On the Gödel class with identity

1981 ◽  
Vol 46 (2) ◽  
pp. 354-364 ◽  
Author(s):  
Warren D. Goldfarb
Keyword(s):  
Recursive Function ◽  
Decision Problem ◽  
Decision Procedure ◽  
Finite Model ◽  
Quantification Theory ◽  
Free Variables ◽  
Primitive Recursive

The Gödel Class is the class of prenex formulas of pure quantification theory whose prefixes have the form ∀y1∀y2∃x1 … ∃xn. The Gödel Class with Identity, or GCI, is the corresponding class of formulas of quantification theory extended by inclusion of the identity-sign “ = ”. Although the Gödel Class has long been kndwn to be solvable, the decision problem for the Gödel Class with Identity is open. In this paper we prove that there is no primitive recursive decision procedure for the GCI, or, indeed, for the subclass of the GCI containing just those formulas with prefixes ∀y1∀y2∃x.Throughout this paper we take quantification theory to include, aside from logical signs, infinitely many k-place predicate letters for each k > 0, but no function signs or constants. Moreover, by “prenex formula” we include only those without free variables. A decision procedure for a class of formulas is a recursive function that carries a formula in the class to 0 if the formula is satisfiable and to 1 if not. A class is solvable iff there exists a decision procedure for it. A class is finitely controllable iff every satisfiable formula in the class has a finite model. Since we speak only of effectively specified classes, finite controllability implies solvability (but not conversely).The GCI has a curious history. Gödel showed the Gödel Class (without identity) solvable in 1932 [4] and finitely controllable in 1933 [5].

The complexity of decision procedures in relevance logic II

1999 ◽  
Vol 64 (4) ◽  
pp. 1774-1802 ◽  
Author(s):  
Alasdair Urquhart
Keyword(s):  
Lower Bound ◽  
Decision Problem ◽  
Decision Procedure ◽  
Linear Logic ◽  
Decision Procedures ◽  
Search Tree ◽  
Saul Kripke ◽  
Decision Method ◽  
Relevant Logics ◽  
Primitive Recursive

In this paper, we show that there is no primitive recursive decision procedure for the implication-conjunction fragments of the relevant logics R, E and T, as well as for a family of related logics. The lower bound on the complexity is proved by combining the techniques of an earlier paper on the same subject [20] with a method used by Lincoln, Mitchell, Scedrov and Shankar in proving that propositional linear logic is undecidable.The decision problem for the pure implicational fragments of E and R were solved by Saul Kripke in a tour de force of combinatorial reasoning, published only as an abstract [9]. Belnap and Wallace extended Kripke's decision procedure to the implication-negation fragment of E in [3]; an account of their decision method is to be found in [1, pp. 124–139]. The decision method extends immediately to the implication/negation fragment of R. In fact, in the case of R we can go farther: Meyer in his thesis [13] showed how to translate the logic LR, which results from R by omitting the distribution axiom, into R→⋀, so that the decision procedure can be extended to all of LR. This decision procedure has been implemented as a program Kripke by Thistlewaite, McRobbie and Meyer [17]. The program is not simply a straightforward implementation of the decision procedure; finite matrices are used extensively to prune invalid nodes from the search tree.

Linear sampling and the ∀∃∀ case of the decision problem

1974 ◽  
Vol 39 (3) ◽  
pp. 519-548 ◽  
Author(s):  
Stål O. Aanderaa ◽  
Harry R. Lewis
Keyword(s):  
Decision Problem ◽  
Combinatorial Problem ◽  
Finite Model ◽  
Unsolvable Problem ◽  
Sampling Problem ◽  
Solvable Class ◽  
Full Class ◽  
Infinite Model ◽  
Intricate Argument ◽  
Linear Sampling

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 Q2 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 Pi be a dyadic predicate letter and let Ri be a monadic predicate letter.

scholarly journals ON THE FINITENESS PROBLEM FOR AUTOMATON (SEMI)GROUPS

2012 ◽  
Vol 22 (06) ◽  
pp. 1250052 ◽  
Author(s):  
ALI AKHAVI ◽  
INES KLIMANN ◽  
SYLVAIN LOMBARDY ◽  
JEAN MAIRESSE ◽  
MATTHIEU PICANTIN
Keyword(s):  
Decision Problem ◽  
Decision Procedure ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Mealy Automata ◽  
Necessary And Sufficient ◽  
New Criteria

This paper addresses a decision problem highlighted by Grigorchuk, Nekrashevich, and Sushchanskiĭ, namely the finiteness problem for automaton (semi)groups. For semigroups, we give an effective sufficient but not necessary condition for finiteness and, for groups, an effective necessary but not sufficient condition. The efficiency of the new criteria is demonstrated by testing all Mealy automata with small stateset and alphabet. Finally, for groups, we provide a necessary and sufficient condition that does not directly lead to a decision procedure.

On primitive recursive permutations and their inverses

1970 ◽  
Vol 34 (4) ◽  
pp. 634-638 ◽  
Author(s):  
Frank B. Cannonito ◽  
Mark Finkelstein
Keyword(s):  
Recursive Function ◽  
Primitive Recursive ◽  
First Time

It has been known for some time that there is a primitive recursive permutation of the nonnegative integers whose inverse is recursive but not primitive recursive. For example one has this result apparently for the first time in Kuznecov [1] and implicitly in Kent [2] or J. Robinson [3], who shows that every singularly recursive function ƒ is representable aswhere A, B, C are primitive recursive and B is a permutation.

Skolem reduction classes

1975 ◽  
Vol 40 (1) ◽  
pp. 62-68 ◽  
Author(s):  
Warren D. Goldfarb ◽  
Harry R. Lewis
Keyword(s):  
Decision Problem ◽  
Atomic Formula ◽  
Complex Situation ◽  
Quantification Theory ◽  
Reduction Class

Among the earliest and best-known theorems on the decision problem is Skolem's result [7] that the class of all closed formulas with prefixes of the form ∀···∀∃···∃ is a reduction class for satisfiability for the whole of quantification theory. This result can be refined in various ways. If the Skolem prefix alone is considered, the best result [8] is that the ∀∀∀∃ class is a reduction class, for Gödel [3], Kalmár [4], and Schütte [6] showed the ∀∀∃···∃ class to be solvable. The purpose of this paper is to describe the more complex situation that arises when (Skolem) formulas are restricted with respect to the arguments of their atomic subformulas. Before stating our theorem, we must introduce some notation.Let x, y1, y2, be distinct variables (we shall use v1, v2, ··· and w1, w2, ··· as metavariables ranging over these variables), and for each i ≥ 1 let Y(i) be the set {y1, ···, yi}. An atomic formula Pv1 ··· vk will be said to be {v1, ···, vk}-based. For any n ≥ 1, p ≥ 1, and any subsets Y1, ··· Yp of Y(n), let C(n, Y1, ···, Yp) be the class of all those closed formulas with prefix ∀y1 ··· ∀yn∃x such that each atomic subformula not containing the variable x is Yi-based for some i, 1 ≤ i ≤ p.

Recursively presentable prime models

1974 ◽  
Vol 39 (2) ◽  
pp. 305-309 ◽  
Author(s):  
Leo Harrington
Keyword(s):  
Decision Procedure ◽  
Positive Answer ◽  
Prime Model ◽  
Property A ◽  
Categorical Theory ◽  
Decidable Theory ◽  
Free Variables ◽  
Closed Fields ◽  
Countable Theory ◽  
Algebraically Closed Fields

It is well known that a decidable theory possesses a recursively presentable model. If a decidable theory also possesses a prime model, it is natural to ask if the prime model has a recursive presentation. This has been answered affirmatively for algebraically closed fields [5], and for real closed fields, Hensel fields and other fields [3]. This paper gives a positive answer for the theory of differentially closed fields, and for any decidable ℵ1-categorical theory.The language of a theory T is denoted by L(T). All languages will be presumed countable. An x-type of T is a set of formulas with free variables x, which is consistent with T and which is maximal in this property. A formula with free variables x is complete if there is exactly one x-type containing it. A type is principal if it contains a complete formula. A countable model of T is prime if it realizes only principal types. Vaught has shown that a complete countable theory can have at most one prime model up to isomorphism.If T is a decidable theory, then the decision procedure for T equips L(T) with an effective counting. Thus the formulas of L(T) correspond to integers. The integer a formula φ(x) corresponds to is generally called the Gödel number of φ(x) and is denoted by ⌜φ(x)⌝. The usual recursion theoretic notions defined on the set of integers can be transferred to L(T). In particular a type Γ is recursive with index e if {⌜φ⌝.; φ ∈ Γ} is a recursive set of integers with index e.

The finite controllability of the Maslov case

1974 ◽  
Vol 39 (3) ◽  
pp. 509-518 ◽  
Author(s):  
Stål Aanderaa ◽  
Warren D. Goldfarb
Keyword(s):  
Atomic Formula ◽  
Finite Model ◽  
Quantification Theory ◽  
Maslov Class

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 A1 ⋁ A2, where A1 and A2 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].

Prefix classes of Krom formulas

1973 ◽  
Vol 38 (4) ◽  
pp. 628-642 ◽  
Author(s):  
Stål O. Aanderaa ◽  
Harry R. Lewis
Keyword(s):  
Decision Problem ◽  
Decision Problems ◽  
Quantification Theory

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 ∀ or an ∃, then let Q1 … Qn be the class of those closed prenex formulas with prefixes of the form (Q1x1)… (Qnxn). Our results may then be stated as follows:Theorem 1. The decision problem for satisfiability is solvable for the class ∀∃∀ ∩ Kr.Theorem 2. The classes ∀∃∀∀ ∩ Kr and ∀∀∃∀ ∩ Kr are reduction classes for satisfiability.Maslov [11] showed that the class ∃…∃∀…∀∃…∃ ∩ Kr is solvable, while the first author [1, Corollary 4] showed ∃∀∃∀ ∩ Kr and ∀∃∃∀ ∩ Kr to be reduction classes. Thus the only prefix subclass of Kr for which the decision problem remains open is ∀∃∀∃…∃∩ Kr.The class ∀∃∀ ∩ Kr, though solvable, contains formulas whose only models are infinite (e.g., (∀x)(∃u)(∀y)[(Pxy ∨ Pyx) ∧ (¬ Pxy ∨ ¬Pyu)], which can be satisfied over the integers by taking P to be ≥). This is not the case for Maslov's class ∃…∃∀…∀∃…∃ ∩ 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].

Solution of a problem of Leon Henkin

1955 ◽  
Vol 20 (2) ◽  
pp. 115-118 ◽  
Author(s):  
M. H. Löb
Keyword(s):  
Number Theory ◽  
Recursive Function ◽  
Formal System ◽  
Free Variable ◽  
Formal Proof ◽  
Predicate Calculus ◽  
First Order ◽  
Free Variables ◽  
Order Predicate Calculus

If Σ is any standard formal system adequate for recursive number theory, a formula (having a certain integer q as its Gödel number) can be constructed which expresses the proposition that the formula with Gödel number q is provable in Σ. Is this formula provable or independent in Σ? [2].One approach to this problem is discussed by Kreisel in [4]. However, he still leaves open the question whether the formula (Ex)(x, a), with Gödel-number a, is provable or not. Here (x, y) is the number-theoretic predicate which expresses the proposition that x is the number of a formal proof of the formula with Gödel-number y.In this note we present a solution of the previous problem with respect to the system Zμ [3] pp. 289–294, and, more generally, with respect to any system whose set of theorems is closed under the rules of inference of the first order predicate calculus, and satisfies the subsequent five conditions, and in which the function (k, l) used below is definable.The notation and terminology is in the main that of [3] pp. 306–326, viz. if is a formula of Zμ containing no free variables, whose Gödel number is a, then ({}) stands for (Ex)(x, a) (read: the formula with Gödel number a is provable in Zμ); if is a formula of Zμ containing a free variable, y say, ({}) stands for (Ex)(x, g(y)}, where g(y) is a recursive function such that for an arbitrary numeral the value of g() is the Gödel number of the formula obtained from by substituting for y in throughout. We shall, however, depart trivially from [3] in writing (), where is an arbitrary numeral, for (Ex){x, ).

Nonstandard theories of quantification and identity

1970 ◽  
Vol 35 (2) ◽  
pp. 267-294 ◽  
Author(s):  
A. Trew
Keyword(s):  
Predicate Logic ◽  
Type A ◽  
The Other ◽  
Standard System ◽  
Free Logic ◽  
Quantification Theory ◽  
Singular Terms ◽  
Free Variables

In this paper a number of nonstandard systems of predicate logic with or without identity, are translated with subsystems of applied standard system of predicate logic with identity. There are nonstandard theories of quantification which, following [16], are described as inclusive systems; their theorems are valid in all domains, including the empty domain. Theories of quantification which allow for the substitution of denotationless terms for free variables, are described, following [21], as systems of free logic; they are said to be free of the requirement that all singular terms must have denotations. Free logics and inclusive logics may each be of the other type. A nonstandard theory of identity, described, following [12] as a theory of nonreflexive identity, may be combined with a standard or with a nonstandard theory of quantification. Another kind of nonstandard system of predicate logic examined is a nonstandard version of a system of monadic predicate logic in which a distinction is made between sentence and predicate negation, and which is nonstandard in the sense that the laws relating sentence and predicate negation diverge from the standard ones. In the systems examined, this is combined with an inclusive quantification theory.

Sign in / Sign up

Export Citation Format

Share Document