arXiv subjects

Besik Dundua

Publications and source records attributed to Besik Dundua.

6 recordsLinked to original sources

arXiv2026

Sensible Intersection Type Theories

Finitary/static semantics in the form of intersection type assignments have become a paradigm for analysing the fine structure of all sorts of lambda-models. The key step is the construction of a filter model isomorphic to a given lambda-model. A property of great interest of filter lambda-models is sensibility, i.e. the interpretation of all unsolvable terms is the least element. The flexibility of intersection type assignments derives from their parametrisation on intersection type theories. We construe intersection type theories as special meet-semilattices and show that appropriate morphisms, in the opposite category of meet-semilattices, preserve sensibility of the induced lambda-models. Interestingly the set of saturated sets together with the set of lambda-terms is such a meet-semilattice, thus showing that arguments based on Tait-Girards's computability amount to the construction of a morphism. We characterise two classes of intersection type theories which induce sensible filter models. The first is non-effective while the second is effective and it amounts to the generalisation of Mendler's criterion to intersection types and head normalising terms. The complete characterisation of sensible filter models however still escapes.

cs.LO↗

arXiv2025

Higher-Order Pattern Unification Modulo Similarity Relations

The combination of higher-order theories and fuzzy logic can be useful in decision-making tasks that involve reasoning across abstract functions and predicates, where exact matches are often rare or unnecessary. Developing efficient reasoning and computational techniques for such a combined formalism presents a significant challenge. In this paper, we adopt a more straightforward approach aiming at integrating two well-established and computationally well-behaved components: higher-order patterns on one side and fuzzy equivalences expressed through similarity relations based on minimum T-norm on the other. We propose a unification algorithm for higher-order patterns modulo these similarity relations and prove its termination, soundness, and completeness. This unification problem, like its crisp counterpart, is unitary. The algorithm computes a most general unifier with the highest degree of approximation when the given terms are unifiable.

cs.AI↗

arXiv2022

Regular matching problems for infinite trees

We study the matching problem of regular tree languages, that is, "$\exists σ:σ(L)\subseteq R$?" where $L,R$ are regular tree languages over the union of finite ranked alphabets $Σ$ and $\mathcal{X}$ where $\mathcal{X}$ is an alphabet of variables and $σ$ is a substitution such that $σ(x)$ is a set of trees in $T(Σ\cup H)\setminus H$ for all $x\in \mathcal{X}$. Here, $H$ denotes a set of "holes" which are used to define a "sorted" concatenation of trees. Conway studied this problem in the special case for languages of finite words in his classical textbook "Regular algebra and finite machines" published in 1971. He showed that if $L$ and $R$ are regular, then the problem "$\exists σ\forall x\in \mathcal{X}: σ(x)\neq \emptyset\wedge σ(L)\subseteq R$?" is decidable. Moreover, there are only finitely many maximal solutions, the maximal solutions are regular substitutions, and they are effectively computable. We extend Conway's results when $L,R$ are regular languages of finite and infinite trees, and language substitution is applied inside-out, in the sense of Engelfriet and Schmidt (1977/78). More precisely, we show that if $L\subseteq T(Σ\cup\mathcal{X})$ and $R\subseteq T(Σ)$ are regular tree languages over finite or infinite trees, then the problem "$\exists σ\forall x\in \mathcal{X}: σ(x)\neq \emptyset\wedge σ_{\mathrm{io}}(L)\subseteq R$?" is decidable. Here, the subscript "$\mathrm{io}$" in $σ_{\mathrm{io}}(L)$ refers to "inside-out". Moreover, there are only finitely many maximal solutions $σ$, the maximal solutions are regular substitutions and effectively computable. The corresponding question for the outside-in extension $σ_{\mathrm{oi}}$ remains open, even in the restricted setting of finite trees.

cs.FL↗

arXiv2021

P$ρ$Log: a system for rule-based programming

In this paper, we give a brief overview of the rule-based programming system, called P$ρ$Log and illustrate its capabilities.

cs.LO↗

arXiv2015

CLP(H): Constraint Logic Programming for Hedges

CLP(H) is an instantiation of the general constraint logic programming scheme with the constraint domain of hedges. Hedges are finite sequences of unranked terms, built over variadic function symbols and three kinds of variables: for terms, for hedges, and for function symbols. Constraints involve equations between unranked terms and atoms for regular hedge language membership. We study algebraic semantics of CLP(H) programs, define a sound, terminating, and incomplete constraint solver, investigate two fragments of constraints for which the solver returns a complete set of solutions, and describe classes of programs that generate such constraints.

cs.LO↗

arXiv2010

Strategies in PRholog

PRholog is an experimental extension of logic programming with strategic conditional transformation rules, combining Prolog with Rholog calculus. The rules perform nondeterministic transformations on hedges. Queries may have several results that can be explored on backtracking. Strategies provide a control on rule applications in a declarative way. With strategy combinators, the user can construct more complex strategies from simpler ones. Matching with four different kinds of variables provides a flexible mechanism of selecting (sub)terms during execution. We give an overview on programming with strategies in PRholog and demonstrate how rewriting strategies can be expressed.

cs.PL↗