Order Sentences Research Articles

A Normative Model of Classical Reasoning in Higher Order Languages

The present paper is concerned with a ramified type theory (cf. (Lorenzen 1955), (Russell), (Schutte), (Weyl), e.g.,) in a cumulative version. §0 deals with reasoning in first order languages. $$\mathbb{N}$$ is introduced as a first order set. In §1 and §2 we introduce an extension of a union of higher order languages by means of variables x, y, . . . for constants of arbitrary order, and variables for tuples (c1, . . ., c j ) of arbitrary order and arbitrary length $$j \in \mathbb{N}^{+}$$ . So we may simply identify types with orders. In that language ‘type-free’ equations x = y are definable. We extend the assertion games introduced in §0, called the ‘primary game’ and the ‘classical game’, to that language, show that their sentences are non-circular, and that their formulas are invariant under (=). So we may argue classically even with higher order sentences in the classical game. In §3 we introduce singular description terms. In §4 we deal with expansions of higher order languages which also contain formulas with indicators and objectual variables, and enable a quantification which combines both substitutional and objectual quantification. One may commute consecutive existential quantifiers that occur in formulas of those expanded languages.

Separations of first and second order theories in bounded arithmetic

We prove that PTCN(n) (the polynomial time closure of the nonstandard natural number n in the model N of S2.) cannot be a model of U12. This implies that there exists a first order sentence of bounded arithmetic which is provable in U12 but does not hold in PTCN(n).

Succinct definitions in the first order theory of graphs

We say that a first order sentence A defines a graph G if A is true on G but false on any graph non-isomorphic to G . Let L ( G ) (resp. D ( G ) ) denote the minimum length (resp. quantifier rank) of such a sentence. We define the succinctness function s ( n ) (resp. its variant q ( n ) ) to be the minimum L ( G ) (resp. D ( G ) ) over all graphs on n vertices. We prove that s ( n ) and q ( n ) may be so small that for no general recursive function f we can have f ( s ( n ) ) ≥ n for all n . However, for the function q ∗ ( n ) = max i ≤ n q ( i ) , which is the least nondecreasing function bounding q ( n ) from above, we have q ∗ ( n ) = ( 1 + o ( 1 ) ) log ∗ n , where log ∗ n equals the minimum number of iterations of the binary logarithm sufficient to lower n to 1 or below. We show an upper bound q ( n ) < log ∗ n + 5 even under the restriction of the class of graphs to trees. Under this restriction, for q ( n ) we also have a matching lower bound. We show a relationship D ( G ) ≥ ( 1 − o ( 1 ) ) log ∗ L ( G ) and prove, using the upper bound for q ( n ) , that this relationship is tight. For a non-negative integer a , let D a ( G ) and q a ( n ) denote the analogs of D ( G ) and q ( n ) for defining formulas in the negation normal form with at most a quantifier alternations in any sequence of nested quantifiers. We show a superrecursive gap between D 0 ( G ) and D 3 ( G ) and hence between D 0 ( G ) and D ( G ) . Despite this, for q 0 ( n ) we still have a kind of log-star upper bound: q 0 ( n ) ≤ 2 log ∗ n + O ( 1 ) for infinitely many n .

Decomposable graphs and definitions with no quantifier alternation

Let $D(G)$ be the minimum quantifier depth of a first order sentence $\Phi$ that defines a graph $G$ up to isomorphism in terms of the adjacency and the equality relations. Let $D_0(G)$ be a variant of $D(G)$ where we do not allow quantifier alternations in $\Phi$. Using large graphs decomposable in complement-connected components by a short sequence of serial and parallel decompositions, we show examples of $G$ on $n$ vertices with $D_0(G) \leq 2 \log^{\ast}n+O(1)$. On the other hand, we prove a lower bound $D_0(G) \geq \log^{\ast}n-\log^{\ast}\log^{\ast}n-O(1)$ for all $G$. Here $\log^{\ast}n$ is equal to the minimum number of iterations of the binary logarithm needed to bring $n$ below $1$.

Primeness Conditions on the Lattice of Hereditary Pretorsion Classes#

A lattice ordered monoid is a structure ⟨L;⊕, 0 L ; ≤⟩ where ⟨L;⊕, 0 L ⟩ is a monoid, ⟨L; ≤ ⟩ is a lattice and the binary operation ⊕ distributes over finite meets. If R is an arbitrary ring with identity, then the set ptors R of all hereditary pretorsion classes in the category of right R-modules is an example of a lattice ordered monoid. It is known that right strongly prime and right strongly semiprime rings are characterizable by means of first order sentences in the language of ptors R. A notion of weak primeness and weak semiprimeness is introduced. These are defined by means of first order sentences in ptors R. It is shown that in the presence of a weak commutativity condition, which is also defined in the language of ptors R, the notions of weak primeness [resp. weak semiprimeness] and strong primeness [resp. strong semiprimeness] coincide. It is also proved that within the class of commutative rings, Quasi–Frobenius rings are characterizable by means of a simple sentence in ptors R.

Algorithmic uses of the Feferman–Vaught Theorem

The classical Feferman–Vaught Theorem for First Order Logic explains how to compute the truth value of a first order sentence in a generalized product of first order structures by reducing this computation to the computation of truth values of other first order sentences in the factors and evaluation of a monadic second order sentence in the index structure. This technique was later extended by Läuchli, Shelah and Gurevich to monadic second order logic. The technique has wide applications in decidability and definability theory. Here we give a unified presentation, including some new results, of how to use the Feferman–Vaught Theorem, and some new variations thereof, algorithmically in the case of Monadic Second Order Logic MSOL. We then extend the technique to graph polynomials where the range of the summation of the monomials is definable in MSOL. Here the Feferman–Vaught Theorem for these polynomials generalizes well known splitting theorems for graph polynomials. Again, these can be used algorithmically. Finally, we discuss extensions of MSOL for which the Feferman–Vaught Theorem holds as well.

Syntactic word order and memory for Lithunian sentences

Being a case-based language, Lithuanian is very flexible with respect to word order (Klimas, 2002). For example, the sentence „The girl threw the ball“, which in English can only be expressed as presented, in Lithuanian can be stated in all possible permutations: subject-verb-object (SVO, SOV, VSO, VOS, OSV, &amp; OVS). The goal of this study was to explore the relationship between the different syntactic arrangements and memory for sentence content and reconstruction of syntactic word order.Two hypotheses were taken into account with respect to memory for sentence content. The first may be called the word order proficiency hypothesis. Since Lithuanian shows a preference for SVO and SOV word order sentences (Ambrazas, 1994), one would expect these sentences to occur more frequently, appear less complex, and possibly involve a reduced memory load (see Clifton &amp; Duffy, 2001; Hemforth &amp; Konieczny, 2000). If that is the case, SVO and SOV sentences should be recalled significantly better than VSO, VOS, OSV, and OVS sentences. The second is the semantic identity hypothesis. Research has consistently shown that sentences are recalled mainuly in terms of their semantic content, whereas surface information is lost very rapidly (Brewer &amp; Hay, 1984; Sachs, 1967). Since in Lithuanian the different word order sentences carry the same underlying semantic content, their recall should thus be the same. Regarding reconstruction of word order, subjects tend to reconstruct sentences according to prevailing forms (Brewer &amp; Hay, 1984). Hence, the prediction was made that the non-canonical syntactic order sentences (VSO, VOS, OSV, &amp; OVS) would tend to mirgate over to the canonical forms (SVO &amp; SOV) during the process of recall.The participants in the study were 72 native Lithuanian speakers who had recently immigrated to the United States. They were told that they would be asked to read a list of sentences and afterwards try to remember as many of them as possible. During the course of individual testing, the participant was presented a list of 24 sentences printed on separate cards. Six of the sentences served as a permanent buffer, three at the beginning and three at the end of the list. The 18 critical sentences that stood in the middle were divided into six groups of three sentences. The respective six syntactic word orders (SVO, SOV, VSO, VOS, OSV, and OVS) were counterbalanced across the six groups of three sentences. The participants had 5 sec. to look at each card, read the sentence, and then flip over to the next sentence. After seeing the whole list, the participant was asked to recall as many of the sentences as possible by writing them down on a lined sheet of paper. Following free recall the participant was given a cued recall test. Each participant was handed an 18-page booklet with one of the critical sentence words: subject, verb, or object, printed in either the first, second, or third position with the task to write in the two missing words. Each of the three sentence words was counterbalanced across the three positions.For both free recall and cued recall, a repeated measures ANOVA of correct recall revealed no significant difference amongst the six types of presented sentences, F(5, 355) &lt; 1 for both. Hence, the results support the semantic identity hypothesis over the word order proficiency hypothesis. Additional support for the semantic identity hypothesis came from the results of cued recall by word type. Apparently, subject, verb, and object were equally effective in facilitating recall of the remaining sentence, F(2, 142) = 1,22, p &gt; .05. However, as to the kind of syntactic word order used in recalling the sentences, the difference amongst the six sentence orders was highly significant, F(5, 355) = 20.51, p &lt; .001. Specifically, the two subjectbefore-object syntactic forms (SVO &amp; SOV) were used far more often than the non-canonical forms (VSO, VOS, OSV, &amp; OVS). The results of cued recall revealed a similar pattern.In conclusion, different syntactic arrangements do not seem to affect the semantic retention of sentences. However, as they are being retrieved, the terms of the non-canonical syntactic order sentences are recast into their natural thematic relations, thus giving prominence to the subject as the agent of the underlying argument structure.

On the admittance of forensic patients to general psychiatric wards

In Germany, due to an increasing number of "hospital order sentences," the capacities of forensic hospitals are exhausted. In the late 1990s,general psychiatric hospitals admitted a remarkable number of mentally disturbed offenders. In this study,data of 140 patients treated in general psychiatric hospitals in the German lower Rhine region are presented. These patients clearly differ from those treated in forensic hospitals. Approximately 60% of them suffer from schizophrenic psychosis. Only a few reveal a long antisocial background, a severe personality disorder, or a sexual offence as an index crime. Though the staff in the general psychiatric units often complain about the circumstances of patients' admittance to the hospitals, in about half of the cases they do not agree with a transferring of their patients to a forensic hospital. The results of this study are discussed in regard to general questions of organizing forensic psychiatric treatment.

The class of groups which have a subgroup of index 2 is not elementary

F. Oger proved that if A is a finite group, then the class of groups which are abelian-by-A can be axiomatized by a single first order sentence. It is established here that, in Oger's result, the word abelian cannot be replaced by group.

Primeness described in the language of torsion preradicals

The objects of study in this paper are lattice ordered monoids. These are structures 〈L;⊕, 0L; ≤〉 where 〈L⊕, 0L〉 is a monoid, 〈L;≤〉 is a lattice and the binary operation ⊕ distributes over finite meets. If R is an arbitrary ring with identity then the set Id R of all ideals of R and the set torsp R of all torsion preradicals on the category of right R -modules, are examples of lattice ordered monoids. Primeness and semiprimeness are ring theoretic notions which are characterizable as first order sentences in the language of Id R . The main results show that the notions right strong primeness and right strong semiprimeness may be characterized in a variety of ways as first order sentences in the language of torsp R.

Ups and Downs of First Order Sentences on Random Graphs

whenever is a fixed positive irrational. This raises the question what zero-one valued functions on the positive irrationals arise as the limit probability of a first order sentence on these graphs. Here we prove two necessary conditions on these functions, a number-theoretic and a complexity condition. We hope to prove in a subsequent paper that these conditions together with two simpler and previously proved conditions are also sufficient and thus they constitute a characterization.

The theory of ${\vec Z}C(2)^2$ -lattices is decidable

For arbitrary finite group $G$ and countable Dedekind domain $R$ such that the residue field $R/P$ is finite for every maximal $R$ -ideal $P$ , we show that the localizations at every maximal ideal of two $RG$ -lattices are isomorphic if and only if the two lattices satisfy the same first order sentences. Then we investigate generalizations of the above results to arbitrary $R$ -torsion-free $RG$ -modules and we apply the previous results to show the decidability of the theory of ${\vec Z}C(2)^2$ -lattices. Eventually, we show that ${\vec Z} [i] C(2)^2$ -lattices have undecidable theory.

A note on the Kolmogorov data complexity and nonuniform logical definitions

We define a new Kolmogorov complexity based measure of complexity of logics. Then we use this new tool to prove a sharp estimate of the length of first order sentences defining nonuniformly more complicated Lindström quantifiers in terms of simpler ones.

Counting finite models

AbstractLetφbe a monadic second order sentence about a finite structure from a classwhich is closed under disjoint unions and has components. Compton has conjectured that if the number ofnelement structures has appropriate asymptotics, then unlabelled (labelled) asymptotic probabilitiesν(φ)(μ(φ)respectively) forφalways exist. By applying generating series methods to count finite models, and a tailor made Tauberian lemma, this conjecture is proved under a mild additional condition on the asymptotics of the number of single component-structures. Prominent among examples covered, are structures consisting of a single unary function (or partial function) and a fixed number of unary predicates.

Coloring rules for finite trees, and probabilities of monadic second order sentences

A system of coloring rules is a set of rules for coloring the vertices of any finite rooted tree, starting at its leaves, which has the property that the color assigned to each vertex depends only on how many of its immediate predecessors there are of each color. The asymptotic behavior of the fraction of n vertex trees with a given root color is investigated. As a primary application, if φ is any monadic second-order sentence, then the fractions μn(φ), νn(φ) of labeled and unlabeled rooted trees satisfying φ, converge to limiting probabilities. An extension of the idea shows that for a particular model of Boolean formulas in k variables, k fixed, the fraction of such formulas of size n which compute a given Boolean function approaches a positive limit as n→∞. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 10, 453–485, 1997

Coloring rules for finite trees, and probabilities of monadic second order sentences

Coloring rules for finite trees, and probabilities of monadic second order sentences

On the very weak 0-1 law for random graphs with orders

Let us draw a graph R on {0,1,...,n-1} by having an edge {i,j} with probability p_(|i-j|), where sum_i p_i is finite and let M_n=(n,<,R). For a first order sentence psi let a^n_psi be the probability of ``M_n satisfies psi''. We prove that the limit of a^n_psi-a^{n+1}_psi is 0, as n goes to infinity.

On the Very Weak 0–1 Law for Random Graphs with Orders

Let us draw a graph R on {0, 1, . . . , n− 1} by having an edge {i, j} with probability p|i−j|, where ∑ i pi < ∞, and let Mn = (n,<,R). For a first order sentence ψ let aψ be the probability of Mn |= ψ. We know that the sequence aψ, a 2 ψ, . . . , a n ψ, . . . does not necessarily converge. But here we find a weaker substitute which we call the very weak 0-1 law. We prove that limn→∞(a n ψ − a n+1 ψ ) = 0. For this we need a theorem on the (first order) theory of distorted sum of models. Saharon: 1) Check: line before rest of the proof 3.4A, should it be 6=? 2) Check the numerical bounds. Research partially supported by the Binational Science Foundation and partially sponsored by the Edmund Landau Center for research in Mathematical Analysis, supported by the Minerva Foundation (Germany). Done fall 91, Publication 463.

Latin Syntax and Semantics

Nuclear predication periphery 1 - adjuncts periphery 2 - disjuncts (sentence adverbials) and theme constituents relators the (internal) structure of noun phrases complex sentences praedicativum word order sentence type, illocutionary force and mood the Latin tense system beyond the sentence.

A zero‐one law for a random subset

AbstractWe consider first order sentences about two logical structures. First we consider 1,…, n with the successor relation and a random unary relation that points satisfy with probability p(n). We then replace the successor relation with less than. For both structures we characterize those p(n) for which a zero‐one law holds.