ABSTRACT. Interpolation is an important technique in computer aided verification and static analysis of programs. In particular, interpolants extracted from so-called local proofs are used in invariant generation and bounded model checking. An interpolant extracted from such a proof is a boolean combination of formulas occurring in the proof. In this talk we first describe a technique of generating and optimizing interpolants based on transformations of what we call the “grey area” of local proofs. Local changes in proofs can change the extracted interpolant. Our method can describe properties of extracted interpolants obtained by such proof changes as a pseudo-boolean constraint. By optimizing solutions of this constraint we also improve the extracted interpolants. Unlike many other interpolation techniques, our technique is very general and applies to arbitrary theories. Our approach is implemented in the theorem prover Vampire and evaluated on a large number of benchmarks coming from first-order theorem proving and bounded model checking using logic with equality, uninterpreted functions and linear integer arithmetic. Our experiments demonstrate the power of the new techniques: for example, it is not unusual that our proof transformation gives more than a tenfold reduction in the size of interpolants. While local proofs admit efficient interpolation algorithms, standard complete proof systems, such as superposition, for theories having the interpolation property are not necessarily complete for local proofs. In this talk we therefore also investigate interpolant extraction from non-local proofs in the superposition calculus and prove a number of general results about interpolant extraction and complexity of extracted interpolants. In particular, we prove that the number of quantifier alternations in first-order interpolants of formulas without quantifier alternations is unbounded. This result has far-reaching consequences for using local proofs as a foundation for interpolating proof systems - any such proof system should deal with formulas of arbitrary quantifier complexity.

This talk is based on  joint work with Andrei Voronkov (U. of Manchester, TU Vienna and EasyChair).

ABSTRACT. 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 expressive 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.

ABSTRACT. If we track atom occurrences in classical propositional proofs in deep inference, we see that they can form cyclic structures between cuts and identity steps. These cycles are an obstacle to a very natural form of normalisation, that simply unfolds all the contractions in a proof. This mechanism, which we call 'decomposition', has many points of contact with explicit substitutions in lambda calculi. In the presence of cycles, decomposition does not terminate, and this is an obvious drawback if we want to interpret proofs computationally. One way of eliminating cycles is eliminating cuts. However, we could ask ourselves whether it is possible to eliminate cycles independently of (general) cut elimination. This paper shows an efficient way to do so, therefore establishing the independence of decomposition from cut elimination. In other words, cut elimination in propositional logic can be separated into three separate procedures: 1) cycle elimination, 2) unfolding of contractions, 3) elimination of cuts in the linear fragment.

ABSTRACT. We show that the class of chordal claw-free graphs admits LREC=-definable canonization. LREC= is a logic that extends first-order logic with counting by an operator that allows it to formalize a limited form of recursion. This operator can be evaluated in logarithmic space. It follows that there exists a logarithmic-space canonization algorithm for the class of chordal claw-free graphs, and that LREC= captures logarithmic space on this graph class. Since LREC= is contained in fixed-point logic with counting, we also obtain that fixed-point logic with counting captures polynomial time on the class of chordal claw-free graphs.

ABSTRACT. Given a graph F, let I(F) be the class of graphs containing F as an induced subgraph. Let W[F] denote the minimum k such that I(F) is definable in k-variable first-order logic. The recognition problem of I(F), known as Induced Subgraph Isomorphism (for the pattern graph F), is solvable in time O(n^W[F]). Motivated by this fact, we are interested in determining or estimating the value of W[F]. Using Olariu's characterization of paw-free graphs, we show that I(K_3+e) is definable by a first-order sentence of quantifier depth 3, where K_3+e denotes the paw graph. This provides an example of a graph F with W[F] strictly less than the number of vertices in F. On the other hand, we prove that W[F]=4 for all F on 4 vertices except the paw graph and its complement. If F is a graph on t vertices, we prove a general lower bound W[F]>(1/2-o(1))t, where the function in the little-o notation approaches 0 as t increases. This bound holds true even for a related parameter W^*[F]\le W[F], which is defined as the minimum k such that I(F) is definable in the infinitary logic L^k_{\infty\omega}. We show that W^*[F] can be strictly less than W[F]. Specifically, W^*[P_4]=3 for P_4 being the path graph on 4 vertices.

ABSTRACT. We begin by revisiting partiality in univalent type theory via the notion of dominance. We then perform first steps in constructive computability theory, discussing the consequences of working with computability as property or structure, without assuming countable choice or Markov's principle. A guiding question is what, if any, notion of partial function allows the proposition ``all partial functions on natural numbers are Turing computable'' to be consistent.

ABSTRACT. In this paper, we analyze and compare three of the many algebraic structures that have been used for modeling dependent type theories: categories with families, split type-categories, and representable maps of presheaves. We study these in the setting of univalent foundations, where the relationships between them can be stated more transparently. Specifically, we construct maps between the different structures and show that these maps are equivalences under suitable assumptions.

We then analyze how these structures transfer along (weak and strong) equivalences of categories, and, in particular, show how they descend from a category (not assumed univalent/saturated) to its Rezk completion. To this end, we introduce relative universes, generalizing the preceding notions, and study the transfer of such relative universes along suitable structure.

We work throughout in (intensional) dependent type theory; some results, but not all, assume the univalence axiom. All the material of this paper has been formalized in Coq, over the UniMath library.

ABSTRACT. Differential categories are now an established abstract setting for differentiation. The paper presents the parallel development for integration by axiomatizing an integral transformation, $\mathsf{s}_A: \oc A \to \oc A \ox A$, in a symmetric monodical category with a coalgebra modality. When integration is combined with differentiation, the two fundamental theorems of calculus are expected to hold (in a suitable sense): a differential category with integration which satisfies these two theorem is called a {\em calculus category\/}.

Modifying an approach to antiderivatives by T. Ehrhard, it is shown how examples of calculus categories arise as differential categories with antiderivatives in this new sense. Having antiderivatives amounts to demanding that a certain natural transformation, $\mathsf{K}: \oc A \to \oc A$, is invertible. We observe that a differential category having antiderivatives, in this sense, is always a calculus category and we provide examples of such categories.

ABSTRACT. In this paper we relate two generalisations of the finite monoid recognisers of automata theory for the study of circuit complexity classes: Boolean spaces with internal monoids and typed monoids. Using the setting of stamps, this allows us to generalise a number of results from algebraic automata theory as it relates to Büchi's logic on words. We obtain an Eilenberg theorem, a substitution principle based on Stone duality, a block product principle for typed stamps and, as our main result, a topological semidirect product construction, which corresponds to the application of a general form of quantification. These results provide tools for the study of language classes given by logic fragments such as the Boolean circuit complexity classes.

ABSTRACT. Girard’s Geometry of Interaction (GoI), a semantics of linear logic proofs, has been applied to program semantics. One style of application produces abstract machines that pass a token on a fixed graph along a path indicated by GoI. These token-passing abstract machines are known to be space efficient, because they handle duplicated computation by repeating the same moves of a token on a fixed graph. They can be adapted to implement different evaluation strategies with linear logic as a back end. However time efficiency of such implementation has attracted little attention. We aim at a token-passing abstract machine that can implement evaluation strategies with certified time efficiency. Our abstract machine, called Dynamic GoI Machine (DGoIM), interleaves token passing with graph rewriting, using a history of token passing. Employing graph rewriting enables the DGoIM to avoid repeated moves of a token and to reduce time cost, whereas it can expand a graph and increase space cost. The flexibility of interleaving lets the DGoIM balance this tradeoff of space and time cost. This work presents a case study of the flexibility; namely, we prove that the DGoIM can implement the call-by- need evaluation by interleaving token passing with as much graph rewriting as possible. The quantitative analysis confirms that DGoIM with this way of interleaving can be classified as “efficient” following Accattoli’s taxonomy of abstract machines.

ABSTRACT. We study the complexity of the valued CSP (VCSP, for short) over arbitrary templates, taking the general framework of integral bounded linearly order monoids as valuation structures. The class of problems considered here subsumes and generalizes the most common one in VCSP literature, since both monoidal and lattice conjunction operations are allowed in the formulation of constraints. Restricting to locally finite monoids, we introduce a notion of polymorphism that captures the pp-definability in the style of Geiger's result. As a consequence, sufficient conditions for tractability of the classical CSP, related to the existence of certain polymorphisms, are shown to serve also for the valued case. Finally, we establish the dichotomy conjecture for the VCSP, modulo the dichotomy for classical CSP.