negative occurrence of a subformula
9
xF
0
in
F
is replaced by
F
0
[
t=x
]
for
some term
t
in which all variables of
t
appear universal ly quantied elsewhere
in
F
outside the scope of
9
x
.
2. Every negative occurrence of a subformula
F
1
_
F
2
in
F
is replaced by one of
F
i
,
i
= 1
;
2
We denote by defconc
(
F
)
the set of all denite consequences of
F
.
If
?
is a set of formulae, then
G
is a denite consequence of
?
if
G
is a conjunction
of denite consequences of each element of
?
.
Note that a denite condition of
9
xF
cannot contain any occurrence of
x
, and
hence must produce a \ground witness" for
x
. For example, the only denite condi-
tions of
9
x p
(
x
) are atoms of the form
p
(
t
) where
t
is a ground term.
For existentially quantied variables appearing within the scope of a universally
quantied variable, we may use the universally quantied variable to construct the
witness. For example, one of the denite conditions of
8
x
9
y p
(
x; y
) is
8
x p
(
x; f
(
x
)).
It should be clear that for negation-free formulae, denite conditions and denite
consequences are denite and goal formulae resp ectively.
It is not hard to show that denite conditions and denite consequences behave
in the expected manner.
Proposition 4.1
Let
F
be an negation-free formula. Then for any denite condition
D
of
F
and denite consequence
G
of
F
1.
D
`
I
F
2.
F
`
I
G
We may think of this as stating that
D
has more explicit information than
F
, so
that if we were to consider an ordering of formulae in which
F
1

F
2
i
F
2
`
I
F
1
, then
the above proposition ensures that for any
F
, there is always a denite formula
D
such that
F

D
. Similar remarks apply to
G
, in that there is always a
G
such that
G

F
. In this way if we think of a lattice of formulae in which the partial order is
(intuitionistic) provability, then any chain has a least upper bound which is a denite
formula, and a greatest lower bound which is a goal formula. Thus we extrapolate
from the formula to a more denite statement.
It is not hard to show that denite conditions and denite consequences preserve
uniform provability.

Proposition 4.2
Let
F
be a negation-free formulae, and let
?
be a set of negation-
free formulae. Then for any denite condition
D
of
?
and any denite consequence
G
of
F
1.
?
`
u
F
)
D
`
u
F
2.
?
`
u
F
)
?
`
u
G
We may think of the above proposition as a form of \extrapolation" result, in that
given a uniform proof of ?
?!
F
we can nd a denite formula
D
and a goal formula
G
such that
D
`
I
V
?,
D
`
u
F
,
F
`
I
G
and ?
`
u
G
, and as a consequence,
D
`
u
G
.
Thus given any uniform proof, we can nd a denite formula
D
and a goal formula
G
which preserve the appropriate provability relationships. Hence we may conclude
that this result supports our contention that hereditary Harrop formulae are maximal
with respect to uniform pro ofs, in that any sequent which has a uniform proof may
be thought of as an incomplete specication of a sequent
D
?!
G
which preserves
the provability properties of the original sequent.
5 Conclusions and Further Work
We have seen how restricting rst-order intuitionistic proofs in certain ways leads to
some results which ensure that the task of searching for a proof is made more feasible
than in the general case. We may think of the restrictions as ensuring that the
information contained in the formulae is presented in a maximal way, so that we do
not need to waste time discovering this information during the computation process.
This may be thought of as requiring that we only consider proofs in a \normal form".
One way to think of this maximal class is to consider it as a \denite" or \uni-
form" sub-logic of intuitionistic logic, with a more restricted notion of provability. In
particular, this gives us a notion of
constructive consequence
, i.e. that the following
properties hold:
?
` 9
xF
,
?
`
F
[
t=x
]
?
`
F
1
_
F
2
,
?
`
F
1
or ?
`
F
2
Note that intuitionistic logic alone is not sucient to guarantee these equiva-
lences (unless, of course, ? is empty). However the following equivalences do hold in
intuitionistic logic:
?
` 8
xF
,
?
`
F
[
y=x
]
?
`
F
1
^
F
2
,
?
`
F
1
and ?
`
F
2
?
`
F
1

F
2
,
?
; F
1
`
F
2

where
y
is not free in ? or
F
.
Hence we see that imposing constructive consequence on intuitionistic logic gives
us precisely goal-directed provability. Alternatively, imposing goal-directed provabil-
ity on intuitionistic logic gives us constructive consequence. Thus we may think of
hereditary Harrop formulae as an important fragment of intuitionistic logic, in that
they seem to be the largest class of formulae for which the notion of constructive
consequence, and hence goal-directed provability, can be guaranteed. In fact, the
natural logic in which to interpret hereditary Harrop formulae is slightly stronger
than intuitionistic logic; see
5
;
7)
for details.
We have also seen some relationship b etween the restricted classes of pro ofs and
formulae and the more general classes, and in particular how denite formulae and
goals may be extracted from an arbitrary uniform proof, and that the extracted
formulae preserve uniform provability. It is possible that this result may be useful
for program specication, in that if a specication is given as a rst-order formula
(without negation), then the extraction process described above may be thought of
as nding a denite formula (i.e. a program) which satises the specication.
6 Acknowledgements
My thanks go to Dale Miller for many interesting discussions and weighty delibera-
tions. Discussions with David Pym have also been enlightening, and comments from
some anonymous referees were very helpful.
This work has been made possible by a grant of the Australian Research Council
through the Machine Intelligence Project.
7 References
1. G.S. Boolos and R.C. Jerey,
Computability and Logic
, Cambridge University
Press, 1980.
2. W.F. Clocksin and C.S. Mellish,
Programming in Prolog
, Springer-Verlag, 1984.
3. H.B. Curry, The Permutability of Rules in the Classical Inferential Calculus,
Journal of Symbolic Logic
17, 245-8, 1952.
4. D. Gabbay and U. Reyle, N-Prolog: An Extension of Prolog with Hypothetical
Implications. I.,
Journal of Logic Programming
1:319-355, 1984.
5. J. Harland, An Intermediate Logic for Logic Programs, Technical Report 90/29,
Department of Computer Science, University of Melb ourne, 1990.
6. J. Harland, A Pro of-Theoretic Analysis of Logic Programming, Technical Re-
port 90/21, Department of Computer Science, University of Melbourne, 1990.

7. J. Harland,
On Hereditary Harrop Formulae as a Basis for Logic Programming
,
Ph.D. Thesis, Department of Computer Science, University of Edinburgh, July,
1991.
8. J. Harland and D. Pym, The Uniform Proof-theoretic Foundation of Linear
Logic Programming,
Proceedings of the International Logic Programming Sym-
posium
, San Diego, October, 1991.
9. J. Harland and D. Pym, The Uniform Proof-theoretic Foundation of Linear
Logic Programming, Report ECS-LFCS-90-124. University of Edinburgh, 1990.
Also published as Technical Report 90/26, Department of Computer Science,
University of Melbourne.
10. S.C. Kleene,
Introduction to Metamathematics
, North-Holland, 1952.
11. S.C. Kleene, Permutability of Inferences in Gentzen's Calculi LK and LJ,
Mem-
oirs of the American Mathematical Society
10, 1952.
12. J.W. Lloyd,
Foundations of Logic Programming
, Springer-Verlag, Berlin, 1984.
13. L.T. McCarty, Clausal Intuitionistic Logic I. Fixed Point Semantics,
Journal of
Logic Programming
5:1:1-32, 1988.
14. L.T. McCarty, Clausal Intuitionistic Logic II. Tableau Proof Procedures,
Jour-
nal of Logic Programming
5:2:93-132, 1988.
15. D.A. Miller, A Logical Analysis of Modules in Logic Programming,
Journal of
Logic Programming
6:79-108, 1989.
16. D.A. Miller, G. Nadathur, F. Pfenning and A. Scedrov, Uniform Proofs as a
Foundation for Logic Programming,
Annals of Pure and Applied Logic
51:125-
157, 1991.
17. G. Nadathur and D.A. Miller, Higher-Order Horn Clauses
Journal of the Asso-
ciation for Computing Machinery
37:4:777-814, October, 1990.
18. P. Schroeder-Heister, Hypothetical Reasoning and Denitional Reection in
Logic Programming,
Extensions of Logic Programming: International Work-
shop, Tubingen FRG, December 1989
, P. Schroeder-Heister (ed.), Lecture Notes
In Articial Intelligence 475, 327-340, Springer-Verlag, 1991.
19. L. Sterling and E. Shapiro,
The Art of Prolog
, MIT Press, 1986.
20. S.-A. Tarnlund, Horn Clause Computability,
BIT
17:215-226, 1977.

A Intuitionistic Sequent Calculus
B; C;
?
?!
F
B
^
C;
?
?!
F
^
-L
?
?!
B
?
?!
C
?
?!
B
^
C
^
-R
B;
?
?!
F C;
?
?!
F
B
_
C;
?
?!
F
_
-L
?
?!
B
?
?!
B
_
C
?
?!
C
?
?!
B
_
C
_
-R
?
?!
B C;
?
?!
F
B

C;
?
?!
F

-L
B;
?
?!
C
?
?!
B

C

-R
?
; B
[
t=x
]
?!
F
?
;
8
xB
?!
F
8
-L
?
?!
B
[
y=x
]
?
?! 8
xB
8
-R
?
; B
[
y=x
]
?!
F
?
;
9
xB
?!
F
9
-L
?
?!
B
[
t=x
]
?
?! 9
xB
9
-R
?
?! ?
?
?!
B
?
-R
The rules
8
-R and
9
-L have the side condition that
y
is not free in ?,
B
or
F
.
An
initial sequent
is a sequent ?
?!
F
where
F
is either an atomic formula or
?
and
F
2
?. A
proof
for the sequent ?
?!
F
is a nite tree, constructed using the
above rules, whose root is ?
?!
F
and whose leaves are initial sequents.
As is done in
15)
, we omit the interchange and contraction rules by considering
the antecedents of sequents to be sets. Note also that thinning is not necessary due
to the way an initial sequent is dened.