CDM FOL Theories Klaus Sutner Carnegie Mellon University 42-fol-theories 2017/12/15 23:21 1 Theories and Models Decidability and Completeness Derivations and Proofs Compactness Incompleteness and Undecidability Structures and Models 3 Recall from last time that we have a notion of a structure being a model of a sentence, or a whole set of sentences. A j= Γ To really make sense out of this, we need some background theory that allows us to talk about the corresponding first-order structures. Typically this is handled via Zermelo-Fraenkel set theory. Semantic Consequence 4 Definition Given a set Γ of sentences and a sentence ', we say that ' is valid in Γ or a semantic consequence of Γ if any structure A that satisfies all formulae in Γ also satisfies ': A j= Γ implies A j= '. Notation: Γ j= ' For example, for a ground term t '(t) j= 9 x '(x) '; ' ! j= Less Obvious 5 The last two examples are purely logical, but semantic entailment is not always so easy to see. For example, the standard group axioms x ∗ (y ∗ z) = (x ∗ y) ∗ z x ∗ e = e ∗ x = x x ∗ x−1 = x−1 ∗ x = e have the semantic consequence (x ∗ y)−1 = y−1 ∗ x−1 The insistence on sentences (formulae without free variables) should be taken with a grain of salt, one can deal with the general situation at some modest cost. Theories 6 Fix some first-order language L once and for all. As always, we are only interested in languages with a finite or at most countable supply of non-logical symbols, and decidable syntax. Definition A theory (over L) is an arbitrary collection of sentences (over L). A theory T is satisfiable if there is some structure A such that A j= T . Warning: This is the skinny definition of theory, some authors insist that a theory be closed under semantic consequence: Γ j= ' implies ' 2 Γ. Satisfiability 7 Obviously, theories that have no models at all are of little interest, they are self-contradictory. Alas, it is surprisingly difficult to avoid contradictions when one constructs some logical system (not necessarily first-order logic). For Hilbert, satisfiability was the key to existence: If the arbitrarily given axioms do not contradict each other through their consequences, then they are true, then the objects defined through the axioms exist. That, for me, is the criterion of truth and existence. Constructivists recoil at this idea, but is hard to see how one could convince mainstream mathematicians to let go of Hilbert's dream. Example: Monoid Axioms 8 Here is a closer look at a standard example from algebra: axioms for semigroups, monoids and groups. Initially, let's use the language L(∗) of type (2) So there is only one binary function symbol ∗, and no constants. We will write ∗ in infix notation to keep things readable and drop the universal quantifiers up front. Definition A semigroup is any structure satisfying the following associativity axiom: x ∗ (y ∗ z) ≈ (x ∗ y) ∗ z (1) If, in addition, there is a neutral element 9 y 8 x (x ∗ y ≈ x ^ y ∗ x ≈ x) (2) then we have a monoid. A Better Language 9 The quantifiers in the second monoid axiom are unnecessarily complicated, we can change our language to L(∗; e) of type (2; 0) and replace the second axiom by x ∗ e ≈ x ^ e ∗ x ≈ x (3) Much easier to read. Example Monoids 10 In order to specify a (small) monoid, we only need to write down a Cayley table (multiplication table) for the operation, and the neutral element. Here are the Cayley tables of two simple examples that we are already familiar with: ∗ 0 1 ∗ 0 1 0 0 1 0 0 0 1 1 1 1 0 1 This is just logical disjunction and conjunction and usually written as h B; _; 0 i and h B; ^; 1 i . Note that it is easy to determine the whole atomic diagram from the Cayley tables. E.g., (1 ∗ 1) ∗ (1 ∗ 1) = (1 ∗ 1) is in the diagram. Six Elements 11 A slightly larger Cayley table with elements f1; 2; 3; 4; 5; 6g. ∗ 1 2 3 4 5 6 1 1 2 3 4 5 6 2 2 4 6 1 3 5 3 3 6 2 5 1 4 4 4 1 5 2 6 3 5 5 3 1 6 4 2 6 6 5 4 3 2 1 Clearly we can test in time cubic in the size of the structure whether is satisfies the associativity axiom. To check for the neutral element only requires linear time. Alas, a Cayley table obfuscates the structure of the monoid, there are other descriptions that are much more useful. Some Infinite Monoids 12 For infinite carrier sets things become more interesting. Again, we have to specify the carrier set, the operation ∗, and the constant e. Here are some monoids: h N; +; 0 i + N ; ∗; 1 + R ; ∗; 1 h strings; concat; empty word i h lists; join; empty list i Also note that it is not at all clear how one would go about verifying that these are indeed models of the monoid axioms: a brute force test is not possible here. We have to argue within a more powerful system. Two Strange Monoids 13 Here is a less obvious example. Let F be the collection of propositional formulae modulo equivalence. Then the following two structures are monoids. h F; ^; > i h F; _; ? i Note that for this to work we must identify equivalent formulae: p ^ > 6= p but certainly p ^ > ≡ p. Exercise What is the size of these monoids if we consider only formulae with n propositional variables? Non-Models 14 Not just any old set with a binary operation is a monoid. + N ; + + R ; exp h full binary trees ; join i In the first example there is no neutral element. The second operation, exponentiation, is not associative. Likewise, joining together full binary trees is not associative (figure out what exactly this means). C 6= A A B B C Example: Groups 1 15 To hammer this home, here is another standard example from algebra: groups. First, we use the language L(∗) of signature (2) of one binary operation. Groups can then be specified as monoids that have inverse elements. x ∗ (y ∗ z) ≈ (x ∗ y) ∗ z 9 z 8 x (x ∗ z ≈ x ^ z ∗ x ≈ x) ^ 8 x 9 y (x ∗ y ≈ z ^ y ∗ x ≈ z) Note that our minimalist language makes the formula for the the existence of a neutral element and corresponding inverses a bit complicated. Example: Groups 2 16 It is better to have a language with a constant for the neutral element: L(∗; e) of signature (2; 0) The theory of groups then looks like x ∗ (y ∗ z) ≈ (x ∗ y) ∗ z x ∗ e ≈ x ^ e ∗ x ≈ x 9 y (x ∗ y ≈ e ^ y ∗ x ≈ e) We are down to one quantifier for the existence of an inverse element. Example: Groups 3 17 If we extend our language one more time by a unary function symbol for inverses we get an even clearer description over L(∗; −1; e) of signature (2; 1; 0) Group theory now is x ∗ (y ∗ z) ≈ (x ∗ y) ∗ z x ∗ e ≈ x ^ e ∗ x ≈ x x ∗ x−1 ≈ e ^ x−1 ∗ x ≈ e This is arguable a better representation: all the formulae are now universal; in fact, we have a purely equational characterization. Example: Groups 3.5 18 The last step is to give up on the pedantic ≈ and write = instead, and to allow for chains of identities. x ∗ (y ∗ z) = (x ∗ y) ∗ z x ∗ e = e ∗ x = x x ∗ x−1 = x−1 ∗ x = e This is the form that is familiar from any (modern) algebra textbook. But note that the models are essentially the same as for the first form. Equivalence 19 Definition Two theories T1 and T2 are equivalent if they have the same models. Note that T1 and T2 have to be over the same language (essentially, extensions are fine). All the examples for groups can be considered as theories over L(∗; −1; e). One can check that they are all equivalent. Structures and Theories 20 Definition Given a class C of structures we define the theory of C to be the collection of sentences valid in all the structures in C: Th(C) = f ' j for all A 2 C : A j= ' g If there is only one structure A in the class we write Th(A). For example, Th(N ) is the theory of arithmetic. Th(Grp) is the theory of groups, Th(BoolAlg) is the theory of Boolean algebras and so on. This is quite different from trying to understand the theory of a specific group, or a specific Boolean algebra. Theory Closure 21 Definition Let T be a theory. The semantic closure of T is defined by Th(T ) = f ' j T j= ' g In other words, if M is the class of all models of T , then Th(T ) = Th(M). It is entirely possible that Th(T ) is much more complicated than T itself: in general, we know little about the class of all possible models. Models 22 Definition For any theory T , the class of models of T is defined to be Mod(T ) = f A j A j= T g: where all the structures have the appropriate signature. It follows that Th(T ) = Th(Mod(T )). For example, the collection of all groups is described concisely by the theories from above. Similarly we can describe Abelian groups or infinite groups (warning: this does not work for the collection of all finite groups).