### Quantified Propositional Logspace Reasoning

, 2008

"... In this paper, we develop a quantified propositional proof systems that corresponds to logarithmic-space reasoning. We begin by defining a class ΣCNF(2) of quantified formulas that can be evaluated in log space. Then our new proof system GL ∗ is defined as G ∗ 1 with cuts restricted to ΣCNF(2) formu ..."

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In this paper, we develop a quantified propositional proof systems that corresponds to logarithmic-space reasoning. We begin by defining a class ΣCNF(2) of quantified formulas that can be evaluated in log space. Then our new proof system GL ∗ is defined as G ∗ 1 with cuts restricted to ΣCNF(2) formulas and no cut formula that is not quantifier free contains a free variable that does not appear in the final formula. To show that GL ∗ is strong enough to capture log space reasoning, we translate theorems of V L into a family of tautologies that have polynomial-size GL ∗ proofs. V L is a theory of bounded arithmetic that is known to correspond to logarithmic-space reasoning. To do the translation, we find an appropriate axiomatization of V L, and put V L proofs into a new normal form. To show that GL ∗ is not too strong, we prove the soundness of GL ∗ in such a way that it can be formalized in V L. This is done by giving a logarithmic-space algorithm that witnesses GL ∗ proofs. 1

### Isomorphic Data Encodings and their Generalization to Hylomorphisms on Hereditarily Finite Data Types

"... Abstract. This paper is an exploration in a functional programming framework of isomorphisms between elementary data types (natural numbers, sets, multisets, finite functions, permutations binary decision diagrams, graphs, hypergraphs, parenthesis languages, dyadic rationals, primes, DNA sequences e ..."

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Abstract. This paper is an exploration in a functional programming framework of isomorphisms between elementary data types (natural numbers, sets, multisets, finite functions, permutations binary decision diagrams, graphs, hypergraphs, parenthesis languages, dyadic rationals, primes, DNA sequences etc.) and their extension to hereditarily finite universes through hylomorphisms derived from ranking/unranking and pairing/unpairing operations. An embedded higher order combinator language provides any-to-any encodings automatically. Besides applications to experimental mathematics, a few examples of “free algorithms ” obtained by transferring operations between data types are shown. Other applications range from stream iterators on combinatorial objects to self-delimiting codes, succinct data representations and generation of random instances. The paper covers 60 data types and, through the use of the embedded combinator language, provides 3660 distinct bijective transformations between them. The self-contained source code of the paper, as generated from a literate Haskell program, is available at

### unknown title

"... “Everything is everything ” revisited: shapeshifting data types with isomorphisms and hylomorphisms ..."

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“Everything is everything ” revisited: shapeshifting data types with isomorphisms and hylomorphisms

### Comments on Beckmann’s Uniform Reducts

, 2006

"... These comments refer to Arnold Beckmann’s paper [Bec05]. That paper introduces the notion of the uniform reduct of a propositional proof system, which consists of a collection of ∆0(α) formulas, where α is a unary relation symbol. Here I will define essentially the same thing, but make it a collecti ..."

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These comments refer to Arnold Beckmann’s paper [Bec05]. That paper introduces the notion of the uniform reduct of a propositional proof system, which consists of a collection of ∆0(α) formulas, where α is a unary relation symbol. Here I will define essentially the same thing, but make it a collection of Σ B 0 formulas instead. The ΣB 0 formulas (called Σp 0 by Zambella) are two-sorted formulas which are the same as bounded formulas of Peano arithmetic, except that they are allowed free “string ” variables X, Y, Z,... which range over finite sets of natural numbers. Terms of the form |X | are allowed, which denote the “length” of the string X (more precisely 1 plus the largest element of X, or 0 if X is empty). The atomic formula X(t) means t is a member of X. Each Σ B 0 formula ϕ(X) translates into a family 〈ϕ(X)[n] : n ∈ N 〉 of propositional formulas (see [Coo05, CN]) in the style of the Paris-Wilkie translation. The difference is that now X has a length |X|, and this affects the semantics of ϕ(X) and the resulting translation. For each n ∈ N the propositional translation ϕ(X)[n] of ϕ(X) has atoms p X 0, · · · , p X n−2 representing the bits of the string X, and ϕ(X)[n] is a tautology iff ϕ(X) holds for all strings X of length n. If ϕ ( X) has several string variables X = X1, · · · , Xk then the translation is the family ϕ ( X)[n] of formulas, where ni is intended to be the length of Xi. In terms of Σ B 0 Definition: (Beckmann) formulas, the definition of uniform reduct in [Bec05] becomes Uf = {ϕ ( X) ∈ Σ B 0: 〈ϕ ( X)[n] : n ∈ N 〉 has polysize f-proofs} Problem 2 in [Bec05] asks (in our teminology) whether there is a proof system f such that

### Expressing vs. proving: relating forms of complexity in logic

"... Abstract. Complexity in logic comes in many forms. In finite model theory, it is the complexity of describing properties, whereas in proof complexity it is the complexity of proving properties in a proof system. Here we consider several notions of complexity in logic, the connections among them, and ..."

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Abstract. Complexity in logic comes in many forms. In finite model theory, it is the complexity of describing properties, whereas in proof complexity it is the complexity of proving properties in a proof system. Here we consider several notions of complexity in logic, the connections among them, and their relationship with computational complexity. In particular, we show how the complexity of logics in the setting of finite model theory is used to obtain results in bounded arithmetic, stating which functions are provably total in certain weak systems of arithmetic. For example, the transitive closure function (testing reachability between two given points in a directed graph) is definable using only NL-concepts (where NL is the non-deterministic log-space complexity class), and its totality (and, thus, the closure of NL under complementation) is provable within NL-reasoning. Lastly, we will touch upon the topic of formalizing complexity theory using logic, and the meta-question of complexity of logical reasoning about complexity-theoretic statements. This is intended to be a high-level overview, suitable for readers who are not familiar with complexity theory and complexity in logic. 1

### Prague

"... We prove the following results: (i) PV proves NP ⊆ P/poly iff PV proves coNP ⊆ NP/O(1). (ii) If PV proves NP ⊆ P/poly then PV proves that the Polynomial Hierarchy collapses to the Boolean Hierarchy. (iii) S1 2 proves NP ⊆ P/poly iff S12 proves coNP ⊆ proves that NP/O(log n). (iv) If S1 2 proves NP ⊆ ..."

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We prove the following results: (i) PV proves NP ⊆ P/poly iff PV proves coNP ⊆ NP/O(1). (ii) If PV proves NP ⊆ P/poly then PV proves that the Polynomial Hierarchy collapses to the Boolean Hierarchy. (iii) S1 2 proves NP ⊆ P/poly iff S12 proves coNP ⊆ proves that NP/O(log n). (iv) If S1 2 proves NP ⊆ P/poly then S12 the Polynomial Hierarchy collapses to PNP [log n]. (v) If S2 2 proves NP ⊆ P/poly then S2 2 proves that the Polynomial Hierarchy col-lapses to P NP. Motivated by these results we introduce a new concept in proof complexity: proof systems with advice, and we make some initial observations about them. 1

### Abstract Logical Approaches to the Complexity of Search Problems: Proof Complexity, Quantified Propositional Calculus, and Bounded Arithmetic

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### unknown title

"... We introduce a second-order theoryÎ-Krom of bounded arithmetic for nondeterministic log space. This system is based on Grädel’s characterization ofÆÄby second-order Krom formulae with only universal first-order quantifiers, which in turn is motivated by the result that the decision problem for 2-CNF ..."

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We introduce a second-order theoryÎ-Krom of bounded arithmetic for nondeterministic log space. This system is based on Grädel’s characterization ofÆÄby second-order Krom formulae with only universal first-order quantifiers, which in turn is motivated by the result that the decision problem for 2-CNF satisfiability is complete for coÆÄ(and hence forÆÄ). This theory has the style of the authors’ theoryÎ-Horn [APAL 124 (2003)] for polynomial time. Both theories use Zambella’s elegant second-order syntax, and are axiomatized by a set 2-BASIC of simple formulae, together with a comprehension scheme for either secondorder Horn formulae (in the case ofÎ-Horn), or secondorder Krom (2CNF) formulae (in the case ofÎ-Krom). Our main result forÎ-Krom is a formalization of the Immerman-Szelepcsényi theorem thatÆÄis closed under complementation. This formalization is necessary to show that the ÆÄfunctions are¦�-definable inÎ-Krom. The only other theory forÆÄin the literature relies on the Immerman-Szelepcsényi’s result rather than proving it. 1.

### The Equivalence of Theories that Characterize

, 2007

"... A number of theories have been developed to characterize ALogTime (or uniform NC 1, or just NC 1), the class of languages accepted by alternating logtime Turing machines, in the same way that Buss’s theory S 1 2 characterizes polytime functions. Among these, ALV ′ (by Clote) is particularly interest ..."

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A number of theories have been developed to characterize ALogTime (or uniform NC 1, or just NC 1), the class of languages accepted by alternating logtime Turing machines, in the same way that Buss’s theory S 1 2 characterizes polytime functions. Among these, ALV ′ (by Clote) is particularly interesting because it is developed based on Barrington’s theorem that the word problem for the permutation group S5 is complete for ALogTime. On the other hand, ALV (by Clote), T 0 NC 0 (by Clote and Takeuti) as well as Arai’s theory AID + Σ B 0-CA and its two-sorted version VNC 1 (by Cook and Morioka) are based on the circuit characterization of ALogTime. While the last three theories have been known to be equivalent, their relationship to ALV ′ has been an open problem. Here we show that ALV ′ is indeed equivalent to the other theories. 1