Reconstruction Proofs at the Assertion Level

  title={Reconstruction Proofs at the Assertion Level},
  author={Xiaorong Huang},
Most automated theorem provers suffer from the problem that they can produce proofs only in formalisms difficult to understand even for experienced mathematicians. Effort has been made to reconstruct natural deduction (ND) proofs from such machine generated proofs. Although the single steps in ND proofs are easy to understand, the entire proof is usually at a low level of abstraction, containing too many tedious steps. To obtain proofs similar to those found in mathematical textbooks, we… 
Translating Machine-Generated Resolution Proofs into ND-Proofs at the Assertion Level
This paper first study resolution proofs in terms of meaningful operations employed by human mathematicians, and thereby establishes a correspondence between resolution proofs and ND proofs at a more abstract level, and shows that if its unit initial clauses are CNFs of literal premises of a problem, a unit resolution corresponds directly to a well-structured ND proof segment that mathematicians intuitively understand as the application of a definition or a theorem.
PROVERB– A System Explaining Machine-Found Proofs
  • Xiaorong Huang
  • Computer Science
    Proceedings of the Sixteenth Annual Conference of the Cognitive Science Society
  • 2019
An implemented system called PROVERB is outlined that explains machine -found natural deduction proofs in natural language using the concept of so-called assertion level inference rules and combines standard hierarchical text planning and techniques that locally organize argumentative texts based on the derivation relation under the guidance of a focus mechanism.
Finding Small Proofs for Description Logic Entailments: Theory and Practice (Extended Technical Report)
An approach for generating proofs for expressive DLs based on a non-standard reasoning task called forgetting is developed and implemented and compared the obtained proofs with proofs generated by the DL reasoner ELK, finding that forgetting-based proofs are often better w.r.t. different measures of proof complexity.
Finding Good Proofs for Description Logic Entailments Using Recursive Quality Measures (Extended Technical Report)
Results are provided for a class of measures called recursive, which yields lower complexities and also encompasses proof depth and closes some gaps left open in the previous work, providing a comprehensive picture of the complexity landscape.
Finding Good Proofs for Description Logic Entailments Using Recursive Quality Measures (Extended Abstract)
The complexity of finding proofs of a given quality among all possible alternative proofs is investigated, based on the notion of a deriver, which generates a so-called derivation structure consisting of possible proof steps, from which all proofs of the given consequence can be constructed.
Presenting Proofs in a Human-Oriented Way
An approach is described that successively enhances a logically self-contained proof at the assertion level through communicatively justified modifications of the original line of reasoning, which includes expansion of involved theorem applications, omission of trivial justifications, compactification of intermediate inference steps, and broadening the scope of justifications to support focused argumentation.
Harpoon: Mechanizing Metatheory Interactively - (System Description)
Harpoon is an interactive proof engine built on top of Beluga that allows users to develop proofs interactively using a small, fixed set of high-level actions that safely transform a subgoal.
Principles of Superdeduction
This paper presents and studies the dual concept where the theory is used to enrich the deduction system with new deduction rules in a systematic, correct and complete way, and introduces a proof-term language and a cut-elimination procedure both based on Christian Urban's work on classical sequent calculus.
Interactive Proof Construction at the Task Level
A human oriented interaction layer is obtained that improves and combines ideas underlying the window inference technique [RS93], the proof by pointing approach [BKT94], and the focus windows of [PB02].


Presenting Intuitive Deductions via Symmetric Simplification
In automated deduction systems that are intended for human use, the presentation of a proof is no less important than its discovery, and too often the rule of indirect proof is used where the introduction of a lemma would result in a shorter and more intuitive proof.
Human oriented proof presentation - a reconstructive approach
A reconstructive architecture is presented which substantially abstracts, reorganizes and finally translates machine-foundproofs into natural language and finds their basis in computational models for informal mathematical reasoning and for proof presentation.
Omega-MKRP: A Proof Development Environment
A system architecture is proposed that combins the positive features of different types of theorem-proving systems, most notably the advantages of human-oriented systems based on methods and the deductive strength of traditional automated theorem provers.
The Use of Explicit Plans to Guide Inductive Proofs
  • A. Bundy
  • Computer Science, Mathematics
  • 1988
We propose the use of explicit proof plans to guide the search for a proof in automatic theorem proving. By representing proof plans as the specifications of LCF-like tactics, [Gordon et al 79], and
Proofs in Higher-Order Logic
This work resolves the open question of what is a sound definition of skolemization in higher-order logic but also provides a direct, syntactic proof of its correctness.
An Explanatory Framework for Human Theorem Proving
A computational theory accounting for human formal deductive competence is presented, which cast the cognitive activities involved in theorem proving as an interleaving process of metalevel planning and object level verification.
Transforming Matings into Natural Deduction Proofs
A procedure is given for transforming refutation matings into natural deduction proofs, which illuminates the close relationship between matings and proofs, and serves as a step toward a synthesis between apparently quite different approaches to automated theorem proving.
The Translation of Formal Proofs into English
  • D. Chester
  • Computer Science, Philosophy
    Artif. Intell.
  • 1976