Lecture 24 nov 2022:
Modelling schnapsen in propositional logics:
You can use logic for games in two ways; reasoning over models / states, or modeling strategies.
Reasoning over states in phase 2:
- ‘Would I be safe when playing the king of hearts.’
- f.e. -O10H ^ -OAsH ^ (OQH v OJH)
- The opponent does not have a higher card, but he does have a lower card of the same color
- Forgetful Schnapsen
- The way human play, up to now we have the assumption that perfect information in phase 2,
but:
- Normal players forget the cards that were played earlier
- I can remember Trumps (on good days)
- I can sometimes remember Asses (on good days)
- I can remember 10s on exceptional days
- We need to use logical reasoning to deal with this imperfect information
Modelling schnapsen cards:
- Use the representation of the cards of the practical assignments to give a numerical value to
each of the cards that’s in the game.
- This information can be modeled to propositions: If the card value is true it means that it exists
in the representation, if the color value is true it means that it’s black.
- In logic you want to evaluate the properties of a card given a deck.
- ‘Is a card entailed by KB w.r.t. PLayNoTrumpJackStrategy?
- We would have to pair all the propositions:
- FOR ALL x: PNTJS <-> J(x) ^ -Trump(x)
- This needs to be checked for each card, but it will work (huge kb)
- Pros:
- Propositional logic is declarative: you write out the properties that something should have
- Unlike in machine learning
- It allows partial/disjunctive/negated information
- Unlike most data structures and databases
- Compositional
- Meaning in propositional logic is context-independent
- In natural language humans can use the context that something was was said in to nd
the meaning
- Meaning does not depend on context and is always the same
- Cons:
- Propositional logic has very limited expressive power (unlike natural language), there are
very many things we cannot say in propositional logic
- Usage for Schnapsen is rather complicated (aka impossible)
Rule based reasoning:
An inference rule is a logical form consisting of a function which takes premises, analyzes their
syntax and returns a conclusion:
- Rules are often abbreviated as A -> B or A over B, this is not the same as ((A->B) ^ A) ->B
- Or with -> (written as ==>) (alpha ==> beta, alpha) over beta (models ponens)
Very speci c with propositional logic language, they are di erent to the object language to which
you model your domain.
- And elimination: form a conjunction any conjunct can be inferred: alpha ^ beta over alpha
- Logical equivalences can be used as rules: (alpha <==> beta) over ((alpha ==> beta ^ beta ==>
alpha)).
Searching for proofs: monotonicity, the set of entailed sentences can only increase as information
is added to the kb. The higher the input gets the higher the output gets. This can be applied to
logic. For any sentence alpha and beta: if KB |= alpha then KB ^ beta |= alpha
Forward chaining is data driven
Backward chaining is goal driven
fi ff fi
Modelling schnapsen in propositional logics:
You can use logic for games in two ways; reasoning over models / states, or modeling strategies.
Reasoning over states in phase 2:
- ‘Would I be safe when playing the king of hearts.’
- f.e. -O10H ^ -OAsH ^ (OQH v OJH)
- The opponent does not have a higher card, but he does have a lower card of the same color
- Forgetful Schnapsen
- The way human play, up to now we have the assumption that perfect information in phase 2,
but:
- Normal players forget the cards that were played earlier
- I can remember Trumps (on good days)
- I can sometimes remember Asses (on good days)
- I can remember 10s on exceptional days
- We need to use logical reasoning to deal with this imperfect information
Modelling schnapsen cards:
- Use the representation of the cards of the practical assignments to give a numerical value to
each of the cards that’s in the game.
- This information can be modeled to propositions: If the card value is true it means that it exists
in the representation, if the color value is true it means that it’s black.
- In logic you want to evaluate the properties of a card given a deck.
- ‘Is a card entailed by KB w.r.t. PLayNoTrumpJackStrategy?
- We would have to pair all the propositions:
- FOR ALL x: PNTJS <-> J(x) ^ -Trump(x)
- This needs to be checked for each card, but it will work (huge kb)
- Pros:
- Propositional logic is declarative: you write out the properties that something should have
- Unlike in machine learning
- It allows partial/disjunctive/negated information
- Unlike most data structures and databases
- Compositional
- Meaning in propositional logic is context-independent
- In natural language humans can use the context that something was was said in to nd
the meaning
- Meaning does not depend on context and is always the same
- Cons:
- Propositional logic has very limited expressive power (unlike natural language), there are
very many things we cannot say in propositional logic
- Usage for Schnapsen is rather complicated (aka impossible)
Rule based reasoning:
An inference rule is a logical form consisting of a function which takes premises, analyzes their
syntax and returns a conclusion:
- Rules are often abbreviated as A -> B or A over B, this is not the same as ((A->B) ^ A) ->B
- Or with -> (written as ==>) (alpha ==> beta, alpha) over beta (models ponens)
Very speci c with propositional logic language, they are di erent to the object language to which
you model your domain.
- And elimination: form a conjunction any conjunct can be inferred: alpha ^ beta over alpha
- Logical equivalences can be used as rules: (alpha <==> beta) over ((alpha ==> beta ^ beta ==>
alpha)).
Searching for proofs: monotonicity, the set of entailed sentences can only increase as information
is added to the kb. The higher the input gets the higher the output gets. This can be applied to
logic. For any sentence alpha and beta: if KB |= alpha then KB ^ beta |= alpha
Forward chaining is data driven
Backward chaining is goal driven
fi ff fi
