A Categorical Critical-pair Completion Algorithm

AbstractWe introduce a general critical-pair/completion algorithm, formulated in the language of category theory. It encompasses the Knuth–Bendix procedure for term rewriting systems (also modulo equivalence relations), the Gröbner basis algorithm for polynomial ideal theory, and the resolution procedure for automated theorem proving. We show how these three procedures fit in the general algorithm, and how our approach relates to other categorical modeling approaches to these algorithms, especially term rewriting

Efficient Interpolation for the Theory of Arrays

Existing techniques for Craig interpolation for the quantifier-free fragment of the theory of arrays are inefficient for computing sequence and tree interpolants: the solver needs to run for every partitioning $(A, B)$ of the interpolation problem to avoid creating $AB$-mixed terms. We present a new approach using Proof Tree Preserving Interpolation and an array solver based on Weak Equivalence on Arrays. We give an interpolation algorithm for the lemmas produced by the array solver. The computed interpolants have worst-case exponential size for extensionality lemmas and worst-case quadratic size otherwise. We show that these bounds are strict in the sense that there are lemmas with no smaller interpolants. We implemented the algorithm and show that the produced interpolants are useful to prove memory safety for C programs.Comment: long version of the paper at IJCAR 201

A Novel Family of Toxoplasma IMC Proteins Displays a Hierarchical Organization and Functions in Coordinating Parasite Division

Apicomplexans employ a peripheral membrane system called the inner membrane complex (IMC) for critical processes such as host cell invasion and daughter cell formation. We have identified a family of proteins that define novel sub-compartments of the Toxoplasma gondii IMC. These IMC Sub-compartment Proteins, ISP1, 2 and 3, are conserved throughout the Apicomplexa, but do not appear to be present outside the phylum. ISP1 localizes to the apical cap portion of the IMC, while ISP2 localizes to a central IMC region and ISP3 localizes to a central plus basal region of the complex. Targeting of all three ISPs is dependent upon N-terminal residues predicted for coordinated myristoylation and palmitoylation. Surprisingly, we show that disruption of ISP1 results in a dramatic relocalization of ISP2 and ISP3 to the apical cap. Although the N-terminal region of ISP1 is necessary and sufficient for apical cap targeting, exclusion of other family members requires the remaining C-terminal region of the protein. This gate-keeping function of ISP1 reveals an unprecedented mechanism of interactive and hierarchical targeting of proteins to establish these unique sub-compartments in the Toxoplasma IMC. Finally, we show that loss of ISP2 results in severe defects in daughter cell formation during endodyogeny, indicating a role for the ISP proteins in coordinating this unique process of Toxoplasma replication

Ordered Sets in the Calculus of Data Structures

Our goal is to identify families of relations that are useful for reasoning about software. We describe such families using decidable quantifier-free classes of logical constraints with a rich set of operations. A key challenge is to define such classes of constraints in a modular way, by combining multiple decidable classes. Working with quantifierfree combinations of constraints makes the combination agenda more realistic and the resulting logics more likely to be tractable than in the presence of quantifiers. Our approach to combination is based on reducing decidable fragments to a common class, Boolean Algebra with Presburger Arithmetic (BAPA). This logic was introduced by Feferman and Vaught in 1959 and can express properties of uninterpreted sets of elements, with set algebra operations and equicardinality relation (consequently, it can also express Presburger arithmetic constraints on cardinalities of sets). Combination by reduction to BAPA allows us to obtain decidable quantifierfree combinations of decidable logics that share BAPA operations. We use the term Calculus of Data Structures to denote a family of decidable constraints that reduce to BAPA. This class includes, for example, combinations of formulas in BAPA, weak monadic second-order logic of k-successors, two-variable logic with counting, and term algebras with certain homomorphisms. The approach of reduction to BAPA generalizes the Nelson-Oppen combination that forms the foundation of constraint solvers used in software verification. BAPA is convenient as a target for reductions because it admits quantifier elimination and its quantifier-free fragment is NP-complete. We describe a new member of the Calculus of Data Structures: a quantifier-free fragment that supports 1) boolean algebra of finite and infinite sets of real numbers, 2) linear arithmetic over real numbers, 3) formulas that can restrict chosen set or element variables to range over integers (providing, among others, the power of mixed integer arithmetic and sets of integers), 4) the cardinality operators, stating whether a given set has a given finite cardinality or is infinite, 5) infimum and supremum operators on sets. Among the applications of this logic are reasoning about the externally observable behavior of data structures such as sorted lists and priority queues, and specifying witness functions for the BAPA synthesis problem. We describe an abstract reduction to BAPA for our logic, proving that the satisfiability of the logic is in NP and that it can be combined with the other fragments of the Calculus of Data Structures

Towards a Cooperating Robots Demonstrator

and automation for robot scenarios and areas like CIM. This led us to set up a database for scenario construction, cf. [DPSS91], which we used to compose robotics scenarios of industrial relevance that we tried to model MEDLAR II Deliverable V.2 2 with our methods. In [Pfa95] the interplay of AI and symbolic mathematical computation and traditional mathematics was discussed on the basis of some selected aspects from geometric, topological and logical reasoning in the classical AI field of robotics. It considered the kinematics model of a robot arm in detail and applied methods from symbolic computation to inverse kinematics and singularity problems. It showed how methods from classical geometry and topology can give support to work on typical robotics questions like the existence of singular configurations. Finally, it contained a sketch of a novel approach for logical modeling in robotics based on so-called logical fiberings. After trying out a small test example w

Representation Theorems and the Semantics of Non-Classical Logics , and Applications to Automated Theorem Proving

We give a uniform presentation of representation and decidability results related to the Kripke-style semantics of several nonclassical logics. We show that a general representation theorem (which has as particular instances the representation theorems as algebras of sets for Boolean algebras, distributive lattices and semilattices) extends in a natural way to several classes of operators and allows to establish a relationship between algebraic and Kripke-style models. We illustrate the ideas on several examples. We conclude by showing how the Kripkestyle models thus obtained can be used (if rst-order axiomatizable) for automated theorem proving by resolution for some non-classical logics

On Deciding Functional Lists with Sublist Sets

Abstract. Motivated by the problem of deciding verification conditions for the verification of functional programs, we present new decision procedures for automated reasoning about functional lists. We first show how to decide in NP the satisfiability problem for logical constraints containing equality, constructor, selectors, as well as the transitive sublist relation. We then extend this class of constraints with operators to compute the set of all sublists, and the set of objects stored in a list. Finally, we support constraints on sizes of sets, which gives us the ability to compute list length as well as the number of distinct list elements. We show that the extended theory is reducible to the theory of sets with linear cardinality constraints, and therefore still in NP. This reduction enables us to combine our theory with other decidable theories that impose constraints on sets of objects, which further increases the potential of our decidability result in verification of functional and imperative software.

Interpolation, amalgamation and combination : the non-disjoint signatures case

In this paper, we study the conditions under which existence of interpolants (for quantifier-free formulae) is modular, in the sense that it can be transferred from two first-order theories T1, T2 to their combination T1 \ue2\u88\uaa T2. We generalize to the non-disjoint signatures case the results from [3]. As a surprising application, we relate the Horn combinability criterion of this paper to superamalgamability conditions known from propositional logic and we use this fact to derive old and new results concerning fusions transfer of interpolation properties in modal logic