Online research in philosophy

We consider the problem about the length of proofs of the sentences $\operatorname{Con}_S(\underline{n})$ saying that there is no proof of contradiction in S whose length is ≤ n. We show the relation of this problem to some problems about propositional proof systems.

We give resource bounded versions of the Completeness Theorem for propositional and predicate logic. For example, it is well known that every computable consistent propositional theory has a computable complete consistent extension. We show that, when length is measured relative to the binary representation of natural numbers and formulas, every polynomial time decidable propositional theory has an exponential time (EXPTIME) complete consistent extension whereas there is a nondeterministic polynomial time (NP) decidable theory which has no polynomial time complete consistent extension when length is measured relative to the binary representation of natural numbers and formulas. It is well known that a propositional theory is axiomatizable (respectively decidable) if and only if it may be represented as the set of infinite paths through a computable tree (respectively a computable tree with no dead ends). We show that any polynomial time decidable theory may be represented as the set of paths through a polynomial time decidable tree. On the other hand, the statement that every polynomial time decidable, relative to the tally representation of natural numbers and formulas, is equivalent to P = NP. For predicate logic, we develop a complexity theoretic version of the Henkin construction to prove a complexity theoretic version of the Completeness Theorem. Our results imply that that any polynomial space decidable theory △ possesses a polynomial space computable model which is exponential space decidable and thus △ has an exponential space complete consistent extension. Similar results are obtained for other notions of complexity.

LK is a natural modification of Gentzen sequent calculus for propositional logic with connectives ¬ and $\bigwedge, \bigvee$ (both of bounded arity). Then for every d ≥ 0 and n ≥ 2, there is a set Td n of depth d sequents of total size O(n3 + d) which are refutable in LK by depth d + 1 proof of size exp(O(log2 n)) but such that every depth d refutation must have the size at least exp(nΩ(1)). The sets Td n express a weaker form of the pigeonhole principle.

We consider small-weight Cutting Planes (CP * ) proofs; that is, Cutting Planes (CP) proofs with coefficients up to $\operatorname{Poly}(n)$ . We use the well known lower bounds for monotone complexity to prove an exponential lower bound for the length of CP * proofs, for a family of tautologies based on the clique function. Because Resolution is a special case of small-weight CP, our method also gives a new and simpler exponential lower bound for Resolution. We also prove the following two theorems: (1) Tree-like CP * proofs cannot polynomially simulate non-tree-like CP * proofs. (2) Tree-like (CP * proofs and Bounded-depth-Frege proofs cannot polynomially simulate each other. Our proofs also work for some generalizations of the CP * proof system. In particular, they work for CP * with a deduction rule, and also for any proof system that allows any formula with small communication complexity, and any set of sound rules of inference.

We describe a general method how to construct from a propositional proof system P a possibly much stronger proof system iP. The system iP operates with exponentially long P-proofs described "implicitly" by polynomial size circuits. As an example we prove that proof system iEF, implicit EF, corresponds to bounded arithmetic theory $V_{2}^{1}$ and hence, in particular, polynomially simulates the quantified propositional calculus G and the $\pi_{1}^{b}-consequences$ of $S_{2}^{1}$ proved with one use of exponentiation. Furthermore, the soundness of iEF is not provable in $S_{2}^{1}$ . An iteration of the construction yields a proof system corresponding to $T_{2} + Exp$ and, in principle, to much stronger theories.

Cook and Reckhow defined a propositional formulation of the pigeonhole principle. This paper shows that there are Frege proofs of this propositional pigeonhole principle of polynomial size. This together with a result of Haken gives another proof of Urquhart's theorem that Frege systems have an exponential speedup over resolution. We also discuss connections to provability in theories of bounded arithmetic.

We consider tautologies formed form a pseudo-random number generator, defined in Krajicek [11] and in Alekhnovich et al. [2]. We explain a strategy of proving their hardness for Extended Frege systems via a conjecture about bounded arithmetic formulated in Krajicek [11]. Further we give a purely finitary statement, in the form of a hardness condition imposed on a function, equivalent to the conjecture. This is accompanied by a brief explanation, aimed at non-specialists, of the relation between prepositional proof complexity and bounded arithmetic.

Sandu and Pietarinen [Partiality and Games: Propositional Logic. Logic J. IGPL 9 (2001) 101] study independence friendly propositional logics. That is, traditional propositional logic extended by means of syntax that allow connectives to be independent of each other, although the one may be subordinate to the other. Sandu and Pietarinen observe that the IF propositional logics have exotic properties, like functional completeness for three-valued functions. In this paper we focus on one of their IF propositional logics and study its properties, by means of notions from computational complexity. This approach enables us to compare propositional logic before and after the IF make-over. We observe that all but one of the best-known decision problems experience a complexity jump, provided that the complexity classes at hand are not equal. Our results concern every discipline that incorporates some notion of independence such as computer science, natural language semantics, and game theory. A corollary of one of our theorems illustrates this claim with respect to the latter discipline.

A proof of the (propositional) Craig interpolation theorem for cut-free sequent calculus yields that a sequent with a cut-free proof (or with a proof with cut-formulas of restricted form; in particular, with only analytic cuts) with k inferences has an interpolant whose circuit-size is at most k. We give a new proof of the interpolation theorem based on a communication complexity approach which allows a similar estimate for a larger class of proofs. We derive from it several corollaries: (1) Feasible interpolation theorems for the following proof systems: (a) resolution (b) a subsystem of LK corresponding to the bounded arithmetic theory S 2 2 (α) (c) linear equational calculus (d) cutting planes. (2) New proofs of the exponential lower bounds (for new formulas) (a) for resolution ([15]) (b) for the cutting planes proof system with coefficients written in unary ([4]). (3) An alternative proof of the independence result of [43] concerning the provability of circuit-size lower bounds in the bounded arithmetic theory S 2 2 (α). In the other direction we show that a depth 2 subsystem of LK does not admit feasible monotone interpolation theorem (the so called Lyndon theorem), and that a feasible monotone interpolation theorem for the depth 1 subsystem of LK would yield new exponential lower bounds for resolution proofs of the weak pigeonhole principle.

Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes.