### The Model-Theoretic Expressiveness of Propositional Proof Systems

Erich Grädel, Benedikt Pago, Wied Pakusa
2017
We establish new, and surprisingly tight, connections between propositional proof complexity and finite model theory. Specifically, we show that the power of several propositional proof systems, such as Horn resolution, bounded width resolution, and the polynomial calculus of bounded degree, can be characterised in a precise sense by variants of fixed-point logics that are of fundamental importance in descriptive complexity theory. Our main results are that Horn resolution has the same
more » ... e power as least fixed-point logic, that bounded width resolution captures existential least fixed-point logic, and that the (monomial restriction of the) polynomial calculus of bounded degree solves precisely the problems definable in fixed-point logic with counting. The Model-Theoretic Expressiveness of Propositional Proof Systems in such systems, and usually have completeness for relevant fragments of propositional logic, such as Horn-logic or 2-CNF. We can now try to solve algorithmic problems by reducing them to provability (or refutability) in some specific polynomial-time proof system, which, if it works successfully for all inputs, would give us a polynomial-time algorithm for the problem. Our goal is to understand how powerful this approach can be, depending on the specific proof system that we use. Let us illustrate this by two concrete problems. First we consider graph isomorphism, a problem which is not known to be solvable in polynomial time although there is strong evidence that it is not NP-complete. Given two graphs G = (V, E) and H = (W, F ) we ask whether there is a bijection π : V → W such that π(E) = F . Of course, this can easily be encoded as the satisfiability problem of a propositional CNF-formula. First, for each pair of vertices v ∈ V and w ∈ W we introduce a variable X vw with the intended meaning that X vw = 1 if π(v) = w. We add clauses w∈W X vw for every v ∈ V and v∈V X vw for every w ∈ W to ensure that every v ∈ V has an image and every w ∈ W has a preimage. is not a partial isomorphism. The resulting CNF-formula, denoted by Iso (G, H) , is satisfiable if, and only if, the two graphs G and H are isomorphic. Following our reasoning from above, we can now use an efficient variant of resolution, or of a stronger proof system, and try to refute the satisfiability of the formula Iso(G, H). If this is possible, then G are H are not isomorphic. Unfortunately, if we do not find a proof, then we are stuck, because it might still be the case that G and H are not isomorphic, but our proof system is not strong enough to show this. Hence, we get an efficient, sound, but not necessarily complete graph isomorphism test. The question how successful this approach is when based on resolution was studied by Toran in [26] . Unfortunately, he proved that shortest resolution proofs for graph non-isomorphism can be of exponential size (even for graphs with colour class size four). More recently, Grohe and Berkholz showed that also in the stronger system polynomial calculus (PC) one cannot obtain small proofs for graph non-isomorphism [9, 10]. Our second example is directed graph reachability: Given a directed graph G = (V, E) with two distinguished vertices s, t ∈ V , we want to know whether there is a path from s to t in G. Again, it is easy to encode this as a satisfiability problem in propositional logic, by taking the conjunction of all implication clauses X v → X w , for all edges (v, w) ∈ E, together with the two clauses 1 → X s and X t → 0. Clearly the resulting formula NonReach(G, s, t) is unsatisfiable if, and only if, t is reachable from s in G. However, in clear contrast to the formulas Iso(G, H) from above, we can easily prove unsatisfiability for the formulas NonReach (G, s, t) in efficient variants of resolution such as Horn-Res and k-Res for k ≥ 2. Our two examples demonstrate the following: while certain problems, such as directed graph reachability, allow for small and efficient resolution proofs, other problems, such as the graph isomorphism problem, provably require proofs of super-polynomial size even in quite strong proof systems. This leads to the main question that we want to address in this paper: is there a classification for those problems which can be solved in fundamental polynomialtime propositional proof systems such as Horn-Res, k-Res and degree-k (monomial)-PC, denoted by mon-PC k . It came as a surprise to us that there is, indeed, a very clear and tight classification of the power of all of these proof systems in terms of definability in important fixed-point logics which are well-studied in the area of descriptive complexity theory. Before we can state our results in detail, we have to explain what we mean by saying that a problem, such as directed graph reachability, can be solved by a propositional proof system Prop. As usual, each decision problem can be identified with a membership problem "A ∈ K?" for some class of structures K. For instance, the graph reachability problem from above is