Higher Order Unification via Explicit Substitutions

AbstractHigher order unification is equational unification for βη-conversion. But it is not first order equational unification, as substitution has to avoid capture. Thus, the methods for equational unification (such as narrowing) built upon grafting (i.e., substitution without renaming) cannot be used for higher order unification, which needs specific algorithms. Our goal in this paper is to reduce higher order unification to first order equational unification in a suitable theory. This is achieved by replacing substitution by grafting, but this replacement is not straightforward as it raises two major problems. First, some unification problems have solutions with grafting but no solution with substitution. Then equational unification algorithms rest upon the fact that grafting and reduction commute. But grafting and βη-reduction do not commute in λ-calculus and reducing an equation may change the set of its solutions. This difficulty comes from the interaction between the substitutions initiated by βη-reduction and the ones initiated by the unification process. Two kinds of variables are involved: those of βη-conversion and those of unification. So, we need to set up a calculus which distinguishes between these two kinds of variables and such that reduction and grafting commute. For this purpose, the application of a substitution of a reduction variable to a unification one must be delayed until this variable is instantiated. Such a separation and delay are provided by a calculus of explicit substitutions. Unification in such a calculus can be performed by well-known algorithms such as narrowing, but we present a specialised algorithm for greater efficiency. At last we show how to relate unification in λ-calculus and in a calculus with explicit substitutions. Thus, we come up with a new higher order unification algorithm which eliminates some burdens of the previous algorithms, in particular the functional handling of scopes. Huet's algorithm can be seen as a specific strategy for our algorithm, since each of its steps can be decomposed into elementary ones, leading to a more atomic description of the unification process. Also, solved forms in λ-calculus can easily be computed from solved forms in λσ-calculus

Congruence Closure with Free Variables (Work in Progress)

International audienceThis paper presents preliminary work on the definition of a general framework for handling quantified formulas in SMT solving. Its focus is on the derivation of instances conflicting with a ground context, redefining the approach introduced in [11]. An enhanced version of the classical congruence closure algorithm, able to handle free variables, is presented

Alchemical normal modes unify chemical space

In silico design of new molecules and materials with desirable quantum properties by high-throughput screening is a major challenge due to the high dimensionality of chemical space. To facilitate its navigation, we present a unification of coordinate and composition space in terms of alchemical normal modes (ANMs) which result from second order perturbation theory. ANMs assume a predominantly smooth nature of chemical space and form a basis in which new compounds can be expanded and identified. We showcase the use of ANMs for the energetics of the iso-electronic series of diatomics with 14 electrons, BN doped benzene derivatives (C$_{6-2x}$(BN)$_{x}$H$_6$ with $x = 0, 1, 2, 3$), predictions for over 1.8 million BN doped coronene derivatives, and genetic energy optimizations in the entire BN doped coronene space. Using Ge lattice scans as reference, the applicability ANMs across the periodic table is demonstrated for III-V and IV-IV-semiconductors Si, Sn, SiGe, SnGe, SiSn, as well as AlP, AlAs, AlSb, GaP, GaAs, GaSb, InP, InAs, and InSb. Analysis of our results indicates simple qualitative structure property rules for estimating energetic rankings among isomers. Useful quantitative estimates can also be obtained when few atoms are changed to neighboring or lower lying elements in the periodic table. The quality of the predictions often increases with the symmetry of system chosen as reference due to cancellation of odd order terms. Rooted in perturbation theory the ANM approach promises to generally enable unbiased compound exploration campaigns at reduced computational cost

Integration of graph iterators into LEDA

This paper explains some implementation details of graph iterators and data accessors in LEDA. It shows how to create new iterators for new graph implementations such that old algorithms can be re--used with new graph implementations as long as they are based on graph iterators and data accessors

Stratification, Tracking and Course-Taking Patterns: An Examination of the Impact of Mathematics Course Placement on Achievement in a Regional High School District

This study examined the impact of mathematics track placement on the academic achievement of students as measured by student performance on the 2014 New Jersey High School Proficiency Assessment (HSPA) in mathematics, and math course attainment in high school as measured by the level of math class completed by the end of grade 12. A substantive empirical base (Goodlad & Oakes, 1998; Slavin, 1995; Oakes, 2005; Darling-Hammond, 2010; Gamoran 2009) exists that outlines the inequalities often associated within tracked school systems. Despite widespread criticism, tracking remains dominant as a strategy to group students in American high schools (Loveless 2013). Regional high schools accept students from multiple sending districts and must by nature make placement decisions based upon, in part, middle school performance. As the student data utilized in this study was culled from sending districts that all use traditional tracking strategies, the measurement of student performance within the larger regional high school may identify both effective and potentially problematic student grouping practices. The current debate in the field of education regarding the passage of standardized assessments as a requirement for graduation demands that the impact of course-taking patterns and tracking decisions on performance on such assessments be more fully investigated. This study indicated that while ninth grade course placement matters as it relates to standardized assessment scores, explaining 27.8 % of the student performance and 17.9% of grade twelve course attainment, other factors may positively alter an individual student academic trajectory regardless of track placement. The evidence indicates that school districts that take specific action to mitigate the impacts of tracking may provide students with opportunities and support that lead to increased academic outcomes. This study contributes to the growing research that indicates the use of standardized assessments as a sole measure of academic achievement is deeply flawed if not taking into consideration other factors such as course-taking atterns, academic opportunities for advancement, and tracking decisions at the local level

Efficient Instantiation Techniques in SMT (Work In Progress)

International audienceIn SMT solving one generally applies heuristic instantiation to handle quantified formulas. This has the side effect of producing many spurious instances and may lead to loss of performance. Therefore deriving both fewer and more meaningful instances as well as eliminating or dismissing , i.e., keeping but ignoring, those not significant for the solving are desirable features for dealing with first-order problems. This paper presents preliminary work on two approaches: the implementation of an efficient instantiation framework with an incomplete goal-oriented search; and the introduction of dismissing criteria for heuristic instances. Our experiments show that while the former improves performance in general the latter is highly dependent on the problem structure, but its combination with the classic strategy leads to competitive results w.r.t. state-of-the-art SMT solvers in several benchmark libraries