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Fundamenta Informaticae ??? (2009) 1–19 1
IOS Press
Skolem Machines
John Fisher∗
Department of Computer Science
California State Polytechnic University
Pomona, California, USA
jrfisher@csupomona.edu
Marc Bezem
Department of Computer Science
University of Bergen
Bergen, Norway
bezem@ii.uib.no
Abstract. The Skolem machine is a Turing-complete machine model where the instructions are
first-order formulas of a specific form. We introduce Skolem machines and prove their logical cor-
rectness and completeness. Skolem machines compute queries for the Geolog language, a rich
fragment of first-order logic. The concepts of Geolog trees and complete Geolog trees are defined,
and these tree concepts are used to show logical correctness and completeness of Skolem machine
computations. The universality of Skolem machine computations is demonstrated. Lastly, the paper
outlines implementation design issues using an abstract machine model approach.
Keywords: Finitary geometric logic, Skolem machines, Geolog language, Geolog trees, correct-
ness, completeness, universality.
1. The Geolog Language
Geolog is a language for expressing first-order geometric logic in a format suitable for computations
using an abstract machine.Geolog rules are used as machine instructions for an abstract machine that
computes consequences for first-order geometric logic.
Address for correspondence: Please contact either author regarding questions.
∗Thanks to M. Bezem and UIB for hosting me in Bergen, March 2008, to work on this paper.
2J. Fisher, M. Bezem /Skolem Machines
AGeolog rule has the general form
A1, A2, . . . , Am⇒C1|C2|. . . |Cn(1)
where the Aiare atomic expressions and each Cjis a conjunction of atomic expressions, m, n ≥1. The
left-hand side of a rule is called the antecedent of the rule (a conjunction) and the right- hand side is
called the consequent (a disjunction). All atomic expressions can contain variables.
If n= 1 then there is a single consequent for the rule (1), and the rule is said to be definite. Otherwise
the rule is a splitting rule that requires a case distinction (case of C1or case of C2or . . . case of Cn).
The separate cases (disjuncts) Cjmust have a conjunctive form
B1, B2, . . . , Bh(2)
where the Biare atomic expressions, and h≥1varies with j. Any free variables occurring in (2) other
than those which occurred free in the antecedent of the rule are taken to be existential variables and their
scope is this disjunct (2).
As an example, consider the Geolog rule
s(X,Y) => e(X,Y) | dom(Z),r(X,Z),s(Z,Y).
The variables X,Y are universally quantified and have scope covering the entire formula, whereas Z
is existentially quantified and has scope covering the last disjunct in the consequent of rule. A fully
quantified first-order logical formula representation of this Geolog rule would be
(∀X)(∀Y)[s(X, Y )→e(X, Y )∨(∃Z)(dom(Z)∧r(X, Z )∧s(Z, Y ))]
Now we come to two special cases of rule forms, the true antecedent and the goal or false conse-
quents. Rules of the form
true ⇒C1|C2|. . . |Cn(3)
are called factuals. Here ‘true’ is a special constant term denoting the empty conjunction. Factuals are
used to express initial information in Geolog theories. Rules of the form
A1, A2, . . . , Am⇒goal (4)
are called goal rules. Here ‘goal’ is a special constant term. A goal rule expresses that its antecedent is
sufficient (and relevant) for goal. Similarly, rules of the form
A1, A2, . . . , Am⇒false (5)
are called false rules. Here ‘false’ is a special constant term denoting the empty disjunction. A false
rule expresses rejection of its antecedent.
The constant terms true,goal and false can only appear in Geolog rules as just described. All
other predicate names, individual constants, and variable names are the responsibility of the Geolog
programmer.
AGeolog theory (or program) is a finite set of Geolog rules. A theory may have any number of
factuals and any number of goal or false rules.
The logical formulas characterized by Geolog, and the bottom-up approach to reasoning with those
logical formulas, finds its earliest precursor (1920) in a particular paper by Thoralf Skolem [17].
J. Fisher, M. Bezem /Skolem Machines 3
2. Skolem Machines
The rules in Geolog theories serve as instructions for an abstract Skolem machine (SM). Skolem machines
resemble multitape Turing machines and the two machine models have actually the same computational
power. See the discussion in Section 1.
An SM starts with one tape, the initial tape having true written on it. The basic operations of an SM
use the Geolog rules in the instruction set to
•extend a tape (write logical terms at the end)
•create new tapes (for splitting rules)
The tapes are also called states. An SM with more than one tape is said to be in a disjunctive state,
comprised of multiple separate simple states or tapes.
The basic purpose of a particular SM is to compute its instruction set and to halt when all of its tapes
have ‘goal ’ or ‘false’ written on them.
In order to motivate the general definitions for the workings of SM, let us work through a small
example. To this end, consider the Geolog rulebase (SM instructions) in Figure 1.
true => domain(X), p(X). % #1
p(X) => q(X) | r(X) | domain(Y), s(X,Y). % #2
domain(X) => u(X). % #3
u(X), q(X) => false. % #4
r(X) => goal. % #5
s(X,Y) => goal. % #6
Figure 1. Sample instructions
The only instruction that applies to the initial tape is instruction #1. The antecedent of the rule
matches true on the tape, so the tape can be extended using the consequent of the rule. In order to
extend the tape using domain(X), p(X) an instance for the free existential variable Xis first generated
and then substituted, and the resulting terms are written on the tape, as shown in Figure 2.
--------------------------------------
true domain(sk1) p(sk1)
--------------------------------------
Figure 2. After applying rule #1
At this point in machine operation time either of the rules #2 or #3 can apply. The general definition
of SM operation does not specify the order, but we will apply applicable rules in top-down order. So,
applying instruction #2 we get tape splitting, as shown in Figure 3.
Each of the disjuncts in the consequent of rule #2 is used to extend the previous single tape. This
requires that the previous tape be copied to two new tapes and then these tapes are extended.
Now, instruction #3 applies to all three tapes, even twice to the last tape, with total result shown in
Figure 4.
4J. Fisher, M. Bezem /Skolem Machines
--------------------------------------
true domain(sk1) p(sk1) q(sk1)
--------------------------------------
--------------------------------------
true domain(sk1) p(sk1) r(sk1)
--------------------------------------
-----------------------------------------------
true domain(sk1) p(sk1) domain(sk2) s(sk1,sk2)
-----------------------------------------------
Figure 3. After applying rule #2
--------------------------------------
true domain(sk1) p(sk1) q(sk1) u(sk1)
--------------------------------------
--------------------------------------
true domain(sk1) p(sk1) r(sk1) u(sk1)
--------------------------------------
---------------------------------------------------------------
true domain(sk1) p(sk1) domain(sk2) s(sk1,sk2) u(sk1) u(sk2)
---------------------------------------------------------------
Figure 4. After applying rule #3 four times (!)
Instruction #4 now adds false to the top tape, shown in Figure 5.
Now instruction #5 applies to the second tape, and then instruction #6 applies to the third tape, shown
in Figure 6.
At this point the SM halts because each tape has either the term goal or the term false written on
it.
The SM has effectively computed a proof that the disjunction
(∃X)(u(X)∧q(X)) ∨(∃X)r(X)∨(∃X)(∃Y)s(X, Y )
is a logical consequence of the Geolog theory consisting of the first three rules in Figure 1. This is so
because every tape of the halted machine either has q(sk1),u(sk1) written on it or has r(sk1) written
on it or else has s(sk1,sk2) written on it. Note that the three disjuncts correspond to the goal and false
rules in Figure 1. We will continue a discussion of this example (specifically, the role intended for the
false rule) later in this section.
The proof tree displayed in Figure 7 was automatically drawn by the program whose implementation
is described in [6]. The diagram displays the tapes generated by the SM in the form of a directed tree.
Notice that the tree splits where the SM would have copied a tape. It is possible to describe an SM using
trees rather than multiple tapes; see the next section.
J. Fisher, M. Bezem /Skolem Machines 5
--------------------------------------------
true domain(sk1) p(sk1) q(sk1) u(sk1) false
--------------------------------------------
--------------------------------------
true domain(sk1) p(sk1) r(sk1) u(sk1)
--------------------------------------
---------------------------------------------------------------
true domain(sk1) p(sk1) domain(sk2) s(sk1,sk2) u(sk1) u(sk2)
---------------------------------------------------------------
Figure 5. Goal tape, rule #4
--------------------------------------------
true domain(sk1) p(sk1) q(sk1) u(sk1) false
--------------------------------------------
-------------------------------------------
true domain(sk1) p(sk1) r(sk1) u(sk1) goal
-------------------------------------------
-------------------------------------------------------------------
true domain(sk1) p(sk1) domain(sk2) s(sk1,sk2) u(sk1) u(sk2) goal
-------------------------------------------------------------------
Figure 6. After applying rule #5 and then #6, HALTED
DEFI NI TI ON O F SKOLEM MAC HI NE O PE RATIONS
•AGeolog rule ANT ⇒CONS is applicable to an SM tape T, provided that it is the case that
all of the terms of ANT can be simultaneously matched against ground terms (no free variables)
written on T. (It may be that ANT can be matched against Tin more than one way; for example,
rule #3 and the third tape of Figure 3.)
•If the rule ANT ⇒CONS is applicable to tape T, then for some matching substitution σapply σ
to CONS and then expand tape Tusing σ(CONS ).
•In order to expand tape Tby σ(CONS ) = C1|C2|. . . |Ckcopy tape Tmaking k−1new tapes
T2, T3, . . . , Tk, and then extend Tusing C1, extend T2using C2, . . . , and extend Tkusing Ck. (No
copying if k= 1.)
•In order to extend a tape Tusing a conjunction C, suppose that X1, . . . , Xpare all of the free
existential variables in C. Create new constants cj,1≤j≤pand substitute cjfor Xjin C,
obtaining C0, and then write each of the terms of C0on tape T. It is mandatory that the constant is
new with respect to the theory and the tape. 1
1These witnesses are called Skolem constants by some, but we would prefer to view them as eigenvariables in the elimination
of existential quantification.
6J. Fisher, M. Bezem /Skolem Machines
Figure 7. Tree display
Notice that only ground terms ever appear on any SM tape. Thus the matching algorithm does not really
need the full power of general term unification. Simple left-to-right term matching suffices.
Given an SM with tapes T1, . . . , Tt,t≥0, we say that a particular tape Tiis saturated if no applicable
instance of a rule leads to new facts.
A tape is halted if it is either saturated or contains goal or contains false (any of which could occur
at the same time). An SM is called halted if all its tapes are halted, it is halted successfully if it is halted
with all tapes containing either goal or false. If a tape of an SM is saturated with neither goal nor false
on it, then this tape actually constitutes a countermodel: all rules are satisfied, they are consistent (by
absence of false) and yet the goal is false (by absence of goal ).
The set of terms on any saturated tape that is not successfully halted is said to be a counter model.
Suppose that we write a Geolog theory in the form
T=A∪G∪F(6)
where Ais the axioms,Gcontains all of the affirming goal rules and Fcontains all of the rejecting false
rules. It is intended that Acontains all the rules of the theory other than the goal rules and the false rules
and that A,G, and Fare mutually disjoint sets.
The Geolog query Qfor a Geolog theory T=A∪G∪Fis the disjunctive normal form Q=C1|
C2|. . . |Ckconsisting of all of the conjunctions Cisuch that either Ciappears as antecedent of one of
the goal rules (in G) or of one of the false rules (in F). As before, the free variables in Qare taken to be
existential variables. The scope of a variable Xappearing in a particular Ci(within Q) is restricted to
Ci.
We say that a Geolog theory Tsupports its query Qif there is a successfully halted SM such that
each tape satisfies some Ci.
Theorem 1. If theory Tsupports its query Qthen Qis a logical consequence of the axioms.
J. Fisher, M. Bezem /Skolem Machines 7
Theorem 2. Suppose that Qis the query for Geolog theory Gand that Qis a logical consequence
of G. Then Gsupports Q.
Theorem 1 is proved in Section 3 next, as a corollary to a general characterization of Geolog trees.
Theorem 2 is proved in Section 4 using the concept of complete Geolog trees. The references [4], [7],
[8], provide additional theoretical background.
3. Geolog Trees
The splitting of tapes during Skolem machine operations suggests that tree structures can provide an
alternate description. This was depicted in Figure 7 and the concept is quite simple.
Suppose that we are given a Geolog theory G. There is only one Geolog tree with one node, and that
is the tree true. This singular tree corresponds to the initial tape of a Skolem machine for G.
Assume that we have a correspondence between Skolem machine tape configurations and Geolog
trees up to some number kof rule applications. If the Skolem machine would have had btapes then the
Geolog tree Tkhas a total of bbranches. Let us examine a (k+ 1)st rule application. This application
would have been applied to a particular tape of the Skolem machine. For the Geolog tree, the application
is at the leaf of the corresponding branch Bof the tree.
Consider again the general form of a Geolog rule (1). When such a rule (r1say) is applied to the
current tape it splits into ntapes. The corresponding Geolog tree branches instead, as visualized in the
following diagram. The leftmost branch (B1in the diagram) is an extension defined using a Skolem
machine operation corresponding to the first disjunct of the consequent of the rule r1. The other new
branches (if any) are similarly formed.
Figure 8. Branching
By induction, any Skolem machine computation (sequence of operations) can be expressed by a
corresponding Geolog tree.
8J. Fisher, M. Bezem /Skolem Machines
Suppose that Bis a branch (from root to leaf) in a Geolog tree. A branch conjunction is any con-
junction b1, b2, . . . , biof logical terms which appear at the nodes of the tree on branch B. Suppose that
s1, s2, . . . , sjare the distinct eigenvariables appearing in the branch conjunction. Let b0
1, b0
2, . . . , b0
ibe the
branch conjunction expression with the eigenvariables replaced by distinct variables x1, x2, . . . , xjand
then form the logical formula (∃x1, x2, . . . , xj)(b0
1, b0
2, . . . , b0
i). A branch wff is any such well-formed
logical formula, where the ordering of the logical variables is arbitrary and the ordering of the conjuncts
is also arbitrary.
Atree wff is any disjunction c1|c2|. . . |cpsuch that for any branch Bof the Geolog tree, one of
the cjis a branch wff for Band each cjis a branch wff of the tree.
For example, (∃x)p(x)is a tree wff for the Geolog tree in Figure 7, and so is
(∃x)(q(x), u(x)) |(∃y)u(y)|(∃x, y)s(x, y).
Proposition. If w is a tree wff for a tree based on a Geolog theory then w is a logical consequence
of the axioms of the theory.
Proof. The proposition is vacuously true in the case that there are 0 rule applications to build the tree;
this is just the tree true. Suppose that the proposition is true whenever krule applications build the tree.
Assume that k+ 1 rule applications built our tree. The last rule application can again be depicted as
in Figure 8. Consider a tree wff wfor this tree. We can express was w=wk|wrwhere wris the
disjunction of branch wffs from Bexpanded using rule r=r1, and wkconsists of the other branch
wffs. If any branch wff in wris formed using only facts along Bproper, then w=wk|wris a logical
consequence of the axioms, by the induction hypothesis (ignore the application of r). Otherwise, let us
write the instance of the expanding rule ras
a1, a2, . . . , an=> c1|c2|. . . |cm(7)
Here the facts a1, . . . , anoccur along B. We can also now express wras wr=w1|. . . |wmwhere
branch wff wicontains at least one conjunct formed using ci(i= 1, . . . , m).
Let bbe the conjunction of all the facts on branch B, among which are a1, . . . , anand consider
v0
i=∃(b0, c0
i)where the existential quantifier captures all of the eigenvariables (if any), i= 1, . . . , m.
Now each wiis a logical consequence of the corresponding vi. Moreover, v1|. . . |vmis a logical
consequence of ∃b0where the latter closes bwith existential quantification. And so wr=w1|. . . |wm
is a logical consequence of ∃b0and the axioms of the theory. Now this makes w=wk|wra logical
consequence of wk| ∃b0and the axioms of the theory. But wk| ∃b0is itself a logical consequence of
the axioms, by the induction hypothesis, because wk| ∃b0is a tree wff formed inside Tk. Therefore,
w=wk|wris also a logical consequence of the axioms of the theory, as required. 2
Notice that the set of all facts along any branch of the tree is a model for a tree wff. Call these models
branch models of the tree wff.
Proof of Theorem 1. Suppose that the theory supports its query Q. Consider the tree corresponding
to the halted Skolem machine. A subdisjunction Q0of Qis the tree wff for this tree. According to the
Proposition, Q0, and hence Q, is a logical consequence of the axioms of the theory. 2
It is worth noting, in regards to the Proposition, that well-formed formulas constructed by (for ex-
ample) existentially quantifying eigenvariables after disjoining branch facts, ∃(− | − | . . . | −), are
J. Fisher, M. Bezem /Skolem Machines 9
also logical consequences of the axioms of the theory. These consequences constitute possible answers
to the query. These wffs may indeed be stronger consequences, but they do not have the geometric (or
coherent) form that queries have.
The reference [8] defines complete Geolog trees and uses them to prove Theorem 2. Roughly speak-
ing, complete trees require that all applicable rules be used to expand trees in stages, and this is the topic
of the next section.
4. Complete Geolog Trees
To motivate the general definitions, consider first the following simple Geolog theory, G1.
true => a | b . % #1
true => c, d . % #2
a => goal . % #3
b, c => e . % #4
e, d => false . % #5
For the definition of a Geolog trees we consider the Geolog theory itself to be an ordered sequence of
Geolog rules. Reference will be made to the rules of theory G1using their serial order (display notation:
#n).The order will turn out to be irrelevant to the branch sets defined by the branches in these trees, and
the branch sets will be the important semantic objects: They will be partial logical models (or possibly
counter-models).
A complete Geolog tree of level 0, for any ordered Geolog theory, consists of just the root node true.
The level 0 tree is, obviously, independent of the rule order. Figure 9 shows the complete Geolog tree of
level 1 for the ordered theory G1.
Figure 9. Complete tree for G1, level 1
The root of any Geolog tree is the unique atom true, which is the complete Geolog tree of level 0.
The complete level 1 tree expands (and extends) the level 0 tree.
The first applicable rule for level 1 in our example is #1, and this constructs two branches for the
growing tree. The second applicable rule (#2) adds elements to the growing tree along both branches
because true is an ancestor for both branches. Notice that the consequents maintain a similar order of
appearance (specifically, top-down) in the tree, as they appear in the consequence of rule #2 (specifically,
left-to-right).
10 J. Fisher, M. Bezem /Skolem Machines
At level 2, rule #3 applies to the left branch of the complete tree for level 1, and rule #4 applies to the
right branch, so a graphical depiction of the complete Geolog tree for G1for level 2 is given in Figure
10.
Figure 10. Complete tree for G1, level 2
Finally, at level 3, rule #5 applies to the right branch in Figure 10, as shown in Figure 11. At this
stage, level #3, the tree is saturated because each branch contains either goal or false .
Figure 11. Complete tree for G1, level 3, with levels marked
Now let us suppose that the rules in the theory G1are reordered, for example
true => c, d . % #2
true => a | b . % #1
b, c => e . % #4
e, d => false . % #5
a => goal . % #3
In this case the complete Geolog trees of levels 0, 1, 2, and 3 could be depicted as shown in Figure 12.
J. Fisher, M. Bezem /Skolem Machines 11
Figure 12. Complete tree rules reordered, level 3, with levels marked
Notice that the level branch sets are the same. A branch set for level kconsists of the set of all facts
on a branch of the complete level ktree from the root of the tree down to the leaf of the branch. The
branch sets for either tree, Figure 11 or 12 are
level 0: {true}
level 1: {a,c,d}, {b,c,d}
level 2: {a,c,d,goal}, {b,c,d,e}
level 3: {a,c,d,goal}, {b,c,d,e,false}
The query for theory G1is Q=a|d, e and the level 3 branch sets also represent successfully halted
tapes for a Skolem machine for G1.
Another example is afforded by the following Geolog theory, G2.G2does not support its query.
Figure 13 shows some of the complete trees for G2.
true => p(a) . % #1
p(X) => q(f(X)) | p(f(X)) . % #2
q(X) => goal . % #3
For G2the complete trees are unbounded, meaning simply that the number of nodes in the tree
grows without bound as the level increases. A corresponding Skolem machine would have an unbounded
number of possible tapes.
More formally, suppose that Gis an arbitrary Geolog theory. We define a complete Geolog tree for
Gof level kby induction on k. The unique complete Geolog tree T0of level 0 for Gis just the root tree,
already describe. The single branch set for T0is {true}. Suppose that Tkis the complete Geolog tree
for Gof level khaving branch sets Bi. It is assumed that any branch of Tkwhich contains either goal
or false has that node as a leaf of the branch. Then Tk+1 is defined as follows. The branches having leaf
goal of false are not extended; they are considered to be saturated. For any branch Bof Tknot having
leaf goal nor false let us assume that r1,r2, ...,rzis a complete ordered list of all possible applicable
instances of geolog rules which are not already satisfied on B. We assume that the specific order is
determined by the order that the rules are given in G. These ground instances may have arisen from
12 J. Fisher, M. Bezem /Skolem Machines
Figure 13. The infinite tree for G2...
the same or from different rules, but there are only finitely many such instances. Use r1to extend Bin
the same way as if Bwere a corresponding Skolem machine tape, as described in the previous section.
However, if r1is a splitting rule, then split the branch Bof Tkrather than reproduce the tape Band then
extend the copies. (If r1is not a splitting rule then Bhas a unique extension.) Assume that this produces
mbranches B1, ..., Bm, as shown in Figure 14. (Figure 8 uses the same graphic, but the tree in Figure
8 was not required to be complete. In the earlier figure, krepresents the kth rule application, and in the
current figure, krepresents a “level” for possibly many rule applications.)
Figure 14. Expanding branch Bof complete tree using first applicable rule
If any of the extended branches Bjhas leaf goal of false , that branch is considered to be saturated,
and it is not extended (or expanded) any further. Continuing, we now apply r2to each of the new branches
not having leaf goal of false, then r3to the resulting branches, until all of the rules . . .rzhave been used
to expand all of the previous branches not having leaf goal of false, using the process described for r1,
J. Fisher, M. Bezem /Skolem Machines 13
corresponding to Figure 8. The tree Tk+1 is the result of this double induction for all branches Bof Tk
and all resulting applicable rules for each B(but never expand leaf goal of false).
Theorem 2. Suppose that Qis the query for Geolog theory Gand that Qis a logical consequence
of G. Then Gsupports Q.
Proof Sketch. The collection of all complete Geolog trees Tkfor k= 0,1,2, . . . defines a (possibly
infinite) tree T. Each node in Thas finitely many children. Branch sets correspond to Herbrand models
(closed term models) in the usual sense [9], but with the Herbrand basis based on the signature plus the
generated constants. Note that, by construction, for each branch set Bof Tk, any false instance of any
rule is applicable and hence satisfied in all extensions of Bin Tk+1.
If Tis a finite tree and some branch set Bdoes not satisfy any of the disjuncts of Qthen Bwould
satisfy the axioms of Gbut not Q. Since Qis a logical consequence of the axioms of Gthis case is
not possible and so if Tis finite then the branch sets of Tcorrespond to a successfully halting Skolem
machine and so Gsupports Q.
If Tis infinite then, by K¨
onig’s lemma [11], Thas an infinite branch. If none of the branch sets
corresponding to this infinite branch satisfies any disjunct of Qthen the set of nodes on this branch is a
counter model. Since Qis a logical consequence of the axioms of Gthis case is not possible. Thus Tis
in fact a finite tree, and every disjunct of Qis satisfied on one of the branch sets of T.2
5. A Universal Skolem Machine
A machine Uis universal for a class Cof machines if for every M∈ C and every input Ifor M, when
given input dMe,dIe, it mimicks the behaviour of Mwhen given input I. Here ‘mimicks’ means in
particular that Uwith input dMe,dIeterminates if and only if Mwith input Idoes so. Moreover, it is
required that if Mwith input Ireturns output O, then Uwith input dMe,dIereturns output dOe. (We
use dMe,dIe,dOeto stress the difference between M, I , O and their representation in the format that U
uses.)
Of course the most famous universal machine is the one for the class of Turing machines, used by
Turing to prove the undecidability of the halting problem. If the class Cof machines is Turing-complete,
then the halting problem for any machine that is universal for Cis undecidable. Therfore the construction
of a universal machine is still important, theoretically as well as in practice, where the concept of a
universal machine plays a role as a so-called interpreter.
In this section we will construct a universal Skolem machine (USM). It would be possible, analo-
gous to what Turing did, to construct a Skolem machine which is universal for the class of all Skolem
machines, but this would involve very many technical details. In order to minimize the amount of detail
we take as class Cthe class of so-called 2-counter machines which is known to be Turing-complete [13].
Using a USM (designed below) for Cone immediately infers that Skolem machines have the same com-
putational power as Turing machines and that, consequently, even tiny fragments of geometric logic are
undecidable.
A counter machine is a device with counters x1, . . . , xm, each capable of storing an arbitrarily large
natural number, together with a program. A counter machine program is a finite enumeration of instruc-
tions from the following instruction set: inc(xi), dec(xi), jpz(xi, l, l0). These instructions lead to the
14 J. Fisher, M. Bezem /Skolem Machines
following respective actions: increment counter xi,decrement counter xi, jump to instruction lif xiis
zero and to l0otherwise. Decrementing a counter which has value 0is not allowed and can be prevented
by using conditional jumps preceeding any decrement instruction.
The execution model for counter machines uses one additional counter, the so-called program counter,
which references the current instruction. Execution of the program starts at the first instruction. The pro-
gram counter is incremented after each instruction inc(xi), dec(xi), its value is changed to either lor l0
in case of a conditional jump. Execution terminates when the program counter gets a value not corre-
sponding to an instruction of the program.
In [13], Minsky proved that this simple machine model is already Turing-complete when only two
counters are used. As an example consider the following program:
1jpz(x2,5,2)
2dec(x2)
3inc(x1)
4jpz(x1,1,1)
This program obviously adds the contents of x2to x1and terminates by jumping to 5, beyond the last
instruction. Instruction 4exhibits an unconditional jump. (The program would also work correctly with
instructions 4jpz(x13,1,1) or 4jpz(x2,13,2) instead of 4jpz(x1,1,1).)
AGeolog theory that is universal for any 2-counter program is displayed in Figure 15. It is important
to observe that there are no function symbols in the theory so Skolem Machines are universal without
function symbols (unlike Datalog programs which are not universal, but are universal if one is allowed the
use of but a single function symbol.) Intuitively the justification for this is that Geolog allows existential
quantification in the consequent of a rule. Notice that there is exactly one existential quantification in the
theory of Figure 15, in the very last rule which is used to generate the natural numbers!
6. Implementation Design
In definite logic programming theory, SLD deduction provides top-down derivations of top-level goals.
(Select goal - use Linear resolution - for Definite rules). See [12] for concise characterization of SLD
deduction for definite logic programs.
For Skolem machine operations on a Geolog theory, the STG acronym has meanings reminiscent of
those SLD deduction for definite logic, but with significant modifications.
STG deductions for Geometric logic theories provide bottom-up derivations from bottom-level facts,
but the STG operations build Geolog trees top-down, using the operations of a Skolem machine.
•S - Select an applicable rule instance (A single rule can have multiple applicable instances.)
•T - use facts on the current branch of the Geolog tree and extend until saturation
•G - using geometric logic rules
J. Fisher, M. Bezem /Skolem Machines 15
STG deductions grow Geolog trees as described in Section 3. Facts on the current branch are used
to create applicable instances of rules. The first disjunct of the consequent of the selected rule is used
to extend the tree using the leftmost disjunct, and save the other disjuncts for subsequent branching only
after the current branch becomes saturated. If the current branch is saturated and has a goal or false leaf,
then branch the tree at the deepest remaining branch point above the leaf, continuing to grow the new
branch using the next disjunct not previously used.
Figures 16-19 illustrates a step-by-step STG deduction for the sample theory of Figure 1. The steps
in the STG deduction correspond in a natural way to the steps used in the Skolem machine operations in
Section 2. The facts in the trees with the red dashed outline indicate the current focus of the deduction.
Figures17(f) and 18(j) illustrate later branching after completing a branch corresponding to a previous
disjunct in the consequent of a rule. The focus is an internal (non-leaf) node in these cases.
Provided that we can use arbitrary rule application instances, STG deduction is correct and complete.
Theorem 3. Suppose that Geolog theory Gsupports its query Qwith tree T. Then there is an STG
deduction which builds T.
Proof. A inorder traversal of T will construct a selection function for which particular rule applications
to make in order to reproduce the same tree with an STG deduction. 2
We do not expand upon the topic in this paper, but an important concept is the extraction of answers
from a successful STG deduction.
There is, of course, no effective algorithm for computing effective rule instance selection functions
in general.
One selection strategy that is easy to implement is the first applicable rule instance selection function:
Find the first applicable rule, in their given order, whose antecedent can be matched against facts on the
current branch of the tree, from the top down, and whose consequent is not already satisfied using the
matching bindings for the antecedent.
Implementations of STG with the first selection function, using Prolog procedures can be very
straightforward. One can rely of Prolog’s unification for matching facts, and Prolog’s backtracking for
scheduling rule applications.
Each of the Geolog rules in the instruction set is translated into a special kind of Prolog clause. The
implementation that we illustrate is called the Skolem Abstract Machine or SAM for short. The reason
for this name is that the clauses resemble the procedures of the Warren Abstract Machine (WAM), which
is used as a basis for most of the efficient implementations of Prolog itself. In particular, each procedure
tries to match bindings for variables in terms. For the SAM procedures, however, the terms can be in
different states (multiple tapes, different tree branches).
The outline given in the remainder of this section can serve as motivation for lower-level implemen-
tations of the SAM, and for modifications to the selection function.
Some cogent references for the machine model and implementation of the WAM are [1] and [18].
The Prolog translator is basically a one-line program that mimics the STG operations, using the first
selection function. The translate rule has the profile shown in Figure 20.
Figure 21 shows the full Prolog code for translating a Geolog rule.
For example, consider Geolog rule #2 from the sample theory in Figure 1, repeated in Figure 22.
First, any existential variables in the consequent are separated and flagged, as shown in Figure 23, and
then the translated Prolog clause is displayed in Figure 24.
16 J. Fisher, M. Bezem /Skolem Machines
The code for the translator in Figure 21 mimics the definition of how STG deduction applies would
use a first applicable rule instances to extend tree branches. The Prolog code implements the tree branch
using a Prolog list, and terms are added to the beginning of the list (end of the branch). (Faster imple-
mentations use memoing rather than lists, but the code presented here may be easier to understand.)
Each of the try clauses describes how to try to extend a tape using the corresponding Geolog rule.
The Geolog rules are translated into Prolog clauses in the order in which they appear in the Geolog
instruction sequence. Figure 25 has an outline for all of the try clauses, showing the order in which they
are asserted to memory (and compiled).
In the SWI-Prolog [19] implementation of the SAM, after the Geolog rules are read from file and
translated into Prolog clauses, the Prolog clauses are asserted and then compiled into internal procedures
for the underlying Prolog machine.
The remaining small amount of code for applying Geolog rules, expanding, and extending tapes
(states) is given in the reference [6]. The Prolog interpreter has filename geoprolog.pl. The reference
also provides a user guide and numerous sample Geolog theories to compute.
As it is for Prolog, the ordering of Geolog instructions becomes important for the first applicable rule
selection function implementation of SAM.
As emphasized in [3], it is often best to sequence the instructions so that splitting rules and rules
introducing existential quantifiers are placed at the end of the rulebase, such as the rules 12 and 13 in
Figure 26. Moving these rules higher up in the list is inhibitive for computing the query depth-first.
Using the first selection function and given ordering one produces the proof displayed in Figure 27. (An
interesting detail is that rule 9 is not necessary for proof, but deleting it makes for a longer proof, 111
steps.)
In the presence of function symbols even more trivial examples can be given, such as the wrong order
of the last two of the following rules:
true => p(a).
p(X) => p(f(X)). %alternatively: p(X) => succ(X,Y),p(Y).
p(X) => goal.
7. Conflicted Geolog Theories
Recall that we can write a Geolog theory in the form
T=A∪G∪F(8)
where Ais the axioms,Gcontains all of the goal rules and Fcontains all of the false rules.
Also recall that he geolog query Qfor a Geolog theory T=A∪G∪Fis the disjunctive normal
form Q=C1|C2|. . . |Ckwhere the Ciare the existential closures of the antecedents of all the rules
in Gand F. A Geolog theory Tsupports its query Qprovided there is a successfully halted Skolem
logic machine such that each tape of the halted machine satisfies at least one of the Ci.
Notice that any theory lacking both goal and false rules cannot support its empty query, since there
is no way for a Skolem machine to successfully halt.
We say that a theory T=A∪G∪Fis conflicted provided that G6=∅,Tsupports its query, and the
theory T−G=A∪Falso supports its query. If T−Gsupports its query, then necessarily F6=∅.
J. Fisher, M. Bezem /Skolem Machines 17
Observation. If T=A∪G∪Fis conflicted then T’s query is not a minimal logical consequence
of A.
To say that Q=C1|C2|. . . |Ckis not a minimal logical consequence of Ameans that there is
a properly smaller disjunction that is also a logical consequence. A simple example would be the theory
T:
true => a,b. % A
a => goal. % G
b => false. % F
Here we have that T’s query Q=a|b, and both aand bare logical consequences of A. Notice that
T−Galso supports its query Q0=b, and so Qis not a minimal consequence of the axioms of T.
The converse of the theorem is not true. For example, consider the previous theory with false
replaced by goal.Q=a|bis not a minimal consequence but T−Ghas no goal or false rules, so
cannot support its empty query. That is, both aand bare logical consequences of the axioms, so a|bis
not a minimal logical consequence, but the theory is not conflicted. So aand bwould be better answers
as logical consequences of the theory.
By convention, goal is often thought of as ”affirming” and false as ”rejecting”, so the definition of
conflicted favors goal as the ”positive” concept.
The theory given in Figure 26 is an example where goal affirms and false rejects. This theory
supports its query Q=e(b, c)|(∃z)r(b, z)|(∃z)r(c, z)but T−Gdoes not support its query Q0=
(∃z)r(b, z)|(∃z)r(c, z). Thus, the theory is not conflicted. The programmer’s intention was to show
that
¬(∃z)r(b, z)∧ ¬(∃z)r(c, z)→e(b, c)(9)
and to disprove the negation of the antecedent:
(∃z)r(b, z)∨(∃z)r(c, z)(10)
This is indeed the case. Figure 27 establishes (9). It can be shown additionally that T−Gdoes not
support its query (10), and that T−Ghas a finite counter-model. There is, in fact, a Skolem machine
computation for T−Gwith saturated tape not containing false which was automatically verified by the
implementation similar to the one described in Section 6). Figure 28 illustrates this by displaying the
rest of the facts in a counter-model, when the goal rule is removed from the theory. Explicitly stated, the
counter-model consists of all the facts from 0 to 24 on the left-most branch of Figure 27 together with
the saturating facts 25 through 36 shown in Figure 28.
8. Conclusions
This main intent of this paper was to supply essential definitions and theorems for the concept of a
Skolem machine, and to explain some of the historical connections for the concept. For this we have
defined a simple logical input language, Geolog, for Skolem machines, and provided the basic operations
18 J. Fisher, M. Bezem /Skolem Machines
of the logical machine using the Geolog rules or instructions. A modest definition of a query, defined
solely in terms of the input theory, has been provided, and essential theorems covering correctness and
completeness of Skolem machine operations (with regard to computing the query) are proved in some
detail. We show how logical correctness follows from a detailed analysis of what we call Geolog trees and
how logical completenesss follows from a detailed analysis of complete Geolog trees. This particular tree
analysis has promise for further study regarding answer extraction and proof strategies. Stronger logical
results (not just for the restricted query) have either been stated (as in the case of correctness) or outlined
for further development.
The universality of Skolem machine calculations has been established using arguments based upon
the direct simulation of a universal 2-counter register machine. The arguments established the interesting
result that no function symbols are need for universality.
On the implementation side, we have provided a depth-first deduction procedure, called STG deduc-
tion, which is pseudo-complete (up to selection or rule choice function), and we have described a simple,
but effective translation into Prolog using the so called first choice selection function, or strategy.
And finally, the paper provides a justification for using, or allowing, two kinds of terminal conclu-
sions for Geolog rules: goal or false, and explains that the distinction is merely a handy formalism that
is useful for extending the meaning of Skolem machine computations.
The next steps in this work involve more efficient and effective implementation of theorem provers
or model checkers whose underlying computational mechanism is the Skolem machine. This may even-
tually involve extensions to the Geolog language.
Much more might be said regarding historical connections between what we call Skolem Machines
and the work of Thoralf Skolem. These connections are not pursued here. A good perspective regarding
Skolem’s influence on logic is the reference [14].
References
[1] H. Ait-Kaci, Warrens’s Abstract Machine, A Tutorial Reconstruction, School of Computing Science, February
18, 1999.
[2] M. Bezem and T. Coquand. Newman’s Lemma – a Case Study in Proof Automation and Geometric Logic.
In Y. Gurevich, editor, The Logic in Computer Science Column, Bulletin of the European Association for
Theoretical Computer Science 79:86–100, February 2003. Also in G. Paun, G. Rozenberg and A. Salomaa,
editors, Current trends in Theoretical Computer Science, Volume 2, pp. 267–282, World Scientific, Singapore,
2004.
[3] M.A. Bezem and T. Coquand, Automating Coherent Logic. In G. Sutcliffe and A. Voronkov, editors, Pro-
ceedings LPAR-12, LNCS 3835, pages 246–260, Springer-Verlag, Berlin, 2005.
[4] Marc Bezem. On the Undecidability of Coherent Logic. In Aart Middeldorp e.a., editors, Processes, Terms
and Cycles: Steps on the Road to Infinity, LNCS 3838, pages 6–13, Springer-Verlag, Berlin, 2005.
[5] A. Blass, Topoi and computation. Bulletin of the EATCS 36:57–65, October 1998.
[6] Geolog website: http://www.csupomona.edu/~jrfisher/www/geolog
[7] John Fisher and Marc Bezem, Skolem Machines and Geometric Logic. In C.B. Jones, Z. Liu and J. Woodcock,
Proc. ICTAC 2007 The 4th International Colloquium on Theoretical Aspects of Computing, Macao SAR, China,
September 26-28, 2007. Springer LNCS vol. 4711, pp. 201-215.
J. Fisher, M. Bezem /Skolem Machines 19
[8] John Fisher and Marc Bezem, Query Completeness of Skolem Machine Computations. In J. Durand-Los´
e and
M. Margenstern, editors, Proc. Machines, Computations and Universality ’07, Universite d’Orleans - LIFO,
Orleans, France September 10-14, 2007. Springer LNCS vol. 4664, pp. 182-192.
[9] Jacques Herbrand, Logical Writings, edited by Warren D. Goldfarb, D. Reidel Publishing Company, Springer
edition, 2006.
[10] P. Johnstone, Sketches of an Elephant: a topos theory compendium, Volume 2, Oxford Logic Guides 44,
OUP, 2002.
[11] D´
enes K¨
onig, Theorie der endlichen und unendlichen Graphen, Akademische Verlagsgesellschaft, Leipzig,
1936. Translated from German by Richard McCoart, Theory of finite and infinite graphs, Birkhauser, 1990.
[12] J.W. Lloyd, Foundations of Logic Programming, Springer-Verlag, Berlin, revised edition 1987.
[13] Marvin L. Minsky, Recursive unsolvability of Post’s problem of ‘tag’ and other topics in theory of Turing
machines, Annals of Mathematics 74(3):437–455, 1961.
[14] Nordic Journal of philosophical Logic 1(2) December 1996. Special issue devoted to the influence on logic
of Thoralf Skolem.
[15] R. Manthey and F. Bry, SATCHMO: A Theorem Prover Implemented in Prolog. In: E. Lusk and R. Overbeek,
editors, Proceedings CADE-9, LNCS 310, pages 415-434, Springer-Verlag, Berlin, 1988.
[16] H. de Nivelle and J. Meng, Geometric Resolution: A Proof Procedure Based on Finite Model Search. In:
U. Furbach and N. Shankar, editors, Automated Reasoning, Proceedings IJCAR 2006, LNCS 4130, pages 303–
317, Springer-Verlag, Berlin, 2006.
[17] Thoralf Skolem, Logisch-kombinatorische Untersuchungen ¨
uber die Erf¨
ullbarkeit und Beweisbarkeit math-
ematischen S¨
atze nebst einem Theoreme ¨
uber dichte Mengen, Skrifter I 4:1–36, Det Norske Videnskaps-
Akademi, 1920. Also in Jens Erik Fenstad, editor, Selected Works in Logic by Th. Skolem, pp. 103–136,
Universitetsforlaget, Oslo, 1970.
[18] D.H.D. Warren, Implementation of Prolog. Lecture notes of Tutorial No. 3, 5th International Conference and
Symposium on Logic Programming, Seattle, WA, August 1988.
[19] J. Wielemaker, SWI-Prolog Reference Manual. Link available at: www.swi-prolog.org
20 J. Fisher, M. Bezem /Skolem Machines
%% 2-Counter machine
%% Example: R1 + R2 -> R1
%% 1: jpz(R2,5,2)
%% 2: dec(R2)
%% 3: inc(R1)
%% 4: goto(1)
%% 5: halt
%%
true =>
% program
instruction(1,jpz,2,5,2),
instruction(2,dec,2,3,x),
instruction(3,inc,1,4,x),
instruction(4,goto,1,x,x),
instruction(5,halt,x,x,x),
% data
inc(0,1), inc(1,2), inc(2,3), inc(3,4),
state(1,3,4). // START compute 3 + 4
%% UNIVERSAL INTERPRETATION OF REGISTER INSTRUCTIONS
%-- jpz
state(PC,0,R2), instruction(PC,jpz,1,PA,PB) => state(PA,0,R2).
state(PC,R1,R2), inc(_X,R1), % R1 is NOT zero
instruction(PC,jpz,1,PA,PB) => state(PB,R1,R2).
state(PC,R1,0), instruction(PC,jpz,2,PA,PB) => state(PA,R1,0).
state(PC,R1,R2), inc(_,R2), % R2 is NOT zero
instruction(PC,jpz,2,PA,PB) => state(PB,R1,R2).
%-- dec
state(PC,R1,R2), instruction(PC,dec,1,PB,x), inc(D,R1) => state(PB,D,R2).
state(PC,R1,R2), instruction(PC,dec,2,PB,x), inc(D,R2) => state(PB,R1,D).
%-- inc
state(PC,R1,R2), instruction(PC,inc,1,PB,x), inc(R1,I) => state(PB,I,R2).
state(PC,R1,R2), instruction(PC,inc,2,PB,x), inc(R2,I) => state(PB,R1,I).
%-- goto
state(PC,R1,R2), instruction(PC,goto,PB,x,x) => state(PB,R1,R2).
%-- halt
state(PC,R1,R2), instruction(PC,halt,x,x,x) => goal.
%% Natural number generation via inc
inc(X,Y) => inc(Y,SomeZ).
Figure 15. A Universal Skolem Machine
J. Fisher, M. Bezem /Skolem Machines 21
Figure 16. STG (a,b,c,d)
Figure 17. STG (e,f,g,h)
Figure 18. STG (i,j,k,l)
22 J. Fisher, M. Bezem /Skolem Machines
Figure 19. STG (m,n)
translate(+GeologRuleIn, -PrologRuleOut)
Figure 20. The intended translation, Geolog rule to Prolog clause
%%%%%%%%%%%%%%%%%%%%
%% Geolog Translator
%%%%%%%%%%%%%%%%%%%%
translate((ANT => CONS) , % to the following Prolog clause ...
(try((ANT => CONS),Branch) :-
satisfy(ANT,Branch),
\+satisfied(CONS,Branch),
cases(CONS,[F|R]),
extend(F, Branch,FBranch),
try(_,FBranch), % try again
continue(Branch,R) ) ) . % other cases, if any
Figure 21. Translating Geolog rules to Prolog clauses
p(X) => q(X) | r(X) | domain(Y), s(X,Y).
Figure 22. +GeologRuleIn, sample input term
p(X) => q(X) | r(X) | Y^(domain(Y), s(X,Y)).
Figure 23. +GeologRuleIn, flag existential variable
J. Fisher, M. Bezem /Skolem Machines 23
try((p(A)=>q(A)|r(A)|B^ (domain(B), s(A, B))), C) :-
satisfy(p(A), C),
\+satisfied((q(A)|r(A)|B^ (domain(B), s(A, B))), C),
cases((q(A)|r(A)|B^ (domain(B), s(A, B))), [E|H]),
extend(E, C, F),
try(_, F),
continue(C, H).
Figure 24. -PrologRuleOut, sample output term
% START with initial state
try :- try(_,[true]).
% test for goal on tape (or \f )
try(_,S) :-
member(goal,S), member(false,S), !.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Translated Geolog clauses asserted
%%% here, in user-specified order.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% last clause, must have stuck tape
try(_,S) :-
write(’counter_model(’),
write(S),
writeln(’).’)
Figure 25. Order for the translated Prolog clauses
24 J. Fisher, M. Bezem /Skolem Machines
true => domain(a), domain(b), domain(c). %1 domain elements a,b,c
e(b,c) => goal. %2 the goal is to prove b=c
r(b,Z) => false. %3 for normal form b
r(c,Z) => false. %4 and normal form c
true => s(a,b),s(a,c). %5 both reducts of a
domain(X) => e(X,X). %6 reflexivity of e
e(X,Y) => e(Y,X). %7 symmetry of e
e(X,Y),e(Y,Z) => e(X,Z). %8 transitivity of e
e(X,Y),r(Y,Z) => r(X,Z) . %9 r contains e and r
e(X,Y) => s(X,Y). %10 s contains e
r(X,Y) => s(X,Y). %11 and r,
s(X,Y),s(Y,Z) => s(X,Z). %12 is transitive,
s(X,Y),s(X,Z) => domain(U),s(Y,U),s(Z,U). %13 satisfies diamond, and
s(X,Y) => e(X,Y)|domain(Z),r(X,Z),s(Z,Y). %14 is included in e + r.s
Figure 26. A Geolog theory expressing that confluence of a rewrite relation rimplies uniqueness of normal forms
J. Fisher, M. Bezem /Skolem Machines 25
Figure 27. A proof tree for theory in Figure 26
26 J. Fisher, M. Bezem /Skolem Machines
Figure 28. Generating remainder of counter-model for theory of Figure 26 without goal rule
