Intro CNL PRS ATPs Future Work The Naproche System Daniel K¨ uhlwein University of Nijmegen daniel.kuehlwein@gmail.com http://www.naproche.net 25th Mai 2011 The Naproche System 1 / 27
Intro CNL PRS ATPs Future Work Outline Introduction 1 The Naproche CNL 2 Proof Representation Structures 3 Checking Naproche Texts 4 Future Work 5 The Naproche System 2 / 27
Intro CNL PRS ATPs Future Work The Naproche Project The Naproche project ( Na tural language Pro of Che cking) is a joint project of the University of Bonn and the University of Duisburg-Essen. They study the semi-formal language of mathematics (SFLM) as used in journals and textbooks from the perspectives of linguistics and mathematics. A central goal of Naproche is to develop a controlled natural language (CNL) for mathematical texts and implement a system, the Naproche system , which can check texts written in this CNL for logical correctness using methods from computational linguistics and automatic theorem proving. The Naproche System 3 / 27
Intro CNL PRS ATPs Future Work The Naproche CNL and the Naproche system The Naproche CNL is a controlled natural language for mathematical texts, i.e. a controlled subset of SFLM. The Naproche system translates Naproche CNL texts first into Proof Representation Structures ( PRS s), an adapted version of Discourse Representation Structures. PRSs are further translated into lists of first-order formulas which are used for checking the logical correctness of a Naproche text using automated theorem provers (ATPs). The Naproche System 4 / 27
Intro CNL PRS ATPs Future Work Overview of the (old) Naproche system The Naproche System 5 / 27
Intro CNL PRS ATPs Future Work Outline Introduction 1 The Naproche CNL 2 Proof Representation Structures 3 Checking Naproche Texts 4 Future Work 5 The Naproche System 6 / 27
Intro CNL PRS ATPs Future Work An Example Example (Euclid’s Elements) Theorem 2. Let b and c be distinct points. Assume a is a point such that a � = b and a � = c . Then there is a point v such that d ( b , c ) = d ( a , v ). Proof. By Theorem 1 there is a point d , such that d ( d , a ) = d ( d , b ) = d ( a , b ). Let M be the line such that b and d are on M . Let α be the circle such that b is the center of α , and c is on α . There is a point u such that u is on α , and u is on M and b is between d and u . Let N be the line L such that d and a are on L . Let β be the circle δ such that d is the center of δ , and u is on δ . There is a point v such that v is on β , and v is on N and a is between v and d . Then d ( b , c ) = d ( b , u ). Hence, d ( d , u ) = d ( d , v ). By Lemma A d ( b , u ) = d ( a , v ). Then d ( b , c ) = d ( a , v ). qed. The Naproche System 7 / 27
Intro CNL PRS ATPs Future Work Features Input in L A T EX. Axioms, theorems, definitions, lemmas, assumptions. Nouns, adjectives, verbs, natural language quantification and negation. Subclauses with such that . Natural language connectors like and , or , i.e. , if and iff . Sentences in a proof can start with words like then , hence , therefore etc. Decent formula parser (e.g. d ( d , a ) = d ( d , b ) = d ( a , b )). Plurals and definite descriptions. Basic induction. The Naproche System 8 / 27
Intro CNL PRS ATPs Future Work Collective vs. distributive readings of plurals “ a and b are foo” could mean either foo ( a ) ∧ foo ( b ) or foo ( a , b ). Usually, words have a preferred reading which is incorporated in the dictionary. The Naproche CNL can handle both collective and distributive readings of plurals: Example Let b and c be distinct points. The example is understood as point ( b ) ∧ point ( c ) ∧ distinct ( b , c ). The Naproche System 9 / 27
Intro CNL PRS ATPs Future Work Definite descriptions A definite descriptions is a definite noun phrase referring to a single object by a restricting property whose extension contains exactly one object. abcdef “Let α be the circle such that b is the center of α ” The presupposition of a singular definite description with the restricting property F (“the F”) is that there is a unique object with property F . This presupposition can be divided into two separate presuppositions: An existential presupposition, claiming that there is at least one F A uniqueness presupposition, claiming that there is at most one F Upon encountering a definite description, both presuppositions are verified. The Naproche System 10 / 27
Intro CNL PRS ATPs Future Work Induction The Naproche CNL supports basic induction: Example (Landau’s Grundlagen der Analysis) Theorem 2: For all x x ′ � = x . Proof: By axiom 3, 1 ′ � = 1. Suppose x ′ � = x . Then by theorem 1, ( x ′ ) ′ � = x ′ . Thus by induction, for all x x ′ � = x . Qed. The keywords “by induction ∀ x ϕ ( x )” trigger an induction verification: If nat ( x ), ϕ (1) and ( nat ( n ) ∧ ϕ ( n )) → ϕ ( n + 1) holds, then ∀ x nat ( x ) → ϕ ( x ). The Naproche System 11 / 27
Intro CNL PRS ATPs Future Work Outline Introduction 1 The Naproche CNL 2 Proof Representation Structures 3 Checking Naproche Texts 4 Future Work 5 The Naproche System 12 / 27
Intro CNL PRS ATPs Future Work Quantifiers in natural language How should we translate natural language quantifiers? Example Every farmer is bald. → ∀ x farmer ( x ) → bald ( x ) A farmer is bald. → ∃ x farmer ( x ) ∧ bald ( x ) So “every” denotes universal quantification and “a” denotes existential quantification. Example Every farmer who owns a donkey beats it. → ∀ x ( farmer ( x ) ∧ ∃ y ( donkey ( y )) ∧ owns ( x , y )) → beats ( x , y ) The “right” translation is ∀ x , y ( farmer ( x ) ∧ donkey ( y ) ∧ owns ( x , y )) → beats ( x , y ) The simple approach doesn’t work here. The Naproche System 13 / 27
Intro CNL PRS ATPs Future Work DRT, DRS and PRS Discourse Representation Theory is a way to solve the problem of anaphora resolution. Non-anaphoric noun phrases introduce discourse referents which are used for the binding of anaphoric expressions. In DRT, discourse representation structures (DRS) are used to disambiguate anaphora. DRSs can easily be translated into first order logic. Proof Representation Structures (PRS) are an extension of DRSs for mathematical texts. PRSs are the linguistic format used in the Naproche System. The Naproche System 14 / 27
Intro CNL PRS ATPs Future Work Constituents of PRSs A PRS has five constituents, which we display as “boxes”: i d 1 , . . . , d m m 1 , . . . , m n c 1 . . . c l r 1 , . . . , r k i is the identifier of the PRS. d 1 , . . . , d m are discourse referents. m 1 , . . . , m n are mathematical referents. c 1 , . . . , c l are PRS conditions. r 1 , . . . , r k are textual references. The Naproche System 15 / 27
Intro CNL PRS ATPs Future Work A PRS Example Show example on www.naproche.net. The Naproche System 16 / 27
Intro CNL PRS ATPs Future Work Outline Introduction 1 The Naproche CNL 2 Proof Representation Structures 3 Checking Naproche Texts 4 Future Work 5 The Naproche System 17 / 27
Intro CNL PRS ATPs Future Work Overview of the (old) Naproche system The Naproche System 18 / 27
Intro CNL PRS ATPs Future Work The checking algorithm: Basics The checking algorithm keeps a list of first-order formulas considered to be true, called premises , which gets continuously updated during the checking process. The conditions of a PRS are checked sequentially. Each condition is checked under the currently active premises. According to the kind of condition, the Naproche system creates obligations in the TPTP format which have to be checked by an ATP. The Naproche System 19 / 27
Intro CNL PRS ATPs Future Work An Example Show example on www.naproche.net. The Naproche System 20 / 27
Intro CNL PRS ATPs Future Work Problems and possible solutions Often, ATPs are unable to verify a proof obligation. This might mean that that particular proof step was wrong, or that the ATP was simply too weak to prove it. The Naproche system has several mechanisms to simplify the ATP proof tasks: Conjunction splitting and existential elimination. Skolemization. Premise Selection. The Naproche System 21 / 27
Intro CNL PRS ATPs Future Work Conjunction splitting and existential elimination Two simple ways of simplifying proof tasks are conjunction splitting and existential elimination: If the conjecture is a conjunction ϕ ∧ ψ we split the proof obligation in two obligation c 1 : ϕ and c 2 : ψ . If the conjecture is of the form ∃ x ϕ ( x ) ∧ x = a we can simplify it to ϕ ( a ). (Example: x is a point.) The Naproche System 22 / 27
Intro CNL PRS ATPs Future Work Skolemization Another method to simplify proof tasks is by simplifying the premises: Assumptions can lead to long premises of the form ∀ x ϕ → ∃ y ψ 1 ( y ) ∧ ψ 2 ( y ) ... ∧ ψ n ( y ) By skolemizing the existential variables, we can split this formula into n formulas: ∀ x ϕ → ψ 1 ( skolem ( x )), ∀ x ϕ → ψ 2 ( skolem ( x ))... The Naproche System 23 / 27
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