On the Impossibility of Probabilistic Proofs in Relativized Worlds

We initiate the systematic study of probabilistic proofs in relativized worlds, where the goal is to understand, for a given oracle, the possibility of "non-trivial" proof systems for deterministic or nondeterministic computations that make queries to the oracle. This question is intimately related to a recent line of work that seeks to improve the efficiency of probabilistic proofs for computations that use functionalities such as cryptographic hash functions and digital signatures, by instantiating them via constructions that are "friendly" to known constructions of probabilistic proofs. Informally, negative results about probabilistic proofs in relativized worlds provide evidence that this line of work is inherent and, conversely, positive results provide a way to bypass it. We prove several impossibility results for probabilistic proofs relative to natural oracles. Our results provide strong evidence that tailoring certain natural functionalities to known probabilistic proofs is inherent

Preprints of Proceedings of GWAI-92

This is a preprint of the proceedings of the German Workshop on Artificial Intelligence (GWAI) 1992. The final version will appear in the Lecture Notes in Artificial Intelligence

Paraconsistent First-Order Logic with infinite hierarchy levels of contradiction

In this paper paraconsistent first-order logic LP^{#} with infinite hierarchy levels of contradiction is proposed. Corresponding paraconsistent set theory KSth^{#} is discussed.Axiomatical system HST^{#}as paraconsistent generalization of Hrbacek set theory HST is considere

Structural Average Case Complexity

AbstractLevin introduced an average-case complexity measure, based on a notion of “polynomial on average,” and defined “average-case polynomial-time many-one reducibility” among randomized decision problems. We generalize his notions of average-case complexity classes, Random-NP and Average-P. Ben-Davidet al. use the notation of 〈C, F〉 to denote the set of randomized decision problems (L, μ) such thatLis a set in C andμis a probability density function in F. This paper introduces Aver〈C, F〉 as the class of randomized decision problems (L, μ) such thatLis computed by a type-C machine onμ-average andμis a density function in F. These notations capture all known average-case complexity classes as, for example, Random-NP= 〈NP, P-comp〉 and Average-P=Aver〈P, ∗〉, where P-comp denotes the set of density functions whose distributions are computable in polynomial time, and ∗ denotes the set of all density functions. Mainly studied are polynomial-time reductions between randomized decision problems: many–one, deterministic Turing and nondeterministic Turing reductions and the average-case versions of them. Based on these reducibilities, structural properties of average-case complexity classes are discussed. We give average-case analogues of concepts in worst-case complexity theory; in particular, the polynomial time hierarchy and Turing self-reducibility, and we show that all known complete sets for Random-NP are Turing self-reducible. A new notion of “real polynomial-time computations” is introduced based on average polynomial-time computations for arbitrary distributions from a fixed set, and it is used to characterize the worst-case complexity classesΔpkandΣpkof the polynomial-time hierarchy

On the complexity of resolution-based proof systems

Propositional Proof Complexity is the area of Computational Complexity that studies the length of proofs in propositional logic. One of its main questions is to determine which particular propositional formulas have short proofs in a given propositional proof system. In this thesis we present several results related to this question, all on proof systems that are extensions of the well-known resolution proof system. The first result of this thesis is that TQBF, the problem of determining if a fully-quantified propositional CNF-formula is true, is PSPACE-complete even when restricted to instances of bounded tree-width, i.e. a parameter of structures that measures their similarity to a tree. Instances of bounded tree-width of many NP-complete problems are tractable, e.g. SAT, the boolean satisfiability problem. We show that this does not scale up to TQBF. We also consider Q-resolution, a quantifier-aware version of resolution. On the negative side, our first result implies that, unless NP = PSPACE, the class of fully-quantified CNF-formulas of bounded tree-width does not have short proofs in any proof system (and in particular in Q-resolution). On the positive side, we show that instances with bounded respectful tree-width, a more restrictive condition, do have short proofs in Q-resolution. We also give a natural family of formulas with this property that have real-world applications. The second result concerns interpretability. Informally, we say that a first-order formula can be interpreted in another if the first one can be expressed using the vocabulary of the second, plus some extra features. We show that first-order formulas whose propositional translations have short R(const)-proofs, i.e. a generalized version of resolution with DNF-formulas of constant-size terms, are closed under a weaker form of interpretability (that with no extra features), called definability. Our main result is a similar result on interpretability. Also, we show some examples of interpretations and show a systematic technique to transform some Sigma_1-definitions into quantifier-free interpretations. The third and final result is about a relativized weak pigeonhole principle. This says that if at least 2n out of n^2 pigeons decide to fly into n holes, then some hole must be doubly occupied. We prove that the CNF encoding of this principle does not have polynomial-size DNF-refutations, i.e. refutations in the generalized version of resolution with unbounded DNF-formulas. For this proof we discuss the existence of unbalanced low-degree bipartite expanders satisfying a certain robustness condition

Pravdivost mezi syntaxí a sémantikou

Sir s m c lem t eto pr ace je vyjasnit vztah mezi syntax a s emantikou, zejm ena pokud jde o jazyky s p resn e speci kovanou strukturou. Hlavn ot azky, kter ymi se zab yv ame, jsou: Co cin s emantick y pojem s emantick ym? Co zp usobuje, ze je pouh a s emantick a anal yza takov eho pojmu nedostate cn a? Co je t m rozhoduj c m krokem, kter y mus me u cinit, abychom pronikli k v yznamov e str ance jazyka? T emito ot azkami se nezab yv ame p r mo, ale prost rednictv m anal yzy typick eho s emantick eho pojmu, a sice pravdivosti. Na s hlavn ot azkou tedy je: Jak e pojmov e prost redky jsou nezbytn e pro uspokojivou de nici pravdivosti? Ke zkoum an pojmu pravdivosti a jednotliv ych zp usob u, jak jej lze de- novat, jsme si vybrali t ri konkr etn syst emy: kumulativn verzi Russellovy rozv etven e teorie typ u, Zermelovu druho r adovou teorii mno zin a Carnapovu logickou syntax. Ka zd y syst em je podroben d ukladn emu studiu. P redkl adan a pr ace je tedy souborem t r v ce m en e samostatn ych studi , je z popisuj mo znosti explicitn de nice pravdivosti a nezbytn eho pojmov eho z azem . Poznamenejme, ze na s m c lem nen historicky v ern a prezentace uveden ych syst em u, n ybr z snaha o dal s rozvinut toho cenn eho, co nab zej , ve sv etle sou casn ych poznatk u. Obecn ym z av erem, k n emu z dosp ejeme na z...The broad aim of this thesis is to clarify the relationship between syntax and semantics, mainly in connection with languages with exactly speci ed structure. The main questions we raise are: What is it that makes a semantic concept genuinely semantic? What exactly makes a merely semantic characterization of such a concept inadequate? What is the decisive step we have to make if we want to start speaking about the meaning-side of language? We approach these questions indirectly: via an analysis of a typically semantic concept, namely that of truth. Our principal question then becomes: What conceptual resources are required for a satisfactory de nition of truth? To investigate the concept of truth and di erent ways in which it can be de ned, we have chosen three individual systems: (a cumulative version of) Russell's rami ed theory of types, Zermelo's second-order set theory and Carnap's logical syntax. Each of the systems is studied in considerable detail. The presented thesis is, in e ect, a collection of three case-studies into the ways in which the concept of truth is explicitly de nable and into the requisite conceptual background, each study forming a more or less closed unity. It should be noted that we are not interested in a historically faithful representation of these systems; our goal is to get...Institute of Philosophy and Religious StudiesÚstav filosofie a religionistikyFilozofická fakultaFaculty of Art