Sequent Calculus and Equational Programming

Proof assistants and programming languages based on type theories usually come in two flavours: one is based on the standard natural deduction presentation of type theory and involves eliminators, while the other provides a syntax in equational style. We show here that the equational approach corresponds to the use of a focused presentation of a type theory expressed as a sequent calculus. A typed functional language is presented, based on a sequent calculus, that we relate to the syntax and internal language of Agda. In particular, we discuss the use of patterns and case splittings, as well as rules implementing inductive reasoning and dependent products and sums.Comment: In Proceedings LFMTP 2015, arXiv:1507.0759

Improving Efficiency and Scalability of Sum of Squares Optimization: Recent Advances and Limitations

It is well-known that any sum of squares (SOS) program can be cast as a semidefinite program (SDP) of a particular structure and that therein lies the computational bottleneck for SOS programs, as the SDPs generated by this procedure are large and costly to solve when the polynomials involved in the SOS programs have a large number of variables and degree. In this paper, we review SOS optimization techniques and present two new methods for improving their computational efficiency. The first method leverages the sparsity of the underlying SDP to obtain computational speed-ups. Further improvements can be obtained if the coefficients of the polynomials that describe the problem have a particular sparsity pattern, called chordal sparsity. The second method bypasses semidefinite programming altogether and relies instead on solving a sequence of more tractable convex programs, namely linear and second order cone programs. This opens up the question as to how well one can approximate the cone of SOS polynomials by second order representable cones. In the last part of the paper, we present some recent negative results related to this question.Comment: Tutorial for CDC 201

On Reverse Engineering in the Cognitive and Brain Sciences

Various research initiatives try to utilize the operational principles of organisms and brains to develop alternative, biologically inspired computing paradigms and artificial cognitive systems. This paper reviews key features of the standard method applied to complexity in the cognitive and brain sciences, i.e. decompositional analysis or reverse engineering. The indisputable complexity of brain and mind raise the issue of whether they can be understood by applying the standard method. Actually, recent findings in the experimental and theoretical fields, question central assumptions and hypotheses made for reverse engineering. Using the modeling relation as analyzed by Robert Rosen, the scientific analysis method itself is made a subject of discussion. It is concluded that the fundamental assumption of cognitive science, i.e. complex cognitive systems can be analyzed, understood and duplicated by reverse engineering, must be abandoned. Implications for investigations of organisms and behavior as well as for engineering artificial cognitive systems are discussed.Comment: 19 pages, 5 figure

Macronutrient cycling in surface waters

The levels and relative proportions of macronutrients set the conditions for life in surface waters. Man-made disturbances to macronutrient cycling have caused environmental problems such as eutrophication, acidification and global change. In this thesis, macronutrient cycling was studied by performing spatial and temporal large-scale studies of aquatic, terrestrial and atmospheric national monitoring data. Trophic status was found to have a profound impact on nitrate-nitrogen (NO₃-N) concentrations in surface waters. Lakes and streams of the same trophic status displayed opposite NO₃-N patterns. These findings are of great importance when dealing with environmental assessment on the landscape scale, and an awareness of these patterns may also facilitate the design of sampling programs. Trophic status also seems important for trends in total phosphorus (TP) and total organic carbon (TOC) concentrations in boreal and alpine catchments. A temporal study of TP and TOC concentrations showed decreases in nutrient-poor catchments and increases in more nutrient-rich surface waters. Different responses of terrestrial organic matter production and decomposition to temperature increases may be responsible for the observed patterns. Consequently, continued global warming may lead to a stronger polarization between the nutrient-poor northern and the more nutrient-rich southern catchments. Further studies showed that nutrient conditions in soils and surface waters were strongly affected by atmospheric deposition. By using large data-sets on nutrient content in soils and nutrient concentrations in lakes, it was found that carbon to nitrogen ratios (C:N) in the organic soil layer and in lakes increased from the southern to the northern parts of Sweden, resulting in a strong relationship between soil and lake water C:N. The strong relationship was primarily due to the high correlation between nitrogen (N) in organic soil layer and lake N. Large-scale variations in soil C content were not strongly linked to lake C concentrations whereas soil N seemed to leach in the form of NO₃-N to lakes. By calculating catchment soil, lake and river mouth C stocks, it was estimated that about 10 % of Sweden's total terrestrial net ecosystem production is transported through lakes annually. This indicates that the amount of C exported from soils is substantial and that boreal soils maybe less important as a C sink as previously thought. Furthermore, it was found that the colored portion of C was selectively lost and that the decrease in water color was dependent on water retention time. This implies that under conditions predicted in future climate scenarios of increased precipitation, water reaching the seas will be more colored than today. The results from this thesis highlight the importance of atmospheric N deposition and trophic status to macronutrient cycling in both terrestrial and aquatic ecosystems