Finitely Presented Groups with Word Problems of Arbitrary Degrees of Insolubility

Clapham, C. R. J.

doi:10.1112/plms/s3-14.4.633

Cited by 37 publications

(22 citation statements)

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“…The first arg of the outermost 3 of y(a) is ß2(a). Using Lemma 2, we can show that this is of neither of forms (1), (2). Hence y(a) cannot be the second premise.…”

Section: Lemma 7 For Any Wff Abc We Have Y(a) # (Y(b) Zz> Y(c))mentioning

confidence: 99%

“…The main result of §5 is that y(a) \-(R + S)y(b) iff (1) m = n, and <D¡ hr *Pf for i = l,---,m, and (2) certain absolutely decidable conditions are satisfied. Using this result, we reduce 7)2, 7)3 to each other.…”

Section: Programmementioning

confidence: 99%

“…Numbers in square brackets, as [1], [2], refer to works by other writers, listed at the end of the paper.…”

Section: Relevant Literaturementioning

confidence: 99%

“…Proof. We shall say that b is standard iff, for some i, where 0 z^i i%n, b is obtained by inserting the brackets in such a way that (1) in the sub-expression ay\J ■■■ \Jah association is always to the left, (2) in the sub-expression ai+y\J ■■■\Ja", association is always to the right. "Standard" is similarly defined for c. To clarify this, an example of a standard wff with n = 9 and i = 4, is a((aiVa2)Vct3)Va4)V(ciiy(a6y(a7y(a8\fa9))))y Using (PI), (P2) and a succession of applications of modus ponens, we can easily show that If b and c are both standard wff, then…”

Section: Programmementioning

confidence: 99%

“…Lemma 8. Let modus ponens be applied to two members of the class of those wff which are of either of the two forms, (1) 7(a), (2) (y(b)z,y(c)). Then the first and second premises must be of forms (1) and (2), respectively.…”

Section: Lemma 7 For Any Wff Abc We Have Y(a) # (Y(b) Zz> Y(c))mentioning

confidence: 99%

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Some Ways of Constructing a Propositional Calculus of Any Required Degree of Unsolvability

Gladstone¹

1965

Transactions of the American Mathematical Society

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“…The first arg of the outermost 3 of y(a) is ß2(a). Using Lemma 2, we can show that this is of neither of forms (1), (2). Hence y(a) cannot be the second premise.…”

Section: Lemma 7 For Any Wff Abc We Have Y(a) # (Y(b) Zz> Y(c))mentioning

confidence: 99%

Section: Programmementioning

confidence: 99%

“…Numbers in square brackets, as [1], [2], refer to works by other writers, listed at the end of the paper.…”

Section: Relevant Literaturementioning

confidence: 99%

Section: Programmementioning

confidence: 99%

Section: Lemma 7 For Any Wff Abc We Have Y(a) # (Y(b) Zz> Y(c))mentioning

confidence: 99%

See 3 more Smart Citations

Some Ways of Constructing a Propositional Calculus of Any Required Degree of Unsolvability

Gladstone¹

1965

Transactions of the American Mathematical Society

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Machine Configuration and Word Problems of Given Degree of Unsolvability

Shepherdson

1965

Mathematical Logic Qtrly

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l ) = have a confluence = lead to a coinnion configuretion. l) This is what TURING [l] calls the 'complete configuration'. 2, i.e. in which at any time at most one move is possible. P , X , y , 2) + ( 3 , x , Yt T b , Y ) ) .Similarly by adding a 4th register to hold a number u and changing line 2 thus:we get a machine M,(y) satisfying: Lemma 3. For each r.e. degree a there exists a machine with confluence problem of degree a and solvable halting and derivability problems. Now take any 3 r.e. predicates A , (x), A , ( x ) , A,(z) with corresponding primitive recursive functions pl, v,, p13 and take a machine with 4 registers holding numbers x , y , z , u and a 6 line program consisting of the program M , ( v l ) followed by the program M , ( y 2 ) followed by M,(y3) [i.e. lines 1 , 2 are identical with lines 1 , 2 of M , (p,) , lines 3 , 4 identical with lines 1 , 2 of M , (y,) except that 'go t o 1' is replaced by 'go to 3' and lines 5 , 6 are identical with lines 1 , 2 of M3(p3) except that 'go to l', 'go to 2' are replaced respectively by 'go to B', 'go t o 6'1. We clearly get a program satisfying : if y ( z , y ) + 0 add 1 to z and go to 1. 2.Put u = 0, go to 2

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Word problems and ceers

Rose

Mauro

Sorbi

2020

Mathematical Logic Qtrly

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This note addresses the issue as to which ceers can be realized by word problems of computably enumerable (or, simply, c.e.) structures (such as c.e. semigroups, groups, and rings), where being realized means to fall in the same reducibility degree (under the notion of reducibility for equivalence relations usually called “computable reducibility”), or in the same isomorphism type (with the isomorphism induced by a computable function), or in the same strong isomorphism type (with the isomorphism induced by a computable permutation of the natural numbers). We observe, e.g., that every ceer is isomorphic to the word problem of some c.e. semigroup, but (answering a question of Gao and Gerdes) not every ceer is in the same reducibility degree of the word problem of some finitely presented semigroup, nor is it in the same reducibility degree of some non‐periodic semigroup. We also show that the ceer provided by provable equivalence of Peano Arithmetic is in the same strong isomorphism type as the word problem of some non‐commutative and non‐Boolean c.e. ring.

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Finitely Presented Groups with Word Problems of Arbitrary Degrees of Insolubility

Cited by 37 publications

References 0 publications

Some Ways of Constructing a Propositional Calculus of Any Required Degree of Unsolvability

Some Ways of Constructing a Propositional Calculus of Any Required Degree of Unsolvability

Machine Configuration and Word Problems of Given Degree of Unsolvability

Word problems and ceers

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