Formally Verified Compositional Algorithms for Factored Transition Systems

Artificial Intelligence (AI) planning and model checking are two disciplines that found wide practical applications. It is often the case that a problem in those two fields concerns a transition system whose behaviour can be encoded in a digraph that models the system's state space. However, due to the very large size of state spaces of realistic systems, they are compactly represented as propositionally factored transition systems. These representations have the advantage of being exponentially smaller than the state space of the represented system. Many problems in AI~planning and model checking involve questions about state spaces, which correspond to graph theoretic questions on digraphs modelling the state spaces. However, existing techniques to answer those graph theoretic questions effectively require, in the worst case, constructing the digraph that models the state space, by expanding the propositionally factored representation of the syste\ m. This is not practical, if not impossible, in many cases because of the state space size compared to the factored representation. One common approach that is used to avoid constructing the state space is the compositional approach, where only smaller abstractions of the system at hand are processed and the given problem (e.g. reachability) is solved for them. Then, a solution for the problem on the concrete system is derived from the solutions of the problem on the abstract systems. The motivation of this approach is that, in the worst case, one need only construct the state spaces of the abstractions which can be exponentially smaller than the state space of the concrete system. We study the application of the compositional approach to two fundamental problems on transition systems: upper-bounding the topological properties (e.g. the largest distance between any two states, i.e. the diameter) of the state spa\ ce, and computing reachability between states. We provide new compositional algorithms to solve both problems by exploiting different structures of the given system. In addition to the use of an existing abstraction (usually referred to as projection) based on removing state space variables, we develop two new abstractions for use within our compositional algorithms. One of the new abstractions is also based on state variables, while the other is based on assignments to state variables. We theoretically and experimentally show that our new compositional algorithms improve the state-of-the-art in solving both problems, upper-bounding state space topological parameters and reachability. We designed the algorithms as well as formally verified them with the aid of an interactive theorem prover. This is the first application that we are aware of, for such a theorem prover based methodology to the design of new algorithms in either AI~planning or model checking

A Formal Analysis of RANKING

We describe a formal correctness proof of RANKING, an online algorithm for online bipartite matching. An outcome of our formalisation is that it shows that there is a gap in all combinatorial proofs of the algorithm. Filling that gap constituted the majority of the effort which went into this work. This is despite the algorithm being one of the most studied algorithms and a central result in theoretical computer science. This gap is an example of difficulties in formalising graphical arguments which are ubiquitous in the theory of computing

Formally Verified SAT-Based AI Planning

We present an executable formally verified SAT encoding of classical AI planning. We use the theorem prover Isabelle/HOL to perform the verification. We experimentally test the verified encoding and show that it can be used for reasonably sized standard planning benchmarks. We also use it as a reference to test a state-of-the-art SAT-based planner, showing that it sometimes falsely claims that problems have no solutions of certain lengths

Iron intake and iron deficiency in young children

INTRODUCTION: Iron deficiency anaemia is recognised as a common problem in young children in the UK (Department of Health, 1994), and even more in Saudi Arabia (Al-Fawaz, 1993 and Al-Hifzi, et al. 1996). However, there is a lack of studies showing how food intake affects iron status in young children in these countries. Such studies are urgently needed to develop informed prevention strategies.These studies have sought to assess iron intake in young children (8-36 months), to identify nutritional and other factors that may affect iron intake and iron status and to ascertain whether a food frequency questionnaire can be designed to identify those at risk due to their diet.METHODS: A 4-day weighed food inventory, a semi-quantitative food frequency and social questionnaire and anthropometric measurements were used.Haemoglobin (Hb ), mean corpuscular volume (MCV), serum ferritin (SF), mean corpuscular haemoglobin (MCH), mean corpuscular haemoglobin concentration (MCHC) and haematocrit (Hct) were estimated in blood, Compeat-5, food analysis software was used to calculate nutrient intakes and SPSS and Excel for data analysis.STUDIES IN RIYADH, SAUDI ARABIA: 104 healthy children randomly chosen from eight different health centres have been studied either longitudinally (n=55) or cross-sectionally (n=49). The prevalence of iron deficiency anaemia was 36.3% in all children. Diet including iron intake has been compared to haematological data. Twenty four previously diagnosed iron deficient children from three hospitals were also studied.STUDIES IN EDINBURGH: 62 healthy children aged 9 and 36 months old were studied. They were those whose parents agreed to participate from a larger number chosen randomly from children registered at three health centres in Edinburgh using the Lothian Health Board list. Diet including iron intake has been compared to haematological data.ROYAL HOSPITAL FOR SICK CHILDREN (RHSC): Over a 2 months period, the prevalence of anaemia period in children whose blood samples were analysed in the Haematology Department was 28.3% in children aged between 8 months to 3 years of age. In 59 children, 45 with Hb below 11 g/dl, and 14 with normal Hb whose parents completed a semi-quantitative food frequency and social questionnaire, the iron intake and iron status was studied in detail, and the results related to the haematological parameters previously measured.CONCLUSIONS: Iron intakes less than both the Recommended Nutrient Intake (RNI) and the Lower Recommended Nutrient Intake (LRNI) have been shown to be common in the children studied in both Saudi Arabia and Edinburgh. Comparison of the haematological parameters with the iron intake enables certain definite statements to be made. Fortified breakfast cereals with iron and meat in addition to infant formula are important dietary factors which positively influence iron intake and iron status in this age group who are vulnerable to iron deficiency anaemia. These foods should be strongly recommended to parents for inclusion in the post-weaning diet of children of this age. In contrast, extended exclusive breast feeding, milk and some milk products and eggs have a negative influence on iron intake, and should be avoided as far as possible for at least the first year of life. The importance of haem iron as a component of the diet of children of the ages studied is evident from these studies. It has been shown that a food frequency questionnaire can be used to identify children at risk

An Isabelle/HOL Formalisation of Green’s Theorem

We mechanise a proof of Green’s theorem in Isabelle/HOL. We use a novel proof that avoids the ubiquitous line integral cancellation argument. This eliminates the need to formalise orientations and region boundaries explicitly with respect to the outwards-pointing normal vector. Instead we appeal to a homological argument about equivalences between paths. Contributions include mechanised theories of line integrals and partial derivatives, as well as the first mechanisation of Green’s theorem

TGFβ1-induced cell motility but not cell proliferation is mediated through Cten in colorectal cancer

Cten (C-terminal tensin-like) is a member of the tensin protein family found in complex with integrins at focal adhesions. It promotes epithelial‐mesenchymal transition (EMT) and cell motility. The precise mechanisms regulating Cten are unknown, although we and others have shown that Cten could be under the regulation of several cytokines and growth factors. Since Transforming growth factor beta 1 (TGF-β1) regulates integrin function and promotes EMT / cell motility, we were prompted to investigate whether TGF-β1 induces EMT and cell motility through Cten signalling in colorectal cancer. TGF-β1 signalling was modulated by either stimulation with TGF-β1 or knockdown of TGF-β1 in the CRC cell lines SW620 and HCT116. The effect of this modulation on expression of Cten, EMT markers and on cellular function was tested. The role of Cten as a direct mediator of TGF-β1 signalling was investigated in a CRC cell line in which the Cten gene had been deleted (SW620ΔCten). When TGF-β1 was stimulated or inhibited, this resulted in, respectively, upregulation and downregulation of Cten expression and EMT markers (Snail, Rock, N-Cadherin, Src). Cell migration and cell invasion were significantly increased following TGF-β1 stimulation and lost by TGF-β1 knockdown. TGF-β1 stimulation of the SW620ΔCten cell line resulted in selective loss of the effect of TGF-β1 signalling pathway on EMT and cell motility whilst the stimulatory effect on cell proliferation was retained. These data suggested Cten may play an essential role in mediating TGF-β1 induced EMT and cell motility and may therefore play a role in metastasis in CRC