J. S. Moore. A mechanical proof of the termination of Takeuchi's function. Information Processing Letters, 9:176-181, 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....as input output relations, and if so then prove their inductive invariance. That would result in a method similar to the one developed by Giesl [5] 3. 2 The Method of Alternative Specifications A well known method for proving termination of nested recursive equations was proposed by Moore [8]. Given a functional F as before, it asks the user to provide a concrete function h satisfying two conditions: M 1 ) #x. F h x = h x ( partial correctness ) M 2 ) TC # h, for some wellfounded relation #. Theorem 5 If h satisfies the conditions (M 1 ) and (M 2 ) then h is a fixpoint of F ....

....in (16) is now legitimate and gives us f (z 11) f (f (z 11) 11, so (17) follows by transitivity of . 10 Takeuchi s Tarai Function This is the function of type int defined by the functional F f (x, y, z) y then y else f(f(x 1, y, z) f(y 1, z, x) f(z 1, x, y) Moore [8] proved the termination of F by showing that the function h defined by h(x, y, z) if x y then y else (if y z then z else x) satisfies the conditions (M 1 ) and (M 2 ) above with respect to the wellfounded relation induced by a certain measure #. It turns out that predicates u = ....

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J. S. Moore. A mechanical proof of the termination of Takeuchi's function. Information Processing Letters, 9:176--181, 1979.

.... x; y; z, t(x; y; z) def ( if x y then y (1) else t(t(x Gamma 1; y; z) t(y Gamma 1; z; x) t(z Gamma 1; x; y) John McCarthy proved that this recursion terminates and that t can be computed without any recursion, t(x; y; z) if x y then y (2) else if y z then z else x: 1 J Moore [5] discovered a simpler measure than the one used by McCarthy and used the early Boyer Moore theorem prover, THM, to verify termination and that t satisfies the simpler nonrecursive equation. 2 Knuth s Generalization Knuth generalizes the tarai function to higher dimensions: For integer inputs, x ....

J S. Moore. "A Mechanical Proof of the Termination of Takeuchi's Function ". Information Precessing Letters 9, 4 (1979), 176--181. 16

....public key encryption algorithm [7] 3. WILSON contains definitions and theorems for Wilson s theorem [45] 4. GAUSS contains definitions and theorems for Gauss s law of quadratic reciprocity; 5. ZTAK includes definitions and theorems for the termination of Takeuchi s function [35]. Table 5 2 shows the statistics of contents in each library; Table 5 3 shows the number of theorems in each library by group; Table 5 4 and Table 5 5 presents, by group, statistics for the ratio of pruned backward chainings and for the percentage of (elapsed) time saved by using examples, ....

Moore, J S. "A Mechanical Proof of the Termination of Takeuchi's Function". Information Processing Letters 9 (1979), 176-181.

....predicate depends on and try to treat this smaller program with other norms or more sophisticated methods. One of the benchmark programs to which we tried to apply our system is a program for the Takeuchi function. For this function there is a whole paper devoted to proving its termination (cf. StM 79] If such a predicate appeared in a larger program we would put it in a separate module and try to handle it by more sophisticated methods, while using our automatic system for the rest of the program. Another kind of subprogram to which our system cannot be applied is a program that involves ....

....than termination. Of 24 programs it was clear 4 could not be handled: rdtok and read because their termination depends on getting an end of f ile in the input, tictactoe because it contains assert, tak defining the Takeuchi function had a whole paper devoted to proving its termination ( StM 79] Some programs were too big and caused memory problems (this is denoted by the remark mem in the table) The zebra program had too many variables, so had to be divided to parts. fib and hanoi had to be transformed to successor notation. progeom announced after a very long time that there may ....

J. Strother Moore. A Mechanical Proof of the Termination of Takeuchi's Function. Information Processing Letters, 9:176-181, 1979.

....Thus the solution is unique: by well founded induction on the measure, norm(x) norm 2 (x) for all x. Reasoning about norm is possible 7 because it is already known to be a total function. Moore describes this principle of definition in his paper on the termination of Takeuchi s function [7]. Moore s proof of norm 2 has several stages: norm(norm(x) norm(x) is proved by induction on the measure m(x) The result is used in the At case of the next stage. #yz.norm(If (x, norm(y) norm(z) norm(If (x, y, z) is proved, like the Lemma, by structural induction on x. The ....

J S. Moore, A mechanical proof of the termination of Takeuchi's function, Information Processing Letters 9 (1979), pages 176--181.

....of prime factorizations. Improvements made to the system since the publication of [2] include the addition of the above mentioned decision procedures for equalities and simple arithmetic inequalities, the extension of the definitional principle to include reflexive functions as described in [16], and a metafunction facility permitting the incorporation of new simplifiers after they have been mechanically proved correct [3] Finally, we have added a primitive hint facility so that the user can tell the theorem prover how to prove a theorem when its heuristics lead it down blind alleys. ....

....existing mechanical theorem prover was used to check a recently published proof. Among the other mathematically interesting proofs performed by our theorem prover are: Wilson s Theorem: if p is a prime then (p 1) mod p = p 1; 18] the termination over the integers of the Takeuchi function [16]: tak(x,y,z) if x y then y else tak(tak(x 1,y,z) tak(y 1,z,x) tak(z 1,x,y) the soundness and completeness of a decision procedure for propositional calculus [2] the existence of nonprimitive recursive functions; the Turing completeness of the Pure LISP programming language ....

J S. Moore. "A Mechanical Proof of the Termination of Takeuchi's Function". Information Processing Letters 9, 4 (1979), 176-181.

....of irrelevant splits grows exponentially with the number of irrelevant difference or predecessor expressions in the clause. Unfortunately, irrelevant hypotheses are common in mechanically generated formulas. For example, in our system s first proof of the termination of the Takeuchi function [12] the proof of one lemma involved 412 cases, many of which were irrelevant. One solution to this problems is that adopted for the tail biting problem. If the linear procedure keeps track of which literals are involved in the derivation of each polynomial, it is possible to report which literals are ....

J S. Moore. "A Mechanical Proof of the Termination of Takeuchi's Function". Information Processing Letters 9, 4 (1979), 176-181.

....of prime factorizations. Improvements made to the system since the publication of [2] include the addition of the above mentioned decision procedures for equalities and simple arithmetic inequalities, the extension of the definitional principle to include reflexive functions as described in [16], and a metafunction facility permitting the incorporation of new simplifiers after they have been mechanically proved correct [3] Finally, we have added a primitive hint facility so that the user can tell the theorem prover how to prove a theorem when its heuristics lead it down blind alleys. ....

....existing mechanical theorem prover was used to check a recently published proof. Among the other mathematically interesting proofs performed by our theorem prover are: Wilson s Theorem: if p is a prime then (p 1) mod p = p 1; 18] the termination over the integers of the Takeuchi function [16]: tak(x,y,z) if x y then y else tak(tak(x 1,y,z) tak(y 1,z,x) tak(z 1,x,y) the soundness and completeness of a decision procedure for propositional calculus [2] the existence of nonprimitive recursive functions; 9 . the Turing completeness of the Pure LISP programming language [8] ....

J S. Moore. "A Mechanical Proof of the Termination of Takeuchi's Function". Information Processing Letters 9, 4 (1979), 176-181.

.... program (Boyer and Moore, BM84a] basic tmi.events ) proof of the Turing completeness of Pure Lisp (Boyer and Moore, BM84b] basic unsolv.events ) proof of the unsolvability of the halting problem for Pure Lisp (Russinoff, Rus85] basic wilson.events ) proof of Wilson s theorem (Moore, [Moo79], basic ztak.events ) proof of the termination of Takeuchi s function (Bevier, Bev87] bevier kit.events ) the formalization, implementation and proof that a simple separation kernel (implementing multi processing on a uniprocessor) provides process scheduling, error handling, message ....

J S. Moore. A mechanical proof of the termination of Takeuchi's function. Information Processing Letters, 9(4):176--181, 1979.

.... proved are the invertibility of the Rivest, Shamir, and Adleman public key encryption algorithm [10] the soundness and completeness of a propositional calculus decision procedure [5] the soundness of an arithmetic simplifier now used in the system [8] the termination of Takeuchi s function [42], and the correctness of many elementary list and tree processing functions. 4 Examples of other work in this area include [15, 17, 2] The Edinburgh LCF system (discussed below) has also been used to prove properties of recursive functions; among the theorems proved with LCF are the correctness ....

J S. Moore. "A Mechanical Proof of the Termination of Takeuchi's Function". Information Processing Letters 9, 4 (1979), 176-181.

....with Shankar s checking of the Church Rosser theorem. On pp. 4 9 of ACLH, we enumerate many other applications of NQTHM, including those in list processing, elementary number theory, metamathematics, set theory, and concurrent algorithms. Descriptions of some of these applications may be found in [16, 66, 12, 21, 17, 67, 68, 69, 20, 60, 28, 51, 37, 52, 13, 14, 15, 22, 77] and also in [1, 31, 32, 33, 40, 75, 3, 48, 44, 41, 42, 39, 45, 23, 24, 25] Recently colleagues of ours at Computational Logic, Inc. Bill Young and Bill Bevier, have used NQTHM to construct mechanically checked proofs of properties relating to faulttolerance. A key problem facing the designers ....

J S. Moore. "A Mechanical Proof of the Termination of Takeuchi's Function". Information Processing Letters 9, 4 (1979), 176-181.

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J. S. Moore. A mechanical proof of the termination of Takeuchi's function. Information Processing Letters, 9:176-181, 1979.

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J. S. Moore. A mechanical proof of the termination of Takeuchi's function. Information Processing Letters, 9:176-181, 1979.

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