## On the unusual effectiveness of Logic in computer science

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Venue: | Bulletin of Symbolic Logic |

Citations: | 7 - 0 self |

### BibTeX

@ARTICLE{Halpern_onthe,

author = {Joseph Y. Halpern and Robert Harper and Neil Immerman and Phokion G. Kolaitis},

title = {On the unusual effectiveness of Logic in computer science},

journal = {Bulletin of Symbolic Logic},

year = {},

pages = {2001}

}

### OpenURL

### Abstract

Effectiveness of Mathematics in the Natural Sciences [Wig60]. This paper can be construed as an examination and affirmation of Galileo’s tenet that “The book of nature is written in the language of mathematics”. To this effect, Wigner presented a large number of examples that demonstrate the effectiveness of

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Citation Context ...irected graph can have its vertices colored with three colors so that no two adjacent vertices have the same color—as well as hundreds of other well-known combinatorial problems are NP-complete. (See =-=[17]-=- for a survey of many of these.) This means that not only are they in NP, but they are the “hardest problems” in NP: all problems in NP are reducible (in polynomial time) to each NP-complete problem. ... |

1558 |
Foundations of Databases
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Citation Context ...urrent-write, parallel random-access machines using polynomially many processors in parallel time O(t(n)). FO, the set of first-order expressible properties, exactly captures the complexity class CRAM=-=[1]-=-, i.e., constantON THE UNUSUAL EFFECTIVENESS OF LOGIC IN COMPUTER SCIENCE 217 parallel time. It is possible to increase the power of FO by allowing longer descriptions for longer inputs. Let FO[t(n)]... |

1534 |
Reasoning about Knowledge
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Citation Context ...ls. Of course, this brief discussion has only scratched the surface of the topic. For more details and further references, the interested reader should consult Fagin, Halpern, Moses, and Vardi’s book =-=[FHMV95]-=-. 6 Automated Verification of Semiconductor Designs The recent growth in computer power and connectivity has changed the face of science and engineering, and is changing the way business is being cond... |

1310 | A structural approach to operational semantics
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Citation Context ...stract machine are closed expressions; the final states are the fully-evaluated expressions. The transition relation is given in Figure 3 using Plotkin’s technique of structured operational semantics =-=[46]-=-. These rules constitute an inductive definition of the call-by-value evaluation strategy, in which function arguments are evaluated prior to application, and for which function expressions are fully ... |

1293 | The model checker SPIN
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Citation Context ...TL formula if all computations in satisfy , that is, if �models . One of the major approaches to automated verification is the automata-theoretic approach, which underlies model checkers such as SPIN =-=[Hol97]-=- and Cadence SMV7 �� . The key idea underlying the automatatheoretic approach is that, given an LTL formula , it is possible to construct a finite-state automaton� �� � ��� on infinite words that acce... |

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Citation Context ...ication is to come up with a formal specification of the design, consisting of a description of the desired behavior. One of the more widely used specification languages for designs is temporal logic =-=[Pnu77]-=-. In linear temporal logics, time is treated as if each moment in time has a unique possible future. Thus, linear temporal formulas are interpreted over linear sequences, and we regard them as describ... |

1228 |
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Citation Context ...paper [19]. Furthermore, A. Turing, A. Church, and A. Tarski demonstrated the undecidability of first-order logic. Specifically, the set of all valid first-order sentences was shown to be undecidable =-=[5, 54]-=-, whereas the set of all first-order sentences that are true in arithmetic was shown to be highly undecidable [53]. Today, mathematical logic is a mature and highly sophisticated research area with de... |

1208 | Automatic verification of finite-state concurrent systems using temporal logic specifications
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Citation Context ...s in which mathematical techniques are used to guarantee the correctness of a design with respect to some specified behavior. Algorithmic formal-verification tools, based on model-checking technology =-=[6, 41, 48, 58]-=- have enjoyed a substantial and growing use over the last few years, showing an ability to discover subtle flaws that result from extremely improbable events. While until recently these tools were vie... |

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Citation Context ...t we are not the first ones to single out the effectiveness of logic in computer science. In fact, already back in 1988 M. Davis wrote an eloquent essay on the Influences of Logic in Computer Science =-=[Dav88]-=-, which begins by stating that “When I was a student, even the topologists regarded mathematical logicians as living in outer space. Today the connections between logic and computers are a matter of e... |

794 |
Applied Operating System Concepts
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Citation Context .... FO can achieve linear scaling, given parallel processing resources. Thus, its full potential as a query language remains yet to be realized. A good introduction to the database area may be found in =-=[SKS97]-=-, while [Ull88] provides a more in-depth presentation. The first text on database theory is [Mai83], followed more recently by [AHV95]. The latter text also described database query languages beyond F... |

605 |
An Automata-Theoretic Approach to Automatic Program Verification
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Citation Context ...1 Introduction and Overview In 1960, E.P. Wigner, a joint winner of the 1963 Nobel Prize for Physics, published a paper titled On the Unreasonable Effectiveness of Mathematics in the Natural Sciences =-=[Wig60]-=-. This paper can be construed as an examination and affirmation of Galileo’s tenet that “The book of nature is written in the language of mathematics”. To this effect, Wigner presented a large number ... |

600 | Symbolic model checking 1020 states and beyond
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Citation Context ...ich have highly efficient solutions [57]. Furthermore, using data structures that enable compact representation of very large state space makes it possible to verify designs of significant complexity =-=[4]-=-. 7 http://www-cad.eecs.berkeley.edu/ ∼ kenmcmil/smv/.232 HALPERN, HARPER, IMMERMAN, KOLAITIS, VARDI, AND VIANU The linear-time framework is not limited to using LTL as a specification language. Ther... |

537 |
The complexity of relational query languages
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Citation Context ...) ∧ R(z, y)). It is exciting that the natural descriptive class FO(LFP)—first-order logic extended with the power to define new relations by induction—precisely captures polynomial time. Theorem 2.3 (=-=[29, 30, 56]-=-). A problem is in polynomial time if and only if it is describable in first-order logic with the addition of the least-fixed-point operator. This is equivalent to being expressible by a first-order f... |

504 |
A course in Model Theory
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Citation Context ...O has syntactic variants that are easy to use. These are used as basic building blocks in practical languages like SQL and QBE. 2 This section is based in part on the article [34]. See also the books =-=[12, 35]-=- for much more information about descriptive complexity.ON THE UNUSUAL EFFECTIVENESS OF LOGIC IN COMPUTER SCIENCE 219 • FO can be efficiently implemented using relational algebra, which provides a se... |

502 | Knowledge and common knowledge in a distributed environment
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- 1990
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Citation Context ...of simple operations on relations expressing all FO queries. Relational algebra as used in the context of databases was introduced by Ted Codd in [Cod70]. It is related to Tarski’s Cylindric Algebras =-=[HMT71]-=-. The algebra turns out to yield a crucial advantage when large amounts of data are concerned. Indeed, the realization by Codd that the algebra can be used to efficiently implement FO queries gave the... |

452 | The formulae-as-types notion of construction - Howard - 1980 |

451 |
Knowledge and Belief
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Citation Context ...ck” on the integrity of the compiler itself [52]. §5. Reasoning about knowledge. The formal study of epistemic logic was initiated in the 1950s and led to Hintikka’s seminal book Knowledge and Belief =-=[25]-=-. The 1960s saw a flourishing of interest in the area in the philosophy community. More recently, reasoning about knowledge has been shown to play a key role in such diverse fields as distributed comp... |

441 |
The Theory of Relational Databases
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Citation Context ... query language remains yet to be realized. A good introduction to the database area may be found in [SKS97], while [Ull88] provides a more in-depth presentation. The first text on database theory is =-=[Mai83]-=-, followed more recently by [AHV95]. The latter text also described database query languages beyond FO, including fixpoint logics. An excellent survey of relational database theory is provided in [Kan... |

406 |
Über formal unentscheidbare Sätze der Pricipia mathematica und verwandter Systeme I,Monatshefte für Mathematik und
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Citation Context ...d. Nonetheless, Hilbert’s Program was dealt devastating blows during the 1930s. Indeed, the standard first-order axioms of arithmetic were shown to be incomplete by Gödel in his celebrated 1931 paper =-=[Göd31]-=-. Furthermore, A. Turing, A. Church, and A. Tarski demonstrated the undecidability of first-order logic. Specifically, the set of all valid first-order sentences was shown to be undecidable [Chu36, Tu... |

347 | Intuitionistic Type Theory - Martin-Löf - 1984 |

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Citation Context ...a finite-state automaton� �� � ��� on infinite words that accepts precisely all computations that satisfy [VW94]. The type of finite automata on infinite words we consider is the one defined by Büchi =-=[Büc62]-=-. A Büchi automaton is a tuple ������, where is a finite alphabet, is a finite set of states, �is a set of initial states, is a nondeterministic transition function, and��is a set of accepting states.... |

336 | Foundations for Programming Languages - Mitchell - 1996 |

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Notes on Database Operating Systems
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Citation Context ...ticular, we have all such that� �� . As an example of how this framework can be used in analyzing distributed protocols, consider the coordinated attack problem, from the distributed systems folklore =-=[Gra78]-=-. It abstracts a problem of data recovery management that arises when using standard protocols in database management called commit protocols. The following presentation is taken from [HM90]: �� ���if... |

278 | Descriptive complexity - Immerman - 1999 |

276 |
Generalized first-order spectra and polynomial-time recognizable sets
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Citation Context ...escriptive Complexity began with the following theorem of R. Fagin. Observe that Fagin’s Theorem characterizes the complexity class NP purely by logic, with no mention of machines or time, Theorem 1 (=-=[Fag74]-=-) A set of structures is in NP iff there exists a second-order existential formula, such that ��������. Formally, NP�SO�. Define CRAM� ℄to be the set of properties checkable by concurrent-read, concur... |

273 | Relational queries computable in polynomial time - Immerman - 1986 |

257 | Reasoning about infinite computations - Vardi, Wolper - 1994 |

244 | Specification and verification of concurrent systems in CESAR - Queille, Sifakis - 1982 |

235 | Nondeterministic space is closed under complementation
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Citation Context ...he drinkers who frequent only bars serving Bass. It turns out that each query expressible in FO can be broken down into a sequence of simple subqueries. 2 This section is based in part on the article =-=[Imm95]-=-. See also the books [EF95, Imm99] for much more information about descriptive complexity. 3 Codd received the ACM Turing Award for his work leading to the development of relational systems. 5���� ��... |

231 | TIL: A typedirected optimizing compiler for ML - Tarditi, Morrisett, et al. - 1996 |

222 | An Automata-Theoretic Approach to Linear Temporal Logic, in \Logics for Concurrency { Structure versus Automata
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Citation Context ... the automaton��is constructed, the verification task is reduced to automata-theoretic problems, namely, intersecting automata and testing emptiness of automata, which have highly efficient solutions =-=[Var96]-=-. Furthermore, using data structures that enable compact representation of very large state space makes it possible to verify designs of significant complexity [BCM 92]. The linear-time framework is n... |

214 |
Combinatory Logic
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Citation Context ...uing similarity to the introduction and elimination rules for implication in Gentzen’s system of natural deduction. This similarity is not accidental: according to the propositions-as-types principle =-=[8, 9, 27]-=- there is an isomorphism between propositions and types with the property that the natural deduction proofs of a proposition correspond to the elements of its associated type. This principle extends t... |

207 |
Principles of Database and Knowledge Base Systems
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Citation Context ...ieve linear scaling, given parallel processing resources. Thus, its full potential as a query language remains yet to be realized. A good introduction to the database area may be found in [51], while =-=[55]-=- provides a more in-depth presentation. The first text on database theory is [42], followed more recently by [1]. The latter text also described database query languages beyond FO, including fixpoint ... |

161 |
An application of games to the completeness problem for formalized theories
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Citation Context ... have been used to prove many inexpressibility results. These bounds provide useful insights but they do not separate relevant complexity classes because they are proved without the ordering relation =-=[13, 16, 35]-=-. No such lower bounds were known for separating the classes corresponding to P and PSPACE. Abiteboul and Vianu showed why, thus proving another fundamental relationship between logic and complexity. ... |

148 |
A.: 1963, ‘Semantical Analysis of Modal Logic I: Normal Modal Propositional Calculi’, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 9
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Citation Context ...or a primitive proposition ) iff� � � �� ���������for������� � � iff ��and ���iff ���iff ��for all ���iff 4 Kripke structures are named after Saul Kripke, who introduced them in their current form in =-=[Kri63]-=-, although the idea of possible worlds was in the air in the philosophy community in the 1950s. 10� � ���iff ����for��������, where��is �and�� �����. defined inductively by taking� �� Note how the se... |

142 | Proofs and Types, volume 7 of Cambridge Tracts - Girard, Lafont, et al. - 1989 |

121 |
Generic computation and its complexity
- Abiteboul, Vianu
- 1991
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Citation Context ...ul and Vianu showed why, thus proving another fundamental relationship between logic and complexity. In the following, FO (wo≤) means first-order logic without a given ordering relation. Theorem 2.6 (=-=[2]-=-). The following conditions are equivalent: 1. FO (wo≤)(LFP) = FO (wo≤)(PFP). 2. FO(LFP) = FO(PFP). 3. P = PSPACE. Descriptive complexity reveals a simple but elegant view of computation. Natural comp... |

119 | Categorical Logic and Type Theory - Jacobs - 1999 |

114 |
The unreasonable effectiveness of mathematics in the natural sciences
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- 1960
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Citation Context ...Introduction and overview. In 1960, E. P. Wigner, a joint winner of the 1963 Nobel Prize for Physics, published a paper titled On the Unreasonable Effectiveness of Mathematics in the Natural Sciences =-=[61]-=-. This paper can be construed as an examination and affirmation of Galileo’s tenet that “The book of nature is written in the language of mathematics”. To this effect, Wigner presented a large number ... |

114 |
Specification and verification of concurrent systems
- Queille, Sifakis
- 1982
(Show Context)
Citation Context ...s in which mathematical techniques are used to guarantee the correctness of a design with respect to some specified behavior. Algorithmic formal-verification tools, based on model-checking technology =-=[6, 41, 48, 58]-=- have enjoyed a substantial and growing use over the last few years, showing an ability to discover subtle flaws that result from extremely improbable events. While until recently these tools were vie... |

108 | Elements of Relational Database Theory - Kanellakis - 1990 |

102 |
Correction to a Note on the Entscheidungsproblem
- Church
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Citation Context ...paper [19]. Furthermore, A. Turing, A. Church, and A. Tarski demonstrated the undecidability of first-order logic. Specifically, the set of all valid first-order sentences was shown to be undecidable =-=[5, 54]-=-, whereas the set of all first-order sentences that are true in arithmetic was shown to be highly undecidable [53]. Today, mathematical logic is a mature and highly sophisticated research area with de... |

94 | Symbolic model checking: states and beyond - Burch, Clarke, et al. - 1992 |

82 |
Sur quelques classifications des systèmes de relations, Publ. Scient. de l’Université d’Alger
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Citation Context ...ected graph can have its vertices colored with three colors so that no two adjacent vertices have the same color — as well as hundreds of other well-known combinatorial problems are NP-complete. (See =-=[GJ79]-=- for a survey of many of these.) This means that not only are they in NP, but they are the “hardest problems” in NP: all problems in NP are reducible (in polynomial time) to each NP-complete problem. ... |

78 | Theories of Programming Languages - Reynolds - 1998 |

77 |
DerWahrheitsbegriffin den formalisierten Sprachen
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Citation Context ...lly, the set of all valid first-order sentences was shown to be undecidable [Chu36, Tur37], whereas the set of all first-order sentences that are true in arithmetic was shown to be highly undecidable =-=[Tar35]-=-. Today, mathematical logic is a mature and highly sophisticated research area with deep results and a number of applications in certain areas of mathematics. All in all, however, it is fair to say th... |

77 |
Notes on database operating systems," in Operating Systems: An Advanced Course
- Gray
- 1979
(Show Context)
Citation Context ...m ′ ) such that ri(m) = r ′ i(m ′ ). As an example of how this framework can be used in analyzing distributed protocols, consider the coordinated attack problem, from the distributed systems folklore =-=[21]-=-. It abstracts a problem of data recovery management that arises when using standard protocols in database management called commit protocols. The following presentation is taken from [22]: Two divisi... |

76 |
Computer Aided Verification of Coordinating Processes
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(Show Context)
Citation Context ...ecification language. There are those who prefer to use automata on infinite words as a specification formalism [VW94]; in fact, this is the approach ��� and����� � ��, and ��, where and��� of COSPAN =-=[Kur94]-=-. In this approach, we are given a design represented as a finite transition system and a property represented by a Büchi (or a related variant) automaton . The design is correct if 7 http://www-cad.e... |

71 |
Cylindric Algebras (Part I
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Citation Context ...set of simple operations on relations expressing all FO queries. Relational algebra as used in the context of databases was introduced by Ted Codd in [7]. It is related to Tarski’s Cylindric Algebras =-=[24]-=-. The algebra turns out to yield a crucial advantage when large amounts of data are concerned. Indeed, the realization by Codd that the algebra can be used to efficiently implement FO queries gave the... |

71 |
Three approaches to type structure
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(Show Context)
Citation Context ...hat improve the efficiency and integrity of generated code. Type theory is the study of type systems. Reynolds defines a type system to be a “syntactic discipline for enforcing levels of abstraction” =-=[Rey85]-=-. A type system is a form of context-sensitive grammar that imposes restrictions on the formation of programs to ensure that a large class of errors, those that arise from misinterpretation of values,... |