Applying Ternary Logic to Decisions in Family Housing

1. Applying Ternary Logic to Decisions in Family Housing Michael S. Cokus, msc@mitre.org John W. Dahlgren, dahlgren@mitre.org The authors’ affiliation with The MITRE Corporation is provided for identification purposes only, and is not intended to convey or imply MITRE's concurrence with, or support for, the positions, opinions or viewpoints expressed by the authors.

2. Abstract Ternary (three-valued) logic is an extension of Boolean algebra in which variables may take on a third truth value (e.g., unknown) in addition to true and false. This allows uncertainty to be represented as a single, discrete value verses a continuum of probabilities. Modifications to the Boolean operators (AND, OR, NOT) have been defined in ternary logic such that expressions can be constructed which evaluate to one of the three truth values. “Applying Ternary Logic to Decisions in Family Housing” presents examples of determining the requirements for a home, and performing tradeoff analysis of those requirements. The specific context is the evaluation of family homes during different points in a family’s “life-cycle”. We discuss the examples as an analogy to requirements development and candidate system evaluations. Combinations of desired/undesired home characteristics are specified as ternary logic expressions. We use the examples to illustrate the utility of ternary logic expressions and representing uncertainty as the truth value “unknown” when making what is often the largest private investment for a family. An overview of ternary logic is presented. Lessons for potential application to the system engineering process are also presented.

3. Purpose • To experiment with applying Ternary Logic to determining the requirements for a home, and as a bonus input, choosing a college

4. Hypothesis • Buying a house is often the largest single purchase many of us make in our lives, with choosing a college the likely second costliest purchase • Most of us make these purchases, and find ways to be happy with our decisions, without formal requirements documents and without specifying every single requirement • Essentially, we accept some unknowns in our decision analysis • Therefore, the authors believe that acknowledging unknowns and accepting unknowns through the initial procurement process may have value in decreasing the cost and development time for future systems

5. Definitions • Boolean Algebra: Boolean algebra, developed in 1854 by George Boole in his book An Investigation of the Laws of Thought, is a variant of ordinary algebra as taught in high school. Boolean algebra differs from ordinary algebra in three ways: in the values that variables may assume, which are of a logical instead of a numeric character, prototypically 0 and 1; in the operations applicable to those values; and in the properties of those operations, that is, the laws they obey. 1http://en.wikipedia.org/wiki/Introduction_to_Boolean_algebra • Ternary Logic: A ternary, three-valued or trivalent logic (sometimes abbreviated 3VL) is any of several multi-valued logic systems in which there are three truth values indicating true, false and some third value. This is contrasted with the more commonly known bivalent logics (such as boolean logic) which provide only for true and false. Conceptual form and basic ideas were initially created by Łukasiewicz, Lewis and Sulski. These were then re-formulated by GrigoreMoisil in an axiomatic algebraic form, and also extended to n-valued logics in 1945. 2http://en.wikipedia.org/wiki/Ternary_logic

6. 3-Valued Truth Table

7. Venn Diagram Basics Just for reference Y Y Y X X X

8. House Example

9. Venn Diagram of Simple Solution • Choice of bedroom on 1st floor initially led to choosing a ranch style home • Expanding to accept a multi-floor home with a 1st floor bedroom opened up the solution space One Floor Multi-floor # BR < 3 Multi-floor, 1st floor BR

10. Complicated House Solution All House Solutions Multi-Floor 1st Floor BR One Floor Circuit Breakers Central A/C Garage 2-Car Garage > 3 BR

11. Ternary Logic Equations for House Solutions Simple House: Complicated House:

12. Requirements for a College Education • This thought exercise is meant to consider what we value in a college choice, and how uncertainty plays into our decision making • Whereas the house example could have more discrete requirements (i.e., number of bedrooms, a garage), the college example proved to have far more unknowns • A key point is we didn’t realize the number of unknowns, and the requirements that were truly “Must Haves” until after performing this exercise • While various publications provide rankings of colleges, we chose to look at their publications as a database but not be influenced by their final ranking

13. College Example

14. Questions that Led to Unknowns • Would you reject a school because it costs $30,100 instead of $30,000? Probably not, therefore price limit is unknown • Would you reject a school that is 255 miles away? No, therefore distance is unknown • With non-stop flights out of Patrick Henry Field, is driving distance a factor or is it travel convenience? How can travel convenience be quantified or qualified? • Would you insert unknown as a solution to some requirements and perform a sensitivity analysis to see if that requirement matters? • Most likely, but the answer probably relates to comfort level and is not quantitative • If you have a requirement where the solution cannot be defined, do you really have a requirement?

15. Equation for College Selection Example College Selection

16. Thoughts to Consider on Decision Analysis • There is a group that feels standard Decision Analysis methodology is no better than intuition 3 • Some problems better lend themselves to quantitative analysis where human intuition has not been shown to be optimal • Some problems do not lend themselves to applied mathematical solutions • http://en.wikipedia.org/wiki/Decision_Analysis • Ternary Logic, with the acceptance of “Unknown” as part of the solution set, may help bridge the abyss between these two camps

17. Conclusions • What did we learn? • Truly “Must Have” requirements are linked by an “AND” in the Ternary Logic equation – relate to Key Performance Parameters • The power of “or” in requirements can greatly increase the solution space, and hopefully decrease program risk and cost • “Must Have” requirements should be reviewed a few times to determine if they can be changed to “Or” requirements • Often what appears to be required is actually tradeable • The “Uncertain” delineation can help avoid allowing some requirements to unnecessarily drive decisions

18. Road Ahead • Relate Ternary Logic to an Analysis of Alternatives and a requirements development document • Determine how systems can be designed to allow for uncertainty in some requirements • Work on the following hypotheses: • fiscal limits and acknowledgment of uncertainty drive flexible requirements development • Uncertain can be related to technology readiness levels and can change over time • Maturing technology levels can move requirements from Uncertain to True or False – the users can actually decide • Boolean Algebra, with the focus on True and False, is similar to requirements definitions that are either fully satisfied or not • Ternary Logic with its acceptance of Unknown may provide lessons for future requirements development, especially when you need to optimize a solution within a given constraint such as cost

19. Bibliography • http://en.wikipedia.org/wiki/Introduction_to_Boolean_algebra • http://en.wikipedia.org/wiki/Ternary_logic • http://en.wikipedia.org/wiki/Decision_Analysis