Flexiformality in Proofs: Bridging between Formal and Informal - PowerPoint PPT Presentation

Flexiformality in Proofs: Bridging between Formal and Informal Proofs Michael Kohlhase Professur für Wissensrepräsentation und -verarbeitung Informatik, FAU Erlangen-Nürnberg http://kwarc.info 20. October 2016, Dagstuhl Seminar on “Universality of Proofs” Kohlhase: Flexiformality in Proofs 1 Dagstuhl

Instead of an Intro: Disclosure of my Angle Today ◮ I want to build a universal digital library of mathematics (in theory graph form) ◮ This is a critical public resource for math system interoperability ◮ Catherine told us about a graph of species in Focalize for Coq/Dedukti interoperability ◮ Florian did the same for more theorem provers in OMDoc/MMT [KR16] ◮ In the OpenDreamKit project we use the same idea for CAS interop [DIK + 16] ◮ The International Mathematical Union (IMU) has chartered a group to prepare a semantic “Global Digital Maths Library” based on [DLC + 14]. ◮ Without proofs, all of this will be for nothing, ◮ proofs certify the validity of theorems ◮ modularity/reuse via theory morphisms carry proof obligations ◮ all the aspects that Bill has talked about apply (thanks for the intro) Kohlhase: Flexiformality in Proofs 2 Dagstuhl

1 The Flexiformalist Program (inspired by Hilbert) Kohlhase: Flexiformality in Proofs 3 Dagstuhl

Migration by Stepwise Formalization ◮ Full Formalization is hard (we have to commit, make explicit) ◮ Let’s look at documents and document collections. number formality Kohlhase: Flexiformality in Proofs 4 Dagstuhl

Migration by Stepwise Formalization ◮ Full Formalization is hard (we have to commit, make explicit) ◮ Let’s look at documents and document collections. ◮ Partial formalization allows us to ◮ formalize stepwise, and ◮ be flexible about the depth of formalization. number formality Kohlhase: Flexiformality in Proofs 4 Dagstuhl

What is Informal Mathematical Knowledge ◮ Idea: informal knowledge could be formalized (but isn’t yet!) ◮ Definition 0.1 The meaning of a knowledge item is the set of all its formalizations ◮ Problem: What is the space of formalizations? ◮ Definition 0.2 The formal space is the set F := {� S , e � | S ∈ F , e ∈ L ( S ) } , where F is the class of formal systems and L ( S ) is the language of S . (i.e. every formal expression is a point in F ) ◮ Different Logics correspond to different bands ◮ The meaning of D is a set I ( D ) ⊆ F . ◮ D can be formalized in multiple logics I ( D ) forms a cross-section of logic-bands. Kohlhase: Flexiformality in Proofs 5 Dagstuhl

A Formality Ordering on F s n o i s s e r p x ◮ Stepwise formalization looks like this: E D 3 D 2 Logics D ′ D 1 D 3 D ′ 2 D ′′ 3 c e a p S Document l a m r o Less Formal More Formal F Space ◮ Definition 0.3 D is more formal than D ′ (write D ≪ D ′ ), iff I ( D ) ⊂ I ( D ′ ) . ◮ This partial ordering relation answers the question of “graded formality” or the nature of “stepwise formalization” raised above. Kohlhase: Flexiformality in Proofs 6 Dagstuhl

The Flexiformalist Program (Details in [Koh13]) ◮ The development of a regime of partially formalizing ◮ mathematical knowledge into a modular ontology of mathematical theories (content commons), and ◮ mathematical documents by semantic annotations and links into the content commons (semantic documents), ◮ The establishment of a software infrastructure with ◮ a distributed network of archives that manage the content commons and collections of semantic documents, ◮ semantic web services that perform tasks to support current and future mathematic practices ◮ active document players that present semantic documents to readers and give access to respective ◮ the re-development of comprehensive part of mathematical knowledge and the mathematical documents that carries it into a flexiformal digital library of mathematics. Kohlhase: Flexiformality in Proofs 7 Dagstuhl

Stephen Watt’s understanding of Flexiformality A person who is flexiformal: ◮ flexible (contortionist) ◮ formal (tuxedo) Kohlhase: Flexiformality in Proofs 8 Dagstuhl

A Flexiformal Theory Graph to Rule them All IMPS TWELF MathScheme MMTCS Surface Languages T EX OHTML S MitM Ontology Interface · · · Coq PVS Mizar GAP LMFDB Theories Systems Coq · · · PVS Mizar GAP lmfdb Kohlhase: Flexiformality in Proofs 9 Dagstuhl

2 Flexiformal Theory Graphs axnd Proofs Kohlhase: Flexiformality in Proofs 10 Dagstuhl

Background: Redesigning OMDoc ◮ OMDoc : Open Math Documents [Koh06, OMD] models,p document & knowledge structures level coverage markup objects/phrases presentation/content/text math MathML, OpenMath , XHTML statements narrative, some declarations OMDoc , XHTML theories domain theory graphs OMDoc documents grouping OMDoc notations OpenMath � MathML rewrites OMDoc quiz, code, . . . ad-hoc OMDoc Kohlhase: Flexiformality in Proofs 11 Dagstuhl

Background: Redesigning OMDoc ◮ OMDoc : Open Math Documents [Koh06, OMD] models,p document & knowledge structures level coverage markup objects/phrases presentation/content/text math MathML, OpenMath , XHTML statements narrative, some declarations OMDoc , XHTML theories domain theory graphs OMDoc documents grouping OMDoc notations OpenMath � MathML rewrites OMDoc quiz, code, . . . ad-hoc OMDoc ◮ MMT: Meta Meta Theory [RK13, MMT] redesigns the formal core of OMDoc and gives it meaning level coverage markup objects content math, literals OpenMath + , MMT statements constant decls/defs OMDoc , MMT domain/meta * theories OMDoc , MMT notations one-level pres/parse OMDoc , MMT rules typing, reconstruction, . . . Scala Kohlhase: Flexiformality in Proofs 11 Dagstuhl

Background: Redesigning OMDoc ◮ OMDoc : Open Math Documents [Koh06, OMD] models,p document & knowledge structures level coverage markup objects/phrases presentation/content/text math MathML, OpenMath , XHTML statements narrative, some declarations OMDoc , XHTML theories domain theory graphs OMDoc documents grouping OMDoc notations OpenMath � MathML rewrites OMDoc quiz, code, . . . ad-hoc OMDoc ◮ MMT: Meta Meta Theory [RK13, MMT] redesigns the formal core of OMDoc and gives it meaning level coverage markup objects content math, literals OpenMath + , MMT statements constant decls/defs OMDoc , MMT domain/meta * theories OMDoc , MMT notations one-level pres/parse OMDoc , MMT rules typing, reconstruction, . . . Scala ◮ iMMT: flexiformal MMT designs a formal language for the informal/narrative parts of OMDoc . (under development with Mihnea Iancu (diss)) Kohlhase: Flexiformality in Proofs 11 Dagstuhl

Background: Redesigning OMDoc ◮ OMDoc : Open Math Documents [Koh06, OMD] models,p document & knowledge structures level coverage markup objects/phrases presentation/content/text math MathML, OpenMath , XHTML statements narrative, some declarations OMDoc , XHTML theories domain theory graphs OMDoc documents grouping OMDoc notations OpenMath � MathML rewrites OMDoc quiz, code, . . . ad-hoc OMDoc ◮ MMT: Meta Meta Theory [RK13, MMT] redesigns the formal core of OMDoc and gives it meaning level coverage markup objects content math, literals OpenMath + , MMT statements constant decls/defs OMDoc , MMT domain/meta * theories OMDoc , MMT notations one-level pres/parse OMDoc , MMT rules typing, reconstruction, . . . Scala ◮ iMMT: flexiformal MMT designs a formal language for the informal/narrative parts of OMDoc . (under development with Mihnea Iancu (diss)) ◮ OMDoc2 ˆ = MMT + iMMT (needs more language design) Kohlhase: Flexiformality in Proofs 11 Dagstuhl

How to model Flexiformal Mathematics ◮ I hope to have convinced you: that Math is informal: ◮ foundations unspecified (what a relief) ◮ natural language & presentation formulae (humans can disambiguate) ◮ context references (but math is better than the pack) ◮ Problem: How do we deal with that in our “formal” systems? ◮ Proposed Answer: learn from OpenMath /MathML ◮ referential theory of meaning (by pointing to symbol definitions) ◮ allow opaque content (presentation/natural language) ◮ parallel markup (mix formal/informal recursively at any level) ◮ pluralism at all levels (object/logic/foundation/metalogic) ◮ underspecification of symbol meaning extend to statement/paragraph and theory/discourse levels ( OMDoc ) Kohlhase: Flexiformality in Proofs 12 Dagstuhl

Parallel Markup e.g. in MathML I ◮ Idea: Combine the presentation and content markup and cross-reference 3 @ (x+2) ( x+2 ) / 3 3 @ x x + + 2 2 ◮ use e.g. for semantic copy and paste. (click on presentation, follow link and copy content) Kohlhase: Flexiformality in Proofs 13 Dagstuhl