Logical Reasoning Systems

Logical Reasoning Systems derive sound conclusions from formal declarative knowledge. Such systems are usually deened by abstract rules of inference. For example, the rule of modus ponens states that given P , and \P implies Q" (usually written P ! Q), we can infer Q. Logical systems have a rich history starting with Greek syllogisms and continuing through the work many prominent mathematicians such as D escartes, Leibniz, Boole, Frege, Hilbert, GG odel, and Cohen. A good discussion of the history of logical reasoning systems can be found in 3]. Logical reasoning provides a well understood general method of symbolic computation. Symbolic computation manipulates expressions involving variables. For example, a computer algebra system can simplify the expression x(x + 1) to x 2 + x. The equation x(x + 1) = x 2 + x is true under any interpretation of x as a real number. Unlike numerical computation, symbolic computation derives truths that hold under a wide variety of interpretations and can be used when only partial information is given, e.g., that x is some (unknown) real number. Logical inference systems can be used to perform symbolic computation with variables that range over conceptual domains such as sets, sequences, graphs, and computer data structures. Symbolic computation underlies esentially all modern eeorts to formally verify computer hardware and software. There are at least two ways in which symbolic inference is relevant to cog-nitive science. First, symbolic inference rules have traditionally been used as models of human mathematical reasoning. Second, symbolic computation also provides a model of certain common sense inferences. For example, suppose one is given a bag of marbles and continues removing marbles from the bag as long as it remains nonempty. People easily reach the conclusion that, barring unusual or magical circumstances, the bag will eventually become empty. They reach this conclusion without being told any particular number of marbles | they reach a conclusion about an arbitrary set s. When building computer systems for drawing such conclusions approaches based on \visualization" do not currently work as well as approaches based on symbolic computation 6]. Here I will divide symbolic computation research into ve general types. First, there are the so-called symbolic algebra systems such as Maple or Mathematica 7]. These are designed to manipulate expressions satisfying certain algebraic properties such as those satissed by the real numbers.