The Logic in Computer Science Collumn

ion can overcome the deficit of the power of modern computers and solve the metaprogram problem: let’s consider amounts of coins instead of coin numbers. This idea is natural: when somebody is solving puzzles he/she operates in terms of amounts of coins of different kinds not in terms of their numbers! Let us present this hint in formal terms as an abstract model game(N,M) (N ≥ 1, M ≥ 0). Positions in this parameterized game are tuples (u, l, h, v, q) where • u is the amount of coins in [1..N ] which were not tested against other coins; • l is the amount of coins in [1..N ] which were tested against other coins and turned out to be lighter than other coins; • h is the amount of coins in [1..N ] which were tested against other coins and turned out to be heavier than other coins; • v is an amount of coins in [1..(N + M)] which are currently known to be valid; • q is a balancing query, i.e. a pair of quadruples ((u1, l1, h1, v1) , (u2, l2, h2, v2)) of numbers of [1..(N +M)]. Five constraints are absolutely natural and are closely related to constraints 1–5 for the concrete model: (1)u+l+h ≤ N , (2)u+l+h+v = N+M , (3)u+l+h ≥ 1, (4)u 6= 0 iff l + h = 0, and (5)v1 = 0 or v2 = 0. Additional constraints should be imposed for queries (since we can borrow coins for weighing from available untested, lighter, heavier and valid ones): (6)u1 + u2 ≤ u, (7)l1 + l2 ≤ l, (8)h1 + h2 ≤ h, (9)v1 + v2 ≤ v, (10)u1 + l1 +h1 + v1 = u2 + l2 +h2 + v2. A possible move of a player prog is a query for balancing two sets of coins, i.e. pair of positions (u, l, h, v, ((0, 0, 0, 0), (0, 0, 0, 0))) prog −→ (u, l, h, v, ((u1, l1, h1, v1), (u2, l2, h2, v2))). A possible move of a player user is a reply <, = or > for a query which causes position change (u, l, h, v, ((u1, l1, h1, v1), (u2, l2, h2, v2))) user −→ (u, l, h, v, ((0, 0, 0, 0), (0, 0, 0, 0))) in accordance with the query and a reply: u =    0 if the reply is < , (u− (u1 + u2)) if the reply is = , 0 if the reply is > ,