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Such a proof begins with its assumptions, axioms of the field under study, and zero or more additional hypotheses. In contrast to a forward proof, I cannot recall seeing in any of my graduate courses in mathematics what I term a backward proof. Such a proof begins with the denial of the theorem to be proved and works backward, eventually producing the negation or denial of some axiom or additional assumption for the theorem in focus.

A so-called bidirectional proof also exists; again, I never saw one in my graduate mathematics courses. Such a proof includes statements some of which follow from axioms and, if present, hypotheses of the theorem in focus, while others follow from mc kats dating fille denial or negation of the goal. At this point I will concentrate on forward proofs, but the observations and challenges that follow apply also to so-called backward proofs and bidirectional proofs.

Implicit steps, demodulation, and automated reasoning Often, many steps in a mathematical proof -- especially in a paper intended for publication -- are omitted from the proof.

### From the AAR President, Larry Wos

The referee has to in effect guess at the missing steps, and sometimes the verification by the referee of the proposed proof is indeed difficult. For a trivial example, applying associativity implicitly and repeatedly to obtain a third step from two others in a problem from group theory can present difficulty and can be easily overlooked.

Much subtler is the implicit use of some lemma or theorem proved earlier in the intended publication.

Such implicit and hidden aspects are often treated as obvious; that is, the parents may be cited, but other aspects are not. In automated reasoning, the object that corresponds to such an implicit use in mathematics is a demodulator -- an equality that encodes, for example, associativity.

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And the number of demodulators cited with a single derived equation in the proof can make a huge difference in the reading of the proof. Let me give you an example.

Among his other successes, some of the proofs he obtained with OTTER offered single lines depending on a large number of demodulators.

When I showed that proof to the mathematician Ken Kunen, he commented that the huge set of demodulators, for a single line of the proof was too much to cope with.

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Indeed, he found the resulting equation difficult to verify because of so many implicit moves that were in effect made after the two parents were treated with paramodulation -- moves implicitly made because of many lines of numbers corresponding to demodulators. Challenges in group theory With this background, I now offer shortly several relevant challenges, challenges taken from group theory. Since the axioms for a group are vital to the challenges issued here, I give these and the definition of commutator, where the function f denotes product, the constant e denotes identity, the function g denotes inverse, and the function h denotes commutator.

In the first challenge I ask you to prove the equivalence of all three properties. In other words, find six proofs: prove property 2 from property 1, 3 from 1, 3 from 2, 1 from 2, 2 from 3, and finally 1 from 3.

In the second challenge, I ask you to find a circle of pure proofs for the three.

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Specifically, I am asking you to find a set of three proofs consisting of a proof that the second property follows from the first in which the third property does not occur; a proof that the third is provable from the second with the first not occurring in the proof; and a proof that the mc kats dating fille is provable from the third with the second property excluded from the proof.

When and if you succeed, I offer more difficult challenges.

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In particular, for each of the six proofs you have found, you are to find 1 a forward proof, 2 a mc kats dating fille free of demodulators, and 3 a forward proof for each that is also free of demodulators. In cases in which the proof you may have found is backward or bidirectional, and in the case in which demodulators are present, I believe you will find that two rather disconnected procedures are required, one for producing a forward proof and one for removing demodulators.

## DS-PD1-MC-RS

A research challenge For those of you ready to face a challenge that is more a type of research, I offer the following. Specifically, you are asked to find methods to convert bidirectional or backward proofs to forward and to find a procedure to remove any and all demodulators from proofs that rely on such.

Should you wish some insight into the possible difficulty regarding demodulation removal, I include the following lines taken from a step OTTER proof proving that property 1 implies property 3.

As line indicates, the proof relies on 38 demodulators, where corresponds to the associativity of product, in the function f, and is a step occurring earlier in the proof, a step you might view as a lemma if written out for a detailed paper being submitted for publication.

### Guest Column: on Zohar Manna

If you find six proofs each of which is a forward proof free of demodulation, I would very much enjoy the opportunity to examine that proof. My e-mail address is wos mcs. Guest Column: on Zohar Manna An extended version, including a list of publications, appears in Formal Aspects of Computing Zohar Manna, founding father of the study and application of formal methods for software and hardware verification, died peacefully at his home in Netanya, Israel on August 30,after a long and marvelously productive career.

He is survived by his wife Nitza and their four children and five grandchildren.

He was Over a career spanning nearly half a century, Zohar had profound impact on most aspects of formal methods and automated reasoning.

He was a deep thinker who laid the foundations for tools that are now coming into widespread use.

He pioneered program verification and program synthesis, two fields that were then at the theoretical edge of computing, but which today help form the foundations of artificial intelligence and help assure the reliability of extraordinarily complex software. His work has inspired several generations of researchers.

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His manifold research interests included, in particular, the design and verification of concurrent, reactive and hybrid systems. His students and colleagues dedicated their research careers to the hardest problems in automated reasoning, including program semantics, partial correctness, termination, invariant generation, program synthesis, program transformation, planning, proof methodology, temporal reasoning, natural language understanding, non-clausal proof search, and decision procedures.

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Each of these activities is today a thriving independent field of research. Afterwards, he attended Carnegie Mellon University in Pittsburgh together with one of us, Richardwhere he earned his Ph. Perlis both Turing Award recipients. Zohar returned to Israel in to the Department of Applied Mathematics at the Weizmann Institute of Science in Rehovot, first as an associate professor and from on as full professor where he supervised the dissertation of the other one of us, Nachum.

Inhe was recruited back to Stanford as a full professor taking Nachum along with himdividing his time between Stanford and Weizmann untilat which point he resigned the latter appointment.

He remained at Stanford University until his retirement in All are all models of clarity and comprehensiveness.

## Facts Plus

Mc kats dating fille magnificent textbook, Mathematical Theory of Computationwas extraordinarily influential; it was translated into Bulgarian, Czech, Hungarian, Italian, Japanese, and Russian. It pioneered the logical analysis of programs for correctness vis-à-specifications and for termination properties. The proof is restricted to be sufficiently constructive so that a program that computes the satisfying output can be extracted.

Conditional expressions are introduced via case analysis in the proof; recursion is introduced by application of recursion induction.