## Simplifying proofs in Fitch-style natural deduction systems (2004)

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### BibTeX

@MISC{Arkoudas04simplifyingproofs,

author = {Konstantine Arkoudas},

title = {Simplifying proofs in Fitch-style natural deduction systems},

year = {2004}

}

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### Abstract

We present an algorithm for simplifying Fitch-style natural deduction proofs in classical first-order logic. We formalize Fitch-style natural deduction as a denotational proof language, N DL, with a rigorous syntax and semantics. Based on that formalization, we define an array of simplifying transformations and show them to be terminating and to respect the formal semantics of the language. We also show that the transformations never increase the size or complexity of a deduction—in the worst case, they produce deductions of the same size and complexity as the original. We present several examples of proofs containing various types of superfluous “detours, ” and explain how our procedure eliminates them, resulting in smaller and cleaner deductions. All of the transformations are fully implemented in SML-NJ, and the complete code listing is available. 1.1

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Citation Context ...t draws on contemporary advances in the field of programming language theory. In particular, N DL proofs are succinctly specified by an abstract grammar ([32]), while a big-step operational semantics =-=[21, 28]-=- attaches a rigorous meaning to every proof that is syntactically well-formed. Assumption scope is captured by context-free block structure, obviating 1Also used by Prawitz [30] in his “Natural Deduct... |

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Citation Context ...mputer (e.g., in N DL form or in a Fitch 4 The idea of representing assumption scope by block structure is also present in formalizations of intuitionist natural deduction in higher-order type theory =-=[19, 20, 17]-=-, but there are important differences. Most notably, in the setting of type theory assumption scope is lexical scope, that is, assumptions are represented by bound λ-calculus variables, and hence ther... |

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Citation Context ...alen in his “Logic and structure” [12], and by Troelstra in “Basic Proof Theory” [33]. 2Used in books such as “Mathematical Logic” by Ebbinghaus et al. [14] and in theorem-proving systems such as HOL =-=[18]-=-, Isabelle [18]. 3Indeed, Pelletier [26] claims that sequent-based calculi are simply not natural deduction systems. 1sthe need for diagrams. 4 Semantically, the formal meaning of a DPL proof is speci... |

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Citation Context ...mputer (e.g., in N DL form or in a Fitch 4 The idea of representing assumption scope by block structure is also present in formalizations of intuitionist natural deduction in higher-order type theory =-=[19, 20, 17]-=-, but there are important differences. Most notably, in the setting of type theory assumption scope is lexical scope, that is, assumptions are represented by bound λ-calculus variables, and hence ther... |

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Citation Context ...t draws on contemporary advances in the field of programming language theory. In particular, N DL proofs are succinctly specified by an abstract grammar ([32]), while a big-step operational semantics =-=[21, 28]-=- attaches a rigorous meaning to every proof that is syntactically well-formed. Assumption scope is captured by context-free block structure, obviating 1Also used by Prawitz [30] in his “Natural Deduct... |

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Citation Context ... is now standard practice in the literature to speak of “Fitch systems.” Such systems are the most popular pedagogical choice for teaching symbolic logic, used by numerous influential logic textbooks =-=[22, 31, 15, 11, 8, 23, 9]-=-. They are considered to be the most “natural” of the three main families of proof systems that claim to capture the way in which mathematicians present proofs in practice, the other two being the nat... |

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Citation Context ...aptured by context-free block structure, obviating 1Also used by Prawitz [30] in his “Natural Deduction” [30], by Van Dalen in his “Logic and structure” [12], and by Troelstra in “Basic Proof Theory” =-=[33]-=-. 2Used in books such as “Mathematical Logic” by Ebbinghaus et al. [14] and in theorem-proving systems such as HOL [18], Isabelle [18]. 3Indeed, Pelletier [26] claims that sequent-based calculi are si... |

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Citation Context ...mputer (e.g., in N DL form or in a Fitch 4 The idea of representing assumption scope by block structure is also present in formalizations of intuitionist natural deduction in higher-order type theory =-=[19, 20, 17]-=-, but there are important differences. Most notably, in the setting of type theory assumption scope is lexical scope, that is, assumptions are represented by bound λ-calculus variables, and hence ther... |

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Citation Context ...ing are likely to produce suboptimal proofs with various detours; our procedures should prove useful there as well. In addition, Athena employs resolution-based systems such as Vampire [34] and Spass =-=[36]-=- for proof search [5] and their output proofs also contain many redundancies. Those resolution proofs could be converted to Fitch-style native Athena proofs and simplified with the procedures we descr... |

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Citation Context ...rm of a denotational proof language, N DL, that draws on contemporary advances in the field of programming language theory. In particular, N DL proofs are succinctly specified by an abstract grammar (=-=[32]-=-), while a big-step operational semantics [21, 28] attaches a rigorous meaning to every proof that is syntactically well-formed. Assumption scope is captured by context-free block structure, obviating... |

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Citation Context ...re disjoint iff Supp(θ1) ∩ Supp(θ2) = ∅; and RanVar(θi) ∩ Supp(θj) = ∅ whenever i, j ∈ {1, 2}, i �= j. Any substitution θ can be extended to a homomorphism � θ from terms to terms in the usual manner =-=[35]-=-. Since the extension is unique, we may simply write θ(t) instead of � θ(t). Substitutions can also be applied to formulas. Further overloading our notation, we define θ(F ), the result of applying a ... |

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Citation Context ... is now standard practice in the literature to speak of “Fitch systems.” Such systems are the most popular pedagogical choice for teaching symbolic logic, used by numerous influential logic textbooks =-=[22, 31, 15, 11, 8, 23, 9]-=-. They are considered to be the most “natural” of the three main families of proof systems that claim to capture the way in which mathematicians present proofs in practice, the other two being the nat... |

67 |
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Citation Context ...r deductions and display the results to them. 1.2 Background Our subject is related to proof-tree normalization in the sense of Prawitz [30] (or alternatively, cutelimination in sequent-based systems =-=[16, 13]-=-). In the intuitionist case, the Curry-Howard correspondence means that Prawitz normalization coincides with reduction in the simply typed λ-calculus. Accordingly, the normalization algorithm in that ... |

57 |
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Citation Context ...r deductions and display the results to them. 1.2 Background Our subject is related to proof-tree normalization in the sense of Prawitz [30] (or alternatively, cutelimination in sequent-based systems =-=[16, 13]-=-). In the intuitionist case, the Curry-Howard correspondence means that Prawitz normalization coincides with reduction in the simply typed λ-calculus. Accordingly, the normalization algorithm in that ... |

44 | Efficient representation and validation of logical proofs
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Citation Context ...thena proofs and simplified with the procedures we describe here. Another potential area of application is proof-carrying code (PCC [24]), where proof size is a very important practical consideration =-=[25]-=-. Finally, our algorithms could prove useful for educational purposes. Beginning logic students often write proofs in an immature style, deriving extraneous conclusions, placing an inference in the sc... |

39 |
The anatomy of Vampire: implementing bottom-up procedures with code trees
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(Show Context)
Citation Context ... deduction setting are likely to produce suboptimal proofs with various detours; our procedures should prove useful there as well. In addition, Athena employs resolution-based systems such as Vampire =-=[34]-=- and Spass [36] for proof search [5] and their output proofs also contain many redundancies. Those resolution proofs could be converted to Fitch-style native Athena proofs and simplified with the proc... |

38 |
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(Show Context)
Citation Context ...rational semantics [21, 28] attaches a rigorous meaning to every proof that is syntactically well-formed. Assumption scope is captured by context-free block structure, obviating 1Also used by Prawitz =-=[30]-=- in his “Natural Deduction” [30], by Van Dalen in his “Logic and structure” [12], and by Troelstra in “Basic Proof Theory” [33]. 2Used in books such as “Mathematical Logic” by Ebbinghaus et al. [14] a... |

37 |
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Citation Context ... is now standard practice in the literature to speak of “Fitch systems.” Such systems are the most popular pedagogical choice for teaching symbolic logic, used by numerous influential logic textbooks =-=[22, 31, 15, 11, 8, 23, 9]-=-. They are considered to be the most “natural” of the three main families of proof systems that claim to capture the way in which mathematicians present proofs in practice, the other two being the nat... |

34 |
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Citation Context |

31 |
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23 |
2000, Denotational Proof Languages
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(Show Context)
Citation Context ...efly reviews the syntax and semantics of N DL, along with some basic notions and results that will form the theoretical background for our transformations. Omitted proofs can be found in Chapter 6 of =-=[2]-=-. The following two sections discuss each group of transformations in turn: first the contractions C, P, and U; and then the restructuring transformations MS, A1, A2, and A3. Finally,in Section 1.6 we... |

18 | M.: Integrating model checking and theorem proving for relational reasoning
- Arkoudas, Khurshid, et al.
(Show Context)
Citation Context ...uce suboptimal proofs with various detours; our procedures should prove useful there as well. In addition, Athena employs resolution-based systems such as Vampire [34] and Spass [36] for proof search =-=[5]-=- and their output proofs also contain many redundancies. Those resolution proofs could be converted to Fitch-style native Athena proofs and simplified with the procedures we describe here. Another pot... |

18 |
Don’t Eliminate Cut
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(Show Context)
Citation Context ... fact it may well result in a dramatically (e.g., exponentially) larger proof, even when the original proof is fairly short and simple. (This has led some logicians to caution against cut elimination =-=[10]-=-.) By contrast, the result of our simplification procedure will never be larger than the original, and will indeed often be smaller and simpler. The remainder of this paper is structured as follows. T... |

18 |
a Generic Theorem Prover
- Isabelle
- 1994
(Show Context)
Citation Context ...], and by Troelstra and Schwichtenberg in “Basic Proof Theory” [34]. 2Used in books such as “Mathematical Logic” by Ebbinghaus et al. [14] and in theorem-proving systems such as HOL [18] and Isabelle =-=[26]-=-. 3E.g., Pelletier [27] states that sequent-based calculi are simply not natural deduction systems. 4The idea of representing assumption scope by block structure is also present in formalizations of i... |

12 | A brief history of natural deduction
- Pelletier
- 1999
(Show Context)
Citation Context ...n Fitch-style natural-deduction systems. The hallmark of such systems is the idea of “making arbitrary assumptions and keeping track of where they lead and for how long the assumptions are in effect” =-=[26]-=-. More briefly, we might say that the cornerstone of such systems is the notion of conditional subproof: if at some point in a proof D we wish to establish a conditional F ⇒ G, we postulate F as a pro... |

11 | Deductive runtime certification
- Arkoudas, Rinard
- 2004
(Show Context)
Citation Context ...s. As discussed above, such an interpreter would be a proof checker. In the case of N DL, an interpreter can be written in about one page of SML code, resulting in a very small trusted computing base =-=[6]-=-. • A formal semantics allows us to develop a rigorous theory of observational equivalence for proofs, providing precise answers to questions such as: What does it mean for two proofs to be equivalent... |

11 |
The Logic Book
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- 2004
(Show Context)
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10 | Rational cognition in OSCAR
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- 1999
(Show Context)
Citation Context ...t from the intrinsic theoretical interest of the subject, there are several practical motivations for this work. Automated theorem proving systems based on Fitch-style natural deduction—such as Oscar =-=[29]-=- and Thinker [27]—often output long proofs with many redundancies. The algorithms we describe here could be easily implemented in these systems to “clean up” the proofs. 5 Likewise, method application... |

6 |
n.d.c, Type-ω DPLs
- Arkoudas
(Show Context)
Citation Context ...oofs with many redundancies. The algorithms we describe here could be easily implemented in these systems to “clean up” the proofs. 5 Likewise, method applications in Athena [1] and other type-ω DPLs =-=[4]-=- that perform proof search in a natural deduction setting are likely to produce suboptimal proofs with various detours; our procedures should prove useful there as well. In addition, Athena employs re... |

4 |
2001a, Type-α DPLs
- Arkoudas
(Show Context)
Citation Context ...ven the most recent textbooks continue to resort to the same diagrammatic techniques that were used 70 years ago. An alternative formalization of Fitch-style natural deduction has been given recently =-=[3]-=- in the form of a denotational proof language, N DL, that draws on contemporary advances in the field of programming language theory. In particular, N DL proofs are succinctly specified by an abstract... |

2 |
Logic and Structure
- Dalen
- 1983
(Show Context)
Citation Context ...yntactically well-formed. Assumption scope is captured by context-free block structure, obviating 1Also used by Prawitz [30] in his “Natural Deduction” [30], by Van Dalen in his “Logic and structure” =-=[12]-=-, and by Troelstra in “Basic Proof Theory” [33]. 2Used in books such as “Mathematical Logic” by Ebbinghaus et al. [14] and in theorem-proving systems such as HOL [18], Isabelle [18]. 3Indeed, Pelletie... |

1 |
Hyperproof: for Macintosh
- Barwise, Etchemendy
- 1995
(Show Context)
Citation Context ...ther different form of scope. 5 Thinker already has a “post-processor” that eliminates some redundant claims, but the procedures we describe here are much more aggressive. 2ssystem such as Hyperproof =-=[7]-=-), it would be possible to immediately simplify their deductions and display the results to them. 1.2 Background Our subject is related to proof-tree normalization in the sense of Prawitz [30] (or alt... |

1 |
Proof-carrying code. Computer Science
- Necula, Lee
- 1996
(Show Context)
Citation Context .... Those resolution proofs could be converted to Fitch-style native Athena proofs and simplified with the procedures we describe here. Another potential area of application is proof-carrying code (PCC =-=[24]-=-), where proof size is a very important practical consideration [25]. Finally, our algorithms could prove useful for educational purposes. Beginning logic students often write proofs in an immature st... |

1 | Automated natural deduction in thinker
- Pelletier
- 1998
(Show Context)
Citation Context ...sic theoretical interest of the subject, there are several practical motivations for this work. Automated theorem proving systems based on Fitch-style natural deduction—such as Oscar [29] and Thinker =-=[27]-=-—often output long proofs with many redundancies. The algorithms we describe here could be easily implemented in these systems to “clean up” the proofs. 5 Likewise, method applications in Athena [1] a... |