A characterization of F-complete type assignments

AbstractThe aim of this paper is to investigate the soundness and completeness of the intersection type discipline (for terms of the (untyped λ-calculus) with respect to the F-semantics (F-soundness and F-completeness).As pointed out by Scott, if D is the domain of a γ-model, there is a subset F of D whose elements are the ‘canonical’ representatives of functions. The F-semantics of types takes into account that theintuitive meaning of “σ→τ” is ‘the type of functions with domain σ and range τ’ and interprets σ→τ as a subset of F.The type theories which induce F-complete type assignments are characterized. It follows that a type assignment is F-complete iff equal terms get equal types and, whenever M has a type ϕ∧ωn, where ϕ is a type variable and ϕ is the ‘universal’ type, the term λz1…zn…Mz1…zn has type ϕ. Here we assume that z1…z.n do not occur free in M

A Dual Characterization of Incentive Efficiency

We show that incentive e cient allocations in economies with adverse se- lection and moral hazard can be determined as optimal solutions to a linear programming problem and we use duality theory to obtain a complete charac- terization of the optima. Our dual analysis identi es welfare e ects associated with the incentives of the agents to truthfully reveal their private information. Because these welfare e ects may generate non-convexities, incentive e cient allocations may involve randomization. Other properties of incentive e cient allocations are also derived

Generalized Satisfiability Problems via Operator Assignments

Schaefer introduced a framework for generalized satisfiability problems on the Boolean domain and characterized the computational complexity of such problems. We investigate an algebraization of Schaefer's framework in which the Fourier transform is used to represent constraints by multilinear polynomials in a unique way. The polynomial representation of constraints gives rise to a relaxation of the notion of satisfiability in which the values to variables are linear operators on some Hilbert space. For the case of constraints given by a system of linear equations over the two-element field, this relaxation has received considerable attention in the foundations of quantum mechanics, where such constructions as the Mermin-Peres magic square show that there are systems that have no solutions in the Boolean domain, but have solutions via operator assignments on some finite-dimensional Hilbert space. We obtain a complete characterization of the classes of Boolean relations for which there is a gap between satisfiability in the Boolean domain and the relaxation of satisfiability via operator assignments. To establish our main result, we adapt the notion of primitive-positive definability (pp-definability) to our setting, a notion that has been used extensively in the study of constraint satisfaction problems. Here, we show that pp-definability gives rise to gadget reductions that preserve satisfiability gaps. We also present several additional applications of this method. In particular and perhaps surprisingly, we show that the relaxed notion of pp-definability in which the quantified variables are allowed to range over operator assignments gives no additional expressive power in defining Boolean relations

On the random satisfiable process

In this work we suggest a new model for generating random satisfiable k-CNF formulas. To generate such formulas -- randomly permute all 2^k\binom{n}{k} possible clauses over the variables x_1, ..., x_n, and starting from the empty formula, go over the clauses one by one, including each new clause as you go along if after its addition the formula remains satisfiable. We study the evolution of this process, namely the distribution over formulas obtained after scanning through the first m clauses (in the random permutation's order). Random processes with conditioning on a certain property being respected are widely studied in the context of graph properties. This study was pioneered by Ruci\'nski and Wormald in 1992 for graphs with a fixed degree sequence, and also by Erd\H{o}s, Suen, and Winkler in 1995 for triangle-free and bipartite graphs. Since then many other graph properties were studied such as planarity and H-freeness. Thus our model is a natural extension of this approach to the satisfiability setting. Our main contribution is as follows. For m \geq cn, c=c(k) a sufficiently large constant, we are able to characterize the structure of the solution space of a typical formula in this distribution. Specifically, we show that typically all satisfying assignments are essentially clustered in one cluster, and all but e^{-\Omega(m/n)} n of the variables take the same value in all satisfying assignments. We also describe a polynomial time algorithm that finds with high probability a satisfying assignment for such formulas

Complexity of Grammar Induction for Quantum Types

Most categorical models of meaning use a functor from the syntactic category to the semantic category. When semantic information is available, the problem of grammar induction can therefore be defined as finding preimages of the semantic types under this forgetful functor, lifting the information flow from the semantic level to a valid reduction at the syntactic level. We study the complexity of grammar induction, and show that for a variety of type systems, including pivotal and compact closed categories, the grammar induction problem is NP-complete. Our approach could be extended to linguistic type systems such as autonomous or bi-closed categories.Comment: In Proceedings QPL 2014, arXiv:1412.810