Dictator Dim wants to replace counting, in particular, counting the population of his land. To
keep an accurate population count, Dictator Dim has instructed his secretary to add one + symbol
to a record each time a citizen is born, and one - symbol to the record whenever a citizen dies.
This has been going on ever since his rule began.
Dictator Dim is not concerned so much with the actual total population of his lands, just with the
net change in population since his rule began. He would be forever disgraced if the net
population change were ever negative. Refusing to solve this problem by counting, Dictator Dim
instructs his secretary to find a sound and complete logic for the subset of all records that would
not disgrace him.
More formally, define the following truth concept
WFF\'s = {+; -}, i.e., all strings composed of only plus and minus symbols.
A WFF x is true if every prefix of x contains at least as many + symbols as - symbols.
Find a logic for this truth concept by proving that it is sound and complete. Hint: it may help to
define additional judgement form(s)
Solution
For each + in x, push a symbol onto a stack. For each - pop a symbol. If there is
ever and attempt to pop an empty stack x is rejected. If not, then x is accepted. This can be
implemented in a push-down automaton..
Dictator Dim wants to replace counting, in particular, counting the .pdf
1. Dictator Dim wants to replace counting, in particular, counting the population of his land. To
keep an accurate population count, Dictator Dim has instructed his secretary to add one + symbol
to a record each time a citizen is born, and one - symbol to the record whenever a citizen dies.
This has been going on ever since his rule began.
Dictator Dim is not concerned so much with the actual total population of his lands, just with the
net change in population since his rule began. He would be forever disgraced if the net
population change were ever negative. Refusing to solve this problem by counting, Dictator Dim
instructs his secretary to find a sound and complete logic for the subset of all records that would
not disgrace him.
More formally, define the following truth concept
WFF's = {+; -}, i.e., all strings composed of only plus and minus symbols.
A WFF x is true if every prefix of x contains at least as many + symbols as - symbols.
Find a logic for this truth concept by proving that it is sound and complete. Hint: it may help to
define additional judgement form(s)
Solution
For each + in x, push a symbol onto a stack. For each - pop a symbol. If there is
ever and attempt to pop an empty stack x is rejected. If not, then x is accepted. This can be
implemented in a push-down automaton.
