Formal learning theory studies, among other things, how people can come to as- sociate meanings with words. In this domain quantifiers strongly contrast with say, common nouns. We can imagine a young child learning the meaning ofdog through repeated association of actual dogs with utterances ofdog (“semantic bootstrapping”). It is unclear how the diverse situations across which a particu- lar quantifier may be used in a sentence can contribute to learning its meaning. The idea that quantifier comprehension involves knowing an algorithm to check its proper application makes immense sense in this context.
Van Benthem already suggested the potential utility of the semantic automata framework and Tree of Numbers for quantifier learning in [5]. In 1996, Clark [7] demonstrated the learnability of FO-definable determiners due to the algorithm
L∗proposed by Angluin in 1987, which allows for the learnability ofanyregular set. The algorithm gives both positive and negative information, meaning the procedure is controlled by a “minimally adequate teacher:” basically, the learner is provided with counterexamples. The learner observes strings and builds an “observation table” that can be transformed into the minimal finite automaton accepting the language. L∗is polynomially bounded in the number of states of the minimal DFA and the size of the counterexamples provided by the teacher. Noting that the set of FO quantifiers is indeed a “very proper” subset of the regular languages, in [9] and [10] Clark gives a new proof of their learnability that relies on the nice properties of these quantifiers. Input is in the form of a sequence of pairsDet(A, B) and pairs (m, n)(points in the Tree of Numbers) such that m=∣ A−B ∣ and n=∣ A∩B ∣, e.g. ⟨every(DOG,ANIMAL);(0,3)⟩.
3Of course, every DFA language is recognized by a DPDA that does not make use of its
stack;evencan be recognized by a DPDA that tracks parity with a non-trivial use of its stack,
The output of the algorithm is the automata recognizing the language ofDet: “The learner converges (successfully learns) a quantifier denotation if she posits an automaton that correctly accepts all and only the strings associated with the quantifier’s set of points in the tree of numbers.” Since these quantifiers are uniquely determined by the upper triangle above the Fra¨ıss´e threshold (see Section 3.2), only a finite amount of input is required to distinguishDet from the rest of the possible quantifiers. In fact, there are at most:
n ∑ j=0 j ∑ i=0( j
i), wherenis the Fra¨ıss´e level
strings represented by the top triangle (we can simply enumerate them all); since the positive instances are themselves distinguishing, there are even fewer strings needed; if the learner knows the information is permutation-invariant, there are fewer still. Moreover, there is an upper bound on the number of potential automata4, and these may also be effectively enumerated (see Section
9.2.1) for remarks on this construction). Thus after a finite number of examples, the learner will find the correct automaton.
Learnability of higher-order quantifiers is trickier, since there is not obviously any finite part of the Tree of Numbers which is sufficient to tell them apart from one another. Clark [10] does give the following formulation as a first step: the set of higher-order determiners is learnable if they have finiteVapnik-Chervonenkis dimension, meaning that there is a finite set of points in the Tree of Numbers containing a data point for each determiner in the set that distinguishes it from all the others (“shatters” the set of points).
Gierasimczuk [18] gives two criticisms of utilizing semantic automata in this domain. One we have already mentioned in Section 1.2.3: these methods do not distinguish between the capacity for comprehension and the capacity for production. The former corresponds to model-checking: givenM and φ, de- cide whether M ⊧ φ; the latter corresponds to giving an adquate description of a model: given M, produce φsuch that M ⊧φ [18]. These are equivalent with respect to the descriptive power of the formalisms of grammars and au- tomata, but research shows that verification is easier for people than generation. The second has to do withinferential meaning, another prong of semantic com- petence. Unlike referential meaning, which involves a procedure to decide or directly verify the value ofϕin any situation (like, e.g., a semantic automata), inferential meaning draws on logical relationships among formula to determine their truth value. If ψ⇒ ϕand we know ψ, we may conclude ϕ. The above methods seem to lend no insight in this direction. Szymanik [49] discusses this distinction with respect to complexity concerns: it may be that a sentence is not directly verifiable because it is computationally intractable, yet we can infer its truth indirectly from one or more tractable sentences.5
4The threshold is related to the pumping length of the language, which is the number of
states in the minimal automaton.
5Consider his example: If we know (1)most villagers are communists, (2)most townsmen
Some remaining open questions are:
Question 9.1.3. Can the learnability of higher-order, context-free quantifiers be demonstrated? Is the problem easier if restricted to deterministic context- free quantifiers?
Question 9.1.4. The learnability of regular iterated quantifiers follows from
L∗as in [7], but is there also a Fra¨ıss´e-style argument demonstrating this? Recall that that iterated languages take the form(wi⧈)∗. Using uniform strings
(with all thewiof the same length) is sufficient to tell them apart, so they could
be enumerated following a “zig-zag” pattern in thex−y plane where the point
(x, y)corresponds to strings of the language where #⧈(w) =xand∣wi∣=y. And
as this thesis demonstrates, there is an effective procedure to build the minimal DFA for an iterated quantifier. Is there a point (x, y) in this enumeration, related to the Fra¨ıss´e thresholds forQ1 andQ2, such that once the learner has
seen that much (finite) input, she can identify the correct automaton?