arXiv subjects

Anela Lolic

Publications and source records attributed to Anela Lolic.

7 recordsLinked to original sources

arXiv2026

Schemata, Cyclic Proofs and Herbrand Systems

Inductive proofs can be represented by proof schemata, a formalism that represents infinite sequences of proofs by recursive definitions. Since proof schemata avoid the explicit application of induction rules, they admit novel applications, one of which is the realization of Herbrand's theorem in the presence of induction. In this paper, we develop a new type of proof schema based on point transition systems. For skolemized proof schemata without quantified cuts, so-called Herbrand systems, that is, schemata of Herbrand instances of quantified formulas, can be computed. Herbrand systems also allow the representation of schemata of Herbrand sequents, thereby realizing Herbrand's theorem for proof schemata. We compare proof schemata with cyclic proofs and define a transformation from a large class of cyclic proofs to proof schemata. Finally, we show that proof schemata based on point transition systems are capable of proving the 2-Hydra statement, a well-known example that is provable by the cyclic proof system CLKID\omega but not in LKID.

cs.LO↗

arXiv2025

Skolemization and Decidability of the Bernays-Schoenfinkel Class in Goedel Logics

In 1928, Bernays and Schoenfinkel proved the decidability of prenex sentences whose matrices contain no function symbols, now known as the Bernays-Schoenfinkel (BS) class. We investigate the decidability of the BS class for all Goedel logics. Our validity argument relies on the fact that Skolemization works for prenex Goedel logics, while 1-satisfiability follows from structural properties of prenex formulas. We show that validity and 1-satisfiability for the BS class are decidable in every Goedel logic, and that these properties persist across all infinite Goedel logics.

cs.LO↗

arXiv2024

Herbrand's Theorem in Refutation Schemata

An inductive proof can be represented as a proof schema, i.e. as a parameterized sequence of proofs defined in a primitive recursive way. A corresponding cut-elimination method, called schematic CERES, can be used to analyze these proofs, and to extract their (schematic) Herbrand sequents, even though Herbrand's theorem in general does not hold for proofs with induction inferences. This work focuses on the most crucial part of the schematic cut-elimination method, which is to construct a refutation of a schematic formula that represents the cut-structure of the original proof schema. We develop a new framework for schematic substitutions and define a unification algorithm for resolution schemata. Moreover, we show that this new formalism allows the extraction of a structure from the refutation schema, called a Herbrand schema, which represents its Herbrand sequent.

math.LO↗

arXiv2024

Epsilon Calculus Provides Shorter Cut-Free Proofs

In this paper we show that cut-free derivations in the epsilon format of sequent calculus provide for a non-elementary speed-up w.r.t. cut-free proofs in usual sequent calculi in first-order language.

math.LO↗

arXiv2020

First-Order Interpolation Derived from Propositional Interpolation

This paper develops a general methodology to connect propositional and first-order interpolation. In fact, the existence of suitable skolemizations and of Herbrand expansions together with a propositional interpolant suffice to construct a first-order interpolant. This methodology is realized for lattice-based finitely-valued logics, the top element representing true. It is shown that interpolation is decidable for these logics.

math.LO↗

arXiv2020

Schematic Refutations of Formula Schemata

Proof schemata are infinite sequences of proofs which are defined inductively. In this paper we present a general framework for schemata of terms, formulas and unifiers and define a resolution calculus for schemata of quantifier-free formulas. The new calculus generalizes and improves former approaches to schematic deduction. As an application of the method we present a schematic refutation formalizing a proof of a weak form of the pigeon hole principle.

cs.LO↗

arXiv2018

Proof Schemata for Theories equivalent to $PA$: on the Benefit of Conservative Reflection Principles

Induction is typically formalized as a rule or axiom extension of the LK-calculus. While this extension of the sequent calculus is simple and elegant, proof transformation and analysis can be quite difficult. Theories with an induction rule, for example Peano arithmetic do not have a {\em Herbrand} theorem. In this work we extend an existing meta-theoretic formalism, so called proof schemata, a recursive formulation of induction particularly suited for proof analysis, to Peano arithmetic. This relationship provides a meaningful conservative reflection principle between \PA and an alternative proof formalism. Proof schemata have been shown to have a variant of Herbrand's theorem for classical logic which can be lifted to the subsystem of our new formalism equivalent to primitive recursive arithmetic.

math.LO↗