Polynomial Pigeonhole Principle (PPP) is an important subclass of TFNP with profound connections to the complexity of the fundamental cryptographic primitives: collision-resistant hash functions and one-way permutations. In contrast to most of the other subclasses of TFNP, no complete problem is known for PPP. Our work identifies the first PPP-complete problem without any circuit or Turing Machine given explicitly in the input, and thus we answer a longstanding open question from [Papadimitriou1994]. Specifically, we show that constrained-SIS, a generalized version of the well-known Short Integer Solution problem (SIS) from lattice-based cryptography, is PPP-complete. In order to give intuition behind our reduction for constrained-SIS, we identify another PPP-complete problem with a circuit in the input but closely related to lattice problems. We call this problem BLICHFELDT and it is the computational problem associated with Blichfeldt's fundamental theorem in the theory of lattices. Building on the inherent connection of PPP with collision-resistant hash functions, we use our completeness result to construct the first natural hash function family that captures the hardness of all collision-resistant hash functions in a worst-case sense, i.e. it is natural and universal in the worst-case. The close resemblance of our hash function family with SIS, leads us to the first candidate collision-resistant hash function that is both natural and universal in an average-case sense. Finally, our results enrich our understanding of the connections between PPP, lattice problems and other concrete cryptographic assumptions, such as the discrete logarithm problem over general groups.
Recently, a syntactic analog @math of the semantic class @math has been defined in @cite_38 , and a complete problem for this class has been identified. It has also been shown that all the classes we described above are subsets of @math . Oracle separations between all these classes are known @cite_23 , with the only exception of whether PLS is contained in PPAD .
@cite_38 @cite_23
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Recognizing analogies, synonyms, antonyms, and associations appear to be four distinct tasks, requiring distinct NLP algorithms. In the past, the four tasks have been treated independently, using a wide variety of algorithms. These four semantic classes, however, are a tiny sample of the full range of semantic phenomena, and we cannot afford to create ad hoc algorithms for each semantic phenomenon; we need to seek a unified approach. We propose to subsume a broad range of phenomena under analogies. To limit the scope of this paper, we restrict our attention to the subsumption of synonyms, antonyms, and associations. We introduce a supervised corpus-based machine learning algorithm for classifying analogous word pairs, and we show that it can solve multiple-choice SAT analogy questions, TOEFL synonym questions, ESL synonym-antonym questions, and similar-associated-both questions from cognitive psychology. For certain subclasses of NP, @math P, or #P characterized by local constraints, it is known that if there exist any problems within that subclass that are not polynomial time computable, then all the problems in the subclass are NP-complete, @math P-complete, or #P-complete. Such dichotomy results have been proved for characterizations such as constraint satisfaction problems and directed and undirected graph homomorphism problems, often with additional restrictions. Here we give a dichotomy result for the more expressive framework of Holant problems. For example, these additionally allow for the expression of matching problems, which have had pivotal roles in the development of complexity theory. As our main result we prove the dichotomy theorem that, for the class @math P, every set of symmetric Holant signatures of any arities that is not polynomial time computable is @math P-complete. The result exploits some special properties of the class @math P and characterizes four distinct tractable ... In multi-class categorization tasks, knowledge about the classes' semantic relationships can provide valuable information beyond the class labels themselves. However, existing techniques focus on preserving the semantic distances between classes (e.g., according to a given object taxonomy for visual recognition), limiting the influence to pairwise structures. We propose to model analogies that reflect the relationships between multiple pairs of classes simultaneously, in the form "p is to q, as r is to s". We translate semantic analogies into higher-order geometric constraints called analogical parallelograms, and use them in a novel convex regularizer for a discriminatively learned label embedding. Furthermore, we show how to discover analogies from attribute-based class descriptions, and how to prioritize those likely to reduce inter-class confusion. Evaluating our Analogy-preserving Semantic Embedding (ASE) on two visual recognition datasets, we demonstrate clear improvements over existing approaches, both in terms of recognition accuracy and analogy completion. This paper starts with the project of finding a large subclass of NP which exhibits a dichotomy. The approach is to find this subclass via syntactic prescriptions. While the paper does not achieve this goal, it does isolate a class (of problems specified by) "monotone monadic SNP without inequality" which may exhibit this dichotomy. We justify the placing of all these restrictions by showing, essentially using Ladner's theorem, that classes obtained by using only two of the above three restrictions do not show this dichotomy. We then explore the structure of this class. We show that all problems in this class reduce to the seemingly simpler class CSP. We divide CSP into subclasses and try to unify the collection of all known polytime algorithms for CSP problems and extract properties that make CSP problems NP-hard. This is where the second part of the title, "a study through Datalog and group theory," comes in. We present conjectures about this class which would end in showing the dichotomy.