# Manual Decidability and Generalized Quantifiers

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Lauchl i : A d e c i s i o n procedure f o r the weak second order theory o f 1 inear order, Contributions LO Math. Logic, Proc. Leonard: On the elementary theory o f l i n e a r o r d e r , Fund. Magidor and J. Logic Makowsky and S. Makowsky, S. Shelah and J. L o g i c 10 Mostowski and A. AMS Vol. Russian , Sibirski Math. AMS Schmerl: personal comnunication. Seese and H. On a decidable generalized quantifier logic corresponding to a decidable fragment of first-order logic.

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## Disjunctive Datalog with Existential Quantifiers: Semantics, Decidability, and Complexity Issues

Blackburn, P. Fine, K. Csirmaz, D. Resumption can be applied to any quantifier in the syntax, this means replacing each individual variable by a corresponding k -tuple of variables ; it has logical uses but also, like RECIP , uses in the interpretation of certain sentences in natural languages; see section 16 below. Then Q satisfies Isomorphism Closure , or just Isom , if the following holds:. One easily checks that all the generalized quantifiers exemplified so far are indeed Isom.

We did not include this requirement in the definition of generalized quantifiers however, since there are natural language quantifiers that do not satisfy it; see below. But logic is supposed to be topic-neutral, so Isom is almost always imposed. Then two important things follow. First, as indicated above, sentences in logical languages do not distinguish isomorphic models.

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More precisely, we have the following. Second, Isom takes a particularly interesting form for monadic quantifiers. Now it is not hard to see that only the sizes of the parts determine whether two models of this kind are isomorphic or not:. This shows that monadic and Isom generalized quantifiers indeed deal only with quantities , i. More generally, under Isom , monadic quantifiers can be seen as relations between cardinal numbers.

Every statement involving a generalized quantifier Q takes place within some universe M. Sometimes it is useful to be able to mirror this relativization to a universe inside M. This means defining a new quantifier with one extra set argument which says that Q behaves on the universe restricted to that argument exactly as it behaves on M. We described how generalized quantifiers can be added to FO , resulting in more expressive logics.

### The fragment checker

A logic in this sense roughly consist of a set of sentences, a class of models, and a truth relation or a satisfaction relation between sentences and models. Such logics are often called model-theoretic logics , since they are defined semantically in terms of models and truth, rather than proof-theoretically in terms of a deductive system for deriving theorems.

There is an obvious way to compare the expressive power of model-theoretic logics. How does one establish facts about expressive power? Similarly for other types. For example, the quantifier all is definable in FO , since the following holds:. The latter is an instance of the following observation:. Such facts about definability can be easy or hard to establish, [ 12 ] but they suffice to establish positive facts about expressivity, since we have:. On the other hand, to prove inexpressibility , i. Some properties that are typical of FO , but fail for most stronger logics, are:.

The Tarski property : If a sentence is true in some countably infinite model, it is also true in some uncountable model. The completeness property : The set of valid sentences is recursively enumerable i. However, this way of proving inexpressibility only works for logics with properties like those above. Logicians have developed much more direct and efficient methods of showing undefinability results that work also for finite models. The above properties in fact characterize FO , in the sense that no proper extension of FO can have certain combinations of them. For an accessible proof see, for example, Ebbinghaus, Flum, and Thomas In addition to the truth conditions associated with generalized quantifiers, one may study the computations required to establish the truth of a quantified statement in a model.

Indeed, generalized quantifiers turn up in various places in the part of computer science that studies computational complexity. In this context, we restrict attention to finite universes, and assume Isom throughout. Such models can be coded as words , i. The abstract notion of an automaton gives an answer; automata are machines that accept or reject words, and they are classified according to the complexity of the operations they perform.

The language recognized by an automaton is the set of words it accepts. A finite automaton has a finite number of states , including a start state and at least one accepting state. It starts scanning a word at the leftmost symbol in the start state, and at each step it moves one symbol to the right and enters a possibly new state, according to a given transition function. If it can move along the whole word ending in an accepting state, the word is accepted.

The application of automata theory to generalized quantifiers was initiated in van Benthem Ch. This machine essentially uses cycles of length 2, whereas the first example had only 1-cycles. Call an automaton of the latter kind acyclic. Van Benthem showed that the FO -definable quantifiers are exactly the ones accepted by finite automata that are acyclic and permutation closed. A slightly more complex automaton, the pushdown automaton, has rudimentary memory resources in the form a of stack of symbols that can be pushed or popped from the top, enabling it to keep track to some extent of what went on at earlier steps.

Thus, an algorithmic characterization is matched with a logical one. This is one prominent direction in the study of algorithmic complexity. Consider now the most general abstract automata or computational devices, i. But this time the order does seem to matter, and in fact the Immerman and Vardi result just mentioned only holds for models with a given linear order and a binary predicate symbol standing for that order. Here infinitely many quantifiers may be required, but in some cases a single one suffices. So generalized quantifiers remain a simple and versatile way of adding expressive power to FO.

One natural question was if the logical characterization of PTIME mentioned above could be improved using generalized quantifiers, in particular if one could remove the restriction to ordered structures in this way. In the late s Richard Montague showed how the semantics of significant parts of natural languages could be handled with logical tools.

Montague worked in type theory, but around a number of linguists and logicians began to apply the model-theoretic framework of logics with generalized quantifiers to natural language semantics. The subject NP consists of a determiner and a noun N. Both the noun and the verb phrase VP have sets as extensions, and so the determiner is naturally taken to denote a binary relation between sets, i. An utterance of 17 has a discourse universe in the background say, the set of people at a particular university , but the meaning of most , every , at least five and similar expressions is not tied to particular universes.

For example, the meaning of all in. It simply stands for the inclusion relation, regardless of what we happen to be talking about. Therefore, the generalized quantifier all , which with each universe M associates the inclusion relation over M , is eminently suitable to interpret all , and similarly for other determiners. However, it is characteristic of sentences of the form 17 that the noun argument and the VP argument are not on a par.

## Generalized Quantifiers (Stanford Encyclopedia of Philosophy)

Thus, at least five students denotes the set of subsets of the universe which contain at least five students. This holds also for some NPs that lack determiners, such as proper names. This is in fact well motivated, not only because the interpretation of NPs becomes more uniform, but also because John can combine with quantified NPs:.

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Here it is convenient if John and three professors have the same semantic category. Note that generalized quantifiers—in contrast with individuals! Similarly, the complex NP in. The difference in syntactic status between these two arguments turns out to have a clear semantic counterpart. This can be seen from sentence pairs such as the following, where it is clear that the second sentence is just an awkward way of expressing the first:.

The reason is that it is characteristic of determiner denotations that the restriction argument restricts the domain of quantification to that argument. Actually, the idea of domain restriction has one further ingredient. This in turn is an instance of a more general property, applicable to arbitrary generalized quantifiers:. That is, nothing happens when the universe is extended, or shrunk, as long as the arguments are not changed. We can now see in b below that the combination of Conserv and Ext amounts to exactly the same thing:.

Again, all determiner denotations appear to satisfy Ext. At first sight, nothing in principle would seem to prevent a language from containing a determiner, say evso , which meant every on universes with less than 10 elements and some on larger universes. Indeed, many quantifiers from language and logic are Ext. Yet one is inclined to say for them too that they mean the same on every universe.

The crux here is thing. When Ext holds, we can usually drop the subscript M and write, for example,. That is, a suitable universe can be presupposed but left in the background. Other properties are not shared by all natural language quantifiers but single out important subclasses. We mentioned two already in section 2 above: symmetry and monotonicity. Typical symmetric quantifiers are some, no, at least five, exactly three, an even number of, infinitely many, whereas all, most, at most one-third of the are non-symmetric.

We noted that some of the syllogisms express monotonicity properties. Similarly for left increasing or decreasing, and indeed for monotonicity in any given argument place of a generalized quantifier. Monotonicity is ubiquitous among natural language quantifiers.

Grammars and decidability problems