0-1 Laws for Fragments of SOL

Transcription

1 0-1 Laws for Fragments of SOL Haggai Eran Iddo Bentov Project in Logical Methods in Combinatorics course Winter 2010

2 Outline 1 Introduction Introduction Prefix Classes Connection between the 0-1 Law and Finite Satisfiability Law for Bernays-Schönfinkel Finite Model Property for Bernays-Schönfinkel 0-1 Law for Bernays-Schönfinkel 3 Negative Results The Parity Property Counterexample for the Gödel Prefix Class Law and Finite Satisfiability Solvability of the 0-1 Law, Finite Model Property, Docility Solvability in Fragments Without Equality The 0-1 Law implies the Finite Model Property 5 Conclusion Conclusion References

3 Introduction The 0-1 Law The 0-1 law states that for any sentence ψ over some vocabulary σ in the logic L, the fraction of structures over σ of size n that satisfy ψ converges to either 0 or 1 as n. We use the notation µ n (ψ) n µ(ψ) {0, 1}. The Status of 0-1 Laws in Various Logics The 0-1 law holds for FOL sentences over any relational vocabulary. What about SOL, MSOL, and SOL sentences? For MSOL and even for MSOL the answer is no [Kaufmann and Shelah 1985, Kaufmann 1987], as discussed in Meirav Zehavi s project. In this presentation we focus on whether the 0-1 law holds in fragments of SOL with restricted first-order quantifiers.

4 Introduction The 0-1 Law The 0-1 law states that for any sentence ψ over some vocabulary σ in the logic L, the fraction of structures over σ of size n that satisfy ψ converges to either 0 or 1 as n. We use the notation µ n (ψ) n µ(ψ) {0, 1}. The Status of 0-1 Laws in Various Logics The 0-1 law holds for FOL sentences over any relational vocabulary. What about SOL, MSOL, and SOL sentences? For MSOL and even for MSOL the answer is no [Kaufmann and Shelah 1985, Kaufmann 1987], as discussed in Meirav Zehavi s project. In this presentation we focus on whether the 0-1 law holds in fragments of SOL with restricted first-order quantifiers.

5 SOL Examples Some SOL properties do converge to 0 or 1. Example 1 A bipartite graph can be defined with the following sentence: U x y Exy (Ux Uy) The bipartite property has probability that converges to 0, because there is almost surely a triangle in a random graph. The above property belongs to a prefix class of SOL that is called Σ 1 1 (Bernays-Schönfinkel), for which the 0-1 law holds. X However, it is easy to see that some SOL sentences do not converge. In particular, the Parity property (i.e. structures of even size) can be expressed in SOL. More details will follow.

6 Prefix Classes Any formula is equivalent to a formula in Prenex Normal Form, where all the quantifiers appear at the beginning. Formulas can be partitioned to fragments based on their quantifiers prefix, for example: ( ) ( ) ε = FOL We are going to explore the 0-1 law for fragments of SOL, partitioned according to well-known FOL prefix classes. Definition 2 Σ 1 1 (Ψ) is the SOL fragment with first-order part from the FOL fragment Ψ, and with all the predicates either existentially quantified, or free as part of the vocabulary (i.e. formulas of the form Sψ(R S) where ψ Ψ and R is some relations from the vocabulary).

7 Partition of FOL Prefix Classes Ackermann Bernays-Schönfinkel 2 (minimal) Gödel Kahr-Moore-Wang In yellow: fragments with SOL 0-1 law. This gives the most refined partition according to prefix classes.

8 0-1 Law for Prefix Classes Given a counterexample, i.e. a sentence with FOL prefix from the class Ψ that does not converge, any fragment that all of its prefixes contain Ψ does not have the 0-1 law. A proof of the 0-1 law for the FOL prefix class Ψ implies that any sentence whose prefix is contained in Ψ has probability of either 0 or 1. Example 3 A sentence with the alternating quantifiers prefix belongs to a fragment that contains the prefix, for which we will see a counterexample.

9 Fragments With and Without Equality We can similiarly divide sentences according to whether or not they use the equality relation. Equality A counterexample for a fragment without equality also holds for the same fragment with equality. A 0-1 law proof for a fragment with equality also holds for the same fragment without equality.

10 0-1 Law for SOL and Finite Satisfiability The 0-1 law for SOL classification project was initiated in 1987 with the following words: The Kolaitis and Vardi Hypothesis [KV87] Our results suggest an intriguing connection between the solvability of the finite satisfiability problem for a class Ψ of first order logic sentences and the 0-1 law for the second order class Σ 1 1 (Ψ).

11 Results of the SOL Classification Project Fragment Bernays-Schönfinkel FOL finite satisfiability problem ( ) presented here SOL 0-1 law ( presented here ) Ackermann Kahr-Moore-Wang X X ( presented here ) Godel 2 (with equality) Gödel 2 (w/o equality) X X X ( presented here )

12 Outline 1 Introduction Introduction Prefix Classes Connection between the 0-1 Law and Finite Satisfiability Law for Bernays-Schönfinkel Finite Model Property for Bernays-Schönfinkel 0-1 Law for Bernays-Schönfinkel 3 Negative Results The Parity Property Counterexample for the Gödel Prefix Class Law and Finite Satisfiability Solvability of the 0-1 Law, Finite Model Property, Docility Solvability in Fragments Without Equality The 0-1 Law implies the Finite Model Property 5 Conclusion Conclusion References

13 The Bernays-Schönfinkel Fragment Theorem 4 If ψ is satisfiable, then ψ has a finite model. Proof. Let ψ = x yφ, over a vocabulary τ. There exists a τ-structure S and elements a such that S yφ [x := a] Since this formula is universal, it is satisfied by any substructure of (S, a), containing a. In particular, it is satisfied by the finite substructure with universe a.

14 The Bernays-Schönfinkel Fragment Theorem 4 If ψ is satisfiable, then ψ has a finite model. Proof. Let ψ = x yφ, over a vocabulary τ. There exists a τ-structure S and elements a such that S yφ [x := a] Since this formula is universal, it is satisfied by any substructure of (S, a), containing a. In particular, it is satisfied by the finite substructure with universe a.

15 The Transfer Property Definition 5 (The Countable Random Structure) The countable random structure R over a relational vocabulary σ is the unique countable structure satisfying all the extension axioms. When we deal with structures over a vocabulary that has a single binary predicate, the random structure is the random graph. Definition 6 (The Transfer Property) A logic fragment L satisfies the transfer property if for every sentence ϕ L R ϕ µ (ϕ) = 1 A fragment that satisfies the transfer property has the 0-1 law.

16 Transfer Property on Π 1 1 Lemma 7 If R Sθ where θ is some FOL sentence, then there exists another FOL sentence ψ such that µ (ψ) = 1 and ψ Sθ θ can use predicate symbols from the vocabulary in addition to S. Corollary 8 Every SOL sentence that is false on the countable random structure R has probability 0.

17 Proof of Lemma 7 Proof. Let T be the set of extension axioms and T = T { θ} Assume (towards contradiction) R Sθ but for every ψ such that µ (ψ) = 1, ψ Sθ. Any finite subset of T has a model: If ψ is a finite conjunction of extension axioms, then µ (ψ) = 1, and there is a model A ψ Sθ, so A ψ and A θ (for some assignment to S). By compactness, T has a countable model B. T is ω-categorical, so R B But then R S θ, a contradiction.

18 The Bernays-Schönfinkel Case Lemma 9 Let ϕ = S x yθ(s σ, x, y) be some SOL( ) sentence over a vocabulary σ that is true on the countable random structure R. Then there exists some other FOL sentence ψ such that µ (ψ) = 1 and fin ψ ϕ. In particular, µ (ϕ) = 1. Corollary 10 Lemma 9 and Corollary 8 imply that the transfer property holds for the SOL(Bernays-Schönfinkel) class: R ϕ µ (ϕ) = 1 The 0-1 law for the Bernays-Schönfinkel class follows.

19 Proof of Lemma 9 R R S x yθ(s σ, x, y) Let a be a sequence of elements witnessing the x. Let R 0 be the finite substructure of R with universe {a}.

20 Proof of Lemma 9 S R R, S x yθ(s σ, x, y) Let a be a sequence of elements witnessing the x. Let R 0 be the finite substructure of R with universe {a}.

21 Proof of Lemma 9 a 1 S a 2 R a 3 R, S, a yθ(s σ, a, y) Let a be a sequence of elements witnessing the x. Let R 0 be the finite substructure of R with universe {a}.

22 Proof of Lemma 9 a 1 S a 2 a 3 R, S, a yθ(s σ, a, y) Let a be a sequence of elements witnessing the x. Let R 0 be the finite substructure of R with universe {a}.

23 Proof of Lemma 9 continued a 1 a 2 a 3 There exists a formula ψ, which is a conjunction of a finite number of extension axioms, such that every model of it contains a substructure isomorphic to R 0. Assume B fin ψ.

24 Proof of Lemma 9 continued a 1 a 2 B a 3 There exists a formula ψ, which is a conjunction of a finite number of extension axioms, such that every model of it contains a substructure isomorphic to R 0. Assume B fin ψ.

25 Proof of Lemma 9 continued Find a substructure B of R that is isomorphic to B, and contains R 0. Since universal statements are preserved under substructures: B S x yθ(s σ, x, y) where x is interpreted by a, and S is interpreted by the restriction of the original relations on R to B. a 1 a 2 B B a 3

26 Proof of Lemma 9 continued Find a substructure B of R that is isomorphic to B, and contains R 0. Since universal statements are preserved under substructures: B S x yθ(s σ, x, y) where x is interpreted by a, and S is interpreted by the restriction of the original relations on R to B. a 1 a 2 B B a 3

27 Outline 1 Introduction Introduction Prefix Classes Connection between the 0-1 Law and Finite Satisfiability Law for Bernays-Schönfinkel Finite Model Property for Bernays-Schönfinkel 0-1 Law for Bernays-Schönfinkel 3 Negative Results The Parity Property Counterexample for the Gödel Prefix Class Law and Finite Satisfiability Solvability of the 0-1 Law, Finite Model Property, Docility Solvability in Fragments Without Equality The 0-1 Law implies the Finite Model Property 5 Conclusion Conclusion References

28 The Parity Property Definition 11 The Parity property is satisfied by finite models whose domain consists of an even number of elements. Obviously, µ (Parity) does not converge. Parity can be defined via a function R that is a permutation, where every element is of order 2. Parity in the Prefix Class Σ 1 ( 1 3 with equality ) R x y z t ( Rxt ((Rxy Rxz) y = z) [R is a function] Rtx [R is 1-1, onto, of order 2] x t [R has no self-loops] ) Therefore Σ 1 1 ( 3 ) doesn t have the 0-1 law.

29 The Parity Property Definition 11 The Parity property is satisfied by finite models whose domain consists of an even number of elements. Obviously, µ (Parity) does not converge. Parity can be defined via a function R that is a permutation, where every element is of order 2. Parity in the Prefix Class Σ 1 ( 1 3 with equality ) R x y z t ( Rxt ((Rxy Rxz) y = z) [R is a function] Rtx [R is 1-1, onto, of order 2] x t [R has no self-loops] ) Therefore Σ 1 1 ( 3 ) doesn t have the 0-1 law.

30 Parity in Kahr-Moore-Wang Parity in the Prefix Class Σ 1 1 ( with equality) R x y z ( Rxy (Rxz y = z) [R is a function] (Rxz Rzx) [R is 1-1, onto, of order 2] Rxx [R has no self-loops] ) Notice that since this prefix class (known as Kahr-Moore-Wang) doesn t have the 0-1 law, by proving that the 0-1 law also fails for the minimal Gödel ( 2 ) prefix class, and that the Bernays-Schönfinkel ( ) and Ackermann ( ) prefix classes have the 0-1 law, we thereby obtain the most possibly refined partition of SOL according to FOL prefix classes.

31 Parity Without Equality Almost Surely How to define Parity without equality? Equality Almost Surely The following extension axiom ψ ext has µ(ψ ext ) = 1: x y (x y z (Exz Eyz)) Thus, the formula S x y z (Sxy (Exz Eyz)) implies almost surely that S is interpreted as the equality relation. The following formula [KV88] defines Parity whenever the predicate S is interpreted as equality, and therefore on structures of odd (resp. even) cardinality it is almost surely unsatisfied (resp. satisfied): R S x y z w Rxw ((Rxy Rxz) Syz) (Rxy Ryx) Rxx (Sxy (Exz Eyz)) = The 0-1 law fails for Σ 1 1 ( 3 without equality ).

32 Parity Without Equality Almost Surely How to define Parity without equality? Equality Almost Surely The following extension axiom ψ ext has µ(ψ ext ) = 1: x y (x y z (Exz Eyz)) Thus, the formula S x y z (Sxy (Exz Eyz)) implies almost surely that S is interpreted as the equality relation. The following formula [KV88] defines Parity whenever the predicate S is interpreted as equality, and therefore on structures of odd (resp. even) cardinality it is almost surely unsatisfied (resp. satisfied): R S x y z w Rxw ((Rxy Rxz) Syz) (Rxy Ryx) Rxx (Sxy (Exz Eyz)) = The 0-1 law fails for Σ 1 1 ( 3 without equality ).

33 Counterexample for the Gödel Prefix Class We will now outline the counterexample for the 0-1 law in the minimal Gödel ( 2 ) prefix class. The proof shows a property that doesn t converge, and can be expressed as a Σ 1 1 ( 2 ) sentence without equality. This proof generalizes the Parity counterexample that we have seen for Σ 1 1 ( 3 ). The property that the proof uses is based on a natural property of graphs, called the Kernel property.

34 The Kernel Property Definition 12 A stable (a.k.a. independent) set of vertices U has no edges between its vertices. A set of vertices U is called dominating (a.k.a. absorbing) if from each vertex outside of U there is a directed edge to a vertex in U. A set that is both stable and dominating is called a kernel. A graph has the Kernel property if it contains a kernel. However, a formula that expresses the Kernel property cannot by itself constitute a counterexample for the 0-1 law, because de la Vega [V90] proved that the Kernel property is almost surely true. The proof establishes that a random graph on n vertices with edge probability p = 1 2 has w.h.p. a kernel of size log 2 n log 2 log 2 n.

35 The Kernel Property Definition 12 A stable (a.k.a. independent) set of vertices U has no edges between its vertices. A set of vertices U is called dominating (a.k.a. absorbing) if from each vertex outside of U there is a directed edge to a vertex in U. A set that is both stable and dominating is called a kernel. A graph has the Kernel property if it contains a kernel. However, a formula that expresses the Kernel property cannot by itself constitute a counterexample for the 0-1 law, because de la Vega [V90] proved that the Kernel property is almost surely true. The proof establishes that a random graph on n vertices with edge probability p = 1 2 has w.h.p. a kernel of size log 2 n log 2 log 2 n.

36 The Kernel Property Definition 12 A stable (a.k.a. independent) set of vertices U has no edges between its vertices. A set of vertices U is called dominating (a.k.a. absorbing) if from each vertex outside of U there is a directed edge to a vertex in U. A set that is both stable and dominating is called a kernel. A graph has the Kernel property if it contains a kernel. However, a formula that expresses the Kernel property cannot by itself constitute a counterexample for the 0-1 law, because de la Vega [V90] proved that the Kernel property is almost surely true. The proof establishes that a random graph on n vertices with edge probability p = 1 2 has w.h.p. a kernel of size log 2 n log 2 log 2 n.

37 Formulating Kernel Kernel is SOL Definable The Kernel property is definable in monadic Σ 1 1 with 2 variables: U( ( x y((ux Uy) Exy)) [U is stable] ( x y( Ux (Uy Exy))) ) [U is dominating] Gödel class This can be equivalently defined in monadic Σ 1 1 ( 2 ) : U x y z ((Ux Uy) Exy) ( Ux (Uz Exz)) The same quantifier can be used for x in both parts of the formula, since the conditions in the prefixes of the two parts are exclusive.

38 The Kernel Property - Le Bars Counterexample There is a frail balance between the two parts of the Kernel property: a large set of vertices has a low probability to be stable, while a small set of vertices has a low probability to be dominating. Theorem 13 (Matula 1972) For fixed p, ɛ > 0, the random graph of size n with edge probability p, has almost surely its clique number between δ ɛ and δ + ɛ, where δ = 2 log 1 n 2 log 1 log 1 n log 1 p p p p Theorem 14 (Erdös and Bollobás 1974) ( e 2) Let d = d (n, p) denote the largest integer r such that the expectation of the number of cliques of order r is greater or equal to 1. the clique number is almost surely in the set {d 1, d, d + 1}.

39 Outline of Le Bars Analysis Le Bars proves that for any real α (0, 1) there is an infinite set S α N so that lim n S α,n δ d = α, thereby obtaining two infinite subsequences of structure sizes such that the modified Kernel property converges to 0 in one and to 1 in the other. In both subsequences, the clique number is almost surely d, as jumps to d + 1 or d 1 would contradict Matula s theorem. Notice that a clique is the complement of a stable set. In one subsequence, δ is significantly larger than the clique number, meaning that the dominating sets (which need to be of about size δ) are almost surely not cliques. In the other subsequence, δ d converges to a small enough difference, meaning that cliques of size close to δ almost surely occur, and these sets will also almost surely be dominating, because being a clique (or stable) and being a dominating set are independent events that involve disjoint sets of edges.

40 The Modified Kernel Property Recall that obtaining two such subsequences for the simple random graph isn t possible, because having a kernel in a random graph is almost surely true. Instead, Le Bars fine-tuned the probabilities for stable/dominating sets, by adding multiple binary relations to the vocabulary. U x yϕ 1 (x, y) x yϕ 2 (x, y) where ϕ 1 denotes the modified stability property: ( ϕ 1 (x, y) = Ux ) ( R i xx (Ux Uy) ) R i xy i= i= and ϕ 2 denotes the modified dominance property: ( ϕ 2 (x, y) = ( Ux R 1 xx) Uy ) R i xy i=1...16

41 The Kernel Property - Summary The extra predicates used in Le Bars modified Kernel property entail that we consider structures with random edge probability p = The property defined using these extra predicates constrains the stable and dominating sets in a precise manner, so that the two infinite subsequences with limits 0 and 1 occur, and therefore the resulting Kernel property doesn t converge. Similarly to the normal Kernel property, the modified Kernel property can be equivalently defined in Σ 1 1 ( 2 without equality ). Le Bar s proof thus implies that the 0-1 law fails for the Gödel without equality fragment of SOL.

42 Outline 1 Introduction Introduction Prefix Classes Connection between the 0-1 Law and Finite Satisfiability Law for Bernays-Schönfinkel Finite Model Property for Bernays-Schönfinkel 0-1 Law for Bernays-Schönfinkel 3 Negative Results The Parity Property Counterexample for the Gödel Prefix Class Law and Finite Satisfiability Solvability of the 0-1 Law, Finite Model Property, Docility Solvability in Fragments Without Equality The 0-1 Law implies the Finite Model Property 5 Conclusion Conclusion References

43 Overview In this section we offer a more detailed view into the connection between the 0-1 law and finite satisfiability. We define the solvability problem for the 0-1 law. We demonstrate that decidable finite satisfiability follows from the finite model property. We then show that solvability of the SOL 0-1 law is at least as hard as the FOL finite satisfiability problem, for sentences without equality. Finally, we show that the SOL 0-1 law implies the finite model property for the corresponding first-order fragment, which explains Kolaitis and Vardi s hypothesis in one direction.

44 Classification and Solvability of 0-1 Laws Definition 15 For fragments of logic in which the 0-1 law holds, we say that the solvability of the 0-1 law also holds in case there is a recursive algorithm that outputs 0 if µ (ϕ) = 0 and outputs 1 if µ (ϕ) = 1. Definition 16 Fragments of logic in which the finite satisfiability problem is solvable are called docile. Classifying the SOL 0-1 Law According to Prefix Classes The classification of SOL was initiated by Kolaitis and Vardi in [KV87], where they raised the question of whether the 0-1 law holds in Σ 1 1 (Ψ) iff the FOL prefix class Ψ is docile, and the question of whether the SOL prefix classes for which the 0-1 law holds are always accompanied by solvability.

45 Finite Model Property Definition 17 A fragment Ψ has the finite model property if for every ϕ Ψ, ϕ is satisfiable there is a finite model A such that A ϕ. Theorem 18 Any set of sentences for which the finite model property holds has a decidable (finite) satisfiability problem. Proof. Given ϕ, search in parallel for a model of ϕ and for a proof of ϕ. Enumerate all the finite models, and for each one check whether it is a model for ϕ. Enumerate all the valid sentences by generating all the possible proofs (cf. Gödel s completeness theorem), and check whether ϕ is derived.

46 The Finite Satisfiability Problem Classifying FOL Prefix Classes According to Docility In 1932 Gödel proved that 2 without equality has the finite model property and is therefore docile, and conjectured that 2 with equality is also docile. The classification of FOL prefix classes according to docility was completed in 1984 by Goldfarb, answering Gödel s conjecture in the negative. Fragment Docility with equality w/o equality Bernays-Schönfinkel Ackermann Gödel 2 X Kahr-Moore-Wang X X Kalmár-Surányi 3 X X Surányi 3 X X

47 Solvability of the 0-1 Law in Fragments Without Equality Theorem 19 If ψ(s) is a FOL sentence without equality over a vocabulary S, then µ ( Sψ(S)) = 1 ψ(s) is finitely satisfiable. Notice that the resulting SOL sentence is over the empty vocabulary. Corollary Solvability of the 0-1 law for Σ 1 1 (Ψ), where Ψ is any FOL fragment without equality, can hold only if Ψ is docile.

48 Solvability of 0-1 Laws Without Equality Proof. The vocabulary of the existential SOL sentence Sψ(S) is empty, thus the density function of Sψ(S) can only be either 0 or 1, and µ ( Sψ(S)) = 1 iff there exists n such that all structures of size larger than n satisfy Sψ(S). To see that an additional element a n+1 can be added to any structure A n over a vocabulary S so that the larger structure A n+1 satisfies the same formulas, we use structural induction.

49 Solvability of 0-1 Laws Without Equality continued Proof (cont.) Pick any element a 0 of A n and define A n+1 so that R(..., a n+1,...) holds iff R(..., a 0,...) held in A n, for every predicate R S. For atomic formulas, a variable assignment that satisfies A n+1 can be modified to satisfy A n by assigning a 0 in place of a n+1. By induction, φ, φ 1 φ 2 hold in A n iff they hold in A n+1. For, suppose that xφ is satisfied in A n. Because there is no equality symbol, we have that φ(x := a n+1 ) holds in A n+1 iff φ(x := a 0 ) holds in A n+1, and by induction φ(x := a 0 ) holds in A n+1 iff φ(x := a 0 ) holds in A n. Since φ(x := a 0 ) indeed holds in A n, we have that φ(x := a n+1 ) and xφ hold in A n+1. For, if xφ is satisfied in A n+1 by φ(x := a n+1 ), then xφ is also satisfied in A n by φ(x := a 0 ).

50 The 0-1 Law implies the Finite Model Property Theorem 20 (Due to Kolaitis and Vardi, cf. [Fag93]) The transfer property for Σ 1 1 (Ψ) implies the finite model property for the fragment Ψ. Observation This theorem partially explains the Kolaitis and Vardi hypothesis. It shows that for SOL fragments that have the 0-1 law, the corresponding FOL finite satisifiability problem is decidable.

51 Proof of Theorem 20 Proof. Let ϕ(s) be an FOL formula of the given fragment. If ϕ(s) is satisfiable, by Löwenheim-Skolem it has a countable model A. R contains A as a sub-model. The relativized sentence ψ(a, S) = A Sϕ A (S) holds for R. Intuitively, ϕ A means restricting quantifiers in ϕ to A. By the transfer property, µ (ψ(a, S)) = 1, and ψ has a finite model. ϕ(s) has a finite model.

52 Outline 1 Introduction Introduction Prefix Classes Connection between the 0-1 Law and Finite Satisfiability Law for Bernays-Schönfinkel Finite Model Property for Bernays-Schönfinkel 0-1 Law for Bernays-Schönfinkel 3 Negative Results The Parity Property Counterexample for the Gödel Prefix Class Law and Finite Satisfiability Solvability of the 0-1 Law, Finite Model Property, Docility Solvability in Fragments Without Equality The 0-1 Law implies the Finite Model Property 5 Conclusion Conclusion References

53 The Classification Project for the SOL 0-1 Law Results of the SOL classification project For all prefix classes other than Gödel, the classification is identical to the classification according to docility, and solvability always accompanies all the fragments in which the 0-1 law holds. Fragment Docility SOL 0-1 law Bernays-Schönfinkel Ackermann Kahr-Moore-Wang Gödel 2 (with equality) Gödel 2 (w/o equality) X X X X X

54 Completion of the SOL Classification Project The Completion of the 0-1 Law SOL Classification Project As a result of Le Bar s counterexample, we have that docility does not always coincide with the 0-1 law for SOL. Le Bar s counterexample generalizes all the previous negative results, including a result by Pacholski and Szwast from 1993 that modified Goldfarb s unsolvability proof to show that the 0-1 law fails for Σ 1 1 ( 2 with equality ). The positive results for the Ackermann prefix class ( ) and the Bernays-Schönfinkel prefix class ( ) were proved by Kolaitis and Vardi [KV88]. The classification according to prefix classes is now complete.

55 References R. Fagin: Finite-model theory a personal perspective. Theoretical Computer Science (1993), vol. 116 pp W.D. Goldfarb: The Gödel class with equality is unsolvable Bull. Amer. Math. Soc. (New Series) 1984, vol. 10, pp P. Kolaitis and M. Y. Vardi: The decision problem for the probabilities of higher-order properties. Proc. 19th ACM Symp. on Theory of Comp. (1987), pp P. Kolaitis and M. Y. Vardi: 0-1 Laws and Decision Problems for Fragments of Second-Order Logic. LICS 1988, pp J.M. Le Bars: Fragments of Existential Second-Order Logic Without 0-1 Laws. LICS 1998, pp W. Fernandez De La Vega: Kernel on random graphs. Discrete Mathematics 82: , 1990 version 5