T-resolution: refinements and model elimination

T-resolution is a binary rule, proposed by Policriti and Schwartz in 1995 for theorem proving in first-order theories (T-theorem proving) that can be seen - at least at the ground level - as a variant of Stickel's theory resolution. In this paper we consider refinements of this rule as well as the model elimination variant of it. After a general discussion concerning our viewpoint on theorem proving in first-order theories and a brief comparison with theory resolution, the power and generality of T-resolution are emphasized by introducing suitable linear and ordered refinements, uniformly and in strict analogy with the standard resolution approach. Then a model elimination variant of T-resolution is introduced and proved to be sound and complete; some experimental results are also reported. In the last part of the paper we present two applications of T-resolution: to constraint logic programming and to modal logic

Expressing Infinity without Foundation

The axiom of infinity can be expressed by stating the existence of sets satisfying a formula which involves restricted universal quantifiers only, even if the axiom of foundation is not assumed

Special Issue on Algorithms and Data-Structures for Compressed Computation

As the production of massive data has outpaced Moore’s law in many scientific areas, the very notion of algorithms is transforming [...

Modelling concurrent systems specified in a temporal concurrent constraint language -I

In this paper we present an approach to model concurrent systems specified in a temporal concurrent constraint language. Our goal is to construct a framework in which it is possible to apply the Model Checking technique to programs specified in such language. This work is the first step to the framework construction. We present a formalism to transform a specification into a tcc Structure. This structure is a graph representation of the program behavior. Our basic tool is the Timed Concurrent Constraint Programming (tcc) framework defined by Saraswat et al. to describe reactive systems. With this language we take advantage of both the natural properties of the declarative paradigm and of the fact that the notion of time is built into the semantics of the programming language. In fact, on this ground it becomes reasonable to introduce the idea of applying the technique of Model Checking to a finite time interval (introduced by the user). With this restriction we naturally force the space representing the behavior of the program to be finite and hence Model Checking algorithms to be applicable. The graph construction is a completely automatic process that takes as input the tcc specification

Modeling Concurrent systems specified in a Temporal Concurrent Constraint language-I

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The Automation of Syllogistic II. Optimization and Complexity Issues

In the first paper of this series it was shown that any unquantified formula p in the collection MLSSF (multilevel syllogistic extended with the singleton operator and the predicate Finite) can be decomposed as a disjunction of set-theoretic formulae called syllogistic schemes. The syllogistic schemes are satisfiable and no two of them have a model in common, therefore the previous result already implied the decidability of the class MLSSF by simply checking if the set of syllogistic schemes associated with the given formula is empty. In the first section of this paper a new and improved searching algorithm for syllogistic schemes is introduced, based on a proof of existence of a 'minimum effort' scheme for any given satisfiable formula in MLSF. The algorithm addressed above can be piloted quite effectively even though it involves backtracking. In the second part of the paper, complexity issues are studied by showing that the class of ( 00)o1-simple prenex formulae (an extension of MLS) has a decision problem which is NP-complete. The decision algorithm that proves the membership of this decision problem to NP can be seen as a different decision algorithm for ML

Decidability results for classes of purely universal formulae and quantifiers elimination in Set Theory

A general mechanism to extend decision algorithms to deal with additional predicates is described. The only conditions imposed on the predicates is stability with respect to some transitive relations

String attractors : Verification and optimization

String attractors [STOC 2018] are combinatorial objects recently introduced to unify all known dictionary compression techniques in a single theory. A set γ ⊆ [1.n] is a k-attractor for a string S ∈ Σn if and only if every distinct substring of S of length at most k has an occurrence crossing at least one of the positions in γ. Finding the smallest k-attractor is NP-hard for k ≥ 3, but polylogarithmic approximations can be found using reductions from dictionary compressors. It is easy to reduce the k-attractor problem to a set-cover instance where the string's positions are interpreted as sets of substrings. The main result of this paper is a much more powerful reduction based on the truncated suffix tree. Our new characterization of the problem leads to more efficient algorithms for string attractors: we show how to check the validity and minimality of a k-attractor in near-optimal time and how to quickly compute exact solutions. For example, we prove that a minimum 3-attractor can be found in O(n) time when |Σ| ∈ O(3+ϵ√log n) for some constant ϵ > 0, despite the problem being NP-hard for large Σ. © Dominik Kempa, Alberto Policriti, Nicola Prezza, and Eva Rotenberg.Peer reviewe

Model building and model checking for biochemical processes

A central claim of computational systems biology is that, by drawing on mathematical approaches developed in the context of dynamic systems, kinetic analysis, computational theory and logic, it is possible to create powerful simulation, analysis, and reasoning tools for working biologists to decipher existing data, devise new experiments, and ultimately to understand functional properties of genomes, proteomes, cells, organs, and organisms. In this article, a novel computational tool is described that achieves many of the goals of this new discipline. The novelty of this system involves an automaton-based semantics of the temporal evolution of complex biochemical reactions starting from the representation given as a set of differential equations. The related tools also provide ability to qualitatively reason about the systems using a propositional temporal logic that can express an ordered sequence of events succinctly and unambiguously. The implementation of mathematical and computational models in the Simpathica and XSSYS systems is described briefly. Several example applications of these systems to cellular and biochemical processes are presented: the two most prominent are Leibler et al.'s repressilator (an artificial synthesized oscillatory network), and Curto-Voit-Sorribas-Cascante's purine metabolism reaction model