## The Complexity of Resource-Bounded First-Order Classical Logic (1994)

Venue: | 11th Symposium on Theoretical Aspects of Computer Science |

Citations: | 8 - 1 self |

### BibTeX

@INPROCEEDINGS{Goubault94thecomplexity,

author = {Jean Goubault},

title = {The Complexity of Resource-Bounded First-Order Classical Logic},

booktitle = {11th Symposium on Theoretical Aspects of Computer Science},

year = {1994},

pages = {59--70},

publisher = {Springer Verlag LNCS}

}

### Years of Citing Articles

### OpenURL

### Abstract

. We give a finer analysis of the difficulty of proof search in classical first-order logic, other than just saying that it is undecidable. To do this, we identify several measures of difficulty of theorems, which we use as resource bounds to prune infinite proof search trees. In classical first-order logic without interpreted symbols, we prove that for all these measures, the search for a proof of bounded difficulty (i.e, for a simple proof) is \Sigma p 2 -complete. We also show that the same problem when the initial formula is a set of Horn clauses is only NP-complete, and examine the case of first-order logic modulo an equational theory. These results allow us not only to give estimations of the inherent difficulty of automated theorem proving problems, but to gain some insight into the computational relevance of several automated theorem proving methods. Topics: computational complexity, logics, computational issues in AI (automated theorem proving). 1 Introduction First-order ...

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Citation Context ...n a given amount of resources corresponding to an a priori bound on the difficulty of the input logical sentence. We assume that the reader is familiar with basic logic and abstract complexity theory =-=[13]-=-. The plan of the paper is as follows. In Section 2, we review related approaches to the analysis of the complexity of theorem proving problems and procedures. In Section 3, we define some measures of... |

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Citation Context ...eads to fragments of linear logic [14]. Propositional linear logic with exponentials, i.e, unbounded uses of these rules, is undecidable [23], though classical propositional logic is only NP-complete =-=[8]-=-. Bounding the uses of these rules regains decidability, but even the small multiplicative propositional fragment without variables is NPcomplete [25]. The decidability is maintained when lifting the ... |

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Citation Context ...nential time, and the decision problem for Ackermann's class is PSPACE-hard. Kozen [21] has shown that first-order logic without negation, even with equality, in NP-complete, and Stockmeyer and Meyer =-=[33]-=- have shown that pure first-order logic with equality is PSPACEcomplete. Decidability, in particular, does not mean tractability. Unfortunately, this approach gives no intuition on the complexity of u... |

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Citation Context ... the original formula M , yielding the \Sigma p 2 bound by direct enumeration. It is interesting to confront this to existing proof search methods. First-order tableaux [12], or the connection method =-=[3, 1] are based-=- on a textual decomposition of M : to refute M 1sM 2 , refute M 1 or M 2 non-deterministically (in "NP style"); to refute M 1sM 2 , refute M 1 , and if it worked, refute M 2 (nondeterminism ... |

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Citation Context ...y a substitution oe of minimal size fi min (M ). The size of the graph representing oe is not larger than that of the common m.g.u. of E. But a property of Paterson and Wegman's unification procedure =-=[28]-=- is that, in an m.g.u. , every variable is bound to a strict subterm of an atom. The size of this subterm is less than s(M ). Therefore fi min (M )ss(M )ffi min (M ). 2 To relate the various measures,... |

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Citation Context ...th a bounded number of steps in second-order Peano arithmetic to that of second-order unifiability. The latter problem, unfortunately, turned out to be undecidable, making the construction unfruitful =-=[15]-=-. Bounding the number of uses of the contraction and weakening rules of classical logic leads to fragments of linear logic [14]. Propositional linear logic with exponentials, i.e, unbounded uses of th... |

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Citation Context ... the original formula M , yielding the \Sigma p 2 bound by direct enumeration. It is interesting to confront this to existing proof search methods. First-order tableaux [12], or the connection method =-=[3, 1] are based-=- on a textual decomposition of M : to refute M 1sM 2 , refute M 1 or M 2 non-deterministically (in "NP style"); to refute M 1sM 2 , refute M 1 , and if it worked, refute M 2 (nondeterminism ... |

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Citation Context ... Topics: computational complexity, logics, computational issues in AI (automated theorem proving). 1 Introduction First-order classical logic proofs are hard to find. The undecidability of this logic =-=[6, 7]-=- if only a superficial justification. It has never deterred researchers from inventing new automated deduction methods. Moreover, it is generally felt that some undecidable problems are harder than ot... |

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Citation Context ...ses of the contraction and weakening rules of classical logic leads to fragments of linear logic [14]. Propositional linear logic with exponentials, i.e, unbounded uses of these rules, is undecidable =-=[23]-=-, though classical propositional logic is only NP-complete [8]. Bounding the uses of these rules regains decidability, but even the small multiplicative propositional fragment without variables is NPc... |

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Citation Context ...ce all systems E of interest can be described in at most this number of equations. The final formula is still in Horn clausal form, and a polynomial algorithm can be used to test its unsatisfiability =-=[26, 9]-=-. Conversely, we transform every instance of SAT into an instance of EQV COVER(\Upsilon ), with M in Horn clausal form; this will prove NPcompleteness. SAT is the problem of knowing whether a clausal ... |

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Citation Context ... alternations of quantifiers, and Q 1 is 9 (resp. 8); and that the problem where the alphabet is restricted to the booleans and \Phi is a propositional formula is complete for the corresponding class =-=[35]-=-. Now, measure the size of the input by ns. EQV COVER(C) is a problem of the form 9E \Delta E 2 Cs(E ) :M k is propositionally valid), where E ranges over all systems of equations between atoms of M k... |

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Citation Context ...nal theory. Section 5 is the conclusion. 2 Related works One approach to the analysis of proof search complexity is to determine the complexity of the so-called decidable classes of first-order logic =-=[10, 18]-=-, which has been worked out by Lewis [22]. These classes provide hard decision problems: the monadic, Godel, and Schonfinkel-Bernays classes cannot be decided in less than deterministic subexponential... |

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Citation Context ... Theorem says), it is also in \Sigma p k for the measure fi T min (M ). But \Sigma p k - completeness is not conserved in general, unless k = 2. As an example, although AC-unifiability is NP-complete =-=[2, 19]-=-, proving modulo associativity and commutativity is not really more complex than without. However, using AC-unification to solve it, as is customary, is a bad idea: ACunification is doubly exponential... |

44 |
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(Show Context)
Citation Context ...M 1 M 2s: : : M k . Notice that Herbrand's theorem just says that F is unsatisfiable iff there is a k such that M k has a propositionally unsatisfiable instance (this idea is basically due to Prawitz =-=[31, 32]-=-). Moreover, the set of such k is upward-closed, so there is a least one (if F is unsatisfiable) or none at all (if F is satisfiable). Therefore, a first possible measure of the difficulty of F is: De... |

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Citation Context ...ce all systems E of interest can be described in at most this number of equations. The final formula is still in Horn clausal form, and a polynomial algorithm can be used to test its unsatisfiability =-=[26, 9]-=-. Conversely, we transform every instance of SAT into an instance of EQV COVER(\Upsilon ), with M in Horn clausal form; this will prove NPcompleteness. SAT is the problem of knowing whether a clausal ... |

36 | The Classical Decision Problems
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Citation Context ...nal theory. Section 5 is the conclusion. 2 Related works One approach to the analysis of proof search complexity is to determine the complexity of the so-called decidable classes of first-order logic =-=[10, 18]-=-, which has been worked out by Lewis [22]. These classes provide hard decision problems: the monadic, Godel, and Schonfinkel-Bernays classes cannot be decided in less than deterministic subexponential... |

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Citation Context ...lausal form. This stems from the fact that every (quantified) propositional formula can be transformed in deterministic polynomial time into a (quantified) clausal form, its definitional clausal form =-=[34]-=-. The proof we have given above is actually overly complicated if the only logic of interest is L!! . In this case, it is enough to know that refuting substitutions can be represented as maps from var... |

30 | Constant-Only Multiplicative Linear Logic is NP-Complete
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Citation Context ... classical propositional logic is only NP-complete [8]. Bounding the uses of these rules regains decidability, but even the small multiplicative propositional fragment without variables is NPcomplete =-=[25]-=-. The decidability is maintained when lifting the multiplicativeadditive fragment to first-order logic, although the problem is NEXPTIME-hard, thus provably intractable [24]. Finally, a complementary ... |

27 |
Complexity of solvable cases of the decision problem for the predicate calculus
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Citation Context ...lated works One approach to the analysis of proof search complexity is to determine the complexity of the so-called decidable classes of first-order logic [10, 18], which has been worked out by Lewis =-=[22]-=-. These classes provide hard decision problems: the monadic, Godel, and Schonfinkel-Bernays classes cannot be decided in less than deterministic subexponential or exponential time, and the decision pr... |

26 |
Some results on the length of proofs
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Citation Context ...es. To get a finer analysis, the idea of bounding resources (lengths of authorized proofs, number of times that certain logical rules can be used, etc.) has been used by others. A precursor is Parikh =-=[27]-=-, who reduced the problem of finding proofs with a bounded number of steps in second-order Peano arithmetic to that of second-order unifiability. The latter problem, unfortunately, turned out to be un... |

18 |
Relative Complexities of First-Order Calculi
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Citation Context ...E-hard, thus provably intractable [24]. Finally, a complementary approach to our work is to compare the minimal lengths of proofs that different calculi allow for, which is the subject of Eder's work =-=[11]-=- in first-order logic. As Eder notes in his introduction, the length of a shortest proof always gives a lower bound on the complexity of any theoremproving search strategy (usually an exponential one)... |

17 | Double-exponential complexity of computing a complete set of AC-unifiers
- Kapur, Narendran
- 1992
(Show Context)
Citation Context ...odulo associativity and commutativity is not really more complex than without. However, using AC-unification to solve it, as is customary, is a bad idea: ACunification is doubly exponential in nature =-=[20]-=-. The solution is to delay the computation of AC-unifiers by propagating AC-unifiability constraints that are tested simultaneously at the end of the proof [4]. 5 Conclusion We have presented an appro... |

15 | First Order Linear Logic Without Modalities is NEXPTIME-Hard
- Lincoln, Scedrov
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(Show Context)
Citation Context ...ut variables is NPcomplete [25]. The decidability is maintained when lifting the multiplicativeadditive fragment to first-order logic, although the problem is NEXPTIME-hard, thus provably intractable =-=[24]-=-. Finally, a complementary approach to our work is to compare the minimal lengths of proofs that different calculi allow for, which is the subject of Eder's work [11] in first-order logic. As Eder not... |

14 |
Positive first-order logic is NP-complete
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(Show Context)
Citation Context ...e monadic, Godel, and Schonfinkel-Bernays classes cannot be decided in less than deterministic subexponential or exponential time, and the decision problem for Ackermann's class is PSPACE-hard. Kozen =-=[21]-=- has shown that first-order logic without negation, even with equality, in NP-complete, and Stockmeyer and Meyer [33] have shown that pure first-order logic with equality is PSPACEcomplete. Decidabili... |

7 |
D'emonstration automatique en logique classique : complexit'e et m'ethodes
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Citation Context ...the logic L is a classical first-order logic, with some symbols interpreted in an equational theory T . A more developed study of proof search complexity modulo a first-order theory T can be found in =-=[16]-=-. Let T be an equational theory, that is, a finite collection of equalities; it is assumed that equality itself does not appear in the theorems we want to prove. Refuting a formula modulo the theory T... |

5 |
Matching, unification and complexity
- Kapur, Narendran
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(Show Context)
Citation Context ... Theorem says), it is also in \Sigma p k for the measure fi T min (M ). But \Sigma p k - completeness is not conserved in general, unless k = 2. As an example, although AC-unifiability is NP-complete =-=[2, 19]-=-, proving modulo associativity and commutativity is not really more complex than without. However, using AC-unification to solve it, as is customary, is a bad idea: ACunification is doubly exponential... |

4 |
Advances and problems in mechanical proof procedures
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- 1969
(Show Context)
Citation Context ...M 1 M 2s: : : M k . Notice that Herbrand's theorem just says that F is unsatisfiable iff there is a k such that M k has a propositionally unsatisfiable instance (this idea is basically due to Prawitz =-=[31, 32]-=-). Moreover, the set of such k is upward-closed, so there is a least one (if F is unsatisfiable) or none at all (if F is satisfiable). Therefore, a first possible measure of the difficulty of F is: De... |

3 |
Deduction with Shannon Graphs or: How to Lift BDDs to First-order Logic
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- 1993
(Show Context)
Citation Context ...form; this will prove NPcompleteness. SAT is the problem of knowing whether a clausal propositional formula \Phi with variablessi , 1sisn, is satisfiable. We build a Prolog program to this end, as in =-=[30]-=-. Let T be a unary predicate symbol, cons a binary function symbol, a and b two constants, and for each subformula \Phi 0 of \Phi, a new variable x \Phi 0 . For each subformula \Phi 0 , we build the f... |

1 |
A new combination technique for AC-unification
- Boudet
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(Show Context)
Citation Context ...ication is doubly exponential in nature [20]. The solution is to delay the computation of AC-unifiers by propagating AC-unifiability constraints that are tested simultaneously at the end of the proof =-=[4]-=-. 5 Conclusion We have presented an approach to the practical analysis of the computational complexity of an undecidable problem, classical first-order theorem proving. For this, we needed to define s... |

1 |
A note on the Entscheidungsproblem (correction
- Church
- 1936
(Show Context)
Citation Context ... Topics: computational complexity, logics, computational issues in AI (automated theorem proving). 1 Introduction First-order classical logic proofs are hard to find. The undecidability of this logic =-=[6, 7]-=- if only a superficial justification. It has never deterred researchers from inventing new automated deduction methods. Moreover, it is generally felt that some undecidable problems are harder than ot... |

1 |
Syntax independent connections. In Workshop on Theorem Proving with Analytic Tableaux and Related Methods, number MPI-I-93-213
- Goubault
- 1993
(Show Context)
Citation Context ... proof without any explanations of why it is actually a solution, i.e, without any deduction steps. The multiplicity can be thought as a minimal number of cases we have to examine to prove the result =-=[17]-=-, and is thus a less natural difficulty measure; we show in the next section that they are nonetheless related. This sheds some light on what kind of proof a refuting substitution is: it is a list of ... |