Propositional logic: Horn clauses
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1 ecture 6 ropositional logic: Horn clauses ilos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square dministration Homework assignment 2 ropositional logic exercises rogramming a resolution solver
2 ogical inference problem ogical inference problem: Given: a knowledge base K (a set of sentences) and a sentence α (called a theorem), oes a K semantically entail α? K = α? In other words: In all interpretations in which sentences in the K are true, is also true? α Solving logical inference problem In the following: How to design the procedure that answers: = α K? Three approaches: Truth-table approach Inference rules Conversion to the inverse ST problem Resolution-refutation
3 K in restricted forms If the sentences in the K are restricted to some special forms some of the sound inference rules may become complete Example: Horn form (Horn normal form) ( ) ( C ) Can be written also as: ( ) (( C ) ) Two inference rules that are sound and complete with respect to propositional symbols for Ks in the Horn normal form: Resolution (positive unit resolution) odus ponens K in Horn form Horn form: a clause with at most one positive literal ( ) ( C ) Not all sentences in propositional logic can be converted into the Horn form K in Horn normal form: Three types of propositional statements: Rules K ) ( 1 2 k ( ( 1 2 K k ) Facts Integrity constraints ) ( 1 2 K k K 1 K k False ) ( 2 ( 1 2 k ) )
4 K in Horn form Horn form: a clause with at most one positive literal ( ) ( C ) Not all sentences in propositional logic can be converted into the Horn form K in Horn normal form: Three types of propositional statements: Rules K ) ( 1 2 k ( ( 1 2 K k ) Facts Integrity constraints ) ( 1 2 K k K 1 K k False ) ( 2 ( 1 2 k ) ) K in Horn form odus ponens:, ore general version of the rule: K k ), 1,, K ( odus ponens is sound and complete with respect to propositional symbols for the Ks in the Horn normal form We assume only logical inference problems for which the theorem α is a propositional symbol: Note: no negation of a propositional symbol is allowed k
5 Inferences: Resolution rule: K in Horn form, C C Resolution is sound and complete with respect to propositional symbols for the Ks in the Horn normal form Complexity of inferences for Ks in HNF uestion: How efficient the inferences in the HNF wrt propositional symbols can be? nswer: rocedures linear in the size of the K in the Horn form exist. Size of a clause: the number of literals it contains. Size of the K in the HNF: the sum of the sizes of its elements. Example:,, ( C ), ( C ), ( C E ), ( E F G ) or,, ( C ), ( C ), ( C E ), ( E F G ) The size is: 12
6 Complexity of inferences for Ks in HNF How to do the inference? If the HNF (is in clausal form) we can apply resolution.,, ( C ), ( C ), ( C E ), ( E F G ) C C E Complexity of inferences for Ks in HNF Features: Every resolution is a positive unit resolution; that is, a resolution in which one clause is a positive unit clause (i.e., a proposition symbol).,, ( C ), ( C ), ( C E ), ( E F G ) C C E
7 Complexity of inferences for Ks in HNF Features: t each resolution, the input clause which is not a unit clause is a logical consequence of the result of the resolution. (Thus, the input clause may be deleted upon completion of the resolution operation.),, ( C ), ( C ), ( C E ), ( E F G ) C C E Complexity of inferences for Ks in HNF Features: t each resolution, the input clause which is not a unit clause is a logical consequence of the result of the resolution. (Thus, the input clause may be deleted upon completion of the resolution operation.),, ( C ), ( C ), ( C E ), ( E F G ) C C E
8 Complexity of inferences for Ks in HNF Features: Following this deletion, the size of the K (the sum of the lengths of the remaining clauses) is one less than it was before the operation.),, ( C ), ( C ), ( C E ), ( E F G ) C C E Complexity of inferences for Ks in HNF Features: If n is the size of the K, then at most n positive unit resolutions may be performed on it.,, ( C ), ( C ), ( C E ), ( E F G ) C C E
9 Complexity of inferences for Ks in HNF linear time resolution algorithm: The number of positive unit resolutions is limited to the size of the formula (n) ut to assure overall linear time we need to access each proposition in a constant time: ata structures indexed by proposition names may be accessed in constant time. (This is possible if the proposition names are number in a range (e.g., 1..n), so that array lookup is the access operation. If propositions are accessed by name, then a symbol table is necessary, and the algorithm will run in time O(n log(n)). Forward and backward chaining Two inference procedures based on modus ponens for Horn Ks: Forward chaining Idea: Whenever the premises of a rule are satisfied, infer the conclusion. Continue with rules that became satisfied. ackward chaining (goal reduction) Idea: To prove the fact that appears in the conclusion of a rule prove the premises of the rule. Continue recursively. oth procedures are complete for Ks in the Horn form!!!
10 Forward chaining example Forward chaining Idea: Whenever the premises of a rule are satisfied, infer the conclusion. Continue with rules that became satisfied. ssume the K with the following rules and facts: K: R1: R2: R3: C C E C F G F1: F2: F3: Theorem: E? Theorem: E Forward chaining example K: R1: C R2: C E R3: C F G F1: F2: F3:
11 Theorem: E Forward chaining example K: R1: C R2: C E R3: C F G F1: F2: F3: Rule R1 is satisfied. F4: C Theorem: E Forward chaining example K: R1: C R2: C E R3: C F G F1: F2: F3: Rule R1 is satisfied. F4: C Rule R2 is satisfied. F5: E
12 Forward chaining Efficient implementation: linear in the size of the K Example: Forward chaining Runs in time linear in the number of literals in the Horn formulae
13 Forward chaining ** genda (facts) Forward chaining inferred
14 Forward chaining inferred add to agenda Forward chaining
15 Forward chaining Forward chaining
16 Forward chaining ackward chaining example roblem: K: R1: C R2: C E R3: C F G F1: F2: F3: Theorem: E Goal: prove the theorem, try to be more theorem driven
17 ? ackward chaining example C R2 ackward chaining tries to prove a theorem rocedure idea: Checks if the theorem is true If not, find the rule with the theorem in its conclusion and try to prove its premises E K: R1: C R2: C E R3: C F G F1: F2: F3: Theorem: E ackward chaining example R1 C R2 E K: R1: C R2: C E R3: C F G F1: F2: F3: ackward chaining is theorem driven
18 ackward chaining ackward chaining
19 ackward chaining ackward chaining
20 ackward chaining ackward chaining
21 ackward chaining ackward chaining
22 ackward chaining ackward chaining
23 Forward vs ackward chaining FC is data-driven, automatic, unconscious processing, e.g., object recognition, routine decisions ay do lots of work that is irrelevant to the goal C is goal-driven, appropriate for problem-solving, e.g., Where are my keys? How do I get into a h program? Complexity of C can be much less than linear in size of K K agents based on propositional logic ropositional logic allows us to build knowledge-based agents capable of answering queries about the world by infering new facts from the known ones Example: an agent for diagnosis of a bacterial disease Facts: Rules: The stain of the organism is gram-positive The growth conformation of the organism is chains (If) (Then) The stain of the organism is gram-positive The morphology of the organism is coccus The growth conformation of the organism is chains The identity of the organism is streptococcus
24 imitations of the HNF The HNF works with propositional symbols: Only non-negated propositional symbols may occur in the premise and the conclusion of the rule Only non-negated propositions can be used as facts ilemma: how to represent the negation so that we can express sentences like: If it is not raining we will go swimming Solution 1: make an explicit proposition for Not_Raining Solution 2: negation as the failure The negation of the propositional symbol will become true if we fail to prove it is true Negation as the failure The negation of the propositional symbol will become true if we fail to prove it is true Caveats: not In terms of logic we have possibilities: =True, or =True roblem is if we use the failure to prove idea: is never on the right hand side of the rule so how we can prove it is not true Solution: stable models Each atom (proposition) has a rule
25 Example. nimal identification system. I1. If the animal has hair then it is a mammal I2. If the animal gives milk then it is a mammal I3. If the animal has feathers then it is a bird I4. If the animal flies and it lays eggs then it is a bird I5. If the animal is a mammal and it eats meat then it is a carnivore I6. If the animal is a mammal and it has pointed teeth and it has claws and its eyes point forward then it is a carnivore I7. If the animal is a mammal and it has hoofs then it is an ungulate 18. If the animal is a mammal and it chews cud then it is an ungulate and it is even-toed I9. If the animal is a carnivore and it has a tawny color and it has dark spots then it is a cheetah I10. If the animal is a carnivore and it has a tawny color and it has black strips then it is a tiger I11. If the animal is an ungulate and it has long legs and it has a long neck and it has a tawny color and it has dark spots then it is a giraffe I12. If the animal is an ungulate and it has a white color and it has black stripes then it is a zebra Il3. If the animal is a bird and it does not fly and it has long legs and it has a long neck and it is black and white then it is an ostrich, Il4. If the animal is a bird and it does not fly and it swims and it is black and white then it is a penguin Il5. If the animal is a bird and it is a good flyer then it is an albatross.
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