Alternatives

Two ways out of Paris, and three ways of asking about them.

alternatives.js · output · proof · check · try it in the playground

The question

From Paris you can take a train to Brussels, or a bus to Lille.

Everyday questions about choices come in different shapes:

• Where can I go? (every option)
• Can I go to Brussels or Lille? (is at least one option open?)
• Just give me one route. (the first that works, and stop)

This example asks all three, and shows the reasoning behind each answer.

What we tell eyecode

Two facts, and two rules that both define a route:

fact(train('paris', 'brussels'))
fact(bus('paris', 'lille'))
// Two alternative clauses: a route is a train or a bus connection.
implied_by(route(From, To), train(From, To))
implied_by(route(From, To), bus(From, To))

Two rules with the same name are alternatives: a route is a train connection, or a bus connection. The implied_by means “work this out when someone asks”.

The three questions

query(route('paris', To))
query(or(train('paris', 'brussels'), bus('paris', 'lille')))
query(once(route('paris', To)))

• Line 1 asks for every destination To reachable from Paris.
• Line 2 uses or: is either connection there?
• Line 3 wraps the question in once: find the first route and commit to it, without looking for more.

A line of the form query(Question) tells eyecode to answer that question.

What eyecode concludes

route('paris', 'brussels')
route('paris', 'lille')
or(train('paris', 'brussels'), bus('paris', 'lille'))
once(route('paris', 'brussels'))

• Both routes, one from each alternative rule.
• The “or” question holds, and is reported just as it was asked.
• once picked Brussels, because the train rule comes first, and stopped there.

Why: the proof in plain words

1. Paris to Brussels by train — fact 1.
2. So there is a route Paris to Brussels — rule 3, the train alternative.
3. Paris to Lille by bus — fact 2.
4. So there is a route Paris to Lille — rule 4, the bus alternative.
5. “Train to Brussels or bus to Lille” holds — because the first half does (step 1). One true side is enough for an “or”.
6. “Once, a route” gives Brussels — because of step 2.

Checked, not just claimed

A separate checker reads the proof against the program. The report says:

• 6 steps: 4 verified as exact instances of the program lines they cite, and 2 (the “or” and the once) checked by confirming the step they rest on.
• No circular reasoning, every claim is justified, and nothing in the proof is extra.
• 0 steps taken on trust.

Verdict: checked.

Try it

eyecode examples/alternatives.js            # the answers
eyecode --proof examples/alternatives.js    # answers with their proof

Or open it in the playground.

Add fact(train('paris', 'amsterdam')) after the bus line and run again: a third route, route('paris', 'amsterdam'), appears, but once still answers Brussels, because that train fact still comes first.

Takeaway

“All of them”, “at least one” and “just the first” are different questions. eyecode lets you say which one you mean, and its proof shows exactly which option answered.