Is there a logic without induction that captures much of P?

Question

The Immerman-Vardi theorem states that PTIME (or P) is precisely the class of languages that can be described by a sentence of First-Order Logic together with a fixed-point operator, over the class of ordered structures. The fixed-point operator can be either least fixed-point (as considered by Immerman and by Vardi), or inflationary fixed-point. (Stephan Kreutzer, Expressive equivalence of least and inflationary fixed-point logic, Annals of Pure and Applied Logic 130 61–78, 2004).

Yuri Gurevich conjectured that there is no logic capturing PTIME (Logic and the Challenge of Computer Science, in Current Trends in Theoretical Computer Science, ed. Egon Boerger, 1–57, Computer Science Press, 1988), while Martin Grohe has stated he is less sure (The Quest for a Logic Capturing PTIME, FOCS 2008).

The fixed-point operator is meant to capture the power of recursion. Fixed-points are powerful, but it isn't obvious to me that they are necessary.

Is there an operator X that is not based on fixed-points, such that FOL+X captures a (large) fragment of PTIME?

Edit: As far as I understand, linear logic can only express statements about structures that have quite restrictive form. I would ideally like to see a reference to, or a sketch of, a logic that can express properties of arbitrary sets of relational structures, while still avoiding fixed points. If I am wrong about the expressive power of linear logic then a pointer or hint would be welcome.

Answer 1

You want to have a look at what some people call Grädel's Theorem. You can find it in Papadimitriou's book "Computational Complexity" (it's Theorem 8.4 in page 176) or in Grädel's original paper.

In a nutshell, Grädel's Theorem is to P what Fagin's Theorem is to NP. It states that on the class of finite structures with a successor relation, the collection of polynomial-time decidable properties coincides with those expressible in the Horn-fragment of existential second-order logic. These are the sentences of second-order logic of the form $$ (\exists R)(\forall x)(\phi) $$ where $R$ is a sequence of second-order relation variables, $x$ is a sequence of first-order variables, and $\phi$ is a quantifier-free formula that, when written in CNF form, is a conjunction of $R$-Horn clauses (i.e. clauses that have at most one non-negated atom involving the variables in $R$).

Answer 2

There are recent exciting results concerning the search for a logic capturing PTIME. The famous example by Cai, Fürer and Immerman showing that LFP+C does not capture PTIME was based on a seemingly artificial class of graphs, though. Of course, it was constructed for the particular task of demonstrating the restrictions of LFP+C. Only recently it was shown by Dawar that the class is not artificial at all. It can rather be seen as an example for the fact that LFP+C cannot solve linear equation systems!

Hence Dawar, Grohe, Holm and Laubner extended logics by operators from linear algebra, for example by an operator to define the rank of a definable matrix. The resulting logic LFP+rank can express strictly more than LFP+C, in fact, there is no known PTIME property that LFP+rank cannot express.

Even FO+rk is surprisingly powerful, it can express deterministic and symmetric transitive closure. It is still open whether it can express the general transitive closure of a graph.

Answer 3

Depending on what you mean by "capture," Soft Linear Logic and Polynomial Time by Yves Lafont may be of interest. There is a 1-1 correspondence to proofs in this logic and PTIME algorithms that take a string as input and output 0 or 1.

The Wikipedia article on Linear Logic is here. It's not a fixpoint logic. The intuition of "classical logic over $C^*$ algebras instead of boolean algebras" is the easiest for me to grasp.

Answer 4

As far as I understand, linear logic can only express statements about structures that have quite restrictive form. I would ideally like to see a reference to, or a sketch of, a logic that can express properties of arbitrary sets of relational structures, while still avoiding fixed points. If I am wrong about the expressive power of linear logic then a pointer or hint would be welcome.

This isn't right: all residuated commutative monoidal lattices are models of linear logic. Here's an easy way to create such a lattice from finite graphs. Start with the set

$$M = \{(g, n) \;|\; g\mbox{ is a finite graph and }n \subseteq \mathrm{nodes}(g)\}$$

So our forcing relation will be $(g,n) \models \phi$, and the intuition is that $n$ is the set of nodes "owned" by the formula $\phi$. There is a partial operation $(\cdot) : M \times M \to M$, defined as: $$ (g,n) \cdot (g',n') = \left\{\begin{array}{ll} (g, n \uplus n') & \mbox{when } g = g' \land n \cap n' = \emptyset \\ \mathrm{undefined} & \mbox{otherwise} \end{array}\right. $$

This combines two elements by mergining their owned sets, if the graphs are equal and the owned sets are disjoint.

Now, we can give a model of linear logic as follows:

$$ \begin{array}{lcl} (g,n) \models I & \iff & n = \emptyset \\ (g,n) \models \phi \otimes \psi & \iff & \exists n_1,n_2.\; n = n_1 \uplus n_2 \mbox{ and } (g,n_1) \models \phi \mbox{ and } (g,n_2) \models \psi \\ (g,n) \models \phi \multimap \psi & \iff & \forall n'.\; \mbox{if } n \cap n' = \emptyset \mbox{ and } (g,n') \models \phi \mbox{ then } (g,n \uplus n') \models \psi \\ (g, n) \models \top & \iff & \mbox{always} \\ (g,n) \models \phi \land \psi & \iff & (g,n) \models \phi \mbox{ and } (g,n) \models \psi \end{array} $$

This model is actually a variant of the ones used in separation logic, which is widely used in the verification of heap-manipulating programs. (If you like, think of the graph as the pointer structure of the heap, and the analogy is exact!)

This isn't really the right way to think about linear logic, though: its real intuitions are proof-theoretic, and the connection to complexity comes via the computational complexity of the cut-elimination theorem. The model theory of linear logic is the shadow cast by its proof theory.

Answer 5

Some older work on this problem, again in the Linear Logic vein is, Jean-Yves Girard, Andre Scedrov, and Philip Scott. Bounded linear logic: A modular approach to polynomial-time computability. Theoretical Computer Science, 97(1):1–66, 1992.

More recent work includes Bounded Linear Logic, Revisited by Ugo Dal Lago and Martin Hofmann.

Answer 6

Let $L \subset K$ be two signatures and $\mathcal{L,K}$ be a nonempty classes of $L$-structures and $K$-structures respectively such that

• every structure in $\mathcal{L}$ has an expansion in $\mathcal{K}$, and
• the $L$-reduct of every structure in $\mathcal{K}$ is contained in $\mathcal{L}$.

Let $\varphi$ be a $K$-sentence. (Think first-order, but really it could be in any language.) We say that $\varphi$ is $\mathcal{(L,K)}$-invariant in case any two structures $A,B \in \mathcal{K}$ agree on $\varphi$ if their $L$-reducts are isomorphic. This means that, morally speaking, $\varphi$ is meaningful over structures in $\mathcal{L}$. We might define a quantifier $\mathfrak{Q}$ by $$ M \models \mathfrak{Q}.\varphi \iff \text{for some/for any expansion $N$ of $M$ in $\mathcal{L}$, } N \models \varphi, $$ where $M \in \mathcal{L}$. So $\mathfrak{Q}$ is a sort of weak second-order quantifier that bounds the symbols in $K \setminus L$. Now we can study $(\mathcal{L,K})$-invariant FO definability, i.e., subsets of $\mathcal{L}$ carved out by sentences of the form $\mathfrak{Q}.\varphi$.

For applications to descriptive complexity theory, we want $\mathcal{L}$ to be the set of all finite $L$-structures. (This means, note, that $\mathcal{K}$ is a set of arbitrarily large finite $K$-structures.) For example, if $\mathcal{K}$ is the set of structures in $\mathcal{L}$ expanded by any linear order, we get order-invariant first-order logic. There is some hope of capturing (large) fragments of PTIME in this way. For example, traversal-invariant and breadth-first traversal-invariant FO logic capture L and NL respectively. (It's not perfect: we have to close under some very simple reductions.) But generally, the smaller $\mathcal{K}$ is, the more powerful $(\mathcal{L,K})$-invariant definability is.

It's not a logic, properly speaking: we can't decide in general whether a formula is $(\mathcal{L,K})$-invariant. (So I'm not sure whether this counts as a "generalized quantifier.") But at least it's a candidate for capturing PTIME over arbitrary signatures, using something of the form FOL + X, where X is "not based on fixed points."