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On the learnability of Boolean formulae

M. Kearns, M. Li, L. Pitt, L. Valiant
1987 Proceedings of the nineteenth annual ACM conference on Theory of computing - STOC '87  
Acknowledgements We w ould like to thank Umesh Vazirani for many helpful ideas and discussions on learning under uniform distributions.  ...  In section 3 w e discuss closure under boolean operations on the members of the learnable classes.  ...  Since learnability m a y depend on the representation chosen, we de ne learnability of a class of representations of boolean concepts, as opposed to de ning learnability of the concepts themselves.  ... 
doi:10.1145/28395.28426 dblp:conf/stoc/KearnsLPV87 fatcat:q7pi5rwh2vhcvmmljaklerwcay

Understanding Boolean Function Learnability on Deep Neural Networks [article]

Anderson R. Tavares, Pedro Avelar, João M. Flach, Marcio Nicolau, Luis C. Lamb, Moshe Vardi
2021 arXiv   pre-print
Computational learning theory states that many classes of boolean formulas are learnable in polynomial time.  ...  Specifically, we analyse boolean formulas associated with the decision version of combinatorial optimisation problems, model sampling benchmarks, and random 3-CNFs with varying degrees of constrainedness  ...  Moshe Vardi is supported in part by NSF grants IIS-1527668, CCF-1704883, IIS-1830549, DoD MURI grant N00014-20-1-2787, and an award from the Maryland Procurement Office.  ... 
arXiv:2009.05908v2 fatcat:ry2evqrirnefzizzhppafjtx5m

Can complexity theory benefit from Learning Theory? [chapter]

Tibor Hegedüs
1993 Lecture Notes in Computer Science  
Using known results on efficient query-learnability of some Boolean concept classes, we prove several (co-NP-completeness) results on the complexity of certain decision problems concerning representability  ...  of general Boolean functions in special forms.  ...  One can show that f ~ 1 if and only if g can be expressed as a read-once DNF. The case of read-twice DNF formulas is handled analogously.  ... 
doi:10.1007/3-540-56602-3_150 fatcat:6owkpchugbcvbcnv4rviw2u2fy

Almost all monotone Boolean functions are polynomially learnable using membership queries

Ilya Shmulevich, Aleksey D. Korshunov, Jaakko Astola
2001 Information Processing Letters  
It is shown that almost all monotone Boolean functions are polynomially identifiable in the input number of variables as well as the output being the sum of the sizes of the CNF and DNF representations  ...  We consider exact learning or identification of monotone Boolean functions by only using membership queries.  ...  In the context of Boolean formulae, the algorithm queries an oracle for the value of the formula f on a particular variable assignment a.  ... 
doi:10.1016/s0020-0190(00)00225-8 fatcat:lluk3uo3h5cx5a44oiekxsk6zm

Exact Learning Boolean Functions via the Monotone Theory

N.H. Bshouty
1995 Information and Computation  
We study the learnability of boolean functions from membership and equivalence queries.  ...  We develop the Monotone Theory that proves 1 Any boolean function is learnable in polynomial time in its minimal DNF size, its minimal CNF size and the number of variables n.  ...  On the other hand, many other results gave subclasses of boolean functions that are learnable in their DNF representations.  ... 
doi:10.1006/inco.1995.1164 fatcat:bjug3shh2ffbtnyzcwaxvbhi4m

A Dichotomy Theorem for Learning Quantified Boolean Formulas

Víctor Dalmau
1999 Machine Learning  
We consider the following classes of quantified boolean formulas. Fix a finite set of basic boolean functions.  ...  We prove the following dichotomy theorem: For any set of basic boolean functions, the resulting set of formulas is either polynomially learnable from equivalence queries alone or else it is not PAC-predictable  ...  Later on, we will use this property to show the learnability of quantified boolean formulas constructed using these families of basis. Let us state the main result of this section: Lemma 7.  ... 
doi:10.1023/a:1007582729656 dblp:journals/ml/Dalmau99 fatcat:c36p2kzlgbc7xg2cc47izw7q6e

Learning decision lists

Ronald L. Rivest
1987 Machine Learning  
normal form with at most k literals per term), and decision trees of depth k, our result strictly increases the set of functions that are known to be polynomially learnable, in the sense of Valiant (1984  ...  Since k-DL properly includes other well-known techniques for representing Boolean functions such as h-CNF (formulae in conjunctive normal form with at most k literals per clause), k-DNF (formulae in disjunctive  ...  the referees for their very helpful comments on an earlier version of this paper.  ... 
doi:10.1007/bf00058680 fatcat:7m4sbslcm5bsflben4w6regl6m

On the learnability of Zn-DNF formulas (extended abstract)

Nader H. Bshouty, Zhixiang Chen, Scott E. Decatur, Steven Homer
1995 Proceedings of the eighth annual conference on Computational learning theory - COLT '95  
Although many learning problems can be reduced to learning Boolean functions, in many  ...  This shows that the difficult y of learning Boolean DNF formulas lies in the fact that the domain is small.  ...  In Blum and Singh [BS90] it was proved that for any constant k, Boolean functions of k monomials are pat-learnable by the more expressive hypothesis class of general DNF formulas.  ... 
doi:10.1145/225298.225322 dblp:conf/colt/BshoutyCDH95 fatcat:hyakjt5uanhtfdoutcko5gqleu

Computational learning theory

Dana Angluin
1992 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing - STOC '92  
Learning in the presence of malicious errors. In Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, pages 267-280. ACM Press, 1988. learnability of boolean formulae.  ...  6.1 Classes of boolean formulas Valiant [131] shows monomials and k-CNF formulas are properly PAC-learnable using only positive examples.  ... 
doi:10.1145/129712.129746 dblp:conf/stoc/Angluin92 fatcat:7aw3cnd745bellyhu7phywpul4

Learning from recursive, tree structured examples [chapter]

P. Jappy, M. C. Daniel-Vatonne, O. Gascuel, C. Higuera
1994 Lecture Notes in Computer Science  
We show that in the PAC framework def'med by Valiant [10], the extensions to this model of two Boolean formula classes: k-DNF and k-DL, remain polynomially learnable.  ...  A signature enables us to define the set of all a/lowed (partial or complete) representations. This model properly contains Boolean representations.  ...  This is why we focus on PAC learning and show (in Section 3) that the natural extensions of two of the main Boolean formula classes, k-DNF and k-DL, stay polynomially learnable.  ... 
doi:10.1007/3-540-57868-4_75 fatcat:o67oadxwxzhlnkt2etsddda6iq

Exact Learning when Irrelevant Variables Abound [chapter]

David Guijarro, Víctor Lavín, Vijay Raghavan
1999 Lecture Notes in Computer Science  
Any Boolean function of O(logn) relevant variables can be exactly learned with a set of non-adaptive membership queries alone and a minimum sized decision tree representation of the function constructed  ...  We show that truth-table minimization of decision trees can be done in polynomial time, complementing the well-known result of Masek that truth-table minimization of DNF formulas is NP-hard.  ...  Domingo for communicating the problem of exact learning Boolean functions of few relevant variables with membership queries. We also thank L.  ... 
doi:10.1007/3-540-49097-3_8 fatcat:cuf62jnrsfhdpcj4gnj4o45ona

Nonuniform learnability

Gyora M. Benedek, Alon Itai
1994 Journal of computer and system sciences (Print)  
Some examples (Boolean formulae, recursive, and r.e. sets) are shown to be nonuniformly learnable by a polynomial (in the size of the representation of the concept and in the error parameters) number of  ...  The learning model of Valiant is extended to allow the number of examples required for learning to depend on the particular concept to be learned, instead of requiring a uniform bound for all concepts  ...  ., "is the set of Boolean formulae polynomially learnable?" We show that Boolean formulae are learnable by polynomially many examples. However, the learning time of our algorithm is not polynomial.  ... 
doi:10.1016/s0022-0000(05)80005-4 fatcat:4xlsvlxn25fvrbgd74x3u2xen4

A dichotomy theorem for learning quantified Boolean formulas

Víctor Dalmau
1997 Proceedings of the tenth annual conference on Computational learning theory - COLT '97  
We consider the following classes of quanti ed boolean formulas. Fix a nite set of basic boolean functions.  ...  We prove the following dichotomy theorem: For any set of basic boolean functions, the resulting set of formulas is either polynomially learnable from equivalence queries alone or else it is not PAC-predictable  ...  If S satis es one of the conditions (a)-(d) below, then C 98-Formula(S) is polynomially exactly learnable with improper equivalence queries.  ... 
doi:10.1145/267460.267496 dblp:conf/colt/Dalmau97 fatcat:nakyr3lzbzaebdame7nfuijd5q

A necessary condition for learning from positive examples

Haim Shvaytser
1990 Machine Learning  
The criterion is applied to several types of Boolean formulae in conjunctive and disjunctive normal form, to the majority function, to graphs with large connected components, and to a neural network with  ...  We present a simple combinatorial criterion for determining concept classes that cannot be learned in the sense of Valiant from a polynomial number of positive-only examples.  ...  With no restrictions on the form of the learned concepts the learnability of a concept class C implies the learnability of all concept classes C' C C.  ... 
doi:10.1007/bf00115896 fatcat:uanjanh4rvauld7hawr57purti

On the limits of proper learnability of subclasses of DNF formulas

Krishnan Pillaipakkamnatt, Vijay Raghavan
1997 Machine Learning  
Second, we show that read-thrice DNF formulas are not properly learnable in the extended PAC model, assuming RP 5& NP.  ...  As a further application of these techniques, we consider read-thrice DNF formulas.  ...  On the other hand, it has been shown that for any constant k, k-term DNF formulas are learnable as k-CNF formulas (Valiant, 1984) , and k-term DNF formulas are learnable as k-terra DNF formulas when the  ... 
doi:10.1007/bf00114011 fatcat:6c66z2yse5ff3gx7nmdqmsrum4
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