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Set theory for verification: I. From foundations to functions
 J. Auto. Reas
, 1993
"... A logic for specification and verification is derived from the axioms of ZermeloFraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higherord ..."
Abstract

Cited by 45 (17 self)
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A logic for specification and verification is derived from the axioms of ZermeloFraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higherorder syntax supports the definition of new binding operators. Unknowns in subgoals can be instantiated incrementally. The paper describes the derivation of rules for descriptions, relations and functions, and discusses interactive proofs of Cantor’s Theorem, the Composition of Homomorphisms challenge [9], and Ramsey’s Theorem [5]. A generic proof assistant can stand up against provers dedicated to particular logics. Key words. Isabelle, set theory, generic theorem proving, Ramsey’s Theorem,
Set Theory for Verification: II  Induction and Recursion
 Journal of Automated Reasoning
, 2000
"... A theory of recursive definitions has been mechanized in Isabelle's ZermeloFraenkel (ZF) set theory. The objective is to support the formalization of particular recursive definitions for use in verification, semantics proofs and other computational reasoning. ..."
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Cited by 42 (20 self)
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A theory of recursive definitions has been mechanized in Isabelle's ZermeloFraenkel (ZF) set theory. The objective is to support the formalization of particular recursive definitions for use in verification, semantics proofs and other computational reasoning.
Automated theorem proving: mapping logic into AI
 Proceedings of the International Symposium on Methodologies for Intelligent Systems
, 1986
"... ABSTRACT. Logic can be defined as the formal study of reasoning; if we replace "formal " by "mechanical " we can place almost the entire set of methodologies used in the field of automated theorem proving (ATP) within the scope of logic. Because of the goals of ATP, if not alway ..."
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Cited by 1 (0 self)
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ABSTRACT. Logic can be defined as the formal study of reasoning; if we replace "formal " by "mechanical " we can place almost the entire set of methodologies used in the field of automated theorem proving (ATP) within the scope of logic. Because of the goals of ATP, if not always the methodologies, ATP has been considered to be within the domain of AI. We explore the methodologies of ATP, including the logics that underlie the theorem provers, and discuss some of the mechanisms that utilize these logics. These include term rewriting systems, mathematical induction, inductionless induction and even mixed integer programming. The ATP field, via resolution, has even provided the foundation for an exciting AI and database programming language, PROLOG. We conclude with a new method for extending the PROLOG system to work with nonHorn clause sets within a positive logic format, particularly simple for "slightly nonHorn " clause sets.
unknown title
"... Prior SO discussing what 4. see as desirable and achievable features of the neXe generation of interactive theorem proverss 1 vant to say something about the history of my o w work and that of my colleagues, which forms the basis for the wfew of the future I sketch in the remainder of this paper. Si ..."
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Prior SO discussing what 4. see as desirable and achievable features of the neXe generation of interactive theorem proverss 1 vant to say something about the history of my o w work and that of my colleagues, which forms the basis for the wfew of the future I sketch in the remainder of this paper. Simple uses Q Pateractive theorem prover for the teaching of elementary hgfc began more than twenty years ago. I remember well our first denonstrations with elementaryschool chSBdren in 1963. For a number of years we concentrated on teaching elementary logic and algebra to bright elementary and middleschool children. We felt at. the time that. this was the rlght Iewel of difficulty to reach capacity and for in terms of compter T~SQU~CSS thae could be devoted to the endeavor. A%B of this early work was done QX ~ one of the lowserialnumber PDPlasa which John McCarthy and I, 30htly purchased fr~s grants at Stanford in 1963.
user interfaces
"... Large online courses often assign problems that are gradable by simple checks such as multiple choice, but these checks are inappropriate for domains in which students may produce an infinity of correct solutions. One such domain is derivations: sequences of logical steps commonly used in assignment ..."
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Large online courses often assign problems that are gradable by simple checks such as multiple choice, but these checks are inappropriate for domains in which students may produce an infinity of correct solutions. One such domain is derivations: sequences of logical steps commonly used in assignments for technical, mathematical and scientific subjects. We present DeduceIt, a system for creating, grading, and analyzing derivation assignments across arbitrary formal domains. DeduceIt supports assignments in any logical formalism, provides students with incremental feedback, and aggregates student paths through each proof to produce instructor analytics. DeduceIt benefits from checking thousands of derivations on the web: it introduces a proof cache, a novel data structure which leverages a crowd of students to decrease the cost of checking derivations and providing realtime, constructive feedback. We evaluate DeduceIt with 990 students in an online compilers course, finding students take advantage of its incremental feedback and instructors benefit from its structured insights into confusing course topics. Our work suggests that automated reasoning can extend online assignments and largescale education to many new domains. Author Keywords MOOC, theorem prover, formal logic, online education