## Looking for an analogue of Rice's Theorem in circuit complexity theory (1989)

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Venue: | Mathematical Logic Quarterly |

Citations: | 5 - 1 self |

### BibTeX

@ARTICLE{Borchert89lookingfor,

author = {Bernd Borchert and Frank Stephan},

title = {Looking for an analogue of Rice's Theorem in circuit complexity theory},

journal = {Mathematical Logic Quarterly},

year = {1989},

pages = {2000}

}

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### Abstract

Abstract. Rice’s Theorem says that every nontrivial semantic property of programs is undecidable. In this spirit we show the following: Every nontrivial absolute (gap, relative) counting property of circuits is UP-hard with respect to polynomial-time Turing reductions. For generators [31] we show a perfect analogue of Rice’s Theorem. Mathematics Subject Classification: 03D15, 68Q15. Keywords: Rice’s Theorem, Counting problems, Promise classes, UP-hard, NP-hard, generators.

### Citations

2346 | Computational Complexity
- Papadimitriou
- 1994
(Show Context)
Citation Context ...roblem for A is p-m-hard for SPP. 2 Preliminaries The standard notions of Theoretical Computer Science like words, languages, polynomial-time reductions, P, NP, : : : follow the book of Papadimitriou =-=[2-=-2]. AB is the join of two languages A and B, namely AB = f0x : x 2 Ag[ f1x : x 2 Bg. Quite important for the present work is the notion of promise classes. These are classes where the number of accept... |

476 |
The complexity of theorem proving procedures, in
- Cook
- 1971
(Show Context)
Citation Context ...olute-Counting(A): Let a language L in co-NP be given and let M be a machine for L in the sense that x 2 L i no path of M(x) is accepting. Let m(x) denote the number of accepting paths of M(x). Cook [=-=11]-=- (see also Ladner [20]) established a method to construct in polynomial time a circuit Cook M x with inputs y 1 ; : : : ; y n (n depends on x and is bounded by a polynomial in the length of x) such th... |

142 | Complexity measures for public-key cryptosystems
- Grollmann, Selman
- 1988
(Show Context)
Citation Context ...ning on input x has an accepting path. Such machines are called unambiguous. By denition UP is a subset of NP. A classical result tells that UP equals P if and only if one-way functions do not exist [=-=14]-=-. The primality problem is a typical example of a problem in UP not known to be in P [12]. Below we use the notion of a class C being reducible to a language L, this is just a short way of saying that... |

122 | Gap-definable counting classes
- Fenner, Fortnow, et al.
- 1994
(Show Context)
Citation Context ... accepting paths compared with the total number of paths) of M on input x is in A. This denition of gap countable classes equals the denition of nice gap denable classes in Fenner, Fortnow & Kurtz [13=-=]-=-. As a special case of the main result in [8, 30] it follows that an absolute (gap, relative) counting problem is p-mcomplete for the corresponding absolute (gap, relative) counting class. In other wo... |

83 |
Relative complexity of checking and evaluation
- Valiant
- 1976
(Show Context)
Citation Context ... we will state and prove the theorems which show UP-hardness of all nontrivial counting problems of the three types dened in the previous chapter. Remember that the class UP, which was denedsrst in [2=-=8]-=-, is the promise class consisting of the languages L such that there is a polynomial-time nondeterministic machine M such that on every input the machine M has at most one accepting path and an input ... |

68 |
Chee-Keng: Some consequences of non-uniform conditions on uniform classes
- Yap
- 1983
(Show Context)
Citation Context ...ring reductions. In Section 5, this result will be extended in terms of approximable sets and also in terms of randomized reductions. In Section 6, we consider generators which were introduced by Yap =-=[31]-=- as a way of representing Boolean functions. For generators a perfect analogue of Rice's Theorem can be found: Let A be any nonempty proper subset of the set of Boolean functions which not only depend... |

62 |
Classes of Recursively Enumerable Sets and their Decision Problems
- Rice
- 1953
(Show Context)
Citation Context ...tion: 03D15, 68Q15. Keywords: Rice's Theorem, counting problems, promise classes, UP-hardness, NP-hardness, generators. 1 Introduction One of the nicest theorems in Recursion Theory is Rice's Theorem =-=[24, 25]-=-. Informally speaking it says: Any nontrivial semantic property of programs is undecidable. More formally it can be stated this way: Theorem 1.1 [Rice 1953] Let A be any nonempty proper subset of the ... |

53 | The Quantum Challenge to Structural Complexity Theory
- Berthiaume, Brassard
(Show Context)
Citation Context ... # 0 + 2 BPP # 1 # 0 # 0 # 1 # 1 > # 0 ^ # 0 > # 1 In this table, is asxed constant with 0s1 and p(n) is a value polynomial in the length n of the input. Berthiaume and Brassard [5] introduced the class HalfP using the name C== P[half]. A central notion of this paper is the notion of a Boolean function which is dened to be a mapping from f0; 1g n to f0; 1g for some integer n ... |

52 | F.Stephan: Approximable Sets
- Beigel
- 1994
(Show Context)
Citation Context ...o B or B, respectively. Theorem 5.2. SATor its complement is randomized polynomial-time reducible to any nontrivial absolute and gap counting problem. 5.2 Approximable sets AsetA is approximable (see =-=[4]-=-) iff there is a constant j and an algorithm which computes for each input x1, ...,xj in polynomial time j bits y1, ...,yj such that A(xh) =yh for some h. Beigel [2, 3] analyzed the notion of approxim... |

47 |
Complexity classes without machines: On complete languages for UP
- Hartmanis, Hemachandra
- 1988
(Show Context)
Citation Context ... m Ag. On the other hand, it is still possible tosnd a set A such that BPP = fL : L is recursive and L p m Ag. This A can { at least in the relativized worlds considered by Hartmanis und Hemachandra [=-=17]-=- { not be recursive. So an interesting question is what kind of properties such a set A still can satisfy. Within the context of the present work, it is quite natural to ask whether such a set can hav... |

47 |
On the unique satisfiability problem
- BLASS, GUREVICH
- 1982
(Show Context)
Citation Context ... is the set 1-SAT consisting of the circuits with exactly one satisfying assignment, that is, 1-SAT = Abs-Count({1}), the complexity class for which this problem is complete is usually called US, see =-=[6]-=-, or also 1-NP. Examples of gap counting problems. The set C=SAT of circuits which have as many satisfying as non-satisfying assignments, is a gap counting problem: C=SAT = Gap-Count({0}). The set PSA... |

46 | A complexity theory of feasible closure properties
- Ogiwara, Hemachandra
- 1993
(Show Context)
Citation Context ...a relativized world where SPP is not p-m-reducible to Absolute-Counting(A). Furthermore, if A is P-constructibly bi-innite then SPP p m Absolute-Counting(A). Gupta [16] and Ogiwara and Hemachandra [2=-=1]-=- introduced the class SPP as the promise class consisting of the languages L such that there is a polynomial-time nondeterministic machine M such that for every input x for the machine M either half o... |

28 |
A structural theorem that depends quantitatively on the complexity of SAT
- Beigel
- 1987
(Show Context)
Citation Context ... A is approximable [4] i there is a constant j and an algorithm which computes for each input x 1 ; : : : ; x j in polynomial time j bits y 1 ; : : : ; y j such that A(x h ) = y h for some h. Beigel [=-=2, 3]-=- analyzed the notion of approximable sets and showed that no NP-hard set is approximable unless P = UP. This result can be transferred to the following theorem which is an extension of Conclusion 4.6.... |

28 |
NP is as easy as detecting unique solutions, Theoret
- Valiant, Vazirani
- 1986
(Show Context)
Citation Context ...it is pointed out that it is unlikely that Theorem 4.5 holds with polynomial-time many-one-reduction in place of the polynomial-time Turing reduction. 5.1 Randomized computations Valiant and Vazirani =-=[29]-=- showed, that using randomized reductions, detecting unique solutions is as hard as solving the satisfiability problem. In particular they showed that every algorithm f which satisfies the following s... |

27 | NP-hard sets are P-superterse unless R = NP
- Beigel
- 1988
(Show Context)
Citation Context ... A is approximable [4] i there is a constant j and an algorithm which computes for each input x 1 ; : : : ; x j in polynomial time j bits y 1 ; : : : ; y j such that A(x h ) = y h for some h. Beigel [=-=2, 3]-=- analyzed the notion of approximable sets and showed that no NP-hard set is approximable unless P = UP. This result can be transferred to the following theorem which is an extension of Conclusion 4.6.... |

27 |
NP is as easy as detecting unique solutions, Theoretical Computer Science 47
- Valiant, Vazirani
- 1986
(Show Context)
Citation Context ...it is pointed out that it is unlikely that Theorem 4.5 holds with polynomial-time many-one-reduction in place of the polynomial-time Turing reduction. 5.1 Randomized Computations Valiant and Vazirani =-=[2-=-9] showed, that using randomized reductions, detecting unique solutions is as hard as solving the satisability problem. In particular they showed that every algorithm f which satises the following spe... |

22 | The Boolean isomorphism problem
- Agrawal, Thierauf
- 1996
(Show Context)
Citation Context ...(stating UP-hardness) if we restrict ourselves to stronger semantic properties. We tried several approaches, for example by considering the equivalence relations like Boolean isomorphism presented in =-=[1, 9]-=-. Note that the above counterexample is also a counterexample under Boolean isomorphism. The restriction for which we couldsnd the intended hardness result is the restriction to the counting propertie... |

22 |
The circuit value problem is log-space complete for P
- Ladner
- 1975
(Show Context)
Citation Context ...ey are encoded as words in some standard way, for the details concerning circuits we refer for example to [22]. Remember that a given circuit can be evaluated on a given assignment in polynomial time =-=[20]-=-. Each circuit c describes a Boolean function F (x 1 ; : : : ; x n ). Note that here we have an example of the classical syntax/semantics dichotomy like we have it for programs: The circuits (programs... |

20 | An observation on probability versus randomness with applications to complexity classes. Mathematical Systems Theory
- Book, Lutz, et al.
(Show Context)
Citation Context ...s for each circuit Cook M x is above 1 2 + > r for x = 2 L and below 1 2 sfor x 2 L. So the mapping x ! :Cook M x is an p-m-reduction from L to Relative-Counting(A). 2 Remark. Book, Lutz and Wagner [7=-=]-=- showed some analogous result for polynomial-time Turing reducibility and random sets in place of p-m-reducibility and counting problems: There is some Martin-Loef random set A such that a recursive s... |

20 |
A connotational theory of program structure
- Royer
- 1987
(Show Context)
Citation Context ...em we tried tosnd some sister of it in Complexity Theory. We will present a concept based on Boolean circuits instead of programs, therefore our approach is dierent from the one of Kozen [19], Royer [=-=26]-=- and Case [10] who study problems on programs with given resource bounds. The main idea behind our approach is that for circuits versus Boolean functions there is a similar syntax/semantics dichotomy ... |

20 |
Self-Witnessing Polynomial-Time Complexity and Prime Factorization, in
- Fellows, Koblitz
- 1992
(Show Context)
Citation Context ...bset of NP. A classical result tells that UP equals P if and only if one-way functions do not exist (see [14]). The primality problem is a typical example of a problem in UP not known to be in P (see =-=[12]-=-). BelowweusethenotionofaclassCbeingreducibletoalanguage L, thisisjusta short way of saying that every language in the class C is reducible to L. The following theorem implies that any nontrivial abso... |

19 | On the computational complexity of some classical equivalence relations on Boolean functions
- Borchert, Ranjan, et al.
- 1998
(Show Context)
Citation Context ...(stating UP-hardness) if we restrict ourselves to stronger semantic properties. We tried several approaches, for example by considering the equivalence relations like Boolean isomorphism presented in =-=[1, 9]-=-. Note that the above counterexample is also a counterexample under Boolean isomorphism. The restriction for which we couldsnd the intended hardness result is the restriction to the counting propertie... |

13 |
Succinct representation, leaf languages, and projection reductions
- Veith
- 1998
(Show Context)
Citation Context ...r of paths) of M on input x is in A. This denition of gap countable classes equals the denition of nice gap denable classes in Fenner, Fortnow & Kurtz [13]. As a special case of the main result in [8,=-= 30]-=- it follows that an absolute (gap, relative) counting problem is p-mcomplete for the corresponding absolute (gap, relative) counting class. In other words, absolute (gap, relative) counting problems a... |

12 |
The power of witness reduction
- Gupta
- 1991
(Show Context)
Citation Context ...rarchy over NP and thus there is a relativized world where SPP is not p-m-reducible to Absolute-Counting(A). Furthermore, if A is P-constructibly bi-innite then SPP p m Absolute-Counting(A). Gupta [1=-=6]-=- and Ogiwara and Hemachandra [21] introduced the class SPP as the promise class consisting of the languages L such that there is a polynomial-time nondeterministic machine M such that for every input ... |

11 | Succinct circuit representations and leaf language classes are basically the same concept
- Borchert, Lozano
- 1996
(Show Context)
Citation Context ...r of paths) of M on input x is in A. This denition of gap countable classes equals the denition of nice gap denable classes in Fenner, Fortnow & Kurtz [13]. As a special case of the main result in [8,=-= 30]-=- it follows that an absolute (gap, relative) counting problem is p-mcomplete for the corresponding absolute (gap, relative) counting class. In other words, absolute (gap, relative) counting problems a... |

11 |
Indexing of subrecursive classes
- Kozen
- 1978
(Show Context)
Citation Context ...Rice's Theorem we tried tosnd some sister of it in Complexity Theory. We will present a concept based on Boolean circuits instead of programs, therefore our approach is dierent from the one of Kozen [=-=19]-=-, Royer [26] and Case [10] who study problems on programs with given resource bounds. The main idea behind our approach is that for circuits versus Boolean functions there is a similar syntax/semantic... |

6 |
On completely recursively enumerable classes and their key arrays
- Rice
- 1956
(Show Context)
Citation Context ...tion: 03D15, 68Q15. Keywords: Rice's Theorem, counting problems, promise classes, UP-hardness, NP-hardness, generators. 1 Introduction One of the nicest theorems in Recursion Theory is Rice's Theorem =-=[24, 25]-=-. Informally speaking it says: Any nontrivial semantic property of programs is undecidable. More formally it can be stated this way: Theorem 1.1 [Rice 1953] Let A be any nonempty proper subset of the ... |

4 |
On the unique satis problem
- Blass, Gurevich
- 1982
(Show Context)
Citation Context ...set 1-SAT consisting of the circuits with exactly one satisfying assignment, that is, 1-SAT = Absolute-Counting(f1g), the complexity class for which this problem is complete is usually called US, see =-=[6]-=-, or also 1-NP. Examples of gap counting problems. The set C=SAT of circuits which have as many satisfying as non-satisfying assignments, is a gap counting problem: C=SAT = Gap-Counting(f0g). The set ... |

4 |
Relativized questions involving probablistic algorithms
- Rackoff
- 1982
(Show Context)
Citation Context ...ized world where it fails as follows: Theorem 4.7 shows that either HalfP ≤ p m Rel-Count(A) or co-HalfP ≤ p m Rel-Count(A) for any A ⊆ D. The theorem that P = UP ⊂ NP for some relativized world (see =-=[23]-=-) can be modified such that P = UP ⊂ HalfP (and therefore also UP ⊂ co-HalfP). In this relativized world, there is, for any non-trivial A ⊆ D, a recursive set L outside UP with L ≤ p m Rel-Count(A). T... |

3 |
Eectivizing Inseparability, Zeitschrift fur Mathematische Logik und Grundlagen der
- Case
- 1991
(Show Context)
Citation Context ...snd some sister of it in Complexity Theory. We will present a concept based on Boolean circuits instead of programs, therefore our approach is dierent from the one of Kozen [19], Royer [26] and Case [=-=10]-=- who study problems on programs with given resource bounds. The main idea behind our approach is that for circuits versus Boolean functions there is a similar syntax/semantics dichotomy like there is ... |

2 |
Neal Koblitz, Self-witnessing polynomial-time complexity and prime factorization
- Fellows
- 1992
(Show Context)
Citation Context ... UP is a subset of NP. A classical result tells that UP equals P if and only if one-way functions do not exist [14]. The primality problem is a typical example of a problem in UP not known to be in P =-=[12]-=-. Below we use the notion of a class C being reducible to a language L, this is just a short way of saying that every language in the class C is reducible to L. The following theorem implies that any ... |

2 |
Gerd Wechsung, A survey on counting classes
- Gundermann, Nasser
- 1990
(Show Context)
Citation Context ...wing way. Let a sequence (A n ) be given for which A n is a subset of f0; : : : ; 2 n g. The counting problem for (A n ) is the set of all circuits c(x 1 ; : : : ; x n ) such that # 1 (c) 2 A n , see =-=[15-=-] for an analogous denition of (general) counting classes. In this way, absolute, gap and relative counting problems are counting problems. It is easy to give an example of a (general) counting proble... |

2 |
Relativized questions involving probablistic algorithms
- Racko
- 1982
(Show Context)
Citation Context ...here it fails as follows: Theorem 4.7 shows that either HalfP p m Relative-Counting(A) or co-HalfP p m Relative-Counting (A) for any A D. The theorem that P = UP NP for some relativized world [23] can be modied such that P = UP HalfP (and therefore also UP co-HalfP). In this relativized world, there is, for any non-trivial A D, a recursive set L outside UP with L p m Relative-Counting... |

2 | Effectivizing inseparability
- Case
- 1991
(Show Context)
Citation Context ...nd some sister of it in Complexity Theory. We will present a concept based on Boolean circuits instead of programs, therefore our approach is different from the one of Kozen [19], Royer [20] and Case =-=[10]-=- who study problems on programs with given resource bounds. 1) The authors are grateful for discussions with Klaus Ambos-Spies, John Case, Wolfgang Merkle and André Nies. Furthermore the authors appre... |

1 |
Jorg Rothe, A second step towards circuit complexity theoretic analogs of Rice's Theorem
- Hemaspaandra
- 1998
(Show Context)
Citation Context ...y proper subset of the set of Boolean functions which not only depends on arity. Then the following problem is NP-hard: Given a generator g, does it describe a function from A? Hemaspaandra and Rothe =-=[18]-=- continue our line of research and study the absolute counting problem. Given any nontrivial set A, they show that the absolute counting problem for A is p2 btt(1)-hard for UPO(1) . If A is furthermor... |

1 |
The computational complexity of some classical equivalence relations on Boolean functions. Theory of Computing Systems 31
- Stephan
- 1998
(Show Context)
Citation Context ...(stating UP-hardness) if we restrict ourselves to stronger semantic properties. We tried several approaches, for example by considering the equivalence relations like Boolean isomorphism presented in =-=[1, 9]-=-. Note that the above counterexample is also a counterexample under Boolean isomorphism. The restriction for which we could find the intended hardness result is the restriction to the counting propert... |

1 |
A survey on counting classes
- Wechsung
- 1990
(Show Context)
Citation Context ...sense we define in the following way: Let a sequence (An) begivenforwhichAnisasubset of {0, ..., 2n }. The counting problem for (An) is the set of all circuits c(x1, ...,xn) such that #1(c) ∈ An, see =-=[15]-=- for an analogous definition of (general) counting classes. In this way, absolute, gap and relative counting problems are counting problems. It is easy to give an494 Bernd Borchert and Frank Stephan ... |

1 |
A second step towards circuit complexity theoretic analogs of Rice’s Theorem
- Rothe
- 1998
(Show Context)
Citation Context ... proper subset of the set of Boolean functions which not only depends on arity. Then the following problem is NP-hard: Given a generator g, does it describe a function from A ? Hemaspaandra and Rothe =-=[18]-=- continue our line of research and study the absolute counting problem. Given any nontrivial set A, they show that the absolute counting problem for A is p-btt(1)-hard for UPO(1). If A is furthermore ... |

1 |
Indexings of subrecursive classes. Theoret
- Kozen
- 1980
(Show Context)
Citation Context ...e’s Theorem we tried to find some sister of it in Complexity Theory. We will present a concept based on Boolean circuits instead of programs, therefore our approach is different from the one of Kozen =-=[19]-=-, Royer [20] and Case [10] who study problems on programs with given resource bounds. 1) The authors are grateful for discussions with Klaus Ambos-Spies, John Case, Wolfgang Merkle and André Nies. Fur... |

1 |
On small generators. Theoret
- Schöning
- 1984
(Show Context)
Citation Context ...is complete for Π p 2 . Such a representation is given by generators which are defined the following way (the first definition by Yap [31] did not have the indicator bit i which was added by Schöning =-=[27]-=- in order to be able to represent, for each n, then-ary constant 0-function). Definition 6.1 (Yap [31]). A generator is a Boolean circuit, with k inputs and n+1 outputs i, r1, ...,rn. The function com... |