C LAUS B RABRAND & J ACOB A NDERSEN © S EMANTICS (Q1,’07) Aug 30, 2007 Adapted version for dSem 2007 Original story by: C LAUS B RABRAND © 2005–2006, University of Aarhus [ ] [ ] S EMANTICS (Q1,’07) Hand-in Exercise 1

C LAUS B RABRAND & J ACOB A NDERSEN S EMANTICS (Q1,’07) [ 2 ] Aug 30, 2007 Purpose Purpose: to describe structure inductively (and recursively) via inference systems to describe relations formally via inference systems and transition systems On: relations inference systems transition systems

C LAUS B RABRAND & J ACOB A NDERSEN S EMANTICS (Q1,’07) [ 3 ] Aug 30, BC It is the year 11,000 BC. The ice age is over and the homo sapiens are rapidly multiplying across the plains of Europe.

C LAUS B RABRAND & J ACOB A NDERSEN S EMANTICS (Q1,’07) [ 4 ] Aug 30, 2007 The Situation However, over-hunting and intense inter-tribal resource competition has left your tribe starving. You have to find new hunting grounds... The island in the horizon off the main land presents itself with a possibility: Scouts from your tribe went there a few seasons ago, but deemed it unfit to sustain a human population. The scouts did however release two rabbits; one old, one young (one male, one female).

C LAUS B RABRAND & J ACOB A NDERSEN S EMANTICS (Q1,’07) [ 5 ] Aug 30, 2007 The Problem Your problem is to figure out whether by now there are enough rabbits on the island to sustain your tribe(?) Being dependent on local wildlife, you have the following knowledge on the reproductive cycle of rabbits: A) a population of rabbits a given season may be divided into two categories: the old and the young B) each passing of a season: kills off the old; ages the young to old; and makes a number of new young ones equal to that of the total size of the population (i.e. the sum of the old and young).

C LAUS B RABRAND & J ACOB A NDERSEN S EMANTICS (Q1,’07) [ 6 ] Aug 30, 2007 The Problem (cont'd) Hence, a population of rabbits may be described by a pair of "numbers":...and the evolution of a rabbit population over time may be described by a sequence of pairs of "numbers": ,   r ,   r ,   r... ,  number of old rabbits number of young rabbits

C LAUS B RABRAND & J ACOB A NDERSEN S EMANTICS (Q1,’07) [ 7 ] Aug 30, 2007 Restrictions However, being the year 11,000 BC, you do not yet have any mathematics at your disposal beyond the capacity to follow simple procedures. i.e. those that can be written down by an inference system or a transition system Also, the tribe will not understand anything beyond stone-age numbers (a structural unary representation of numbers) in which: '0' (zero) is represented as " zero " '1' (one) is represented as " (succ zero) " "...the successor of zero". '2' (two) is repr. as " (succ (succ zero)) " "...the successor of the successor of zero".... () ( ()) ( ( ())) +–  /

C LAUS B RABRAND & J ACOB A NDERSEN S EMANTICS (Q1,’07) [ 8 ] Aug 30, 2007 Your Tasks Thus, you have three tasks: define... "|- san "...stone-age numbers define... "|- add "...addition on stone-age numbers define... "  r "...evolution of rabbits in a season You may do this either: INDUCTIVELY: first, "|- san "; then "|- add "; and finally "  r "...or...: DEDUCTIVELY: first "  r "; then "|- add "; and finally "|- san " (your choice)

C LAUS B RABRAND & J ACOB A NDERSEN S EMANTICS (Q1,’07) [ 9 ] Aug 30, 2007 Task "|- san " Represent stone-age numbers: i.e., define an inference system "|- san " that describes all legal stone-age numbers, such that e.g.: Let the relation "|- san " induce the set of all stone-age numbes, S: S := { n | | _ san n } | _ san  { " ( ", " ) ", " succ ", " zero " }* | _ san zero | _ san (succ zero) | _ san succ zero | _ san wgjkewkl | _ san (succ (succ zero))...

C LAUS B RABRAND & J ACOB A NDERSEN S EMANTICS (Q1,’07) [ 10 ] Aug 30, 2007 Task "|- add " Define stone-age number addition: i.e., define a relation "|- add " adding stone-age numbers: that describes stone-age number addition, such that e.g.: | _ add  S  S  S | _ add  zero, zero, zero  | _ add  (succ zero), (succ zero), (succ (succ zero))  | _ add  zero, (succ zero), (succ (succ zero)) 

C LAUS B RABRAND & J ACOB A NDERSEN S EMANTICS (Q1,’07) [ 11 ] Aug 30, 2007 Task "  r " Define the evolution of the rabbit population: i.e., a transition system  r,  r  describing the evolution of rabbits: such that, e.g.: Using only your newly invented “stone-age arithmetic” calculate how many rabbits are on the island after two seasons. The answer should be a single stone-age number providing the total of young and old rabbits.  (succ zero), (succ zero)   (succ zero), (succ (succ zero))  rr   r,  r 

C LAUS B RABRAND & J ACOB A NDERSEN S EMANTICS (Q1,’07) [ 12 ] Aug 30, 2007 Final Rules You have to define and prove everything formally, including: all arities, signatures, configurations,... without using any abbreviations (cf. [slide #14])...and include a reference to either a presentation slide number or a page in the [SOS] note every time you use something introduced in the course. Of course, these strict rules are only for the first-week... Finally, be sure to explain everything you do carefully. Note: This is an “exam training exercise” – use this chance to train (and demonstrate) your skills according to the Aim & Goal section on the webpage !