Limited Reasoner Demos

Available demos:

About the Reasoner

Representation

The reasoner uses a first-order modal logic as representation language. In their most basic form, formulas are about the equality of terms—functions, variables, or standard names. For instance, we could say fatherOf(Sally) = x, where fatherOf is a unary function, x is a variable, and Sally is a standard name. A standard name is a special constant symbols that denote one specific individual; that is, two distinct standard names are guaranteed to refer to two distinct individuals. Predicates are not an explicit part of the language, but can be easily simulated with functions and a special standard name T that indicates truth: rich(x) = T represents that x is in the relation rich. Such equality or inequality expressions are called literals. From these literals, we can form more complex formulas using logical connectives such as logical negation, disjunction and conjunction, existential and universal quantification, implication, or equivalence. For example, x (fatherOf(Sally) = x ∧ rich(x) = T) means Sally's father is rich. Finally, modal operators allow to expressly refer to what is known or considered possible. These modal operators are indexed with a natural number that indicates how much reasoning effort is put into proving the formula in its scope. A statement such as "I know Sally's father is rich, but I don't know him personally" could be formalised as follows:

K1x (fatherOf(Sally) = x ∧ rich(x) = T ∧ M1 fatherOf(Sally) ≠ x)

The above example is a typical query. Usually we are interested in whether or not such a formula is logically entailed by a knowledge base. For knowledge bases we make a syntactic restriction to so-called proper+ formulas: they need to be conjunctions of clauses without existential quantifiers. Note that every knowledge base can be brought into proper+ form by converting it into prenex-CNF and Skolemizing the existentialy quantified variables. Furthermore, knowledge bases are assumed have no nested modal operators.

To help the reasoner optimise query evaluation, the user can provide a guarantee that the knowledge base is consistent with a special modal operator: G α indicates that when evaluating α, the knowledge base can be assumed to be consistent.

The language furthermore comprises conditional belief to refer to differently plausible beliefs. For instance, we could write Bkl fatherOf(Sally) = Frank ⟹ rich(Frank) = T to say that we believe that if Frank is Sally's father, he presumably is rich.

Reasoning

Reasoning in this language is fundamentally based on case splitting—that is, by testing all possible denotations of a term like fatherOf(Sally)—and on unit propagation and subsumption. The reasoning procedures are sound with respect to classical first-order logic, but sacrifice completeness for the sake of decidability. This means that whenever Kk α can be inferred from a knowledge base, then K α holds in the classical semantics, that is, α follows from the knowledge base. Analogously, whenever Mk α can be inferred from a knowledge base, then M α holds in the classical semantics, that is, α is consistent with the knowledge base.

The reasoner is implemented in C++ and will be available as open source. The demos on this website are compiled to Javascript by emscripten. Note that this does impair performance compared to running the same code natively on your computer. How bad the penalty is depends a lot on the specific example, the browser or its Javascripts engine, and possibly your computer, but as a very rough rule of thumb, the Javascript version appears to be about 2× to 4× slower.

For future work, I plan to integrate sitation calculus-style actions, belief revision for sensing, have a look at some other KR concepts, and improve performance.