Related papers: A Saturation-Based Unification Algorithm for Highe…
Matching Logic is a framework for specifying programming language semantics and reasoning about programs. Its formulas are called patterns and are built with variables, symbols, connectives and quantifiers. A pattern is a combination of…
We study first-order logic over unordered structures whose elements carry a finite number of data values from an infinite domain. Data values can be compared wrt.\ equality. As the satisfiability problem for this logic is undecidable in…
Motivated by applications in automated verification of higher-order functional programs, we develop a notion of constrained Horn clauses in higher-order logic and a decision problem concerning their satisfiability. We show that, although…
Model explainability is crucial for human users to be able to interpret how a proposed classifier assigns labels to data based on its feature values. We study generalized linear models constructed using sets of feature value rules, which…
Logical frameworks provide natural and direct ways of specifying and reasoning within deductive systems. The logical framework LF and subsequent developments focus on finitary proof systems, making the formalization of circular proof…
We prove that the isomorphism of scattered tree automatic linear orders as well as the existence of automorphisms of scattered word automatic linear orders are undecidable. For the existence of automatic automorphisms of word automatic…
We study the problem of explainability-first clustering where explainability becomes a first-class citizen for clustering. Previous clustering approaches use decision trees for explanation, but only after the clustering is completed. In…
We study the question of whether a given regular language of finite trees can be defined in first-order logic. We develop an algebraic approach to address this question and we use it to derive several necessary and sufficient conditions for…
Matching logic is a logical framework for specifying and reasoning about programs using pattern matching semantics. A pattern is made up of a number of structural components and constraints. Structural components are syntactically matched,…
Modular exponentiation is a common mathematical operation in modern cryptography. This, along with modular multiplication at the base and exponent levels (to different moduli) plays an important role in a large number of key agreement…
Algorithms can be used to prove and to discover new theorems. This paper shows how algorithmic skills in general, and the notion of invariance in particular, can be used to derive many results from Euclid's algorithm. We illustrate how to…
Type checking algorithms and theorem provers rely on unification algorithms. In presence of type families or higher-order logic, higher-order (pre)unification (HOU) is required. Many HOU algorithms are expressed in terms of…
Separation Logic is a widely used formalism for describing dynamically allocated linked data structures, such as lists, trees, etc. The decidability status of various fragments of the logic constitutes a long standing open problem. Current…
We give a deterministic algorithm that, given a composite number $N$ and a target order $D \ge N^{1/6}$, runs in time $D^{1/2+o(1)}$ and finds either an element $a \in \mathbb{Z}_N^*$ of multiplicative order at least $D$, or a nontrivial…
The first-order theory of finite and infinite trees has been studied since the eighties, especially by the logic programming community. Following Djelloul, Dao and Fr\"uhwirth, we consider an extension of this theory with an additional…
Recently, symbolic structures were proposed as finite representations of potentially infinite first-order structures, where Linear Integer Arithmetic terms and formulas define the domain and interpretations of a structure. We generalize…
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…
Human reasoning can distill principles from observed patterns and generalize them to explain and solve novel problems. The most powerful artificial intelligence systems lack explainability and symbolic reasoning ability, and have therefore…
This work presents a formalization of the theorem of existence of most general unifiers in first-order signatures in the higher-order proof assistant PVS. The distinguishing feature of this formalization is that it remains close to the…
We present a first-order theorem proving framework for establishing the correctness of functional programs implementing sorting algorithms with recursive data structures. We formalize the semantics of recursive programs in many-sorted…