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Stanford CS 157 - Lecture 06 Computational Logic

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Computational Logic Lecture 6 Relational Logic Semantics Michael Genesereth David Haley Autumn 2007 Propositional Logic Semantics A Propositional logic interpretation is an association between the propositional constants in a propositional language and the truth values T or F 10 9 07 2 1 Relational Logic Semantics The big question what is a relational logic interpretation There are no propositional constants just object constants function constants and relation constants To what do they refer 10 9 07 3 Universe of Discourse The Universe of Discourse is the set of all objects about which we want to say something Primitive a quark Composite an engine this class Real Sun Mike Imaginary a unicorn Sherlock Holmes Physical Earth Moon Sun Abstract Justice 10 9 07 4 2 Blocks World 10 9 07 5 Universe of Discourse 10 9 07 6 3 Relations A relation is a set of objects or tuples of objects each of which manifest a particular property or relationship clear set of all blocks with no blocks on top table set of all blocks on the table on set of pairs of blocks in which first is on the second above all pairs in which first block is above the second below all pairs in which first is below the second stack set of triples of blocks arranged in a stack 10 9 07 7 Arity Each relation has an arity that determines the number of objects that can participate in an instance of the relation 1 ary unary clear 2 ary binary on 3 ary ternary stack 10 9 07 8 4 Relations as Tables on a b b c d e clear a d stack a b c 10 9 07 9 Set Representation of Relations Each row in a table with n columns can be represented as a single n tuple The table can be represented as the set of such tuples a b b c d e a d a b c 10 9 07 a b b c d e a d or a d a b c 10 5 Counting Assume a universe of discourse with 5 objects Number of 2 tuples 52 25 Number of binary tables 225 Assume a universe of discourse with n objects Number of k tuples nk Number of k ary tables 2 nk 10 9 07 11 Functions An n ary function is a relation associating each combination of n objects in a universe of discourse called the arguments with a single object called the value Numerical Examples Unary sqrt log Binary Symbolic Examples Unary mother father Binary grade 10 9 07 12 6 Functions Functions are total and single valued one and exactly one value for each combination of arguments Partial not defined for some combination of arguments Multivalued more than value for some argument combination NB We ignore partial and multi valued functions 10 9 07 13 Unary Function a b c d e 10 9 07 b a d c e 14 7 Binary Function a a b a b a a b b b c c c c a b c a b c c c b a b b b 10 9 07 15 Unary Functions as Binary Relations a b c d e b a d c e a b b a c d d c e e An n ary function can always be viewed as an n 1 ary relation 10 9 07 16 8 Binary Functions as Ternary Relations a a b a b a a b b b c c c c a b c a b c c c b a b b b a a a b b b a b c a b c b a c c b a c a b c b b c c b 10 9 07 17 Set Representation of Functions A function can be represented as a set of associations of arguments and values a b b a c d d e a b b a c d d c e e c e Same as representation of tables except for the use of arrows as a reminder that the table is a function 10 9 07 18 9 Counting Assume a universe of discourse with 5 objects Number of 1 tuples 5 Number of unary functions 55 3125 Number of binary relations 225 33554432 Assume a universe of discourse with n objects Number of k tuples nk Number of k ary functions n nk 2 nk log n Number of k 1 ary relations 2 nk 1 2 nkn 10 9 07 19 Role of Logic Incomplete Information Block a is on block b or it is on block c Block a is not on block b Integrity A block may not be on itself A block may be on only one block at a time Definitions A block is under another iff the second is on the first A block is clear iff there is no block on it A block is on the table iff there is no block under it 10 9 07 20 10 Our World 10 9 07 21 Conceptualization Universe of Discourse a set U of objects Functional Basis Set set f1 fm of functions on U fi Uk U Relational Basis Set set r1 rn of relations on U ri U k 10 9 07 22 11 Interpretations An interpretation is a mapping from the constants of a language into elements of a conceptualization U F R i objconst U i funconst F i relconst R The arity of the function and relation constants must match the arity of their interpretations 10 9 07 23 Example i U i a i b i c i f1 i q1 i r2 i s3 10 9 07 24 12 Ground Value Assignments A ground value assignment si based on interpretation i is a mapping from the ground terms of the language into the universe of discourse that agrees with i on constants and that for functional terms yields the result of applying the interpretation of the functional constant to the values assigned to the argument terms si i si 1 n i si 1 si n 10 9 07 25 Example Interpretation i a i b i f i r Ground Value Assignment si a i a si f a i f si a i f 10 9 07 26 13 Ground Truth Assignments A ground truth assignment ti based on interpretation i is a mapping from the ground sentences of the language into true false ti ground sentences true false The details of the definition are given on the following slides 10 9 07 27 Relational Sentences A ground truth assignment satisfies a ground relational sentence if and only if the tuple of objects denoted by the arguments is a member of the relation denoted by the relation constant ti 1 n true if i 1 i 1 i false otherwise 10 9 07 28 14 Example Interpretation i a i b i f i r Example tiv r a b true since i r tiv r b a false since i r 10 9 07 29 Logical Sentences ti true …


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Stanford CS 157 - Lecture 06 Computational Logic

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