Polynomial Size Analysis of First-Order Shapely Functions

We present a size-aware type system for first-order shapely function definitions. Here, a function definition is called shapely when the size of the result is determined exactly by a polynomial in the sizes of the arguments. Examples of shapely function definitions may be implementations of matrix multiplication and the Cartesian product of two lists. The type system is proved to be sound w.r.t. the operational semantics of the language. The type checking problem is shown to be undecidable in general. We define a natural syntactic restriction such that the type checking becomes decidable, even though size polynomials are not necessarily linear or monotonic. Furthermore, we have shown that the type-inference problem is at least semi-decidable (under this restriction). We have implemented a procedure that combines run-time testing and type-checking to automatically obtain size dependencies. It terminates on total typable function definitions.Comment: 35 pages, 1 figur

Pairwise Stability in Two Sided Market with Strictly Increasing Valuation Functions

This paper deals with two-sided matching market with two disjoint sets, i.e. the set of buyers and the set of sellers. Each seller can trade with at most with one buyer and vice versa. Money is transferred from sellers to buyers for an indivisible goods that buyers own. Valuation functions, for participants of both sides, are represented by strictly increasing functions with money considered as discrete variable. An algorithm is devised to prove the existence of stability for this model.Comment: 10 pages, no figure

Observation of implicit complexity by non confluence

We propose to consider non confluence with respect to implicit complexity. We come back to some well known classes of first-order functional program, for which we have a characterization of their intentional properties, namely the class of cons-free programs, the class of programs with an interpretation, and the class of programs with a quasi-interpretation together with a termination proof by the product path ordering. They all correspond to PTIME. We prove that adding non confluence to the rules leads to respectively PTIME, NPTIME and PSPACE. Our thesis is that the separation of the classes is actually a witness of the intentional properties of the initial classes of programs

Learning Cooperative Games

This paper explores a PAC (probably approximately correct) learning model in cooperative games. Specifically, we are given $m$ random samples of coalitions and their values, taken from some unknown cooperative game; can we predict the values of unseen coalitions? We study the PAC learnability of several well-known classes of cooperative games, such as network flow games, threshold task games, and induced subgraph games. We also establish a novel connection between PAC learnability and core stability: for games that are efficiently learnable, it is possible to find payoff divisions that are likely to be stable using a polynomial number of samples.Comment: accepted to IJCAI 201

Type Checking and Weak Type Inference for Polynomial Size Analysis of First-Order Functions

Abstract. We present a size-aware type system for first-order shapely functions. Here, a function is called shapely when the size of the result is determined exactly by a polynomial in the sizes of the arguments. Examples of shapely functions are matrix multiplication and the Cartesian product of two lists. The type checking problem for the type system is shown to be undecidable in general. We define a natural syntactic restriction such that the type checking becomes decidable, even though size polynomials are not necessarily linear. Furthermore, an algorithm for weak type inference for this system is given

Back to the Future: Economic Self-Organisation and Maximum Entropy Prediction

This paper shows that signal restoration methodology is appropriate for predicting the equilibrium state of certain economic systems. A formal justification for this is provided by proving the existence of finite improvement paths in object allocation problems under weak assumptions on preferences, linking any initial condition to a Nash equilibrium. Because a finite improvement path is made up of a sequence of systematic best-responses, backwards movement from the equilibrium back to the initial condition can be treated like the realisation of a noise process. This underpins the use of signal restoration to predict the equilibrium from the initial condition, and an illustration is provided through an application of maximum entropy signal restoration to the Schelling model of segregation