Jefferson de Barros Santos

arXiv2021

A logical framework with a graph meta-language

We conjecture that the relative unpopularity of logical frameworks among practitioners is partly due to their complex meta-languages, which often demand both programming skills and theoretical knowledge of the meta-language in question for them to be fruitfully used. We present ongoing work on a logical framework with a meta-language based on graphs. A simpler meta-language leads to a shallower embedding of the object language, but hopefully leads to easier implementation and usage. A graph-based logical framework also opens up interesting possibilities in time and space performance by using heavily optimized graph databases as backends and by proof compression algorithms geared towards graphs. Deductive systems can be specified using simple domain-specific languages built on top of the graph database's query language. There is support for interactive (through a web-based interface) and semiautomatic (following user-defined tactics specified by a domain-specific language) proof modes. We have so far implemented nine systems for propositional logic, including Fitch-style and backward-directed Natural Deduction systems for intuitionistic and classical logic (with the classical systems reusing the rules of the intuitionistic ones), and a Hilbert-style system for the K modal logic.

cs.LO↗

arXiv2020

A Sequent Calculus Proof Search Procedure and Counter-model Generation based on Natural Deduction Bounds

In a previously published ENTCS paper (Santos et al. (2016)), we introduced a sequent calculus called $\mathbf{LMT^{\rightarrow}}$ for Minimal Implicational Propositional Logic ($\mathbf{LMT^{\rightarrow}}$). This calculus provides a proof search procedure for $\mathbf{LMT^{\rightarrow}}$ that works in a bottom-up approach. We proved there that $\mathbf{LMT^{\rightarrow}}$ is sound and complete. We also suggested a strategy to guarantee termination of the proof search procedure. In this current paper, we refined this strategy and presented a new strategy for $\mathbf{LMT^{\rightarrow}}$ termination. Considering this new strategy, we also provide a (new) completeness proof for the system, which improves the previous version. Besides that, we present explicit upper bounds on the proof search procedure, derived from this new strategy. We also provide a full soundness proof of the system.

cs.LO↗