An Introduction to Mechanized Reasoning

Mechanized reasoning uses computers to verify proofs and to help discover new theorems. Computer scientists have applied mechanized reasoning to economic problems but -- to date -- this work has not yet been properly presented in economics journals. We introduce mechanized reasoning to economists in three ways. First, we introduce mechanized reasoning in general, describing both the techniques and their successful applications. Second, we explain how mechanized reasoning has been applied to economic problems, concentrating on the two domains that have attracted the most attention: social choice theory and auction theory. Finally, we present a detailed example of mechanized reasoning in practice by means of a proof of Vickrey's familiar theorem on second-price auctions

Formal representation and proof for cooperative games

In this contribution we present some work we have been doing in representing and proving theorems from the area of economics, and mainly we present work we will do in a project in which we will apply mechanised theorem proving tools to a class of economic problems for which very few general tools currently exist. For mechanised theorem proving, the research introduces the field to a new application domain with a large user base; more specifically, the researchers are collaborating with developers working on state-of-the-art theorem provers. For economics, the research will provide tools for handling a hard class of problems; more generally, as the first application of mechanised theorem proving to centrally involve economic theorists, it aims to properly introduce mechanised theorem proving techniques to the discipline.\u

The ForMaRE Project - Formal Mathematical Reasoning in Economics

The ForMaRE project applies formal mathematical reasoning to economics. We seek to increase confidence in economics' theoretical results, to aid in discovering new results, and to foster interest in formal methods, i.e. computer-aided reasoning, within economics. To formal methods, we seek to contribute user experience feedback from new audiences, as well as new challenge problems. In the first project year, we continued earlier game theory studies but then focused on auctions, where we are building a toolbox of formalisations, and have started to study matching and financial risk. In parallel to conducting research that connects economics and formal methods, we organise events and provide infrastructure to connect both communities, from fostering mutual awareness to targeted matchmaking. These efforts extend beyond economics, towards generally enabling domain experts to use mechanised reasoning.Comment: Conference on Intelligent Computer Mathematics, 8--12 July, Bath, UK. Published as number 7961 in Lecture Notes in Artificial Intelligence, Springe

Proving soundness of combinatorial Vickrey auctions and generating verified executable code

Using mechanised reasoning we prove that combinatorial Vickrey auctions are soundly specified in that they associate a unique outcome (allocation and transfers) to any valid input (bids). Having done so, we auto-generate verified executable code from the formally defined auction. This removes a source of error in implementing the auction design. We intend to use formal methods to verify new auction designs. Here, our contribution is to introduce and demonstrate the use of formal methods for auction verification in the familiar setting of a well-known auction

N,N'-dimethylperylene-3,4,9,10-bis(dicarboximide) on alkali halide(001) surfaces

The growth of N,N'-dimethylperylene-3,4,9,10-bis(dicarboximide) (DiMe-PTCDI) on KBr(001) and NaCl(001) surfaces has been studied. Experimental results have been achieved using frequency modulation atomic force microscopy at room temperature under ultra-high vacuum conditions. On both substrates, DiMe-PTCDI forms molecular wires with a width of 10 nm, typically, and a length of up to 600 nm at low coverages. All wires grow along the [110] direction (or [1$\bar{1}$0] direction, respectively) of the alkali halide (001) substrates. There is no wetting layer of molecules: Atomic resolution of the substrates can be achieved between the wires. The wires are mobile on KBr surface but substantially more stable on NaCl. A p(2 x 2) superstructure in brickwall arrangement on the ionic crystal surfaces is proposed based on electrostatic considerations. Calculations and Monte-Carlo simulations using empirical potentials reveal possible growth mechanisms for molecules within the first layer for both substrates, also showing a significantly higher binding energy for NaCl(001). For KBr, the p(2 x 2) superstructure is confirmed by the simulations, for NaCl, a less dense, incommensurate superstructure is predicted.Comment: 5 pages, 5 figure

Fragmentation of ice by low velocity impact

Low velocity impact experiments (0.14 to l km/s) carried out in polycrystalline water ice targets at 257 and 81 K resulted in interactions which can be assigned to four fragmentation classes, cratering, erosion, disruption, and total fragmentation. Specific kinetic energies for the transitions between these classes range from l x 10^5 to 7 x 10^5 ergs/g for 81 K ice and from 3 x 10^5 to ~ 2 x 10^6 ergs/g for 257 K ice. These values are about one to two orders of magnitude below those for silicate rocks. The mass vs. cumulative number distribution of fragments in our experiments can be described by a simple power law, similar to that observed in fragmented rocks in both the laboratory and in nature. The logarithmic slopes of cumulative number vs. fragment weight vary between - 0.9 and - 1.8 decreasing with increasing projectile energy and are approximately independent of target temperature. The shapes of fragments resulting from erosion and disruption of ice targets are significantly less spherical for 257 K targets than for 81 K targets. Fragment sphericity increases with increasing projectile energy at 257 K, but no similar trend is observed for 81 K ice. Our results support the hypothesis that the specific projectile energy is a measure for target comminution for a relatively wide range of projectile energies and target masses. We apply our results to the collisional interaction of icy planetary bodies and find that the complete destruction of a target body with radii between 50 m and 100 km· range from 10^(17) to 10^(27) ergs. Energies corresponding to basaltic bodies of the same size range from 10^(18) to 10^(28) ergs. Our experiments suggest that regolith components on icy planets resemble those on rocky planetary bodies in size and shape. We predict that the initial shapes of icy particles in the Saturnian ring system were roughly spherical. The initial mass distribution of ring particles should follow a power law with a slope of ~ - 1.5