Some Connection Between Mathematical Logic and Complexity Theory

"Tarski initiated a logic-based approach to formal geometry that studies first-order structures with a ternary betweenness relation (\beta) and a quaternary equidistance relation (\equiv). Tarski established, inter alia, that the first-order (FO) theory of (R^2,\beta,\equiv) is decidable. Aiello and van Benthem (2002) conjectured that the FO-theory of expansions of (R^2,\beta) with unary predicates is decidable. We refute this conjecture by showing that for all n>1, the FO-theory of monadic expansions of (R^2,\beta) is \Pi^1_1-hard and therefore not even arithmetical. We also define a natural and comprehensive class C of geometric structures (T,\beta), where T is a subset of R^2, and show that for each structure (T,\beta) in C, the FO-theory of the class of monadic expansions of (T,\beta) is undecidable. We then consider classes of expansions of structures (T,\beta) with restricted unary predicates, for example finite predicates, and establish a variety of related undecidability results. In addition to decidability questions, we briefly study the expressivity of universal MSO and weak universal MSO over expansions of (R^n,\beta). While the logics are incomparable in general, over expansions of (R^n,\beta), formulae of weak universal MSO translate into equivalent formulae of universal MSO. This is an extended version of a publication in the proceedings of the 21st EACSL Annual Conferences on Computer Science Logic (CSL 2012)."

The Bounds of Logic: A Generalized Viewpoint, 1991

The Bounds of Logic presents a new philosophical theory of the scope and nature of logic based on critical analysis of the principles underlying modern Tarskian logic and inspired by mathematical and linguistic development. Extracting central philosophical ideas from Tarski’s early work in semantics, Sher questions whether these are fully realized by the standard first-order system. The answer lays the foundation for a new, broader conception of logic. By generally characterizing logical terms, Sher establishes a fundamental result in semantics. Her development of the notion of logicality for quantifiers and her work on branching are of great importance for linguistics. Sher outlines the boundaries of the new logic and points out some of the philosophical ramifications of the new view of logic for such issues as the logicist thesis, ontological commitment, the role of mathematics in logic, and the metaphysical underpinning of logic. She proposes a constructive definition of logical terms, reexamines and extends the notion of branching quantification, and discusses various linguistic issues and applications.

Asterisque- Societe Mathematique de France

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1980

Theoretical Computer Science, 1987

Two models of proofs of lower bounds on the complexity are introduced. They have very wide applicability. The purpose of the models is to explain the difficulties of establishing high lower bounds on the complexity. * Th? paper is based on the text of two lectures given by the author at the International Banach Mathematical Center (Warsaw, November I985). Russian text of the lectures is submitted to the Banach Center Publications.

Studia Logica, 1981

Logic, Epistemology, and the Unity of Science

Some systems of modal logic, such as S5, which are often used as epistemic logics with the 'necessity' operator read as 'the agent knows that', are problematic as general epistemic logics for agents whose computational capacity does not exceed that of a Turing machine because they impose unwarranted constraints on the agent's theory of non-epistemic aspects of the world, for example by requiring the theory to be decidable rather than merely recursively axiomatizable. To generalize this idea, two constraints on an epistemic logic are formulated: r.e. conservativeness, that any recursively enumerable theory R in the sublanguage without the epistemic operator is conservatively extended by some recursively enumerable theory in the language with the epistemic operator which is permitted by the logic to be the agent's overall theory; the weaker requirement of r.e. quasi-conservativeness

2002

Abstract In [3] the tautology problem for Hájek's Basic Logic BL is proved to be co-NP-complete by showing that if a formula ϕ is not a tautology of BL then there exists an integer m> 0, polynomially bounded by the length of ϕ, such that ϕ fails to be a tautology in the infinite-valued logic mL corresponding to the ordinal sum of m copies of the Lukasiewicz t-norm.

Journal of Combinatorial Optimization, 2010

We show that for all reals c and d such that c 2 d < 4 there exists a positive real e such that tautologies of length n cannot be decided by both a nondeterministic algorithm that runs in time n c , and a nondeterministic algorithm that runs in time n d and space n e. In particular, for every d < 3 √ 4 there exists a positive e such that tautologies cannot be decided by a nondeterministic algorithm that runs in time n d and space n e .

Annals of Pure and Applied Logic, 2012

Justification Logic studies epistemic and provability phenomena by introducing justifications/proofs into the language in the form of justification terms. Pure justification logics serve as counterparts of traditional modal epistemic logics, and hybrid logics combine epistemic modalities with justification terms. The computational complexity of pure justification logics is typically lower than that of the corresponding modal logics. Moreover, the so-called reflected fragments, which still contain complete information about the respective justification logics, are known to be in NP for a wide range of justification logics, pure and hybrid alike. This paper shows that, under reasonable additional restrictions, these reflected fragments are NP-complete, thereby proving a matching lower bound. The proof method is then extended to provide a uniform proof that the corresponding full pure justification logics are Π p 2 -hard, reproving and generalizing an earlier result by Milnikel.