Decidable Theories

In this lecture we work exclusively with first-order logic with equality. Fix a signature σ. A theory T is a set of sentences (closed formulas) that is closed under semantic entailment, i.e., if T |= F then F ∈ T . Given a σ-structure A it is clear that the set of sentences that hold in A is a theory. We denote this theory by Th(A) and call it the theory of A. We say that a theory is complete if for any sentence F , either F ∈ Th(A) or ¬F ∈ Th(A). Clearly the theory of any particular structure is complete. The set of valid σ-formulas is an example of a theory that is not complete. An example of a structure-based theory is Th(Q, 1, <,+, {c · }c∈Q), linear arithmetic over the rationals. Here, + is the binary addition function and c · denotes the unary function “multiply by c” for each c ∈ Q. The theory is defined over a signature σ that has symbols for each component of the structure (Q, 1, <,+, {c · }c∈Q). Specifically, σ has a constant symbol 1, binary function symbol +, binary relation symbol < , and an infinite family of unary function symbols c · , indexed by c ∈ Q. Note that having a family of unary multiplication functions {c · }c∈Q is completely different from having a single binary multiplication function. Under the above definition σ-terms are essentially linear combinations of the the first-order variables, e.g., 12x+ 1 3y+ z+ 5 9 is a σ-term. On the other hand, incorporating binary multiplication in σ would lead to polynomial terms, such as x2y + z4. Atomic formulas have the form t1 = t2 or t1 < t2 for σ-terms. Here are some assertions that can be formalized in linear arithmetic (where A denotes a matrix of rationals, x a vector of variables, and b a vector of rationals):