Extensions in graph normal form

Graph normal form, introduced earlier for propositional logic, is shown to be a normal form also for first-order logic. It allows to view syntax of theories as digraphs, while their semantics as kernels of these digraphs. Graphs are particularly well suited for studying circularity, and we provide some general means for verifying that circular or apparently circular extensions are conservative. Traditional syntactic means of ensuring conservativity, like definitional extensions or positive occurrences guaranteeing exsitence of fixed points, emerge as special cases.

The remaining propositional matrix, if nonatomic, is rewritten using only ¬ and ∧ and processed in an analogous manner, introducing new predicates. The following two definitions give the general construction of GNF − (φ) for an arbitrary formula φ. The reader satisfied with the example can skip them on the first reading and continue now after Definition 1.4.
Function _ _ in Definition 1.3 transforms quantifier prefix of a PNF formula into GNF, sending the quantifier-free matrix for further processing by function pGNF − from Definition 1.4. The numerical argument i in _ i , counting the number of generated symbols, ensures that all introduced predicate symbols are distinct. All ∃x in the initial formula are assumed replaced by ¬∀x¬. A block of quantifiers ∀x 1 ...∀x n not separated by ¬ is abbreviated below as a single quantifier ∀x, with x abbreviating the sequence x 1 ...x n . In the generated formulas, x 1 ... x n replace then x .
The two main cases in Definition 1.3 correspond to φ starting with ¬∀ (1 and 3) or ∀ (2 and 4), each having two subcases, when φ is open (o) or closed (c). Cases (o) show also how GNF − (φ) can be constructed for open φ. (Alternatively, one could construct it for φ's universal closure.) One starts typically with case 1 or 2, then performs step 2o until reaching the last quantifier, to which equation 2o, 3o or 4o is applied. Point 5 sends the remaining quantifier-free matrix ρ to pGNF − ρ j given in Definition 1.4. In case of only one quantifier, there is only one step, which may be any of 1 through 4. DEFINITION 1.3 For a formula φ(X ) in PNF, with X = V(φ), we define GNF − (φ) = φ(X ) 1 , with the recursive function _ _ given by (z is a fresh free variable): when quantifier-free ρ does not start with ¬ : 3o. ¬∀xρ(X , when ρ is quantifier-free, continue with Definition1. 4: 5.
For instance: top predicate of GNF − (φ). For a theory Γ , its GNF − (Γ ) is obtained as the union of GNF − (φ), for each φ ∈ Γ , where distinct GNF − (φ) introduce distinct predicate symbols. GNF − (Γ ) is a definitional extension of Γ , in the following sense. An explicit definition, in an FOL language L, is Bx ↔ φ, where predicate B ∈ L and φ is an L-formula with free variables V(φ) ⊆ x. A definitional extension of (the language of) Γ is a sequence D i i<λ , for an ordinal λ, where (i) D 0 is an explicit definition in the language of Γ , (ii) for j = i + 1 ≤ λ, D j is an explicit definition in the language of all D k , k < j and (iii) D l = i<l D i for limits l ≤ λ. A set of formulas is a definitional extension of Γ if it can be well-ordered into one, and is a definitional extension simpliciter, if it is a definitional extension of some language. PROOF. We view the process from Definitions 1.3 and 1.4 bottom-up, so that for j > i, B j is introduced before B i .
Step i adds the explicit definition x B i (x) ↔ y ¬B i+1 (x, y) of fresh predicate B i in the language of the previous stage i + 1 (containing, in addition to the original symbols, the predicate symbols B j for j > i). Thus, GNF − (A) is a definitional extension of A, for every A ∈ Γ . This yields the claim for finite Γ . If Γ is infinite, then we well-order Γ using axiom of choice. Since distinct GNF − (A i ) introduce distinct predicates, the resulting well-founded chain of explicit definitions is a definitional extension of Γ . Now, theory GNF − (φ) is satisfiable even if φ is not. To obtain equisatisfiability of φ and its GNF, we add one more formula. (1.7) For a theory Γ , GNF(Γ ) = φ∈Γ GNF(φ), where distinct GNF(φ) introduce distinct symbols.
Although GNF(Γ ) is no longer a definitional extension of Γ , due to formula (1.7), it shares its essential feature: every model of Γ has a unique expansion to a model of GNF(Γ ), because for any structure M : M | A z ↔ (¬Az ∧ ¬A z) if and only if M | Az. On the other hand, reduct of every model of GNF(Γ ), forgetting A , is a model of Γ . This gives the following fact, where X Y denotes a bijection. (We show existence of injections X → Y and Y → X , also when X , Y are classes, assuming their theory with Schröder-Bernstein theorem, e.g. NGB.) GNF is thus a normal form for FOL, and we study only theories in GNF. Theory GNF(Γ ), given by Definitions 1.3-1.6, is only one possible GNF for Γ respecting the equivalence from the fact above, and we will occasionally use other, simpler forms.

Graphs as syntax
The syntax of a GNF theory can be represented by a graph, while the semantics amounts to specific properties of this syntax graph. By a 'graph' we mean directed graph, namely, a pair This notation is extended to sets, e.g.
One proviso is needed. Definition 1.1 admits in a GNF theory several equivalences with the same predicate in LSs, say, Pa ↔ ... and Pb ↔ ..., with a, b being constants or even unifiable terms. Such general cases are treated in Section 2.2. Until then, definitions, results and examples are formulated for FOL without equality or function symbols, FOL − , to convey the main ideas undisturbed by more detailed technicalities. Thus, T X = X and the only terms in LS of each GNF formula are variables, LS ∈ A X . We assume also that each predicate symbol P occurs at most once in LS of some formula of a theory, to which we refer as P's equivalence.
We let B i range over predicate symbols from the language L of a GNF theory Γ . Any formula from Γ (with the restrictions just mentioned) is represented schematically by the following pattern (where B may occur in RS as one of B 1 , ..., B n ): For a GNF theory Γ in FOL − and set D, the graph G = G D (Γ ) is given by: For each undefined predicate symbol P, we add a fresh symbol P and the 2-cycle Pd Pd for each d ∈ D ar(P) . Vertices Pd form A D .
A nonempty D is a domain of an FOL-structure interpreting the language L of the theory. We exclude the empty graph, G ∅ (Γ ) = (∅, ∅), from considerations, and denote by Gr(Γ ) the class of graphs G D (Γ ), for all nonempty sets D.
For each instance Bd, d ∈ D ar(B) , of an atomic axiom Bx (with no RS), point 2 gives the empty set of neighbours, including Bd in sinks(G) = {v ∈ V G | E G (v) = ∅}. As will be seen in Section 2.1, sinks are included in every kernel of a graph, which represents their valuation as true. The copies Pd, added along with the 2-cycles for each undefined Pd in point 3, ensure that such atoms are not sinks and can obtain arbitrary values, restricted only by the rest of the theory. Generally, in G D (Γ ), each vertex Ad, for d ∈ D, has edges to |D| copies of a subgraph with a source Sx and edges to Pxy, for all x, y ∈ D. Each pair Pxy, Pxy forms a 2-cycle.

Kernels as models
Given a GNF theory Γ , we denote its usual models by Mod(Γ ), while its models over a given set D by Mod D (Γ ). Graphs from Definition 2.2 provide an equivalent representation of these models.
Graph G D (Γ ) mixes syntax, using the predicate symbols from the language of Γ , with semantics, applying these symbols to the elements of the interpretation domain D. Typically, by 'atoms' we refer to such mixed expressions. One uses such a notational abbreviation when, for a formula φ(x), one writes φ(d) for some d ∈ D. It may denote (i) the formula φ(x) with a new constant-naming the object d-substituted for x, or else, (ii) the interpretation of the formula φ(x) under a valuation of variables assigning the object d to x. Vertices of our graphs come closest to (i)-formulas with names of the objects from D substituted for variables. They obtain truth values, becoming (ii), relatively to the solutions of the graph, which we now define.
A kernel (or a solution [4]) of a graph G is a subset A graph G is solvable when Ker(G) = ∅, where Ker(G) denotes the set of all kernels of G. Vertices of G D (Γ ) contain all D instances of all atomic formulas. An assignment v ∈ D V(φ) to free variables V(φ) of a formula φ, along with a kernel K, determine a valuation of all atoms over V(φ), with atoms in K assigned 1, as given in line 1 below. Satisfaction of arbitrary formulas is defined from this basis in the usual way, with the needed adjustments in the last two lines. (For a structure M (or kernel of a graph) over a set D, a formula φx and d ∈ D, we write M | φd if M satisfies φx with x assigned d.) Repeating the standard definition of satisfaction, this gives Γ | φ coinciding with the standard notion Mod(Γ ) | φ, provided that kernels of graphs in Gr(Γ ) correspond to Mod(Γ ). We establish this now, showing first that kernels of and formula φ of the language of Γ .
PROOF. An injection kr : Mod D (Γ ) → Ker(G D (Γ )) is obtained as follows. Let D + be a model of Γ over a set D, G = G D (Γ ) be as in Definition 2.2, and kr(D where A D is as in Definition 2.2.3. If Bd ∈ K, for Bd instantiating LS Bx of some axiom (2.1), say F, then D + | Bd and D + | ¬Bd. Since D + satisfies F, for some B i xy in its RS and some c ∈ D ar (B i Since for each atom Bd, D + | Bd ⇔ K | Bd, this equivalence extends to arbitrary formula φ. Thus, M satisfies also the right to left implication of B's (2.1), and we conclude that M | Γ .
Since for each atom Bd (also undefined by Γ ), md(K) | Bd ⇔ K | Bd, this equivalence extends to all formulas. Obviously, if two kernels K 1 , K 2 of G are different, then so are md(K 1 ) and md(K 2 ), giving different values to at least one atom Bd. Thus, md is injective.
One sees easily that kr and md are inverses of each other.
Fact 2.5, augmented by the extension from GNF − (Γ ) to GNF(Γ ) in Definition 1.6 and by Fact 1.9, yield the following correspondence between models of any FOL − theory Γ and kernels of graphs from Gr(Γ ). More precisely, a graph model consists of a pair (G, K) with G ∈ Gr(Γ ) and K ∈ Ker(G), and the class Mod

For an arbitrary GNF theory
PROOF.

By Fact 2.5, we have injections Mod D (Γ )
Ker(G D (Γ )) for each set D. This gives the obvious injections 2. By Fact 1.9, Mod(T) Mod(GNF(T)), so the claim follows by point 1.

A Γ with an uncountable model has a countable one, by Skolem-Löwenheim, so
Ker(G ω (Γ )) = ∅ by Fact 2.5. If Γ has a finite model, it also has an infinite one by a standard argument for FOL − . Conversely, if Ker(G ω (Γ )) = ∅, then Fact 2.5 gives a countable model of Γ .

FOL with function symbols and equality
This section shows that graph representation from Definition 2.2 can be generalized from FOL − to full FOL, with function symbols and equality, retaining Fact 2.7.1-2. Following sections, 2.3 and 3, can be read without absorbing the details of this section. Definitions and facts from Section 1 remain unchanged for FOL − with equality. The construction of GNF − (Γ ) follows Definitions 1.3 and 1.4, with equality treated as a binary predicate.
However, introduction of terms complicates the straightforward Definition 2.2 of a theory's graph. For the first, axioms may now have a more specific form than schema (2.1), with terms instead of variables. This general form is abbreviated as Its graph is constructed as in Definition 2.2, except that also equality is needed. Unlike the two formulas in (1), it forces Px = 0 for all x distinct from a and b. To handle situations like (1) we introduce, along with terms, T X = X , also the equality predicate eq(s, t), often abbreviated by eq st , with the standard axioms. Vertices of graph G D (Γ ) contain all atoms Btd ∈ A T D , which are partitioned into two sets. Ins contains all atoms Btd that result from substituting some d ∈ D ar(Bt) for all variables x in the LS Btx of some axiom (2.8). For the remaining atoms, we include dual vertices A brief explanation of the definition follows underneath.

DEFINITION 2.9
For Γ in FOL language L and a set D, the graph G = G D (Γ ) is given by: where Aux are auxiliary vertices used below. 2. For each pair of distinct terms, s, t ∈ T D , we form 2-cycle eq st eq st ; for each pair of distinct a, b ∈ D, we add a vertex with a loop and the edge: → eq(a, b) eq(a, b).
3. For each term t ∈ T D \ D (including constants), we form first the complete digraph C(t) 4. We add the standard equality axioms for eq(_, _), i.e. for all distinct p, q, r ∈ T D : (r) vertex eq pp is a sink -for ref lexivity, (s) the subgraph -for symmetry, (t) the subgraph -for transitivity.
(c) for each function f /predicate P with arity n and pairs of terms we add the congruence subgraph with the sources •/•: If t, s are single terms, edges going out of eq(t, s) can be replaced by the 2-cycle to eq(t, s). 5. For atomic axioms s = t or s = t in Γ , with s, t ∈ T X , we augment each instance sd, td ∈ T D of the 2-cycle from point 2 with a new vertex • with the loop and the edge: Auxiliary vertices Aux are all • and anonymous vertices in the indicated subgraphs. For each kernel K ∈ Ker(G), the subgraphs in respective points above ensure the following properties: 2. For distinct a, b ∈ D, eq(a, b) ∈ K, representing inequality. 3. Unique interpretation in D of every function application. With eq representing equality, these subgraphs ensure that each application of a function to arguments from D returns a unique element of D, in particular, that each constant is interpreted as some unique d ∈ D. This follows because, in a complete graph, the kernels are exactly individual vertices, so that each kernel of C(t) is exactly one eq(t, d). 4. Satisfaction of the standard equality axioms by eq(_, _). Equivalence is ensured by (r), (s) and (t), while in the subgraphs (c), vertex • captures the congruence axiom t = s → ft = fs, and vertex • its predicate version t = s → (Pt ↔ Ps). Vertices Pt, Ps initiate the subgraphs according to point 6. 5. Satisfaction of the atomic nonlogical (in)equality axioms. 6. Satisfaction of other nonlogical axioms (2.8). 7. If a predicate B is only partially defined (like in (1)), then each Bd ∈ Ins can be interpreted arbitrarily, provided eq(d, tc) ∈ K for each Btc ∈ Ins (and the chosen interpretation does not collide with other restrictions).
Saying below that something follows 'by subgraphs...' refers to the points above, applied to any kernel restricted to these subgraphs. For instance, if K ∈ Ker(G D (Γ )) and s = t ∈ Γ then, by subgraphs 5.(a), eq(sd, td) ∈ K for each d ∈ D ar(s,t) , because kernel requires exclusion of vertex • with the loop, which forces eq(sd, td) ∈ K.  eq(a, s) ∈ K then K | Pa ↔ Ps, while if eq(a, b) ∈ K then also K | ¬Qaa ↔ y,z ¬Rayz. But if {eq(d, a), eq(d, b)} ⊆ K, then by subgraphs 7.(b), either Pd ∈ K or Pd ∈ K.

EXAMPLE 2.11
For the axiom (P) Px ↔ ¬Psx, and the set with one element 0, terms T {0} are (isomorphic to) natural numbers and the graph becomes a ray P0 → Ps0 → Pss0 → ..., with subgraphs 2.9.4.(c) for each pair s n 0, s m 0, n = m. Since all these terms are interpreted identically over {0}, for any kernel K ∈ Ker(G {0} (P)), the subgraph 2.9.4.(c) with the source • for P0, Ps0 is such that eq(0, s0) ∈ K, while the edge P0 → Ps0 yields P0 ∈ K ⇔ Ps0 ∈ K: Consequently, this subgraph, and hence the whole graph, has no kernel, ref lecting the nonexistence of models of (P) over one element domain.
Taking as the underlying set the natural numbers N, with the standard interpretation of s as +1, the graph again becomes the ray P0 → P1 → P2 → .... But now no equality eq(p, q) holds except for eq(p, p). The instance of the subgraph above swaps 0 and 1, obtaining two kernels, with Ps n 0 for all even n ≥ 0, or for all odd n > 0.
The following extends Fact 2.5 to FOL, showing that G D (Γ ) captures all models of Γ over a set D.
The second summand is empty in case A D = ∅ and each Bd obtains a value relatively to its outneighbours E G (Bd) determined by the axioms.
Vertices included into kr(D + ) by inducing from K, but not mentioned in the definition of K above, are among the auxiliary vertices in graphs 2.9.4.(c). They do not affect the argument below, so we identify kr(D + ) = K.
i. Equality in D + is ref lected by eq in G. Since each term applied to elements of D yields a unique element of D, K determines a unique solution to each subgraph from point 3 of Definition 2.9 with eq(t, d) = 1 for d ∈ D interpreting the term t ∈ T D , i.e. D + | t = d, which induces 0 to • t . The last two summands of K determine unique solutions to all subgraphs from Definition 2.9.4: (r), since eq(p, p) is a sink, (s), since eq(p, q) ⇔ eq(q, p) both • vertices in subgraph (s) obtain induced value 0 and (t), since eq(p, g) and eq(q, r) imply eq(p, r), so • in subgraph (t) obtains induced value 0.
For (c), if eq(t, s) ∈ K, i.e. D + | t = s, then D + | ft = fs for each function f , and D + | Bte ↔ Bse for each predicate B and e ∈ D ar(Bt) = D ar (Bs) . Thus, if eq(t, s) ∈ K then eq(ft, fs) ∈ K, while Bte ∈ K ⇔ Bse ∈ K, and the subgraph from 4.(c) obtains a solution since inducing from these values ensures The last two summands of K give also unique solutions to the subgraphs from 2.9.5.
ii. Instances Bd ∈ Ins defined by (2.8) are treated as in the proof of Fact 2.5, with a small proviso. It may happen that Bd ∈ Ins, while for some axiom with Btx in LS, D + | d = tc, so D + | Bd ⇔ D + | Btc. Then eq(d, tc) ∈ K and Bd ∈ K ⇔ Btc ∈ K, so K solves the subgraph 2.9.4.(c) with Bd, Btc in place of Ps, Pt. We . This inclusion follows for vertices eq(_, _) and eq(_, _) by i (and for Aux by inducing from K), so we consider atoms A ∈ A T D with A ∈ K, i.e. D + | A.
Let A = Btc ∈ Ins, i.e. Btc is an instance of LS Btx of some axiom (2.8) F. Since D + | Btc, so D + | ¬Btc and D + | B i ce for some instance of some conjunct B i xy in RS of F. Hence B i ce ∈ K, and Btc ∈ E − G (K) since B i ce ∈ E G (Btc) by 2.9.6. If A = Bd ∈ Ins, then E G (Bd) = {Bd} by 2.9.7. Since D + | Bd, so Bd ∈ K and Bd ∈ E − G (K). These two cases, with point i, establish V G \ K ⊆ E − G (K). iii. For the opposite inclusion, assuming A ∈ E − G (K), there are two cases. If A = Btc ∈ Ins then Btc ∈ E − G (K) means that B i ce ∈ K for some instance B i ce of some B i xy in the RS of the axiom Btx ↔ ...B i xy. Then D + | B i ce, so D + | Btc and Btc ∈ K.
If A = Bd ∈ Ins then E G (Bd) = {Bd} by 2.9.7 and since Bd ∈ E − G (K), so Bd ∈ K, which means that D + | Bd, so that Bd ∈ K.
These two cases give iv. Since D + | Bd ⇔ K | Bd for each atom Bd ∈ A T D , and D + | s = t ⇔ K | eq(s, t) for s, t ∈ T D , the equivalence D + | φ ⇔ K | φ holds for arbitrary formula φ of the language of Γ . i. Given a K ∈ Ker(G), we note first that points 2 and 3 of Definition 2.9 ensure well-definedness of the function i : T D → D, given by i(t) = d, for d ∈ D such that eq(t, d) ∈ K. By 2.9.3 such a d is unique for each t ∈ T D , while by 2.9.2, the restriction i| D is the identity on D.
Subgraphs (r), (s) and (t) from 2.9.4 ensure that eq(_, _) is an equivalence on D, while (c) that eq(s, t) entails eq(fs, ft) for each function symbol f . Hence, i is a well-defined quotient mapping. For each atom Bd ∈ A T D , we define M | Bd iff Bd ∈ K. This is well-defined, since subgraphs 2.9.4.(c) ensure that for all t, s ∈ T D , if eq(t, s) ∈ K then Bt ∈ K ⇔ Bs ∈ K. Suppose that Γ contains two distinct axioms (2.8), Btx ↔ ... and Bsy ↔ ..., while eq(ta, d) ∈ K and eq(sb, d) ∈ K for some a ∈ T ar(t) Then eq(ta, sb) ∈ K by the subgraphs (s), (t) from Definition 2.9.3, and Bta ∈ K ⇔ Bsb ∈ K by (c). Hence, iii. For any atomic equality axiom s = t ∈ Γ , subgraph → eq(sd, td) from 2.9.5 forces eq(sd, td) ∈ K for every instance sd, td ∈ T D of s, t. By i, for each such instance i(sd) = i(td), so that M | s = t. Similarly, for the axiom s = t ∈ Γ , the subgraph eq(sd, td) eq(sd, td) ← , for each instance sd, td ∈ T D , forces eq(sd, td) ∈ K, so i(sd) = i(td), giving M | s = t. If M | Btd then Btd ∈ K and Btd ∈ E − G (K), since K ∈ Ker(G). This means that for some B i xy in the right side of F and c ∈ D ar(B i d) , B i dc ∈ K, since such B i dc form E G (Btd) by point 6 of Definition 2.9. Thus, M satisfies also the right to left implication of (2.8), so M | F. v. Since M | Bd ⇔ K | Bd for each atom Bd and, by ii, M | s = t ⇔ K | eq(s, t) for each s, t ∈ T D , the equivalence M | φ ⇔ K | φ holds for arbitrary formula φ of the language of Γ .
vi. Two different kernels K 1 = K 2 of G differ for at least one atom Bd. This follows because membership of atoms in a kernel K determines uniquely this kernel, as can be seen inspecting the subgraphs in Definition 2.9. For instance, restriction of any kernel K to the atoms in any subgraph 2.9.4.(c), i.e. to Pt, Ps and all eq(t i , s i ), induces unique values to all remaining (auxiliary) vertices of this subgraph, which therefore must coincide with their (non)membership in K. By Definition 2.9.7, each K ∈ Ker(G) determines Bd ∈ K or Bd ∈ K also for Bd ∈ Ins.
Consequently, kernels represent exactly models of a theory: Fact 2.13 below follows from Fact 2.12 in the same way Fact 2.7.1-2 follows from Fact 2.5.

Some facts about kernels
We gather some relevant facts about kernels. Since existence of a kernel for some graph in Gr(Γ ) is equivalent to consistency of Γ , we start by quoting a couple of results on kernel existence. A graph is kernel perfect if each induced subgraph has a kernel. Often, establishing solvability, one shows actually kernel perfectness and we, too, will use this stronger notion. The central result in kernel theory is the following theorem of Richardson.

THEOREM 2.14 ([3]).
A graph G without odd cycles is kernel perfect if (a) for each x ∈ V G : E G (x) is finite or (b) there are no rays (infinite, simple, outgoing paths).
In particular, a finite graph without odd cycles is kernel perfect. We will also encounter the following notion and fact. A digraph G is bipartite if so is its underlying undirected graph (forgetting directions of edges), that is, if V G can be partitioned into two independent subsets, so that each E G -edge connects a vertex in one subset to a vertex in the other. Kernels can be transferred from homomorphic images to preimages. A graph homomorphism from G to H is a function h : PROOF. When K is independent then so is and K − ∈ Ker(G). An isomorphism of graphs G and H, denoted by G H, is a bijective homomorphism either way.  β (d))). On the other hand, β(B(d))).
Homomorphisms ref lect also bipartitions, according to the (proof of the) following fact.

FACT 2.19
If h : G → H is a homomorphism and H is bipartite, then so is G.
Finally, we sometimes start with a partial assignment of boolean values (select a part of a kernel) and propagate its consequences, that is, induce values to some other vertices. Brief ly, a vertex must be assigned 0 if it has an edge to a vertex assigned 1, while if all outneighbours are assigned 0, the vertex itself must be 1. More formally, given a partial assignment σ with domain X ⊆ V G , we start , and iterate the following: Depending on σ , this may not be a function, but we induce only when it is. More details can be found in [6].

Extensions
Transforming manually a given theory into GNF may be a cumbersome task worthwhile only in special cases. However, GNF is often encountered directly, as in definitional extensions or fixed point definitions, where a new predicate B is introduced by a form Bx ↔ RS, from which GNF can easily be obtained, transforming RS. Often, extension should be conservative or allow a (unique) expansion of every model of the theory under extension, and various syntactic restrictions are utilized to ensure this. Graph representation gives a new perspective on such situations, in particular, when circularity is involved.
We therefore view now extensions as the primary objects. An extension is simply a GNF theory Δ, with undefined predicates (if any) marking connections to any theory Γ which it may extend. This is how extensions are often used: as a generic possibility of augmenting a wide range of theories. Just like transitive closure can be applied to any binary relation, we can think of an extension as potentially applicable to any theory possessing predicates with the arities of the undefined predicates of the extension.
Given Δ as an independent object, its undefined predicates may need renaming to match the appropriate predicates of Γ , avoiding such an identification of predicates defined in Δ. Ensuring this, when applying Δ to Γ , is straightforward, so we assume the naming details are always resolved and write the result of such an extension as Δ(Γ ). 3 For every (set of) cardinality κ, the graph equality G κ (Δ(Γ )) = G κ (Δ) ∪ G κ (Γ ) holds because of this assumption, so that each sink of G κ (Δ), representing an undefined predicate application Pd, is identified with vertex Pd of G κ (Γ ) (or, if no Pd occurs in G κ (Γ ), acquires 2-cycle in G κ (Δ(Γ ))). Whenever the choice of κ is inessential, we speak about the graph G(Δ) of the extension and the graph G(Γ ) of the extended theory.
Consequently, while sinks for undefined atoms of G(Δ) end up among vertices of G(Γ ), there are no edges from G(Γ ) to G(Δ)-the theory being extended does not use any predicates defined by the extension.
Sinks of G(Δ) obtain thus a dual status. Some may represent LSs of axioms with empty RSs, which are simply true. Belonging to every kernel, they affect the graph in a unique way, inducing some consequences (following (2.3), with σ 1 0 = sinks). Such sinks must be distinguished from those which represent undefined atoms, to be identified with identical atoms of the extended theory. We call the latter u-sinks. Kernels of G(Δ) are to be investigated under arbitrary valuations of u-sinks, as their values are (to be) determined by the theory which is being extended. Formally, this comes closer to equipping them with 2-cycles, but name 'u-sinks' marks that the question concerns now existence of a kernel under arbitrary-and not only some appropriate-valuation of these atoms.  The notions are listed with decreasing strength: every definitional extension is a model unique extension, which is a model extension, and every model extension is conservative. In general, none of these inclusions can be reversed. We begin with a few simple examples. An extension with a predicate B, even if not in GNF, has typically the form Bx ↔ RS. An equivalent extension in GNF can be then obtained more easily, than by following Definitions 1.3 and 1.4, by reformulating appropriately RS. This is typically done below. EXAMPLE 3.2 Definitional extension Δ given by Dx ↔ Fx ∨ ¬Hx, as written in GNF to the left, has graph G(Δ) to the right: An application of Δ to any actual theory Γ amounts to matching F and H to some unary predicates of Γ . For any κ, the graph G κ (Δ) has a copy of the above G(Δ) for each x ∈ κ. The u-sinks Fx and Hx of G κ (Δ) obtain in G κ (Δ(Γ )) the edges which Fx, Hx have in G κ (Γ ).

EXAMPLE 3.3
The extension Nx ↔ Fx ∨ (Hx ∧ ¬Nx), i.e. Nx ↔ ¬(¬Fx ∧ ¬(Hx ∧ ¬Nx)), given in GNF to the left, has graph G(N) to the right: The graph has no solution for Fx = 0, Hx = 1, so (N) does not model extend any Γ consistent with ∃x(¬Fx ∧ Hx). From (N), we can actually prove Fx ∨ ¬Hx, so this extension is not even conservative for any such Γ . 4 EXAMPLE 3.4 Let (N') be as (N) in Example 3.3, but with Bx replaced by the following B x: The new edge B x → Nx makes the induced subgraph {B x, Cx, Nx} kernel perfect. Consequently, it is solvable for every valuation of its u-sinks, so (N') is a model extension of every theory.
The following more complex example illustrates also the effects of quantifiers.

Transitive closure
Given a binary relation E, a natural attempt to define its transitive closure is by adding the axiom (TC) TCxy ↔ Exy ∨ ∃z : Exz ∧ TCzy.
We ask first about the general relation between possible models of E and models of E extended with (TC). In GNF, the definition of TC becomes the four equivalences in (3.5). The dotted edges, marked with ∀z on the sketch of G(TC) below, signal branching to all instances of the target formula, with some marked explicitly on the dotted edges leading to them. The edges Exy → Exy to the u-sinks Exy of G(TC) are left implicit.
ii. Considering irref lexive E, (TC) still does not capture transitive closure, though this is less obvious. The case G ω (TC) of G(TC) has namely a kernel including TCab when there is an infinite chain R = {z 0 , z 1 , z 2 , ...} with z 0 = a, and Ez i b = 0 = Ez i z i+1 for each z i ∈ R. The last condition means Ez i z i+1 = 1, i.e. R is a ray (or enters a cycle), which gives a very specific and unintended meaning to any b being E-reachable from a, as if an infinite walk reached every vertex (then TCz i b for each z i ∈ R). Still, in this situation there is also another model in which TCab = 0.
The aim of this example is not to reexamine undefinability of transitive closure in FOL but to note that the graph above has a kernel for every valuation of E-vertices: (TC) is a model extension of an arbitrary theory of E. This follows from the observation that G(TC) is bipartite. We justify it by the following general argument.
For a GNF theory Γ , a simpler schematic graph S(Γ ) conveys often much information. Its vertices are (labeled by) the predicate symbols alone, ignoring the arguments. Each equivalence gives thus edges from the predicate in its LS, to each predicate occurring in its RS. Each of the graphs in Examples 3.2, 3.3 and 3.4 is isomorphic to such a schematic graph of its theory. For the extension (TC) from (3.5), the schematic graph S(TC) is (3.6) Every path in the graph G κ (Γ ), for any κ, results from unfolding some path in such a schematic graph S(Γ ). The latter is also the homomorphic image of the former under the canonical homomorphism, identifying all vertices with the same predicate symbol.
The graph S(TC) is trivially bipartite (since so is its underlying undirected graph, having no odd undirected cycles). Using the just described canonical homomorphism and Fact 2.19, we conclude that G(TC) is bipartite and, by Fact 2.15, kernel perfect. This holds generally. Whenever the schematic graph S(Γ ) is bipartite, then each graph G κ (Γ ) is kernel perfect.
The simplified graph S(TC) allows thus to conclude that every G(TC) is solvable for every assignment to its u-sinks Exy. Consequently, (TC) is a model extension of any theory of E. This triviality about (TC) becomes a useful fact, when formulated generally. FACT 3.7 An extension Δ, of any theory Γ , is a model (unique) extension of Γ if for every cardinality κ, the graph G κ (Δ) is (uniquely) solvable for every assignment to its u-sinks.
This follows because, given a theory Γ , each kernel M of G κ (Γ ) determines values of (some) u-sinks(G κ (Δ)). Since no edges go from G κ (Γ ) to G κ (Δ), while the latter is solvable for every valuation of its u-sinks, kernel M of G κ (Γ ) can be extended to a kernel of G κ (Δ(Γ )). Section 3.3 addresses the fact that the condition, requiring solvability of G(Δ), is independent of Γ .
In Example 3.4, (N') is a model extension. As observed there, this follows because its graph is kernel perfect so, in particular, has a solution for every valuation of its u-sinks.
In Example 3.2, (D) is a definitional extension, while its graph G(D) is a dag without any rays. By the first result in kernel theory from [4], such a graph has a unique kernel and is actually kernel perfect. The effect of the syntactic restrictions on a definitional extension is, in graph terms, that its graph becomes a rayless dag-uniqueness of its kernel yields model uniqueness. It is a special case of Fact 3.7 which implies, more generally, that Δ is a model extension whenever G(Δ) is kernel perfect. This general statement will be used below.
Before that, it may be useful to point out some limitations of using schematic graph S(Δ) instead of G(Δ), which is attractive whenever applicable. One such limitation is that even though bipartition, and solvability in general, are ref lected by homomorphisms, solvability for all valuations of sinks is not. For instance, graph is unsolvable if s 1 = 1 and s 2 = 0, but its homomorphic image has a solution for each valuation of s. When S(Δ) is solvable for all valuations of its u-sinks, solvability of G(Δ) for all valuations of its u-sinks may fail and requires additional argument, as illustrated also in Example 3.13 further ahead.
A dual problem is exemplified in the language with constants a, b and predicate Q. Its extension with predicate P and axioms Pa ↔ ¬Qa and Pb ↔ ¬Pa is now model unique for every valuation of Q, but the schematic graph P → Q, is solvable only when Q = 1.
In the examples above, schematic graphs of consistent extensions are still solvable, but they need not be. Let Δ be the following extension of a theory having a binary predicate symbol E: xyz ↔ ¬Exy ∧ ¬Eyz ∧ ¬Exz. More comprehensibly: Δ ⇔ GNF({∀x¬Exx, ∀x∃yExy, ∀xyz(Exy ∧ Eyz → Exz)}), which forces domain to be infinite. Although Δ is consistent, its schematic graph S(Δ) is unsolvable: When the only terms are variables, S(Δ) is isomorphic to the graph G 1 (Δ) over domain with one element. As Δ forces here domain to be infinite, it does not appear possible to represent it by such a graph. It does not even seem possible to represent it by retaining an infinite number of u-sinks Exy, while collapsing distinct instances of internal vertices, e.g. identifying B 2 d for all d in actual domain, etc. For each d, B 2 d = 1 must hold; then also B 2a d = 0, requiring some e, distinct from d, with Ede = 1. Multiplicity of distinct vertices Ede may require multiplicity of distinct vertices B 2 d.
In short, simplification offered by the schematic graph is far from universal. It seems highly improbable that any single schema could replace the whole class Gr(Δ), for arbitrary Δ, but the range of Δs for which schema S(Δ) is applicable, or its generalizations, might deserve clarification.

Fixed points and positive occurrences
Restrictions on fixed point definitions provide another example, besides definitional extension, of syntactic means ensuring condition of Fact 3.7. One defines a predicate by (*) Bx ↔ φx, with B occurring in φ (and V(Bx) = x = V(φ)). If M is a structure interpreting the symbols from φ, let (M, X ) denote its expansion with the interpretation of B as X ⊆ M. A model of (*) over a given M is then a fixed point of the operator B M (X ) = {m ∈ M | (M, X ) | φm}. Often, one chooses only least or greatest fixed points, but we address only the consistency conditions, that is, the mere existence of fixed points.
A simple restriction, ensuring monotonicity of B and existence of fixed points, forbids negative occurrences of B in φ. In terms of GNF, this amounts to forbidding any occurrence of B under an odd number of negations, when replacing predicates in the RS of B's equivalence, by the RSs of their equivalences. Such a substitution, performed in Example 1.2 and below, provides a procedure for identifying negative occurrences in GNF. Each equivalence below marks one step of the successive substitution: Nx ↔ ¬Bx ↔ ¬(¬Fx ∧ ¬Cx) ↔ ¬(¬Fx ∧ ¬(¬Hx ∧ ¬Nx)).
In the last formula, N occurs under three negations, displaying thus its negative occurrence.
Each ¬ in GNF amounts to an edge in the corresponding graph, so a negative occurrence of Nx, in the RS of (some such substitution instance of) the equivalence for Nx, amounts to an odd cycle in the corresponding graph. Each odd cycle signals negative occurrence of all predicates in its vertices. In (N) above, N, B and C all have such occurrences. Forbidding negative occurrences amounts to excluding odd cycles.
Strictly speaking, what must not occur negatively is the same atom, say Td, for some d in the domain, and not merely the predicate symbol T. Negative occurrences of T in T1 ↔ ¬T2 or Tx ↔ ¬Tsx may appear circular, but they do not create any cycles in the graph, as long as 1 = 2 and x = sx. An (odd) cycle emerges only from a (negative) occurrence in RS of an atom, like Td, which is an instance of Tx in LS, from which Td is reached in the substitution process.
A universal model unique extension occurs, for instance, when its graph is a rayless dags, as with definitional extensions. Every assignment to u-sinks induces then a unique valuation of all vertices. But model unique extensions occur also in many other situations, as illustrated by the concluding examples. EXAMPLE 3.12 The extension Θ to the left has the simplified graph S(Θ) to the right (for every κ, the graph G κ (Θ) consists of κ copies of this graph): The graph is not kernel perfect, as witnessed by the odd cycle {A, B, C}, showing also negative occurrences of its atoms. Still, S(Θ) is uniquely solvable for every assignment to its u-sink H: H = 1 gives C = D = A = 0 and B = 1, while H = 0 yields D = 1 = B and C = A = 0. In spite of the negative occurrences, Θ is a universal model unique extension. EXAMPLE 3.13 Below, in a more complicated version Δ of Θ from Example 3.12, some predicates have different arities: The graph G κ (Δ) has κ copies of the above graph, one for each x ∈ κ, and in each such copy, Ax (Dx) has edges to κ vertices Bxy i (Hxy i ), for each y i ∈ κ. Each vertex Bxy i starts a copy of the subgraph following Bxy 0 (and Bxy 1 ), with an edge from each Cxy i to the same Ax and Dx. Now, S(Δ) = S(Θ). The canonical homomorphism from any G κ (Δ) onto S(Δ) ref lects kernels of S(Δ) by Fact 2.17. So G κ (Δ) is solvable whenever, for each x ∈ κ, all Hxy i = 0 or all Hxy i = 1. To conclude that Δ is a universal model extension, we have to consider also the case of Hxy i = 1 only for some y i . Then Dx = Cxy i = 0, making Bxy i = 1 and Ax = 0. All other Cxy j , Bxy j are then determined by the respective Hxy j . Thus, Δ is a universal model extension (in fact, unique), all negative occurrences and odd cycles notwithstanding.