Stål Aanderaa

On the Decision Problem for Formulas in which all Disjunctions are Binary

Abstract Let Z 1 be the class of closed formulas of the form ∃ a ∀ yKay & A x ∃ u∀yMxuy where Kay and Mxuy are conjunctions of binary disjunctions of signed atomic formulas of the form F αβ or F αβ where F is a binary predicate symbol, and α and β are one of the variables a , x , u and y . We prove in our paper that there is no recursive set which separates the non-satisfiable formulas in Z 1 from those satisfiable in a finite domain.

A Proof of Higman's Embedding Theorem Using Britton Extensions of Groups

Publisher Summary This chapter discusses a proof of Higmans embedding theorem using Britton extensions of groups. The theorem states that a finitely generated group can be embedded in a finitely presented group if it is recursively presented. As pointed out by Higman, one half of the theorem is trivial. The chapter focuses on the nontrivial half of the theorem.

The Solvability of the Halting Problem for 2-State Post Machines

A Post machine is a Turing machine which cannot both write and move on the same machine step. It is shown that the halting problem for the class of 2-state Post machines is solvable. Thus, there can be no universal 2-state Post machine. This is in contrast with the result of Shannon that there exist universal 2-state Turing machines when the machines are capable of both writing and moving on the same step.

Linear sampling and the |forall |exists |forall case of the decision problem

Let Q be the class of closed quantificational formulas ∀ x ∃ u ∀ yM without identity such that M is a quantifier-free matrix containing only monadic and dyadic predicate letters and containing no atomic subformula of the form Pyx or Puy for any predicate letter P . In [DKW] Dreben, Kahr, and Wang conjectured that Q is a solvable class for satisfiability and indeed contains no formula having only infinite models. As evidence for this conjecture they noted the solvability of the subclass of Q consisting of those formulas whose atomic subformulas are of only the two forms Pxy, Pyu and the fact that each such formula that has a model has a finite model. Furthermore, it seemed likely that the techniques used to show this subclass solvable could be extended to show the solvability of the full class Q , while the syntax of Q is so restricted that it seemed impossible to express in formulas of Q any unsolvable problem known at that time. In 1966 Aanderaa refuted this conjecture. He first constructed a very complex formula in Q having an infinite model but no finite model, and then, by an extremely intricate argument, showed that Q (in fact, the subclass Q 2 defined below) is unsolvable ([Aa1], [Aa2]). In this paper we develop stronger tools in order to simplify and extend the results of [Aa2]. Specifically, we show the unsolvability of an apparently new combinatorial problem, which we shall call the linear sampling problem (defined in §1.2 and §2.3). From the unsolvability of this problem there follows the unsolvability of two proper subclasses of Q , which we now define. For each i ≥ 0, let P i be a dyadic predicate letter and let R i be a monadic predicate letter.

Prefix Classes of Krom Formulas

?0. Prologue. In this paper we consider decision problems for subclasses of Kr, the class of those formulas of pure quantification theory whose matrices are conjunctions of binary disjunctions of signed atomic formulas. If each of Q1, Qn is an V or an 3, then let Q1... Qn be the class of those closed prenex formulas with prefixes of the form (Qlxl)... (QnXn). Our results may then be stated as follows: THEOREM 1. The decision problem for satisfiability is solvable for the class V3V fl Kr. THEOREM 2. The classes VWVV f Kr and VV3V f Kr are reduction classes for satisfiability. Maslov [11] showed that the class 3 ... V3 ...3 n Kr is solvable, while the first author [1, Corollary 4] showed 3V3V f Kr and V33V f Kr to be reduction classes. Thus the only prefix subclass of Kr for which the decision problem remains open is V3V3. . 3 n Kr. The class V3V f Kr, though solvable, contains formulas whose only models are infinite (e.g., (Vx)(3u)(Vy)[(Pxy V Pyx) A (-1Pxy V -1Pyu)], which can be satisfied over the integers by taking P to be >). This is not the case for Maslovs class 3 *... *... Y ... 3 nr Kr, which contains no formula whose only models are infinite ([2], [5]). Theorem 1 was announced in [1, Theorem 4], but the proof sketched there is defective: Lemma 4 (p. 17) is incorrectly stated. Theorem 2 was announced in [9]. Our decision procedure for V3V n Kr has two main stages. First we transform each formula into one whose matrix is a conjunction of binary disjunctions of two special kinds, called monadic and linear. Then, exploiting a natural algebraic interpretation of the linear conjuncts, we solve the decision problem for the special case of formulas having none but monadic and linear conjuncts. For the more general class V3 ... V n Kr, analogues of the monadic and linear conjuncts can be defined, and formulas having only such conjuncts comprise a solvable class. But the decision problem for the full class VV ... V n Kr cannot be reduced to that for the solvable subclass: indeed, Theorem 2 shows that this is impossible even for VWVV n Kr.

A Universal Turing Machine

The aim of this paper is to give an example of a universal Turing machine, which is somewhat small. To get a small universal Turing machine a common constructions would go through simulating tag system (see Minsky 1967). The universal machine here simulate two-symbol Turing machines directly.

Prefix classes of krom formulae with identity

Two small classes of first order formulae without function symbols but with identity, in prenex conjunctive normal form with all disjunctions binary, are shown to have a recursively unsolvable decision problem, whereas for another such class an algorithm is developed which solves the decision problem of that class. This solves the prefix problem for such classes of formulae except for the Gödel-Kalmàr-Schütte case.ZusammenfassungFür zwei Klassen erststufiger Formeln in pränexer konjunktiver Normalform mit Identität aber ohne Funktionssymbole wird das Entscheidungsproblem als rekursiv unlösbar nachgewiesen. Für eine weitere solche Ausdrucksklasse wird ein Algorithmus zur Lösung des Entscheidungsproblems angegeben. Bis auf den Gödel-Kalmàr-Schütte-Fall löst dies das Präfixproblem für derartige Ausdrucksklassen.

The Finite Controllability of the Maslov Case

In this paper we show the finite controllability of the Maslov class of formulas of pure quantification theory (specified immediately below). That is, we show that every formula in the class has a finite model if it has a model at all. A signed atomic formula is an atomic formula or the negation of one; a binary disjunction is a disjunction of the form A 1 ⋁ A 2 , where A 1 and A 2 are signed atomic formulas; and a formula is Krom if it is a conjunction of binary disjunctions. Finally, a prenex formula is Maslov if its prefix is ∃···∃∀···∀∃···∃ and its matrix is Krom. A number of decidability results have been obtained for formulas classified along these lines. It is a consequence of Theorems 1.7 and 2.5 of [4] that the following are reduction classes (for satisfiability): the class of Skolem formulas, that is, prenex formulas with prefixes ∀···∀∃···∃, whose matrices are conjunctions one conjunct of which is a ternary disjunction and the rest of which are binary disjunctions; and the class of Skolem formulas containing identity whose matrices are Krom. Moreover, the following results (for pure quantification theory, that is, without identity) are derived in [1] and [2]: the classes of prenex formulas with Krom matrices and prefixes ∃∀∃∀, or prefixes ∀∃∃∀, or prefixes ∀∃∀∀ are all reduction classes, while formulas with Krom matrices and prefixes ∀∃∀ comprise a decidable class. The latter class, however, is not finitely controllable, for it contains formulas satisfiable only over infinite universes. The Maslov class was shown decidable by Maslov in [11].

Search for good examples of Hall’s conjecture

This paper is a preliminary report on our search for new good examples of Halls Conjecture. We present a new algorithm that will detect all good examples within a given search space. We have implemented the algorithm, and our executions have so far found five new good examples.

Recursive Inseparability in Linear Logic

We first give our version of the register machines to be simulated by proofs in propositional linear logic. Then we look further into the structure of the computations and show how to extract ”finite counter models” from this structure. In that way we get a version of Trakhtenbrots theorem without going through a completeness theorem for propositional linear logic. Lastly we show that the interpolant I in propositional linear logic of a provable formula A ⊸ B cannot be totally recursive in A and B.