Advanced Proof Viewing in ProofTool

Sequent calculus is widely used for formalizing proofs. However, due to the proliferation of data, understanding the proofs of even simple mathematical arguments soon becomes impossible. Graphical user interfaces help in this matter, but since they normally utilize Gentzen's original notation, some of the problems persist. In this paper, we introduce a number of criteria for proof visualization which we have found out to be crucial for analyzing proofs. We then evaluate recent developments in tree visualization with regard to these criteria and propose the Sunburst Tree layout as a complement to the traditional tree structure. This layout constructs inferences as concentric circle arcs around the root inference, allowing the user to focus on the proof's structural content. Finally, we describe its integration into ProofTool and explain how it interacts with the Gentzen layout.Comment: In Proceedings UITP 2014, arXiv:1410.785

Towards Deciding Second-order Unification Problems Using Regular Tree Automata

International audienceThe second-order unification problem is undecidable [5]. While unification procedures, like Huet's pre-unification, terminate with success on unifiable problems, they might not terminate on non-unifiable ones. There are several decidability results for unification problems with infinitely-many pre-unifiers, such as for monadic second-order problems [3]. These results are based on the regular structure of the solutions of these problems and by computing minimal unifiers. Beyond the importance of the knowledge that searching for unifiers of decidable problems always terminates, one can also use this information in order to optimize unification algorithms, such as in the case for pattern unification [10]. Nevertheless, being able to prove that the unification problem of a certain class of unification constraints is decidable is far from easy. Some results were obtained for certain syntactic restrictions on the problems (see Levy [8] for some results and references) or on the unifiers (see Schmidt-Schauß [11], Schmidt-Schauß and Schulz [12, 13] and Je˙ z [7] for some results). Infinitary unification problems, like the ones we are considering, might suggest that known tools for dealing with the infinite might be useful. One such tool is the regular tree automaton. The drawback of using regular automata for unification is, of course, their inability to deal with variables. In this paper we try to overcome this obstacle and describe an ongoing work about using regular tree automata [1] in order to decide more general second-order unification problems. The second-order unification problems we will consider are of the form λz n .x 0 t. = λz n .C(x 0 s) where C is a non-empty context [2] and x 0 does not occur in t or s. We will call such problems cyclic problems. An important result in second-order unification was obtained by Ganzinger et al. [4] and stated that second-order unification is undecidable already when there is only one second-order variable occurring twice. The unification problem they used for proving the undecidability result was an instance of the following cyclic problem. Note that we chose to use in the definition only unary second-order variables but that this restriction should not be essential. x 0 (w 1 , g(y 1 , a)) = g(y 2 , x 0 (w 2 , a)) (1) Our decidability result is obtained by posing one further restriction over cyclic problems which is based on the existence and location of variables other than the cyclic one. A sufficient condition for the decidability of second-order unification problems was given by Levy [8]. This condition states that if we can never encounter, when applying Huet's pre-unification procedure [6] to a problem, a cyclic equation, then the procedure terminates. It follows from this result that deciding second-order unification problems depends on the ability to decide cyclic problems. The rules of Huet's procedure (PUA) are given in Fig. 1. Imitation partial bindings and projection partial bindings are defined in [14] and are denoted, respectively, by PB(f, α) and PB(i, α) where α is a type, Σ a signature f ∈ Σ and i > 0

Effect of Nursery Pen Group Zize on Performance and Weight Variation of Pigs Weaned at 10 to 20 and 20 to 30 Days of Age

Current trends in the swine industry are for management of pigs in larger groups well beyond that which has been recommended. With larger groups there is the potential for higher stocking rates because of more effective pig space per pen, savings in cost of dividing fences, and, in some cases, feeding equipment. However, with larger groups, more initial weight variation within a pen would be expected due to less sorting and more social competition because of more pigs per pen. It is unclear if these potentially negative factors would result in poorer pig performance during the nursery period and/or more weight variation as pigs leave the nursery negating the savings in fixed building costs. The research reported herein was designed to evaluate the effect of doubling nursery pig group size from 24 to 48 on performance, survival, and weight variation for pigs weaned at 1 O to 20 and 20 to 30 days of age and provided the same unit of floor space per pig

Effect of Body Fat Measurement at Breeding and Gestation Energy Level on Farrowing and Lactation Performance of First Parity Sows

Records kept by the swine industry reveal that a large precentage of females entering the sow herd are culled prior to reaching their peak reproductive performance expected in their fourth to sixth parity. In herds with low replacement rates over 30% of the females are culled and in herds with high replacement rates this figure exceeds 50% annually. Of those sows not culled for health or structural problems, most are removed because they fail to return to estrus or fail to maintain pregnancy after their first or second litter. Low feed intake and excessive weight loss during lactation have been associated with this problem, particularly with first parity sows from very productive genetic lines. Many believe that selection of replacement females from lean genotypes and limit feeding the filts during the development stage has exacerbated the problem. Thin sows are often prevalent among the sows that are culled. However, research has demonstrated that absolute body condtion is not of itself associated with reproductive failure. Some feel that it is the amount of fat lost that is the problem, not the resulting body fat level after lactation. Sows with low feed intake and substantial body weight loss during lactation are among those most often culled due to failure to recycle. Sows with low feed intake are thinner than those with more desirable feed intake and have lost both body fat and body protein as they have lost weight. A feeding strategy that would increase lactation feed intake and allow sows to maintain body weight and condition during lactation, reguardless of their starting body fat level, might allow more sows to reach their potential for reproductive efficiency. The research reported herein was designed to evaluate the effect of body backfat level at breeding and the effect gestation energy level on sow lactation feed intake, body weight change, and return to strus of first parity sows

Scrapie

Scrapie is a disease of sheep and goats caused by a very small infectious agent. The agent is smaller than a virus and its exact identity is still unknown. The agent infects the central nervous system (brain) and, after a long incubation period (years) , affected animals show clinical signs of itching, incoordination, stumbling in the hind legs, head tremor, and weight loss. Although not all clinical signs are seen in every affected animal, the disease is ultimately fatal

Effect of Nutrient Deletion from Lactation Diets on Feed Consumption and Weight Change of Sows Intended to be Culled and on Litter Performance During Lactation and Subsequent Nursery Phase

Sows that are destined to be culled from the herd need only produce milk to support adequate litter growth during the last lactation and need not maintain nutrient stores to support subsequent reproduction processes. Most reduction in lactation performance has been associated with nutrient deficiencies during the gestation period, not nutrient deficiences during lactation. Supplemental sources of trace minerals, vitamins, and calcium (Ca) and phosphorus (P) add cost to the lactation diet. If lactation performance is not affected by nutrient reduction, considerable feed savings could be realized for sows to be culled. Pig performance after weaning is also of concern. Pigs with heavier weights within a weaning group tend to maintain heavier weights in subsequent growth periods. However, pigs that have lighter weights due to nutrient deficiencies tend to exhibit compensatory growth in subsequent growth periods. If weaning weight is affected by nutrient reduction in the lactation diet, evaluation of subsequent pic performance is important. Detrimental effects of nutrient deficiencies on return to estrus after weaning need to be documented to demonstrate the importance of proper nutrition for sows intended to remain in the reproduction pool. Also of concern is the weight of the sow at selling time and its effects on salvage value

Creating Artificial Ice States Using Vortices in Nanostructured Superconductors

We demonstrate that it is possible to realize vortex ice states that are analogous to square and kagome ice. With numerical simulations, we show that the system can be brought into a state that obeys either global or local ice rules by applying an external current according to an annealing protocol. We explore the breakdown of the ice rules due to disorder in the nanostructure array and show that in square ice, topological defects appear along grain boundaries, while in kagome ice, individual defects appear. We argue that the vortex system offers significant advantages over other artificial ice systems.Comment: 4 pages, 4 postscript figures; version to appear in Phys. Rev. Let