Bounded Arithmetic in Free Logic

One of the central open questions in bounded arithmetic is whether Buss' hierarchy of theories of bounded arithmetic collapses or not. In this paper, we reformulate Buss' theories using free logic and conjecture that such theories are easier to handle. To show this, we first prove that Buss' theories prove consistencies of induction-free fragments of our theories whose formulae have bounded complexity. Next, we prove that although our theories are based on an apparently weaker logic, we can interpret theories in Buss' hierarchy by our theories using a simple translation. Finally, we investigate finitistic G\"odel sentences in our systems in the hope of proving that a theory in a lower level of Buss' hierarchy cannot prove consistency of induction-free fragments of our theories whose formulae have higher complexity

Interpolation Properties and SAT-based Model Checking

Craig interpolation is a widespread method in verification, with important applications such as Predicate Abstraction, CounterExample Guided Abstraction Refinement and Lazy Abstraction With Interpolants. Most state-of-the-art model checking techniques based on interpolation require collections of interpolants to satisfy particular properties, to which we refer as "collectives"; they do not hold in general for all interpolation systems and have to be established for each particular system and verification environment. Nevertheless, no systematic approach exists that correlates the individual interpolation systems and compares the necessary collectives. This paper proposes a uniform framework, which encompasses (and generalizes) the most common collectives exploited in verification. We use it for a systematic study of the collectives and of the constraints they pose on propositional interpolation systems used in SAT-based model checking

Quantifier-Free Interpolation of a Theory of Arrays

The use of interpolants in model checking is becoming an enabling technology to allow fast and robust verification of hardware and software. The application of encodings based on the theory of arrays, however, is limited by the impossibility of deriving quantifier- free interpolants in general. In this paper, we show that it is possible to obtain quantifier-free interpolants for a Skolemized version of the extensional theory of arrays. We prove this in two ways: (1) non-constructively, by using the model theoretic notion of amalgamation, which is known to be equivalent to admit quantifier-free interpolation for universal theories; and (2) constructively, by designing an interpolating procedure, based on solving equations between array updates. (Interestingly, rewriting techniques are used in the key steps of the solver and its proof of correctness.) To the best of our knowledge, this is the first successful attempt of computing quantifier- free interpolants for a variant of the theory of arrays with extensionality

Efficient Interpolation for the Theory of Arrays

Existing techniques for Craig interpolation for the quantifier-free fragment of the theory of arrays are inefficient for computing sequence and tree interpolants: the solver needs to run for every partitioning $(A, B)$ of the interpolation problem to avoid creating $AB$-mixed terms. We present a new approach using Proof Tree Preserving Interpolation and an array solver based on Weak Equivalence on Arrays. We give an interpolation algorithm for the lemmas produced by the array solver. The computed interpolants have worst-case exponential size for extensionality lemmas and worst-case quadratic size otherwise. We show that these bounds are strict in the sense that there are lemmas with no smaller interpolants. We implemented the algorithm and show that the produced interpolants are useful to prove memory safety for C programs.Comment: long version of the paper at IJCAR 201

Proof Complexity In Algebraic Systems And Bounded Depth Frege Systems With Modular Counting

We prove a lower bound of the form N on the degree of polynomials in a Nullstellensatz refutation of the Count q polynomials over Zm , where q is a prime not dividing m. In addition, we give an explicit construction of a degree N design for the Count q principle over Zm . As a corollary, using Beame et al. (1994) we obtain a lower bound of the form 2 for the number of formulas in a constant-depth Frege proof of the modular counting principle Count q from instances of the counting principle Count m . We discus

Improved Bounds on the Weak Pigeonhole Principle and Infinitely Many Primes from Weaker Axioms

We show that the known bounded-depth proofs of the Weak Pigeonhole Principle PHP 2n n in size n O(log(n)) are not optimal in terms of size. More precisely, we give a size-depth trade-off upper bound: there are proofs of size n O(d(log(n)) 2=d ) and depth O(d). This solves an open problem of Maciel, Pitassi and Woods (2000). Our technique requires formalizing the ideas underlying Nepomnjascij's Theorem which might be of independent interest. Moreover, our result implies a proof of the unboundedness of primes in I \Delta 0 with a provably weaker `large number assumption' than previously needed

A Parametric Interpolation Framework for First-Order Theories

Craig interpolation is successfully used in both hardware and software model checking. Generating good interpolants, and hence automatic understanding of the quality of interpolants is however a very hard problem, requiring non-trivial reasoning in first-order theories. An important class of state-of-the-art interpolation algorithms is based on recursive procedures that generate interpolants from refutations of unsatisfiable conjunctions of formulas. We analyze this type of algorithms and develop a theoretical framework, called a parametric interpolation framework, for arbitrary first-order theories and inference systems. As interpolation-based verification approaches depend on the quality of interpolants, our method can be used to derive interpolants of different structure and strength, with or without quantifiers, from the same proof. We show that some well-known interpolation algorithms are instantiations of our framework

The Grothendieck ring of varieties and piecewise isomorphisms

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