A Sequent Calculus with Implicit Term Representation

@inproceedings{Hetzl2010ASC,
  title={A Sequent Calculus with Implicit Term Representation},
  author={Stefan Hetzl},
  booktitle={CSL},
  year={2010}
}
  • Stefan Hetzl
  • Published in CSL 23 August 2010
  • Computer Science, Mathematics
We investigate a modification of the sequent calculus which separates a first-order proof into its abstract deductive structure and a unifier which renders this structure a valid proof. We define a cutelimination procedure for this calculus and show that it produces the same cut-free proofs as the standard calculus, but, due to the implicit representation of terms, it provides exponentially shorter normal forms. This modified calculus is applied as a tool for theoretical analyses of the… 
Analogy in Automated Deduction: A Survey
TLDR
A general framework for reasoning by analogy based on a constrained sequent calculus in which higher-order variables denote first-order formulae is proposed.
Proof Nets for First-Order Additive Linear Logic
TLDR
Two versions of canonical proof nets are presented, one of which retains explicit witnessing information to existential quantification, and the other which captures sequent calculus cut-elimination as a one-step global composition operation.
Proof Nets for First-Order Additive Linear Logic
TLDR
Two versions of the proof nets for first-order additive linear logic are presented, one of which retains explicit witnessing information to existential quantification, and the other, which captures sequent calculus cut-elimination as a one-step global composition operation.
Proof Generalization in $$\mathrm {LK}$$LK by Second Order Unifier Minimization
TLDR
A lifting theorem shows how a valid LKc-proof can be lifted to a most general proof, yielding a non-trivial constraint together with a solution.

References

SHOWING 1-10 OF 33 REFERENCES
Proof Normalization Modulo
TLDR
It is conjectured that proof normalization and logical consistency always hold over this class of formalisms, provided some minimal conditions over the rewrite system are fulfilled, and this conjecture is proved for some subcases, including Church's higher-order logic (HOL).
On the non-confluence of cut-elimination
TLDR
The construct of a sequence of polynomial-length proofs having a non-elementary number of different cut-free normal forms illustrates that the constructive content of a proof in classical logic is not uniquely determined but rather depends on the chosen method for extracting it.
A Theory of Explicit Substitutions with Safe and Full Composition
  • D. Kesner
  • Computer Science
    Log. Methods Comput. Sci.
  • 2009
TLDR
Very simple technology in named variable-style notation is used to establish a theory of explicit substitutions for the lambda-calculus which enjoys a whole set of useful properties such as full composition, simulation of one-step beta-reduction, preservation of beta-strong normalisation, strong normalisation of typed terms and confluence on metaterms.
Theorem Proving Modulo
TLDR
This paper defines a sequent calculus modulo that gives a proof-theoretic account of the combination of computations and deductions and gives a complete proof search method, called extended narrowing and resolution (ENAR), for theorem proving modulo such congruences.
On the form of witness terms
  • Stefan Hetzl
  • Computer Science, Mathematics
    Arch. Math. Log.
  • 2010
TLDR
A regular tree grammar computing witness terms is given and a class of proofs is shown to have only elementary cut-elimination.
Lambda-Mu-Calculus: An Algorithmic Interpretation of Classical Natural Deduction
TLDR
This paper presents a way of extending the paradigm "proofs as programs" to classical proofs, which can be seen as a simple extension of intuitionistic natural deduction, whose algorithmic interpretation is very well known.
A mechanization of type theory
TLDR
A refutational system of logic for a language of order is presented, in the sense that a refutation of a set of sentences exists if and only if this set does not possess a general Henkin model.
Describing proofs by short tautologies
Uniform Proofs as a Foundation for Logic Programming
Natural deduction via graphs: formal definition and computation rules
  • H. Geuvers, I. Loeb
  • Mathematics, Computer Science
    Mathematical Structures in Computer Science
  • 2007
TLDR
This paper introduces the formalism of deduction graphs as a generalisation of both Gentzen–Prawitz style natural deduction and Fitch style flag deduction, and proposes a translation to a context calculus with lets that faithfully captures the structure of deduction graph.
...
...