Prenex Sentences Research Articles

The Skolemization of prenex formulas in intermediate logics

The skolem class of a logic consists of the formulas for which the derivability of the formula is equivalent to the derivability of its Skolemization. In contrast to classical logic, the skolem classes of many intermediate logics do not contain all formulas. In this paper it is proven for certain classes of propositional formulas that any instance of them by (independent) predicate sentences in prenex normal form belongs to the skolem class of any intermediate logic complete with respect to a class of well-founded trees. In particular, all prenex sentences belong to the skolem class of these logics, and this result extends to the constant domain versions of these logics.

Set-syllogistics meet combinatorics

This paper considers ∃*∀* prenex sentences of pure first-order predicate calculus with equality. This is the set of formulas which Ramsey's treated in a famous article of 1930. We demonstrate that the satisfiability problem and the problem of existence of arbitrarily large models for these formulas can be reduced to the satisfiability problem for ∃*∀* prenex sentences of Set Theory (in the relators ∈, =).We present two satisfiability-preserving (in a broad sense) translations Φ ↦ $\dot{\Phi}$ and Φ ↦ Φσ of ∃*∀* sentences from pure logic to well-founded Set Theory, so that if $\dot{\Phi}$ is satisfiable (in the domain of Set Theory) then so is Φ, and if Φσ is satisfiable (again, in the domain of Set Theory) then Φ can be satisfied in arbitrarily large finite structures of pure logic.It turns out that |$\dot{\Phi}$| = $\mathcal{O}$(|Φ|) and |Φσ| = $\mathcal{O}$(|Φ|2).Our main result makes use of the fact that ∃*∀* sentences, even though constituting a decidable fragment of Set Theory, offer ways to describe infinite sets. Such a possibility is exploited to glue together infinitely many models of increasing cardinalities of a given ∃*∀* logical formula, within a single pair of infinite sets.

ON EXISTENTIAL DECLARATIONS OF INDEPENDENCE IN IF LOGIC

AbstractWe analyze the behaviour of declarations of independence between existential quantifiers in quantifier prefixes of Independence-Friendly (IF) sentences; we give a syntactical criterion to decide whether a sentence beginning with such prefix exists, such that its truth values may be affected by removal of the declaration of independence. We extend the result also to equilibrium semantics values for undetermined IF sentences.The main theorem defines a schema of sound and recursive inference rules; we show more explicitly what happens for some simple special classes of sentences.In the last section, we extend the main result beyond the scope of prenex sentences, in order to give a proof of the fact that the fragment of IF sentences with knowledge memory has only first-order expressive power.

Deciding polynomial-transcendental problems

This paper presents a decision procedure for a certain class of sentences of first order logic involving integral polynomials and a certain specific analytic transcendental function trans ( x ) in which the variables range over the real numbers. The list of transcendental functions to which our decision method directly applies includes exp ( x ) , the exponential function with respect to base e , ln ( x ) , the natural logarithm of x , and arctan ( x ) , the inverse tangent function. The inputs to the decision procedure are prenex sentences in which only the outermost quantified variable can occur in the transcendental function. In the case trans ( x ) = exp ( x ) , the decision procedure has been implemented in the computer logic system REDLOG. It is shown how to transform a sentence involving a transcendental function from a much wider collection of functions (such as hyperbolic and Gaussian functions, and trigonometric functions on a certain bounded interval) into a sentence to which our decision method directly applies. Closely related work is reported by Anai and Weispfenning (2000), Collins (1998), Maignan (1998), Richardson (1991), Strzebonski (in press) and Weispfenning (2000).

Infinity, in short

It is shown that within the language of Set Theory, if membership is assumed to be non-well-founded a la Aczel, then one can state the existence of infinite sets by means of an ∃∃∀∀ prenex sentence. Somewhat surprisingly, this statement of infinity is essentially the one which was proposed in 1988 for well-founded sets, and it is satisfied exclusively by well-founded sets. Stating infinity inside the BSR (Bernays–Schonfinkel–Ramsey) class of the ∃*∀*-sentences becomes more challenging if no commitment is taken as whether membership is well-founded or not: for this case, we produce an ∃∃∀∀∀ -sentence, thus lowering the complexity of the quantificational prefix with respect to earlier prenex formulations of infinity. We also show that no prenex specification of infinity can have a prefix simpler than ∃∃∀∀. The problem of determining whether a BSR-sentence involving an uninterpreted predicate symbol and = can be satisfied over a large domain is then reduced to the satisfiability problem for the set theoretic class BSR subject to the ill-foundedness assumption. Envisaged enhancements of this reduction, cleverly exploiting the expressive power of the set theoretic BSR-class, add to the motivation for tackling the satisfaction problem for this class, which appears to be anything but unchallenging.

Turning decision procedures into disprovers

AbstractA class of many‐sorted polyadic set algebras is introduced. These generalise structure and model in a way that is relevant in regards to the Entscheidungsproblem and to automated reasoning.A downward Löwenheim‐Skolem property is shown in that each satisfiable finite conjunction of purely relational first‐order prenex sentences has a finite generalised model. This property does, together with a construction related to doubling the size of a finite structure, provide several strict generalisations of the strategy of finite model search for disproving. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Strictly Primitive Recursive Realizability, II. Completeness with Respect to Iterated Reflection and a Primitive Recursive $\omega$-Rule

The notion of strictly primitive recursive realizability is further investigated, and the realizable prenex sentences, which coincide with primitive recursive truths of classical arithmetic, are characterized as precisely those provable in transfinite progressions $\{\mathrm{PRA}(b) \vert b \in \underline{\mathrm{O}}\}$ over a fragment $\mbox{PR-}(\Sigma^{0}_{1}\mbox{-IR})$ of intuitionistic arithmetic. The progressions are based on uniform reflection principles of bounded complexity iterated along initial segments of a primitive recursively formulated system $\mathrm{\underline{O}}$ of notations for constructive ordinals. A semiformal system closed under a primitive recursively restricted $\omega$-rule is described and proved equivalent to the transfinite progressions with respect to the prenex sentences.

A NOTE ON THE ASYMPTOTIC PROBABILITIES OF EXISTENTIAL SECOND-ORDER MINIMAL GÖDEL SENTENCES WITH EQUALITY

The minimal Gödel class is the class of first-order prenex sentences whose quantifier prefix consists of two universal quantifiers followed by just one existential quantifier. We prove that asymptotic probabilities of existential second-order sentences, whose first-order part is in the minimal Gödel class, form a dense subset of the unit interval.

Asymptotic probabilities of existential second-order Gödel sentences

In [9] and [10] P. Kolaitis and M. Vardi proved that the 0-1 law holds for the second-order existential sentences whose first-order parts are formulas of Bernays-Schonfinkel or Ackermann prefix classes. They also provided several examples of second-order formulas for which the 0-1 law does not hold, and noticed that the classification of second-order sentences for which the 0-1 law holds resembles the classification of decidable cases of first-order prenex sentences. The only cases they have not settled are the cases of Gödel classes with and without equality.In this paper we confirm the conjecture of Kolaitis and Vardi that the 0-1 law does not hold for the existential second-order sentences whose first-order part is in Gödel prenex form with equality. The proof we give is based on a modification of the example employed by W. Goldfarb [5] in his proof that, contrary to the Gödel claim [6], the class of Gödel prenex formulas with equality is undecidable.

Complexity of the first-order theory of almost all finite structures

A first-order sentence of a relational type L is true almost everywhere if the proportion of its models among the structures of type L and cardinality m tends to 1 when m tends to ∞. It is shown that Th( L ) , the set of sentences (of type L ) true almost everywhere, is complete in PSPACE. Further, various upper and lower bounds of the complexity of this theory are obtained. For example, if the arity of the relation symbols of L is d ⩾ 2 and if Pr Th( L ) is the set of prenex sentences of Th( L ) , then Pr Th( L ) ∈ DSPACE(( n / log n ) d ) and Pr Th( L ) ∉ NTIME ( o ( n / log n ) d ) . If R is a binary relation symbol and L = { R } , ( Th ( L ) is the theory of almost all graphs), then Pr Th ( L ) ∉ NSPACE ( o ( n / log n ) ) . These results are optimal modulo open problems in complexity such as NTIME( T )? DSPACE( T ) and NSPACE( S ) = ? DSPACE( S 2 ).