Instantiation Schemes for Nested Theories

Article 11 - 33 pagesInternational audienceThis article investigates under which conditions instantiation-based proof procedures can be combined in a nested way, in order to mechanically construct new instantiation procedures for richer theories. Interesting applications in the field of verification are emphasized, particularly for handling extensions of the theory of arrays

Fibrational induction rules for initial algebras

This paper provides an induction rule that can be used to prove properties of data structures whose types are inductive, i.e., are carriers of initial algebras of functors. Our results are semantic in nature and are inspired by Hermida and Jacobs’ elegant algebraic formulation of induction for polynomial data types. Our contribution is to derive, under slightly different assumptions, an induction rule that is generic over all inductive types, polynomial or not. Our induction rule is generic over the kinds of properties to be proved as well: like Hermida and Jacobs, we work in a general fibrational setting and so can accommodate very general notions of properties on inductive types rather than just those of particular syntactic forms. We establish the correctness of our generic induction rule by reducing induction to iteration. We show how our rule can be instantiated to give induction rules for the data types of rose trees, finite hereditary sets, and hyperfunctions. The former lies outside the scope of Hermida and Jacobs’ work because it is not polynomial; as far as we are aware, no induction rules have been known to exist for the latter two in a general fibrational framework. Our instantiation for hyperfunctions underscores the value of working in the general fibrational setting since this data type cannot be interpreted as a set

Instantiation of SMT problems modulo Integers

Many decision procedures for SMT problems rely more or less implicitly on an instantiation of the axioms of the theories under consideration, and differ by making use of the additional properties of each theory, in order to increase efficiency. We present a new technique for devising complete instantiation schemes on SMT problems over a combination of linear arithmetic with another theory T. The method consists in first instantiating the arithmetic part of the formula, and then getting rid of the remaining variables in the problem by using an instantiation strategy which is complete for T. We provide examples evidencing that not only is this technique generic (in the sense that it applies to a wide range of theories) but it is also efficient, even compared to state-of-the-art instantiation schemes for specific theories.Comment: Research report, long version of our AISC 2010 pape

Modular Instantiation Schemes

International audienceInstantiation schemes are proof procedures that test the satisfiability of clause sets by instantiating the variables they contain, and testing the satisfiability of the resulting ground set of clauses. Such schemes have been devised for several theories, including fragments of linear arithmetic or theories of data-structures. In this paper we investigate under what conditions instantiation schemes can be combined to solve satisfiability problems in unions of theories

Heterogeneous substitution systems revisited

Matthes and Uustalu (TCS 327(1-2):155-174, 2004) presented a categorical description of substitution systems capable of capturing syntax involving binding which is independent of whether the syntax is made up from least or greatest fixed points. We extend this work in two directions: we continue the analysis by creating more categorical structure, in particular by organizing substitution systems into a category and studying its properties, and we develop the proofs of the results of the cited paper and our new ones in UniMath, a recent library of univalent mathematics formalized in the Coq theorem prover.Comment: 24 page

The emerging structure of the Extended Evolutionary Synthesis: where does Evo-Devo fit in?

The Extended Evolutionary Synthesis (EES) debate is gaining ground in contemporary evolutionary biology. In parallel, a number of philosophical standpoints have emerged in an attempt to clarify what exactly is represented by the EES. For Massimo Pigliucci, we are in the wake of the newest instantiation of a persisting Kuhnian paradigm; in contrast, Telmo Pievani has contended that the transition to an EES could be best represented as a progressive reformation of a prior Lakatosian scientific research program, with the extension of its Neo-Darwinian core and the addition of a brand-new protective belt of assumptions and auxiliary hypotheses. Here, we argue that those philosophical vantage points are not the only ways to interpret what current proposals to ‘extend’ the Modern Synthesis-derived ‘standard evolutionary theory’ (SET) entail in terms of theoretical change in evolutionary biology. We specifically propose the image of the emergent EES as a vast network of models and interweaved representations that, instantiated in diverse practices, are connected and related in multiple ways. Under that assumption, the EES could be articulated around a paraconsistent network of evolutionary theories (including some elements of the SET), as well as models, practices and representation systems of contemporary evolutionary biology, with edges and nodes that change their position and centrality as a consequence of the co-construction and stabilization of facts and historical discussions revolving around the epistemic goals of this area of the life sciences. We then critically examine the purported structure of the EES—published by Laland and collaborators in 2015—in light of our own network-based proposal. Finally, we consider which epistemic units of Evo-Devo are present or still missing from the EES, in preparation for further analyses of the topic of explanatory integration in this conceptual framework

Generic Fibrational Induction

This paper provides an induction rule that can be used to prove properties of data structures whose types are inductive, i.e., are carriers of initial algebras of functors. Our results are semantic in nature and are inspired by Hermida and Jacobs' elegant algebraic formulation of induction for polynomial data types. Our contribution is to derive, under slightly different assumptions, a sound induction rule that is generic over all inductive types, polynomial or not. Our induction rule is generic over the kinds of properties to be proved as well: like Hermida and Jacobs, we work in a general fibrational setting and so can accommodate very general notions of properties on inductive types rather than just those of a particular syntactic form. We establish the soundness of our generic induction rule by reducing induction to iteration. We then show how our generic induction rule can be instantiated to give induction rules for the data types of rose trees, finite hereditary sets, and hyperfunctions. The first of these lies outside the scope of Hermida and Jacobs' work because it is not polynomial, and as far as we are aware, no induction rules have been known to exist for the second and third in a general fibrational framework. Our instantiation for hyperfunctions underscores the value of working in the general fibrational setting since this data type cannot be interpreted as a set.Comment: For Special Issue from CSL 201

Decomposable Theories

We present in this paper a general algorithm for solving first-order formulas in particular theories called "decomposable theories". First of all, using special quantifiers, we give a formal characterization of decomposable theories and show some of their properties. Then, we present a general algorithm for solving first-order formulas in any decomposable theory "T". The algorithm is given in the form of five rewriting rules. It transforms a first-order formula "P", which can possibly contain free variables, into a conjunction "Q" of solved formulas easily transformable into a Boolean combination of existentially quantified conjunctions of atomic formulas. In particular, if "P" has no free variables then "Q" is either the formula "true" or "false". The correctness of our algorithm proves the completeness of the decomposable theories. Finally, we show that the theory "Tr" of finite or infinite trees is a decomposable theory and give some benchmarks realized by an implementation of our algorithm, solving formulas on two-partner games in "Tr" with more than 160 nested alternated quantifiers