We introduce a system of Hyper Natural Deduction for Gödel Logic as an extension of Gentzen’s system of Natural Deduction. A deduction in this system consists of a finite set of derivations which uses the typical rules of Natural Deduction, plus additional rules providing means for communication between derivations. We show that our system is sound and complete for infinite-valued propositional Gödel Logic, by giving translations to and from Avron’s Hypersequent Calculus. We provide conversions for normalization extending usual conversions for Natural Deduction and prove the existence of normal forms for Hyper Natural Deduction for Gödel Logic. We show that normal deductions satisfy the subformula property.
The Cobham Recursive Set Functions (CRSF) provide an analogue of polynomial time computation which applies to arbitrary sets. We give three new equivalent characterizations of CRSF. The first is algebraic, using subset-bounded recursion and a form of Mostowski collapse. The second is our main result: the CRSF functions are shown to be precisely the functions computed by a class of uniform, infinitary, Boolean circuits. The third is in terms of a simple extension of the rudimentary functions by transitive closure and subset-bounded recursion.
We define a new class of total search problems as a subclass of Megiddo and Papadimitriou’s class of total $\NP$ search problems, in which solutions are verifiable in $\AC^0$. We denote this class $\forall\exists\AC^0$. We show that all total $\NP$ search problems are equivalent wrt. $\AC^0$-many-one reductions to search problems in $\forall\exists\AC^0$. Furthermore, we show that $\forall\exists\AC^0$ contains well-known problems such as the Stable Marriage and the Maximal Independent Set problems. We introduce the class of Inflationary Iteration problems in $\forall\exists\AC^0$ and show that it characterizes the provably total $\NP$ search problems of the bounded arithmetic theory corresponding to polynomial-time. Cook and Nguyen introduced a generic way of defining a bounded arithmetic theory $\VC$ for complexity classes $\C$ which can be obtained using a complete problem. For such $C$ we will define a new class $\KPT[C]$ of $\forall\exists\AC^0$ search problems based on Student-Teacher games in which the student has computing power limited to $\AC^0$. We prove that $\KPT[C]$ characterizes the provably total $\NP$ search problems of the bounded arithmetic theory corresponding to $\C$. All our characterizations are obtained via “new-style” witnessing theorems, where reductions are provable in a theory corresponding to $\AC^0$.
We study consistency search problems for Frege and extended Frege proofs, namely the NP search problems of finding syntactic errors in Frege and extended Frege proofs of contradictions. The input is a polynomial time function, or an oracle, describing a proof of a contradiction; the output is the location of a syntactic error in the proof. The consistency search problems for Frege and extended Frege systems are shown to be many-one complete for the provably total NP search problems of the second order bounded arithmetic theories U^1_2 and V^1_2 , respectively.
This paper studies the complexity of constant depth propositional proofs in the cedent and sequent calculus. We discuss the relationships between the size of tree-like proofs, the size of dag-like proofs, and the heights of proofs. The main result is to correct a proof construction in an earlier paper about transformations from proofs with polylogarithmic height and constantly many formulas per cedent.
We show that logics based on linear Kripke frames – with or without constant domains – that have a scattered end piece are not recursively enumerable. This is done by reduction to validity in all finite classical models.
This paper introduces bounded fragments of Kripke Platek set theory which characterise the Cobham Recursive Set Functions.
This paper introduces the Cobham Recursive Set Functions (CRSF) as a version of polynomial time computable functions on general sets, based on a limited (bounded) form of epsilon-recursion. This is inspired by Cobham’s classic definition of polynomial time functions based on limited recursion on notation. We introduce a new set composition function, and a new smash function for sets which allows polynomial increases in the ranks and in the cardinalities of transitive closures. We bootstrap CRSF, prove closure under (unbounded) replacement, and prove that any CRSF function is embeddable into a smash term. When restricted to natural encodings of binary strings as hereditarily finite sets, the CRSF functions define precisely the polynomial time computable functions on binary strings. Prior work of Beckmann, Buss and Friedman and of Arai introduced set functions based on safe-normal recursion in the sense of Bellantoni-Cook. We prove an equivalence between our class CRSF and a variant of Arai’s predicatively computable set functions.
We consider intermediate predicate logics defined by fixed well-ordered (or dually well-ordered) linear Kripke frames with constant domains where the order-type of the well-order is strictly smaller than~$\omega^\omega$. We show that two such logics of different order-type are separated by a first-order sentence using only one monadic predicate symbol. Previous results by Minari, Takano and Ono, as well as the second author, obtained the same separation but relied on the use of predicate symbols of unbounded arity.
We introduce the safe recursive set functions based on a Bellantoni-Cook style subclass of the primitive recursive set functions. We show that the functions computed by safe recursive set functions under a list encoding of finite strings by hereditarily finite sets are exactly the polynomial growth rate functions computed by alternating exponential time Turing machines with polynomially many alternations. We also show that the functions computed by safe recursive set functions under a more efficient binary tree encoding of finite strings by hereditarily finite sets are exactly the quasipolynomial growth rate functions computed by alternating quasipolynomial time Turing machines with polylogarithmic many alternations.
We characterize the safe recursive set functions on arbitrary sets in definability-theoretic terms. In its strongest form, we show that a function on arbitrary sets is safe recursive if, and only if, it is uniformly definable in some polynomial level of a refinement of Jensen’s J-hierarchy, relativised to the transitive closure of the function’s arguments.
We observe that safe-recursive functions on infinite binary strings are equivalent to functions computed by so-called infinite-time Turing machines in time less than ω^ω. We also give a machine model for safe recursion which is based on set-indexed parallel processors and the natural bound on running times.
We introduce a Hyper Natural Deduction system as an extension of Gentzen’s Natural Deduction system. A Hyper Natural Deduction consists of a finite set of derivations which may use, beside typical Natural Deduction rules, additional rules providing means for communication between derivations. We show that our Hyper Natural Deduction system is sound and complete for infinite-valued propositional Gödel Logic, by giving translations to and from Avron’s Hyper sequent Calculus. We also provide conversions for normalisation and prove the existence of normal forms for our Hyper Natural Deduction system.
A propositional proof system is weakly automatizable if there is a polynomial time algorithm which separates satisfiable formulas from formulas which have a short refutation in the system, with respect to a given length bound. We show that if the resolution proof system is weakly automatizable, then parity games can be decided in polynomial time. We give simple proofs that the same holds for depth-1 propositional calculus (where resolution has depth 0) with respect to mean payoff and simple stochastic games. We define a new type of combinatorial game and prove that resolution is weakly automatizable if and only if one can separate, by a set decidable in polynomial time, the games in which the first player has a positional winning strategy from the games in which the second player has a positional winning strategy.
Our main technique is to show that a suitable weak bounded arithmetic theory proves that both players in a game cannot simultaneously have a winning strategy, and then to translate this proof into propositional form.
This paper concerns the second order systems U^1_2 and V^1_2 of bounded arithmetic, which have proof theoretic strengths corresponding to polynomial space and exponential time computation. We formulate improved witnessing theorems for these two theories by using S^1_2 as a base theory for proving the correctness of the polynomial space or exponential time witnessing functions. We develop the theory of nondeterministic polynomial space computation in U^1_2 . Kołodziejczyk, Nguyen, and Thapen have introduced local improvement properties to characterize the provably total NP functions of these second order theories. We show that the strengths of their local improvement principles over U^1_2 and V^1_2 depend primarily on the topology of the underlying graph, not the number of rounds in the local improvement games. The theory U^1_2 proves the local improvement principle for linear graphs even without restricting to logarithmically many rounds. The local improvement principle for grid graphs with only logarithmically rounds is complete for the provably total NP search problems of V^1_2 . Related results are obtained for local improvement principles with one improvement round, and for local improvement over rectangular grids.
A propositional proof system is weakly automatizable if there is a polynomial time algorithm which separates satisfiable formulas from formulas which have a short refutation in the system, with respect to a given length bound. We show that if the resolution proof system is weakly automatizable, then parity games can be decided in polynomial time. We also define a combinatorial game and prove that resolution is weakly automatizable if and only if one can separate, by a set decidable in polynomial time, the games in which the first player has a positional winning strategy from the games in which the second player has a positional winning strategy.
This volume documents the presentations that were given as part of the research programme Semantics and Syntax: A Legacy of Alan Turing, held at the Isaac Newton Institute for the Mathematical Sciences in Cambridge, UK, 9 January – 6 July 2012.
We explore the support of automatic verification via careful design of a domain specific language (DSL) in the context of algebraic specification. Formally a DSL is a loose specification the logical closure of which we regard as implicitly encoded “domain knowledge”. We systematically exploit this “domain knowledge” for automatic verification. We illustrate these ideas within the Railway Domain using the algebraic specification language Casl and an existing DSL, designed by Bjøerner, for modelling railways. Empirical evidence to the benefit of our approach is given in the form of the successful automatic verification of four railway track plans of real world complexity.
The paper corrects earlier upper bounds on the size of free-cut elimination. Free-cut elimination allows cut elimination to be carried out in the presence of non-logical axioms. The correction requires that the notion of a free-cut be modified so that a cut formula is anchored provided that all of its introductions are anchored, instead of only requiring that one of its introductions is anchored. With the correction, the originally proved size upper bounds remain unchanged. These results also apply to partial cut elimination. We also apply these bounds to elimination of cuts in propositional logic.
If the non-logical inferences are closed under cut and infer only atomic formulas, then all cuts can be eliminated. This generalizes earlier results of Takeuti and of Negri and van Plato.
Using appropriate notation systems for proofs, cut-reduction can often be rendered feasible on these notations, and explicit bounds can be given. Developing a suitable notation system for Bounded Arithmetic, and applying these bounds, all the known results on definable functions of certain such theories can be reobtained in a uniform way.
The complexity class of Π^p_k – Polynomial Local Search (PLS) problems with Π^p_l – goal is introduced, and is used to give new characterisations of definable search problems in fragments of Bounded Arithmetic. The characterisations are established via notations for propositional proofs obtained by translating Bounded Arithmetic proofs using the Paris-Wilkie-translation. For l ≤ i ≤ k , the Σ bi – definable search problems of T^k+1_2 are exactly characterised by Π^p_k – PLS problems with Π^p_l – goals. These Π^p_k – PLS problems can be defined in a weak base theory such as S^1_2 , and proved to be total in T^k+1_2 . Furthermore, the Π^p_k – PLS definitions can be Skolemised with simple polynomial time functions. The Skolemised Π^p_k – PLS definitions give rise to a new ∀Σ^b_1(α) – principle conjectured to separate T^k_2(α) and T^k+1_2(α).
The complexity class of Π^p_k – polynomial local search (PLS) problems is introduced and is used to give new witnessing theorems for fragments of bounded arithmetic. For 1 ≤ i ≤ k+1 , the Σ^b_i – definable functions of T^k+1_2 are characterized in terms of Π^p_k – PLS problems. These Π^p_k – PLS problems can be defined in a weak base theory such as S^1_2 , and proved to be total in T^k+1_2 . Furthermore, the Π^p_k – PLS definitions can be skolemized with simple polynomial time functions, and the witnessing theorem itself can be formalized, and skolemized, in a weak base theory. We introduce a new ∀Σ^b_1(α) – principle that is conjectured to separate T^k_2(α) and T^k+1_2(α) .