Applications of Metric Coinduction

Metric coinduction is a form of coinduction that can be used to establish properties of objects constructed as a limit of finite approximations. One can prove a coinduction step showing that some property is preserved by one step of the approximation process, then automatically infer by the coinduction principle that the property holds of the limit object. This can often be used to avoid complicated analytic arguments involving limits and convergence, replacing them with simpler algebraic arguments. This paper examines the application of this principle in a variety of areas, including infinite streams, Markov chains, Markov decision processes, and non-well-founded sets. These results point to the usefulness of coinduction as a general proof technique

Set Constraints and Logic Programming

AbstractSet constraints are inclusion relations between expressions denoting sets of ground terms over a ranked alphabet. They are the main ingredient in set-based program analysis. In this paper we describe a constraint logic programming languageclp(sc) over set constraints in the style of J. Jaffar and J.-L. Lassez (1987, “Proc. Symp. Principles of Programming Languages 1987,” pp. 111–119). The language subsumes ordinary logic programs over an Herbrand domain. We give an efficient unification algorithm and operational, declarative, and fixpoint semantics. We show how the language can be applied in set-based program analysis by deriving explicitly the monadic approximation of the collecting semantics of N. Heintze and J. Jaffar (1992, “Set Based Program Analysis”; 1990, “Proc. 17th Symp. Principles of Programming Languages,” pp. 197–209)

New

We propose a theoretical device for modeling the creation of new indiscernible semantic objects during program execution. The method fits well with the semantics of imperative, functional, and object-oriented languages and promotes equational reasoning about higher-order state

Partial Automata and Finitely Generated Congruences: An Extension of Nerode's Theorem

Let T_Sigma be the set of ground terms over a finite ranked alphabet Sigma. We define partial autornata on T_Sigma and prove that the finitely generated congruences on T_Sigma are in one-to one correspondence (up to isomorphism) with the finite partial automata on Sigma with no inaccessible and no inessential states. We give an application in term rewriting: every ground term rewrite system has a canonical equivalent system that can be constructed in polynomial time

Modularizing the Elimination of r=0 in Kleene Algebra

Given a universal Horn formula of Kleene algebra with hypotheses of the form r = 0, it is already known that we can efficiently construct an equation which is valid if and only if the Horn formula is valid. This is an example of elimination of hypotheses, which is useful because the equational theory of Kleene algebra is decidable while the universal Horn theory is not. We show that hypotheses of the form r = 0 can still be eliminated in the presence of other hypotheses. This lets us extend any technique for eliminating hypotheses to include hypotheses of the form r = 0

Certification of Compiler Optimizations using Kleene Algebra with Tests

We use Kleene algebra with tests to verify a wide assortment of common compiler optimizations, including dead code elimination, common subexpression elimination, copy propagation, loop hoisting, induction variable elimination, instruction scheduling, algebraic simplification, loop unrolling, elimination of redundant instructions, array bounds check elimination, and introduction of sentinels. In each of these cases, we give a formal equational proof of the correctness of the optimizing transformation

On Distance Coloring

Call a connected undirected graph (d,c)-colorable if there is a vertex coloring using at most c colors such that no two vertices of distance d or less have the same color. It is well known that (1,2)-colorability is decidable in linear time, but (1,c)-colorability for c greater than or equal to 3 is NP-complete. Sharp (2007) shows that for fixed d greater than or equal to 2, the (d,c)-colorability problem is solvable in linear time for c less than or equal to 3d/2 and NP-complete otherwise. In this note we give an alternative construction that improves the upper time bound as a function of d for the case c less than or equal to 3d/2. The construction entails a generalization of the notion of tree decomposition and bounded treewidth (Robertson and Seymour 1986) to arbitrary overlay graphs, not just trees, which may be of independent interest

Publication/Citation: A Proof-Theoretic Approach to Mathematical Knowledge Management

There are many real-life examples of formal systems that support constructions or proofs, but that do not provide direct support for remembering them so that they can be recalled and reused in the future. In this paper we examine the operations of publication (remembering a proof) and citation (recalling a proof for reuse), regarding them as forms of common subexpression elimination on proof terms. We then develop this idea from a proof theoretic perspective, describing a simple complete proof system for universal Horn equational logic using three new proof rules, publish, cite, and forget. These rules can provide a proof-theoretic infrastructure for proof reuse in any system

Some Notes on Rational Spaces

Set constraints are inclusions between expressions denoting set of ground terms over a finitely ranked alphabet $\Sigma$. Rational spaces are topological spaces obtained as spaces of runs of topological $\Sigma$-hypergraphs. They were introduced by Kozen in \cite{K95a}, where the topological structure of the spaces of solutions to systems of set constraints was given in terms of rational spaces. In this paper we continue the investigation of rational spaces. We give a Myhill-Nerode like characterization of rational points, which in turn is used to re-derive results about the rational points of finitary rational spaces. We define congruences on $\Sigma$-hypergraphs, investigate their interplay with the Myhill-Nerode characterization, and finally we determine the computational complexity of some decision problems related to rational spaces

On Moessner's Theorem

Moessner's theorem describes a procedure for generating a sequence of n integer sequences that lead unexpectedly to the sequence of nth powers 1^n, 2^n, 3^n, ... Paasche's theorem is a generalization of Moessner's; by varying the parameters of the procedure, one can obtain the sequence of factorials 1!, 2!, 3!, ... or the sequence of superfactorials 1!!, 2!!, 3!!, ... Long's theorem generalizes Moessner's in another direction, providing a procedure to generate the sequence a, (a+d)2^{n-1}, (a+2d)3^{n-1}, ... Proofs of these results in the literature are typically based on combinatorics of binomial coefficients or calculational scans. In this note we give a short and revealing algebraic proof of a general theorem that contains Moessner's, Paasche's, and Long's as special cases. We also prove a generalization that gives new Moessner-type theorems