Kanazawa, M. (1998). Learnable classes of categorial grammars. Stanford, CA: CSLI Publications.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in 2.2. First, we address the problem of learning from structures in the base logic. 2. 1 Solving type equations by hypothetical reasoning The unification perspective on learning type assignments from structures is well understood we refer the reader to the seminal work of [5] and to [9]. Here we present the problem of solving type assignment equations from a Logic Programming perspective in order to highlight the role of hypothetical reasoning in the process. Consider the standard abstract interpreter for logic programs (see for example [16] The resolution algorithm takes as ....

Kanazawa, M., Learnable classes of categorial grammars. PhD Dissertation, Stanford, 1994.

....implementation, in contrast, does not provide complete information about syntactic structure, and it provides no specific information about the particular semantic or syntactic categories which should be used to generate the language. 4 Learnability of optimally unified lexicons Definition 4. 1 [8] Let , S, L# be a grammar frame, consisting of a set# of grammars, a set S of expressions (sentences) and a function L which maps from the grammars to sets of expressions (i.e. languages) A learning function is a partial function # that maps non empty finite sequences of sentences to 8 ....

....let ## A learning function # is said to learn if the following condition holds: for every language # in L(G) for every infinite sequence i## that enumerates the elements of # (i.e. s N = #) there exists some G in such that L(G) # and # converges to G on i## . [8] 4.1 A negative result Definition 4.3 A class of languages is said to have a limit point if there exists an infinite sequence n## of languages in # # L n # (we call this an infinite ascending chain) and there is another language L in L = L n . The language L is said to ....

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Kanazawa, M., "Learnable Classes of Categorial Grammars," Ph.D. thesis, Stanford University (1994).

.... Institute of Mathematics and Computer Science University of Warmia and Mazury Olsztyn Poland Abstract In this paper we present a more general approach to restricted optimal unification, introduced and appplied to learning algorithms for categorial grammars in [1] and further developed in [6, 7, 3, 4]. In particular, we solve a rather general problem of finding minimal unifiers with respect to some preordering relation between substitutions. 1 Introduction and preliminaries The notion of an optimal unifier of a finite family of finite sets of terms has been introduced in [1] as a tool for ....

....given sample of (structured) expressions of a language. In [6, 7] there are defined and studied optimal unifiers restricted to a set of substitutions; the goal is to make the procedure of determining categorial grammars sensitive to certain negative constraints upon the form of final outcomes. In [3, 4] the (unrestricted) procedures from [1] are used to define computable learning functions for categorial grammars. In this paper we simplify and generalize the framework of restricted optimal unification. Instead of finite families of finite sets of terms we consider finite binary relations ....

M. Kanazawa, Learnable Classes of Categorial Grammars, CSLI Publications, FoLLI, Stanford, 1998.

....to CG, such that for any G and e i i#N any enumeration of L(G) there exist a grammar G # and n 0 N such that #n n 0 #( e 0 , e n = G # . After pessimistic unlearnability results in [11] learnability of non trivial classes has been proved in [2,19] Recent works[12,18] following [6] have answered the problem for di#erent sub classes of classical categorial grammars (the whole class of classical categorial grammars and the whole class of (non) associative Lambek grammars are equivalent to context free grammars and thus is not learnable in Gold s model) In ....

....the indication of the head of each sub component. More complex input informations give natural deduction structure or semantics informations. For k valued categorial grammars , classical categorial grammars [3] noted AB grammars, are learnable from strings, the simplest form of informations[12]. Associative Lambek categorial grammars [14] noted L grammars, are learnable from natural deduction structures [4] but not from strings and sub component trees [9,10] Non associative Lambek categorial grammars [15] noted NL grammars, lie between classical categorial grammars and associative ....

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Kanazawa, M., "Learnable classes of categorial grammars," Studies in Logic, Language and Information, FoLLI & CSLI, 1998, distributed by Cambridge University Press.

....for any G 2 CG and e i i2N any enumeration of L(G) there exist a grammar G ) L(G) and n 0 2 N such that 8n n 0 (fe 0 ; e n g) G . After pessimistic unlearnability results in [Gol67] learnability of non trivial classes has been proved in [Ang80, Shi90] Recent works[Kan98, Nic99] following [BP90] have answered the problem for di erent sub classes of classical categorial grammars (the whole class of classical categorial grammars and the whole class of (non) associative Lambek grammars are equivalent to context free grammars and thus is not learnable in Gold s model) In ....

....the indication of the head of each sub component. More complex input informations give natural deduction structure or semantics informations. For k valued categorial grammars , classical categorial grammars [BH53] noted AB grammars, are learnable from strings, the simplest form of informations[Kan98]. Associative Lambek categorial grammars [Lam58] noted L grammars, are learnable from natural deduction structures [RB01] but not from strings and sub component trees [FL02a, FL02b] Non associative Lambek categorial grammars [Lam61] noted NL grammars, lie between classical categorial grammars ....

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Makoto Kanazawa. Learnable classes of categorial grammars. Studies in Logic, Language and Information. FoLLI & CSLI, 1998. distributed by Cambridge University Press.

....of regular languages identi able in the limit from positive data : k reversible languages. The identi cation algorithm she de ned has now been adapted and used in several richer context to learn some classes of context free grammars [Sak92] or to learn some classes of categorial grammars [Kan98], which go against the idea that regular languages are not enough expressive to be of some interest. This set of works shows that the rst path mentioned above can be successful. This paper is following these leads, and present some new classes of regular languages identi able in the limit by ....

....of all nite languages and at least one in nite language can not be identi ed in the limit from positive data. As a consequence, the class of regular languages is not identi able in the limit from positive data. In this paper, we use the following extension of the Gold s result (see [BB75] and [Kan98]) that 8n; L n ( L n 1 and [ n2N L n 2 L, then L is not identi able in the limit from positive data. We also use the following weaker result from [Kap91] that there exists an in nite sequence (S n ) n2N with 8n; S n ( S n 1 , S n L n and [ n2N S n 2 L, then L is not identi able in the ....

M. Kanazawa. Learnable Classes of Categorial Grammars. The European Association for Logic, Language and Information. CLSI Publications, 1998.

....seuls : les langages k r eversibles. L algorithme d apprentissage qu elle a d e ni a depuis et e adapt e et utilis e dans de nombreux contextes plus riches, pour apprendre certaines classes de grammaires context free [Sak92] ou pour apprendre certaines classes de grammaires cat egorielles [Kan98], r efutant l objection selon laquelle les langages r eguliers sont trop peu expressifs pour qu on doive s y int eresser. Cet ensemble de travaux montre que la premi ere voie signal ee plus haut peut etre fructueuse. Le pr esent article se situe dans cette lign ee, et pr esente de nouvelles ....

....un langage in ni ne peut pas etre identi ee a la limite par exemples positifs seuls. En cons equence, la classe des langages r eguliers n est pas identi able a la limite par exemples positifs. Nous utiliserons a plusieurs reprises l extension suivante du r esultat de Gold (voir [BB75] et [Kan98]) Lemme 2. Si une classe de langages L contient une suite in nie (L n ) n2N de langages tels que 8n; L n ( L n 1 et [ n2N L n 2 L, alors L n est pas identi able a la limite par exemples positifs. 3 D e nitions Parmi les langages r eguliers, la famille des langages k r eversibles est l une ....

M. Kanazawa. Learnable Classes of Categorial Grammars. The European Association for Logic, Language and Information. CLSI Publications, 1998.

....Flor encio costa let.uu.nl UiL OTS (Utrecht University) Trans 10, 3512 JK Utrecht, Netherlands Tel. 31.30.253.6178, Fax: 31.30.253.6000 Abstract. In [Bus87] BP90] discovery procedures for CCGs were de ned that accept a sequence of structures as input and yield a set of grammars. In [Kan98] it was shown that some of the classes based on these procedures are learnable (in the technical sense of [Gol67] In [CF00] it was shown that learning some of these classes by means of a consistent learning function is NP hard. The complexity of learning classes from one particular family, G ....

....question whether a sentence belongs to a language generated by a grammar is called the universal membership problem. A triple h ; S; Li satisfying the above conditions is called a grammar system. A class of grammars is denoted G, a class of languages is denoted L. I will adopt notation from [Kan98] and let FL denote a class of structure languages, to be de ned in Section 3. The corresponding naming function is FL(G) Learning functions are written as , their input sequences as or . 1.1 Constraints on Learning Functions The behaviour of learning functions can be constrained in a ....

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M. Kanazawa. Learnable Classes of Categorial Grammars. CSLI Publications, Stanford University, 1998.

....semantics can help syntax learning. Categorial Grammars are well known for their formalized connection with semantics ( Mon74] DWP81] They provide a good compromise between formalism and linguistic expressivity ( OBW88] Previous works have studied the learnability of such grammars ( Adr92] [Kan98], MO98] but neither of them uses the syntax semantics interface to help the syntactic learning process. Links between Kanazawa s learning strategy and semantic information have been shown in [Tel99] This rst approach is still not satisfactory as it does not avoid combinatorial explosion. This ....

....Grammars Learning a Categorial Grammar consists in identifying the categories assigned to each word of its vocabulary. The rewriting schemas are supposed to be known. The formal learnability of Categorial Grammars has been studied in a variant of the PAC model ( Adr92] and in Gold s model ([Kan96,Kan98]) The most powerful result obtained uses the notion of Structural Example. A Structural Example is derived from a syntactic analysis structure by deleting intermediate categories while preserving the terminal symbols and the reduction schemas used. Example 2. The Structural Examples dervived from ....

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M. Kanazawa. Learnable Classes of Categorial Grammars. The European Association for Logic, Language and Information. CLSI Publications, 1998.

....the full length article was published in [8] c Springer Verlag. UiL OTS (Utrecht University) Trans 10, 3512 JK Utrecht, Netherlands costa let.uu.nl Abstract. In [5, 7] discovery procedures for CCGs were de ned that accept a sequence of structures as input and yield a set of grammars. In [12] it was shown that some of the classes based on these procedures are learnable. The complexity of learning them was still left open. In this paper it is shown that learning some of these classes is NP hard under certain restrictions. Keywords: identi cation in the limit, inductive inference, ....

....language generated by (associated with) G. L is also called the naming function. The question whether a sentence belongs to a language generated by a grammar is called the universal membership problem. A class of grammars is denoted G, a class of languages is denoted L. I will adopt notation from [12] and let FL denote a class of structure languages, to be de ned in Section 3. The corresponding naming function is FL(G) Learning functions are written as , their input sequences as or . The classes discussed in this paper are all indexed families of recursive languages. These are quite ....

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M. Kanazawa. Learnable Classes of Categorial Grammars. CSLI Publications, Stanford University, 1998.

....costa let.uu.nl Abstract. In [Bus87] and [BP90] some discovery procedures for classical categorial grammars were defined. These procedures take a set of structures (strings labeled with derivational information) as input and yield a set of hypotheses in the form of grammars. In [Kan98] learning functions based on these discovery procedures were studied, and it was shown that some of the classes associated with these functions can be identified in the limit (i.e. are learnable) from strings, by a computable function. The time complexity of these functions however was still ....

....question whether a sentence belongs to a language generated by a grammar is called the universal membership problem. A triple ##, S, L# satisfying the above conditions is called a grammar system. A class of grammars is denoted G, a class of languages is denoted L. I will adopt notation from [Kan98] and let FL denote a class of structure languages, to be defined in Section 3. The corresponding naming function is FL(G) Learning functions are written as #, their input sequences as # or # . 1.1 Constraints on Learning Functions The behaviour of learning functions can be constrained in a ....

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Makoto Kanazawa. Learnable Classes of Categorial Grammars. CSLI Publications, Stanford University, 1998.

....Avalanche of Hypotheses Christophe Costa Flor encio UiL OTS, Utrecht University 1 Introduction In [Bus87, BP90] discovery procedures for CCGs were de ned that accept a sequence of structures as input and yield a set of grammars. In [Kan98] it was shown that some of the classes based on these procedures are learnable, in the technical sense of identi ability in the limit. It was left an open question just how many hypotheses these discovery procedures can generate. In this paper it will be shown that all but one of these ....

....[Nic99] and [Moo01] and are based on generalizations of uni cation. These seem to have applications in other domains, such as type checking. Therefore the implications may reach beyond the elds of machine learning, inductive inference and the like. 2 Size Complexity Of Grammar Classes In [Kan98] learning functions were proposed that are based on procedures that, given a sample of structures from a language. yield nite sets of grammars consistent with that sample. The learning functions then select a (minimal) grammar from such a set. In algorithmic terms this is a generate and test ....

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M. Kanazawa. Learnable Classes of Categorial Grammars. CSLI Publications, Stanford University, 1998.

....consists in identifying the categories assigned to each word of its vocabulary, when the rewriting schemas (i.e. Forward Application, Backward Application, etc) are supposed to be known. The formal learnability of AB Categorial Grammars in Gold s model has been deeply studied in [Kan96] [Kan98]. Kanazawa has proved that the class Gk of Categorial Grammars assigning at most k di erent categories with any given word is learnable in Gold s model from positive Structural Examples. When k = 1, the grammars are called rigid and they can be eciently learned. When k 1, the learnability is ....

M. Kanazawa. Learnable Classes of Categorial Grammars. The European Association for Logic, Language and Information. CLSI Publications, 1998.

.... and unification in Lambek categorial grammars Annie Foret Email : foret irisa.fr IRISA and University of Rennes1, FRANCE December 18, 2001 Abstract Recently, learning algorithms in Gold s model have been proposed for some particular classes of classical categorial grammars [Kan98] We are interested here in learning Lambek categorial grammars. In general grammatical inference uses unification and substitution. In the context of Lambek categorial grammars it seems appropriate to incorporate an operation on types based both on deduction (Lambek derivation) and on ....

....; e n g) G 2 G with L(G ) L(G) One good reason to use categorial grammars in a learning perspective is that they are fully lexicalized : the rules are already known, only types assigned to words have to be derived from examples. Essential known results on this subject may be found in [Kan98] The learning technique avoids to add a new type each time there is a new use of a word in an example, but applies a unification algorithm instead. One important case is when we limit the number of types per word for example to only one type. Our aim is to explore learning mechanisms for Lambek ....

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Makoto Kanazawa. Learnable classes of categorial grammars. Studies in Logic, Language and Information. FoLLI & CSLI, 1998. distributed by Cambridge University Press.

....the following form (# (#) stands for leftheaded (rightheaded) application structures) # # # # A if # # A B and # # B # # # # A if # # B and # # B A In a language with . one can add a clause for headless structures: # # # # A . B if # # A and # # B [Buszkowski Penn 90, Kanazawa 94] Contents First Last Prev Next # 2.4. Factoring We obtain optimal typing by factoring: contraction of unifiable type assignments. factors( member(X:Type, X:Type1 T] factors( Atom T] Z) unify(Type,Type1) member(Atom,T) member(U, X T] factors(T,Z) member(U,T) ....

Kanazawa, M., Learnable classes of categorial grammars. PhD Dissertation, Stanford, 1994.

....on parsing and generating strategies, but this was not the topic of the paper. Learning is also in the scope of such approaches : we could probably draw many teachings from works already done in the Categorial Grammar framework by M. Kanazawa, W. Buszkowski, G. Penn etc. see for instance [6]) But let us terminate on a deep reason to wish such a kind of mathematical model: is is simply because it allows us to make predictions exactly like it is the case in every experimental science, those predictions being confirmed or not. Correct predictions increase the fiability of the model and ....

M. Kanazawa, Learnable Classes of Categorial Grammars, CSLI & FoLLI, Stanford, 1997.

....from traditional propositional calculus one obtains the Lambek calculus, which can be interpreted as a categorial grammar. This grammatical formalism is known to be equivalent with context free grammars( 12] Various authors have proved learnability results for categorial grammars( 25] 26] [3]) The Lambek calculus can be seen as a variant of the so called substructural logics. Research in logic in the past decennia ( 9] has shown that the landscape between the traditional propositional calculus and the predicate calculus is inhabited by a rich variety of systems: modal logics, ....

Kanazawa, M. Learnable Classes of Categorial Grammars, PhD thesis, University of Stanford, 1994.

....lin guistic data has been studied in Buszkowski [1, 2] as well as van Ben them [9] Further modications of the initial algorithms were presented in Buszkowski Penn [3] and Marciniec [8] All the procedures men tioned make use of unication of nite sets. The learning procedure given by Kanazawa [4, 5] admits as an input innite sequences of structures but only those which constitute the whole categorial language. An attempt to admitting arbitrary innite sets of postulates leads to the need of considering unication with respect to innite sets. Fundamental concepts related to unication maintain ....

....of the whole set. The above fact, employed in the theory of categorial grammar, makes it possible to dene a notion of compact consequence operator on the universe of functor argument structures. Tarski s conditions, in particular compactness, of this operator can be shown using methods of Kanazawa [4, 5], based on an analysis of categorial grammar. However, in the scope of consistent data, it is possible to prove these facts in a dioeerent (more fundamental) way, by reducing them to general properties of unication of innite sets. The paper is organised as follows. At rst we establish the ....

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M. Kanazawa, Learnable Classes of Categorial Grammars, Disserta tion, Stanford University, 1994.

.... grammar : it has been proved that even the class of regular languages is not learnable from positive examples in usual models of learning ( 4, 14] To overcome this difficulty, a recently investigated solution consists in providing Structural Examples to the learner instead of strings of words ([2, 6, 7, 10, 11]) A Structural Example is a more or less simplified version of the syntactic (or analysis) tree. But this solution is not very satisfying from a cognitive point of view, as Structural Examples seem to be very unnatural species. The purpose of this article is to provide a new interpretation of ....

....in identifying a formal grammar from Structural Examples. It has been recently studied and partly solved when the set of rules is partitioned into one class (i.e. with skeletons) in [10, 11] or, like in the example, into two classes for Classical Categorial Grammars (or AB Categorial Grammars) in [2, 6, 7]. Some of the algorithms providing a solution to this new problem are computationally efficient. But, when provided with sentences, trying every possible composition based on these sentences is computationally highly expensive in space and time and the result is a set of many compositions among ....

M. Kanazawa, "Learnable classes of Categorial Grammars", CSLI Publications and FoLLI, Studies in Logic, Language and Information, 1998.

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Kanazawa, M. (1998). Learnable classes of categorial grammars. Stanford, CA: CSLI Publications.

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Kanazawa, M., \Learnable Classes of Categorial Grammars," Ph.D. thesis, Stanford University (1994).

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M. Kanazawa, Learnable classes of categorial grammars, Studies in Logic, Language and Information, FoLLI & CSLI, 1998, distributed by Cambridge University Press.

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Kanazawa, M. Learnable Classes of Categorial Grammars. CSLI Publications and folli, 1988.

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Kanazawa, Makoto (1994), Learnable Classes of Categorial Grammars. Ph.D. Dissertation, Stanford University.

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Kanazawa, M. (1994), Learnable Classes of Categorial Grammars. Ph.D. Dissertation. Stanford.

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