On the Sizes of DPDAs, PDAs, LBAs

Abstract There are languages A such that there is a Pushdown Automata (PDA) that recognizes A which is much smaller than any Deterministic Pushdown Automata (DPDA) that recognizes A. There are languages A such that there is a Linear Bounded Automata (Linear Space Turing Machine, henceforth LBA) that recognizes A which is much smaller than any PDA that recognizes A. There are languages A such that both A and A are recognizable by a PDA, but the PDA for A is much smaller than the PDA for A. There are languages A 1 , A 2 such that A 1 , A 2 , A 1 ∩ A 2 are recognizable by a PDA, but the PDA for A 1 and A 2 are much smaller than the PDA for A 1 ∩ A 2 . We investigate these phenenoma and show that, in all these cases, the size difference is captured by a function whose Turing degree is on the second level of the arithmetic hierarchy. Our theorems lead to infinitely-often results. For example: for infinitely many n there exists a language A n such that there is a small PDA for A n , but any DPDA for A n is large. We look at cases where we can get almost-all results, though with much smaller size differences

On the Sizes of DPDAs, PDAs, LBAs

Abstract There are languages A such that there is a Pushdown Automata (PDA) that recognizes A which is much smaller than any Deterministic Pushdown Automata (DPDA) that recognizes A. There are languages A such that there is a Linear Bounded Automata (Linear Space Turing Machine, henceforth LBA) that recognizes A which is much smaller than any PDA that recognizes A. There are languages A such that both A and A are recognizable by a PDA, but the PDA for A is much smaller than the PDA for A. There are languages A 1 , A 2 such that A 1 , A 2 , A 1 ∩ A 2 are recognizable by a PDA, but the PDA for A 1 and A 2 are much smaller than the PDA for A 1 ∩ A 2 . We investigate these phenomema and show that, in all these cases, the size difference is captured by a function whose Turing degree is on the second level of the arithmetic hierarchy. Our theorems lead to infinitely-often results. For example: for infinitely many n there exists a language A n such that there is a small PDA for A n , but any DPDA for A n is large. We look at cases where we can get almost-all results, though with much smaller size differences

Describing classical spin Hamiltonians as automata: a new complexity measure

We describe classical spin Hamiltonians as automata and use the classification of the latter to obtain a new complexity measure of Hamiltonians. Specifically, we associate a classical spin Hamiltonian to the formal language consisting of pairs of spin configurations and the corresponding energy, and classify this language in the Chomsky hierarchy. We prove that the language associated to (i) effectively zero-dimensional spin Hamiltonians is regular, (ii) local one-dimensional (1D) spin Hamiltonians is deterministic context-free, (iii) local two-dimensional (2D) or higher-dimensional spin Hamiltonians is context-sensitive, and (iv) totally unbounded spin Hamiltonians is recursively enumerable. It follows that only highly non-physical spin Hamiltonians [(iv)] correspond to Turing machines. It also follows that the Ising model without fields is easy or hard if defined on a 1D or 2D lattice, in contrast to the computational complexity of its ground state energy problem, where the threshold is found between planar and non-planar graphs. Our work puts classical spin Hamiltonians at the same level as automata, and paves the road toward a rigorous comparison of universal spin models and universal Turing machines.Comment: v3: more results; 24 pages, 9 figures, 9 tables. v2: More results and extensively rewritten; 18 pages and 7 figures; code of linear bounded automaton also attached. v1: 13 pages, 7 figures, code of deterministic pushdown automaton attache

The Frobenius Problem in a Free Monoid

Given positive integers c1,c2,...,ck with gcd(c1,c2,...,ck) = 1, the Frobenius problem (FP) is to compute the largest integer g(c1,c2,...,ck) that cannot be written as a non-negative integer linear combination of c1,c2,...,ck. The Frobenius problem in a free monoid (FPFM) is a non-commutative generalization of the Frobenius problem. Given words x1,x2,...,xk such that there are only finitely many words that cannot be written as concatenations of words in {x1,x2,...,xk}, the FPFM is to find the longest such words. Unlike the FP, where the upper bound g(c1,c2,...,ck)≤max 1≤i≤k ci2 is quadratic, the upper bound on the length of the longest words in the FPFM can be exponential in certain measures and some of the exponential upper bounds are tight. For the 2FPFM, where the given words over Σ are of only two distinct lengths m and n with 1<m<n, the length of the longest omitted words is ≤g(m, m|Σ|n-m + n - m). In Chapter 1, I give the definition of the FP in integers and summarize some of the interesting properties of the FP. In Chapter 2, I give the definition of the FPFM and discuss some general properties of the FPFM. Then I mainly focus on the 2FPFM. I discuss the 2FPFM from different points of view and present two equivalent problems, one of which is about combinatorics on words and the other is about the word graph. In Chapter 3, I discuss some variations on the FPFM and related problems, including input in other forms, bases with constant size, the case of infinite words, the case of concatenation with overlap, and the generalization of the local postage-stamp problem in a free monoid. In Chapter 4, I present the construction of some essential examples to complement the theory of the 2FPFM discussed in Chapter 2. The theory and examples of the 2FPFM are the main contribution of the thesis. In Chapter 5, I discuss the algorithms for and computational complexity of the FPFM and related problems. In the last chapter, I summarize the main results and list some open problems. Part of my work in the thesis has appeared in the papers