Solving constrained Horn clauses modulo algebraic data types and recursive functions

This work addresses the problem of verifying imperative programs that manipulate data structures, e.g., Rust programs. Data structures are usually modeled by Algebraic Data Types (ADTs) in verification conditions. Inductive invariants of such programs often require recursively defined functions (RDFs) to represent abstractions of data structures. From the logic perspective, this reduces to solving Constrained Horn Clauses (CHCs) modulo both ADT and RDF. The underlying logic with RDFs is undecidable. Thus, even verifying a candidate inductive invariant is undecidable. Similarly, IC3-based algorithms for solving CHCs lose their progress guarantee: they may not find counterexamples when the program is unsafe. We propose a novel IC3-inspired algorithm Racer for solving CHCs modulo ADT and RDF (i.e., automatically synthesizing inductive invariants, as opposed to only verifying them as is done in deductive verification). Racer ensures progress despite the undecidability of the underlying theory, and is guaranteed to terminate with a counterexample for unsafe programs. It works with a general class of RDFs over ADTs called catamorphisms. The key idea is to represent catamorphisms as both CHCs, via relationification , and RDFs, using novel abstractions . Encoding catamorphisms as CHCs allows learning inductive properties of catamorphisms, as well as preserving unsatisfiabilty of the original CHCs despite the use of RDF abstractions, whereas encoding catamorphisms as RDFs allows unfolding the recursive definition, and relying on it in solutions. Abstractions ensure that the underlying theory remains decidable. We implement our approach in Z3 and show that it works well in practice.

Comparing Weak Admissibility Semantics to their Dung-style Counterparts (Extended Abstract)

Semantics based on weak admissibility were recently introduced to overcome a problem with self-defeating arguments that has not been solved for more than 25 years. The recursive definition of weak admissibility mainly relies on the notion of a reduct regarding a set E which only contains arguments which are neither in E, nor attacked by E. At first glance the reduct seems to be tailored for the weaker versions of Dung-style semantics only. In this paper we show that standard Dung semantics can be naturally reformulated using the reduct revealing that this concept is already implicit. We further identify a new abstract principle for semantics, so-called modularization describing how to obtain further extensions given an initial one. Its importance for the study of abstract argumentation semantics is shown by its ability to alternatively characterize classical and non-classical semantics.

The Energy Spectral Density of the Mertens Function

The Mertens function is the summatory Mobius function but the Mertens function can be generated recursively without using this definition. This recursive definition is the basis of autocorrelations that can be done on sequences of Mertens function values. Fourier transforms of the autocorrelations result in the energy spectral density. A likely upper bound of the absolute value of the Mertens function is determined.

Constructing a universe for the setoid model

The setoid model is a model of intensional type theory that validates certain extensionality principles, like function extensionality and propositional extensionality, the latter being a limited form of univalence that equates logically equivalent propositions. The appeal of this model construction is that it can be constructed in a small, intensional, type theoretic metatheory, therefore giving a method to boostrap extensionality. The setoid model has been recently adapted into a formal system, namely Setoid Type Theory (SeTT). SeTT is an extension of intensional Martin-Löf type theory with constructs that give full access to the extensionality principles that hold in the setoid model.Although already a rich theory as currently defined, SeTT currently lacks a way to internalize the notion of type beyond propositions, hence we want to extend SeTT with a universe of setoids. To this aim, we present the construction of a (non-univalent) universe of setoids within the setoid model, first as an inductive-recursive definition, which is then translated to an inductive-inductive definition and finally to an inductive family. These translations from more powerful definition schemas to simpler ones ensure that our construction can still be defined in a relatively small metatheory which includes a proof-irrelevant identity type with a strong transport rule.

Comparing Weak Admissibility Semantics to their Dung-style Counterparts -- Reduct, Modularization, and Strong Equivalence in Abstract Argumentation

Semantics based on weak admissibility were recently introduced to overcome a problem with self-defeating arguments that has not been solved for more than 25 years. The recursive definition of weak admissibility mainly relies on the notion of a reduct regarding a set E which only contains arguments which are neither in E, nor attacked by E. At first glance the reduct seems to be tailored for the weaker versions of Dung-style semantics only. In this paper we show that standard Dung semantics can be naturally reformulated using the reduct revealing that this concept is already implicit. We further identify a new abstract principle for semantics, so-called modularization describing how to obtain further extensions given an initial one. Its importance for the study of abstract argumentation semantics is shown by its ability to alternatively characterize classical and non-classical semantics. Moreover, we tackle the notion of strong equivalence via characterizing kernels and give a complete classification of the weak versions regarding well-known properties and postulates known from the literature.

QUANTILE TRANSFORM IN STRUCTURAL BIOINFORMATICS PROBLEMS

In this paper we study features of the multivariate empirical quantum function implementation for which sample is distributed at the mesh points of the regular grid. We present an algorithm for continuous and discrete quantile transform based on recursive definition of the multivariate quantile function. We perform numerical study of the presented algorithm and demonstrate it computational complexity according to representation of the sample. We present the results of using evolutionary optimization algorithm with quantile transform for solving the problems in structural bioinformatics: protein structure prediction from amino acid sequence and protein-peptide docking with known binding site and linear peptide structure.

A Recursive Definition of Goodness of Space for Bridging the Concepts of Space and Place for Sustainability

Conceived and developed by Christopher Alexander through his life’s work, The Nature of Order, wholeness is defined as a mathematical structure of physical space in our surroundings. Yet, there was no mathematics, as Alexander admitted then, that was powerful enough to capture his notion of wholeness. Recently, a mathematical model of wholeness, together with its topological representation, has been developed that is capable of addressing not only why a space is good, but also how much goodness the space has. This paper develops a structural perspective on goodness of space (both large- and small-scale) in order to bridge two basic concepts of space and place through the very concept of wholeness. The wholeness provides a de facto recursive definition of goodness of space from a holistic and organic point of view. A space is good, genuinely and objectively, if its adjacent spaces are good, the larger space to which it belongs is good, and what is contained in the space is also good. Eventually, goodness of space, or sustainability of space, is considered a matter of fact rather than of opinion under the new view of space: space is neither lifeless nor neutral, but a living structure capable of being more living or less living, or more sustainable or less sustainable. Under the new view of space, geography or architecture will become part of complexity science, not only for understanding complexity, but also for making and remaking complex or living structures.

Contralog: a Prolog conform forward-chaining environment and its application for dynamic programming and natural language parsing

The backward-chaining inference strategy of Prolog is inefficient for a number of problems. The article proposes Contralog: a Prolog-conform, forward-chaining language and an inference engine that is implemented as a preprocessor-compiler to Prolog. The target model is Prolog, which ensures mutual switching from Contralog to Prolog and back. The Contralog compiler is implemented using Prolog's de facto standardized macro expansion capability. The article goes into details regarding the target model. We introduce first a simple application example for Contralog. Then the next section shows how a recursive definition of some problems is executed by their Contralog definition automatically in a dynamic programming way. Two examples, the well-known matrix chain multiplication problem and the Warshall algorithm are shown here. After this, the inferential target model of Prolog/Contralog programs is introduced, and the possibility for implementing the ReALIS natural language parsing technology is described relying heavily on Contralog's forward chaining inference engine. Finally the article also discusses some practical questions of Contralog program development.

Alternative Semantics

This chapter presents the semantics and pragmatics of prosodic focus in alternative semantics. Half a dozen examples are given of empirical phenomena that are to be covered by the theory. Then a syntax marking the locus, scope, and antecedent for focus is introduced. The syntax is interpreted semantically and pragmatically by a presupposition involving alternatives. The alternative sets that are used in the definition are computed compositionally using a recursive definition. Alternatives are also employed in the semantics of questions, and this ties in with the phenomenon of question-answer congruence, where the position of focus in an answer matches questioned positions in the question. A different semantic interpretation for focus is entailment semantics, which uses a generalized entailment condition in place of a condition involving alternatives. The semantic and pragmatic interpretation for contrastive topic uses an additional layer of alternatives. Independent of focus, alternatives are deployed in the semantics of disjunction and of negative polarity items.