Encoding monomorphic and polymorphic types

Most automatic theorem provers are restricted to untyped logics, and existing translations from typed logics are bulky or unsound. Recent research proposes monotonicity as a means to remove some clutter. Here we pursue this approach systematically, analysing formally a variety of encodings that further improve on efficiency while retaining soundness and completeness. We extend the approach to rank-1 polymorphism and present alternative schemes that lighten the translation of polymorphic symbols based on the novel notion of “cover”. The new encodings are implemented, and partly proved correct, in Isabelle/HOL. Our evaluation finds them vastly superior to previous schemes

Encoding monomorphic and polymorphic types

Many automatic theorem provers are restricted to untyped logics, and existing translations from typed logics are bulky or unsound. Recent research proposes monotonicity as a means to remove some clutter when translating monomorphic to un-typed first-order logic. Here we pursue this approach systematically, analysing formally a variety of encodings that further improve on efficiency while retaining soundness and completeness. We extend the approach to rank-1 polymorphism and present alternative schemes that lighten the translation of polymorphic symbols based on the novel notion of “cover”. The new encodings are implemented in Isabelle/HOL as part of the Sledgehammer tool. We include informal proofs of soundness and correctness, and have formalized the monomorphic part of this work in Isabelle/HOL. Our evaluation finds the new encodings vastly superior to previous schemes

Neuronal post-developmentally acting SAX-7S/L1CAM can function as cleaved fragments to maintain neuronal architecture in C. elegans [preprint]

Whereas remarkable advances have uncovered mechanisms that drive nervous system assembly, the processes responsible for the lifelong maintenance of nervous system architecture remain poorly understood. Subsequent to its establishment during embryogenesis, neuronal architecture is maintained throughout life in the face of the animal’s growth, maturation processes, the addition of new neurons, body movements, and aging. The C. elegans protein SAX-7, homologous to the vertebrate L1 protein family, is required for maintaining the organization of neuronal ganglia and fascicles after their successful initial embryonic development. To dissect the function of sax-7 in neuronal maintenance, we generated a null allele and sax-7S-isoform-specific alleles. We find that the null sax-7(qv30) is, in some contexts, more severe than previously described mutant alleles, and that the loss of sax-7S largely phenocopies the null, consistent with sax-7S being the key isoform in neuronal maintenance. Using a sfGFP::SAX-7S knock-in, we observe sax-7S to be predominantly expressed across the nervous system, from embryogenesis to adulthood. Yet, its role in maintaining neuronal organization is ensured by post-developmentally acting SAX-7S, as larval transgenic sax-7S(+) expression alone is sufficient to profoundly rescue the null mutants’ neuronal maintenance defects. Moreover, the majority of the protein SAX-7 appears to be cleaved, and we show that these cleaved SAX-7S fragments together, not individually, can fully support neuronal maintenance. These findings contribute to our understanding of the role of the conserved protein SAX-7/L1CAM in long-term neuronal maintenance, and may help decipher processes that go awry in some neurodegenerative conditions

Functional Requirements for Heparan Sulfate Biosynthesis in Morphogenesis and Nervous System Development in C. elegans

The regulation of cell migration is essential to animal development and physiology. Heparan sulfate proteoglycans shape the interactions of morphogens and guidance cues with their respective receptors to elicit appropriate cellular responses. Heparan sulfate proteoglycans consist of a protein core with attached heparan sulfate glycosaminoglycan chains, which are synthesized by glycosyltransferases of the exostosin (EXT) family. Abnormal HS chain synthesis results in pleiotropic consequences, including abnormal development and tumor formation. In humans, mutations in either of the exostosin genes EXT1 and EXT2 lead to osteosarcomas or multiple exostoses. Complete loss of any of the exostosin glycosyltransferases in mouse, fish, flies and worms leads to drastic morphogenetic defects and embryonic lethality. Here we identify and study previously unavailable viable hypomorphic mutations in the two C. elegans exostosin glycosyltransferases genes, rib-1 and rib-2. These partial loss-of-function mutations lead to a severe reduction of HS levels and result in profound but specific developmental defects, including abnormal cell and axonal migrations. We find that the expression pattern of the HS copolymerase is dynamic during embryonic and larval morphogenesis, and is sustained throughout life in specific cell types, consistent with HSPGs playing both developmental and post-developmental roles. Cell-type specific expression of the HS copolymerase shows that HS elongation is required in both the migrating neuron and neighboring cells to coordinate migration guidance. Our findings provide insights into general principles underlying HSPG function in development

Efficient Certified RAT Verification

Clausal proofs have become a popular approach to validate the results of SAT solvers. However, validating clausal proofs in the most widely supported format (DRAT) is expensive even in highly optimized implementations. We present a new format, called LRAT, which extends the DRAT format with hints that facilitate a simple and fast validation algorithm. Checking validity of LRAT proofs can be implemented using trusted systems such as the languages supported by theorem provers. We demonstrate this by implementing two certified LRAT checkers, one in Coq and one in ACL2

Finding Finite Models in Multi-Sorted First-Order Logic

This work extends the existing MACE-style finite model finding approach to multi-sorted first order logic. This existing approach iteratively assumes increasing domain sizes and encodes the related ground problem as a SAT problem. When moving to the multi-sorted setting each sort may have a different domain size, leading to an explosion in the search space. This paper focusses on methods to tame that search space. The key approach adds additional information to the SAT encoding to suggest which domains should be grown. Evaluation of an implementation of techniques in the Vampire theorem prover shows that they dramatically reduce the search space and that this is an effective approach to find finite models in multi-sorted first order logic.Comment: SAT 201

Poincaré on the Foundation of Geometry in the Understanding

This paper is about Poincaré’s view of the foundations of geometry. According to the established view, which has been inherited from the logical positivists, Poincaré, like Hilbert, held that axioms in geometry are schemata that provide implicit definitions of geometric terms, a view he expresses by stating that the axioms of geometry are “definitions in disguise.” I argue that this view does not accord well with Poincaré’s core commitment in the philosophy of geometry: the view that geometry is the study of groups of operations. In place of the established view I offer a revised view, according to which Poincaré held that axioms in geometry are in fact assertions about invariants of groups. Groups, as forms of the understanding, are prior in conception to the objects of geometry and afford the proper definition of those objects, according to Poincaré. Poincaré’s view therefore contrasts sharply with Kant’s foundation of geometry in a unique form of sensibility. According to my interpretation, axioms are not definitions in disguise because they themselves implicitly define their terms, but rather because they disguise the definitions which imply them

Efficient Certified Resolution Proof Checking

We present a novel propositional proof tracing format that eliminates complex processing, thus enabling efficient (formal) proof checking. The benefits of this format are demonstrated by implementing a proof checker in C, which outperforms a state-of-the-art checker by two orders of magnitude. We then formalize the theory underlying propositional proof checking in Coq, and extract a correct-by-construction proof checker for our format from the formalization. An empirical evaluation using 280 unsatisfiable instances from the 2015 and 2016 SAT competitions shows that this certified checker usually performs comparably to a state-of-the-art non-certified proof checker. Using this format, we formally verify the recent 200 TB proof of the Boolean Pythagorean Triples conjecture

A Unifying Model of Genome Evolution Under Parsimony

We present a data structure called a history graph that offers a practical basis for the analysis of genome evolution. It conceptually simplifies the study of parsimonious evolutionary histories by representing both substitutions and double cut and join (DCJ) rearrangements in the presence of duplications. The problem of constructing parsimonious history graphs thus subsumes related maximum parsimony problems in the fields of phylogenetic reconstruction and genome rearrangement. We show that tractable functions can be used to define upper and lower bounds on the minimum number of substitutions and DCJ rearrangements needed to explain any history graph. These bounds become tight for a special type of unambiguous history graph called an ancestral variation graph (AVG), which constrains in its combinatorial structure the number of operations required. We finally demonstrate that for a given history graph $G$, a finite set of AVGs describe all parsimonious interpretations of $G$, and this set can be explored with a few sampling moves.Comment: 52 pages, 24 figure

Formalizing Bachmair and Ganzinger’s Ordered Resolution Prover

We present a formalization of the first half of Bachmair and Ganzinger’s chapter on resolution theorem proving in Isabelle/HOL, culminating with a refutationally complete first-order prover based on ordered resolution with literal selection. We develop general infrastructure and methodology that can form the basis of completeness proofs for related calculi, including superposition. Our work clarifies several of the fine points in the chapter’s text, emphasizing the value of formal proofs in the field of automated reasoning