Heterogeneous Reasoning

Transcription

1 Heterogeneous Reasoning Stanford University

2 The Openproof Project Developing software for logic education since 1984 Tarski s World The Language of First-order Logic Turing s World Hyperproof LPL: Fitch, Boole, Submit and the Grade Grinder Openbox and Playfair Kripke s World

3 Openproof Day Stanford January 2009 Logic Software from Openproof Project The Openproof Project John Etchemendy & Jon Barwise Albert Liu & Michael Murray Mark Greaves & Alan Bush Gerard Allwein & Richard Cox, Robert Dale, Nik Swoboda, Keith Stenning many graduate and undergraduate students: Mike Mellenthin, Tony Ricciardi, Adeline Wong, Aaron Kalb, Olalere Williams and Alex Romanczuk

4 Overview Formal proof: the sentential case Formal proof: the heterogeneous case Generalizing notions of proof The Openproof framework and Playfair

5 Overview Formal proof: the sentential case Formal proof: the heterogeneous case Generalizing notions of proof The Openproof framework and Playfair

6 Sentential Proof Arguments are presented in textual form Arguments can be translated from text into First-order logic (or some extension) First-order logic can be checked for validity, consequence etc; Arguments can therefore be validated.

7 Fitch Implemented environment for formal proof Information at a step (sentence) Support (information at previous steps.) Inference rule (justification for new information)

8 An Example Natural Deduction Proof

9 Fitch-Style Natural Deduction

10 Fitch-Style Natural Deduction A proof is a linear sequence of steps

11 Fitch-Style Natural Deduction A proof is a linear sequence of steps Steps may contain embedded proofs.

12 Fitch-Style Natural Deduction A proof is a linear sequence of steps Steps may contain embedded proofs. Each step working toward a specific (perhaps subsidiary) goal

13 Fitch-Style Natural Deduction

14 Fitch-Style Natural Deduction Structure gives us a unique collection of information available at each step

15 Fitch-Style Natural Deduction Structure gives us a unique collection of information available at each step Steps are justified by citation of information available at this point in the proof

16 Promotion Rules Sentential rule of proof by cases P Q Q ~P Q

17 Features of the model Global structure of reasoning. Reasoning is. (locally) linear and goal-directed, typically directed at a unique outcome (design, decision, plan) punctuated by reasoning about alternatives (hypothetical reasoning)

18 Assumptions of Sentential Logic Assumption that arguments are presented or are to be constructed textually. Or at least that however the information comes to you, it can be rendered textually. And that such rendering, if possible, does not impact the tractability of determining the validity of the argument.

19 Overview Formal proof: the sentential case Formal proof: the heterogeneous case Generalizing notions of proof The Openproof framework and Playfair

20 Outside the FOL Model Homeowner deciding whether to refinance mortgage (spreadsheet.) Driver planning a route to unfamiliar location (map). Chemist figuring out structure of a protein. Architect designing a home for a client (plan). Engineer designing an integrated circuit (wiring, timing) Graphic artist designing a brochure.

21 Some Features of These Examples Involve non-sentential representations of information, E.g., Maps, Venn diagrams, architectural plans Typically involve multiple representations working together (heterogeneous reasoning) Even some mathematical reasoning has this feature (E.g. set-theoretic reasoning with Venn diagrams).

22 Heterogeneous Reasoning Heterogeneous reasoning is the typical case Multiple sources of information Multiple representations Must work together

23 Does This Matter? We invent these representations for a reason: sentential representations are not working in some way. Representation is important: try dividing CXXIII by IX without converting to Arabic numerals.

24 Formal Heterogeneous Logic Goal: to construct formal logical system(s) for heterogeneous reasoning. We need to account for: logic of non-sentential representations, and interactions between the representations.

25 Hyperproof Implemented system for heterogeneous deduction. Sentential system: First-order logic. Diagrammatic system: Blocks world with abstraction. Heterogeneous inference rules: Observe, Apply. Heterogeneous Goals: Counterexamples, naming.

26 An Example Hyperproof

27 The Content of Reasoning If A then B C if and only if not D A unless not C

28 The Content of Reasoning If A then B C if and only if not D A unless not C Sentences express incremental information

29 The Content of Reasoning If A then B C if and only if not D Diagrams are incrementally modified A unless not C Sentences express incremental information

30 Promotion Rules Diagrammatic Merge rule

31 Overview Formal proof: the sentential case Formal proof: the heterogeneous case Generalizing notions of proof The Openproof framework and Playfair

32 Generalizations New representations: E.g. Venn diagrams. Homogeneous or heterogeneous reasoning Multiple (more than 2) representations in single proof Multiple instances of same representation: E.g, inference from one Venn diagram to another. Generalizing logical inference and justification.

33 Heterogeneity of Representation Traditionally: sentences Information may be represented in multiple forms (sentences but also maps, Venn diagrams, spreadsheets, etc;) Multiple instances of each representation may be used simultaneously. Support the use of new representations.

34 Heterogeneity of Rationales Traditionally: Deduction We may obtain new information also on the basis of abduction, cost, aesthetic reasoning,... or all of the above. Generalize the notion of inference rule.

35 Promotion Rules Generalizing Rationale Because, for example, this Is the cheapest design

36 Heterogeneity of Goals Traditionally: Consequence Goals: Vary in type and structure, often we choose among solutions meeting primary criteria (bridge is structurally sound) on the basis of secondary criteria (most cost-effective design). Support different analyses of proof

37 Heterogeneity of Goals

38 Overview Formal proof: the sentential case Formal proof: the heterogeneous case Generalizing notions of proof The Openproof framework and Playfair

39 Openproof A framework for assembling heterogeneous reasoning applications Component-based architecture Openproof manages the proof tree Components manage the content of the nodes in the tree.

40 What is a component? Representation (model): abstract representation of diagram Editor (view): user interface for creating and modifying representations. Engine (controller): collection of operations using the representation(s). May be developed independently.

41 Three Roles for Openproof for us to build heterogeneous reasoning environments for logic education for you to build heterogeneous reasoning environments for building real world applications.

42 Playfair Collection of Openproof components Aimed at teaching general heterogeneous reasoning techniques. Expose students to reasoning patterns/ strategies in multiple contexts (some very familiar) to promote transfer out of the classroom.

43 Playfair Representation Modules Venn/Euler diagrams: for set theory Position diagrams: SAT/GRE exams Hyperproof diagrams Attribute/Value tables Arbitrary drawings and text (in the future): network/state machine diagrams

44 Venn Diagrams

45 Position diagrams

46 Coincidence Grids

47 Rationale Capture Generalizing the notion of justification allows us to view documents as capturing rationales. Users can write text as the justification for a step Document records, and allows replaying of, reasoning. Building a document requires attention to the structure of the reasoning.

48 Architectural Design Maintain the structure of reasoning Justifications are free text, not verified. Example has only one (diagrammatic) representation.

49 Architectural Design

50 Graphic Design Maintain the structure of reasoning No justifications. Example has only one (diagrammatic) representation.

51 Graphic Design

52 Summary Heterogeneous reasoning is typical It is possible to formalize and implement logics for heterogeneous reasoning. The Openbox allows the construction of complex heterogeneous environments for a variety of representations. Reasoning does not have to be formal, or involve formally specified semantics to be useful.

53 Heterogeneous Reasoning Stanford University