67 research outputs found
A Spectrum of Applications of Automated Reasoning
The likelihood of an automated reasoning program being of substantial
assistance for a wide spectrum of applications rests with the nature of the
options and parameters it offers on which to base needed strategies and
methodologies. This article focuses on such a spectrum, featuring W. McCune's
program OTTER, discussing widely varied successes in answering open questions,
and touching on some of the strategies and methodologies that played a key
role. The applications include finding a first proof, discovering single
axioms, locating improved axiom systems, and simplifying existing proofs. The
last application is directly pertinent to the recently found (by R. Thiele)
Hilbert's twenty-fourth problem--which is extremely amenable to attack with the
appropriate automated reasoning program--a problem concerned with proof
simplification. The methodologies include those for seeking shorter proofs and
for finding proofs that avoid unwanted lemmas or classes of term, a specific
option for seeking proofs with smaller equational or formula complexity, and a
different option to address the variable richness of a proof. The type of proof
one obtains with the use of OTTER is Hilbert-style axiomatic, including details
that permit one sometimes to gain new insights. We include questions still open
and challenges that merit consideration.Comment: 13 page
Double-Negation Elimination in Some Propositional Logics
This article answers two questions (posed in the literature), each concerning
the guaranteed existence of proofs free of double negation. A proof is free of
double negation if none of its deduced steps contains a term of the form
n(n(t)) for some term t, where n denotes negation. The first question asks for
conditions on the hypotheses that, if satisfied, guarantee the existence of a
double-negation-free proof when the conclusion is free of double negation. The
second question asks about the existence of an axiom system for classical
propositional calculus whose use, for theorems with a conclusion free of double
negation, guarantees the existence of a double-negation-free proof. After
giving conditions that answer the first question, we answer the second question
by focusing on the Lukasiewicz three-axiom system. We then extend our studies
to infinite-valued sentential calculus and to intuitionistic logic and
generalize the notion of being double-negation free. The double-negation proofs
of interest rely exclusively on the inference rule condensed detachment, a rule
that combines modus ponens with an appropriately general rule of substitution.
The automated reasoning program OTTER played an indispensable role in this
study.Comment: 32 pages, no figure
Superposition as a logical glue
The typical mathematical language systematically exploits notational and
logical abuses whose resolution requires not just the knowledge of domain
specific notation and conventions, but not trivial skills in the given
mathematical discipline. A large part of this background knowledge is expressed
in form of equalities and isomorphisms, allowing mathematicians to freely move
between different incarnations of the same entity without even mentioning the
transformation. Providing ITP-systems with similar capabilities seems to be a
major way to improve their intelligence, and to ease the communication between
the user and the machine. The present paper discusses our experience of
integration of a superposition calculus within the Matita interactive prover,
providing in particular a very flexible, "smart" application tactic, and a
simple, innovative approach to automation.Comment: In Proceedings TYPES 2009, arXiv:1103.311
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