A calculational approach to mathematical induction

Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence, 2017

In many knowledge representation formalisms, a constructive semantics is defined based on sequential applications of rules or of a semantic operator. These constructions often share the property that rule applications must be delayed until it is safe to do so: until it is known that the condition that triggers the rule will remain to hold. This intuition occurs for instance in the well-founded semantics of logic programs and in autoepistemic logic. In this paper, we formally define the safety criterion algebraically. We study properties of so-called safe inductions and apply our theory to logic programming and autoepistemic logic. For the latter, we show that safe inductions manage to capture the intended meaning of a class of theories on which all classical constructive semantics fail.

School Science and Mathematics, 1993

Journal of Symbolic Computation, 1991

The inductionless induction (also called proof by consistency) approach for proving equations by induction from an equational theory, requires a consistency cheek for equational theories. A new method using test sets for checking consistency of an equational theory is proposed. Using this method, a variation of the Knuth-Bendix completion procedure can be used for automatically proving equations by induction. The method does not suffer from limitations imposed by the methods proposed by Musser as well as by Huet and Hullot, and is as powerful as Jouannaud and Kounalis' method based on ground-reducibility. A theoretical comparison of the test set method with Jouannaud and Kounalis' method is given showing that the test set method is generally much better. Both the methods have been implemented in RRL, Rewrite Rule Laboratory, a theorem proving environment based on rewriting techmques and completion. In practice also, the test set method is faster than Jouannaud and Kounalis' method. The test set construction can also be used to check for the sufficient-completeness property of equational axiomatizations including algebraic specifications of abstract data types as well as for identifying constructors in an algebraic specification.

Journal of Automated Reasoning, 2006

Otter-lambda is Otter modified by adding code to implement an algorithm for lambda unification. Otter is a resolution-based, clause-language first-order prover that accumulates deduced clauses and uses strategies to control the deduction and retention of clauses. This is the first time that such a first-order prover has been combined in one program with a unification algorithm capable of instantiating variables to lambda terms to assist in the deductions. The resulting prover has all the advantages of the proof-search algorithm of Otter (speed, variety of inference rules, excellent handling of equality) and also the power of lambda unification. We illustrate how these capabilities work well together by using Otterlambda to find proofs by mathematical induction. Lambda unification instantiates the induction schema to find a useful instance of induction, and then Otter's first-order reasoning can be used to carry out the base case and induction step. If necessary, induction can be used for those, too. We present and discuss a variety of examples of inductive proofs found by Otter-lambda: some in pure Peano arithmetic; some in Peano arithmetic with defined predicates; some in theories combining algebra and the natural numbers; some involving algebraic simplification (used in the induction step) by simplification code from MathXpert; and some involving list induction instead of numerical induction. These examples demonstrate the feasibility and usefulness of adding lambda unification to a first-order prover.

2006

Developing correct software remains one of the most important subjects in computer science. Bugs can be costly and annoying. One of the most secure ways to eliminate errors in computer programs is by proving them to be correct in a mathematical context. Sparkle is a proof assistant that helps programmers with constructing mathematical proofs about algorithms written in Clean or any other functional language. Proofs on programs that use one of the most important programming techniques, called recursion, usually need proof principles called inductive and co-inductive reasoning. Support for these mathematical proof steps were limited within Sparkle. In order to support for reasoning about a larger class of programs we have extended the proof techniques in Sparkle. A method that supports mutually reasoning on mutually recursive types has been added. A new method has been devised that allows for the derivation of an induction scheme from function definitions. For co-inductive reasoning, ...

2004

These notes are written as a supplement to [1, Sect. 16.1&16.3], but can be read independently. 1 Demystifying Induction Consider a loop of the form while B do C od, and assume that we know 1 ψ is established by the preamble of the loop (1) if with B true, ψ holds prior to C, then ψ also holds after C (2) Then we can infer that ψ is an invariant of the loop. (3) That is, each time control reaches B, ψ holds. this amounts to (1) and (2) below being valid annotations. {ψ} (1) while B do {ψ ∧ B} WhileTrue C {ψ} (2) od

Journal of Automated Reasoning, 1996

We present a procedure for proving inductive theorems which is based on explicit induction, yet supports mutual induction. Mutual induction allows the postulation of lemmas whose proofs use the theorems ex hypothesi while the theorems themselves use the lemmas. This feature has always been supported by induction procedures based on Knuth-Bendix completion, but these procedures are limited by the use of rewriting (or rewriting-like) inferences. Our procedure avoids this limitation by making explicit the implicit induction realized by these procedures. As a result, arbitrary deduction mechanisms can be used while still allowing mutual induction.

Logic, Construction, Computation, 2012

Induction-induction is a principle for mutually defining data types A ∶ Set and B ∶ A → Set. Both A and B are defined inductively, and the constructors for A can refer to B and vice versa. In addition, the constructor for B can refer to the constructor for A. Induction-induction occurs in a natural way when formalising dependent type theory in type theory. We give some examples of inductive-inductive definitions, such as the set of surreal numbers. We then give a new finite axiomatisation of the principle of induction-induction, and prove its consistency by constructing a model.

2011

The purpose of this note is to present a simplification of the system of arithmetical axioms given in previous work; specifically, it is shown how the induction principle can in fact be obtained from the remaining axioms, without the need of explicit postulation. The argument might be of more general interest, beyond the specifics of the proposed axiomatization, as it highlights the interaction of the notion of Dedekind-finiteness and the induction principle.

Theoretical Computer Science, 2000

This paper is part of a long-term e ort to increase expressiveness of algebraic speciÿcation languages while at the same time having a simple semantic foundation on which e cient execution by rewriting and powerful theorem-proving tools can be based. In particular, our rewriting techniques provide semantic foundations for Maude's functional sublanguage, where they have been e ciently implemented. This e ort started in the late 1970s, led by the ADJ group, who promoted equational logic and universal algebra as the semantic basis of program speciÿcation languages. An important later milestone was the work around order-sorted algebras and the OBJ family of languages developed at SRI-International in the 1980s. This e ort has been substantially advanced in the mid-1990s with the development of Maude, a language based on membership equational logic. Membership equational logic is quite simple, and yet quite powerful. Its atomic formulae are equations and sort membership assertions, and its sentences are Horn clauses. It extends in a conservative way both (a version of) order-sorted equational logic and partial algebra approaches, while Horn logic with equality can be very easily encoded. After introducing the basic concepts of the logic, we give conditions and proof rules with which ecient equational deduction by rewriting can be achieved. We also give completion techniques to transform a speciÿcation into one meeting these conditions. We address the important issue of proving that a speciÿcation protects a subspeciÿcation, a property generalizing the usual notion of su cient completeness. Using tree-automata techniques, we develop a test-set-based approach for proving inductive theorems about a parameterized speciÿcation. We brie y discuss a number of extensions of our techniques, including rewriting modulo axioms such as associativity and commutativity, having extra variables in conditions, and solving goals by narrowing. Finally, : S 0 3 0 4 -3 9 7 5 ( 9 9 ) 0 0 2 0 6 -6 36 A. Bouhoula et al. / Theoretical Computer Science 236 (2000) we discuss the generality of our approach and how it extends several previous approaches. →, respectively. Given a relation → on the set S, an element s ∈ S is in normal form if there is no t ∈ S such that s → t. A normal form of s ∈ S is an element in normal form t ∈ S such that s → * t. We denote by s↓ the set of normal forms of s.